### 12th Standard CBSE Maths Study material & Free Online Practice Tests - View Model Question Papers with Solutions for Class 12 Session 2020 - 2021 CBSE [ Chapter , Marks , Book Back, Creative & Term Based Questions Papers - Syllabus, Study Materials, MCQ's Practice Tests etc..]

#### Class 12th Maths - Three Dimensional Geometry Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Two motorcycles A and B are running at the speed more than allowed speed on the road along the lines $\vec{r}=\lambda(\hat{i}+2 \hat{j}-\hat{k}) \text { and } \vec{r}=3 \hat{i}+3 \hat{j}+\mu(2 \hat{i}+\hat{j}+\hat{k})$, respectively.

Based on the above information, answer the following questions.
(i) The cartesian equation of the line along which motorcycle A is running, is

 (a) $\frac{x+1}{1}=\frac{y+1}{2}=\frac{z-1}{-1}$ (b) $\frac{x}{1}=\frac{y}{2}=\frac{z}{-1}$ (c) $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ (d) none of these

(ii) The direction cosines of line along which motorcycle A is running, are

 (a) < 1, -2, 1 > (b) < 1, 2, -1 > (c) $<\frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}, \frac{1}{\sqrt{6}}>$ (d) $<\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{-1}{\sqrt{6}}>$

(iii) The direction ratios of line along which motorcycle B is running, are

 (a) <  1, 0, 2 > (b) <  2, 1, 0 > (c) < 1, 1, 2  > (d) <  2, 1, 1  >

(iv) The shortest distance between the gives lines is

 (a) 4 units (b) 2.$\sqrt 3$ units (c) 3.$\sqrt 2$ units (d) 0 units

(v) The motorcycles will meet with an accident at the point

 (a) (-1, 1, 2) (b) (2, 1, -1) (c) (1, 2, -1) (d) does not exist
• 2)

A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and B are respectively sitting on the plane represented by the equation $\vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=5 \text { and } \vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=6$ ,to cheer up the team of their respective schools.

Based on the above information, answer the following questions.
(i) The cartesian equation of the plane on which students of school A are seated is

 (a) 2x - y +z = 8 (b) 2x + y + z = 8 (c) x + y + 2z = 5 (d) x + y + z = 5

(ii) The magnitude of the normal to the plane on which students of school B are seated, is

 (a) $\sqrt 5$ (b) $\sqrt 6$ (c) $\sqrt 3$ (d) $\sqrt 2$

(iii) The intercept form of the equation of the plane on which students of school B are seated, is

 (a) $\frac{x}{6}+\frac{y}{6}+\frac{z}{6}=1$ (b)  $\frac{x}{3}+\frac{y}{(-6)}+\frac{z}{6}=1$ (c) $\frac{x}{3}+\frac{y}{6}+\frac{z}{6}=1$ (d) $\frac{x}{3}+\frac{y}{6}+\frac{z}{3}=1$

(iv) Which of the following is a student of school B?

 (a) Mohit sitting at (1, 2, 1) (b) Ravi sitting at (0,1,2) (c) Khushi sitting at (3, 1, 1) (d) Shewta sitting at (2, -1, 2)

(v) The distance of the plane, on which students of school B are seated, from the origin is

 (a) 6 units (b) $\frac{1}{\sqrt{6}}$ units (c) $\frac{5}{\sqrt{6}}$ units (d) $\sqrt 6$ units
• 3)

Consider the following diagram, where the forces in the cable are given.

Based on the above information, answer the following questions.
(i) The cartesian equation of line along EA is

 $(a) \ \frac{x}{-4}=\frac{y}{3}=\frac{z}{12}$ $(b) \ \frac{x}{-4}=\frac{y}{3}=\frac{z-24}{12}$ $(c) \ \frac{x}{-3}=\frac{y}{4}=\frac{z-12}{12}$ $(d) \ \frac{x}{3}=\frac{y}{4}=\frac{z-24}{12}$

(ii) The vector $\overline{E D}$ is

 (a) $8 \hat{i}-6 \hat{j}+24 \hat{k}$ (b) $-8 \hat{i}-6 \hat{j}+24 \hat{k}$ (c) $-8 \hat{i}-6 \hat{j}-24 \hat{k}$ (d) $8 \hat{i}+6 \hat{j}+24 \hat{k}$

(iii) The length of the cable EB is

 (a) 24 units (b) 26 units (c) 27 units (d) 25 units

(iv) The length of cable EC is equal to the length of

 (a) EA (b) EB (c) ED (d) All of these

(v) The sum of all vectors along the cables is

 (a) $96 \hat{i}$ (b) $96 \hat{j}$ (c) $-96 \hat{k}$ (d) $96 \hat{k}$
• 4)

Consider the following diagram, where the forces in the cable are given.

(i) The equation of line along the cable AD is

 (a) $\frac{x}{5}=\frac{y}{4}=\frac{z-30}{15}$ (b) $\frac{x}{4}=\frac{y}{5}=\frac{z-30}{15}$ (c) $\frac{x}{5}=\frac{y}{4}=\frac{30-z}{15}$ (d) $\frac{x}{4}=\frac{y}{5}=\frac{30-z}{15}$

(ii) The length of cable DC is

 (a) $4 \sqrt{61} \mathrm{~m}$ (b) $5 \sqrt{61} \mathrm{~m}$ (c) $6\sqrt{61} \mathrm{~m}$ (d) $7 \sqrt{61} \mathrm{~m}$

(iii) The vector DB is

 (a)  $-6 \hat{i}+4 \hat{j}-30 \hat{k}$ (b) $6 \hat{i}-4 \hat{j}-30 \hat{k}$ (c) $6 \hat{i}+4 \hat{j}+30 \hat{k}$ (d) none of these

(iv) The sum of vectors along the cables, is

 (a) $17 \hat{i}+6 \hat{j}+90 \hat{k}$ (b) $17 \hat{i}-6 \hat{j}-90 \hat{k}$ (c) $​​17 \hat{i}+6 \hat{j}-90 \hat{k}$ (d) none of these

(v) The sum of distances of points A, Band C from the origin, i.e., OA + OB + OC, is

 (a) $\sqrt{164}+\sqrt{52}+\sqrt{625}$ (b) $\sqrt{52}+\sqrt{625}+\sqrt{48}$ (c) $\sqrt{164}+\sqrt{625}+\sqrt{49}$ (d) none of these
• 5)

Suppose the floor of a hotel is made up of mirror polished Kota stone. Also, there is a large crystal chandelier attached at the ceiling of the hotel. Consider the floor of the hotel as a plane having equation x - 2y + 2z = 3 and crystal chandelier at the point (3, -2, 1).

Based on the above information, answer the following questions.
(i) The d.r's of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

 (a) < 1,2,2 > (b) < 1, - 2, 2  > (c) < 2,1,2 > (d) < 2, -1, 2 >

(ii) The length of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

 (a) $\frac{2}{3}$units (b) 3 units (c) 2 units (d) none of these

(iii) The equation of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

 (a) $\frac{x-3}{1}=\frac{y-2}{-2}=\frac{z-1}{2}$ (b) $\frac{x-3}{1}=\frac{y+2}{-2}=\frac{z-1}{2}$ (c) $\frac{x+3}{1}=\frac{y+2}{-2}=\frac{z-1}{2}$ (d) none of these

(iv) The equation of plane parallel to the plane x - 2y + 2z = 3, which is at a unit distance from the point (3, -2, 1) is

 (a) x - 2y + 2z = 0 (b) x - 2y + 2z = 6 (c) x - 2y + 2z = 12 (d) Both (b) and (c)

(v) The image of the point (3, -2, 1) in the given plane is

 (a) $\left(\frac{5}{3}, \frac{2}{3}, \frac{-5}{3}\right)$ (b) $\left(\frac{-5}{3}, \frac{-2}{3}, \frac{5}{3}\right)$ (c) $\left(\frac{-5}{3}, \frac{2}{3}, \frac{5}{3}\right)$ (d) none of these

#### Class 12th Maths - Linear Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when relationship is expressed as linear equations or inequations.
Based on the above information, answer the following questions.
(i) The optimal value of the objective function is attained at the points

 (a) on X-axis (b) on Y-axis (c) which are corner points of the feasible region (d) none of these

(ii) The graph of the inequality 3x + 4y < 12 is

 (a) half plane that contains the origin (b) half plane that neither contains the origin nor the points of the line 3x + 4y =12. (c) whole XOY-plane excluding the points on line 3x + 4y = 12 (d) none of these

(iii) The feasible region for an LPP is shown in the figure. Let Z = 2x + 5y be the objective function. Maximum of Z occurs at

 (a) (7,0) (b) (6,3) (c) (0,6) (d) (4,5)

(iv) The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is

 (a) p = q (b) p = 2q (c) q=2p (d) q=3p

(v)  The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20,40), (60,20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in Column A and Column B

 Column A Column B Maximum of Z 325
 (a) The quantity in column A is greater (b) The quantity in column B is greater (c) The two quantities are equal (d) The relationship cannot be determined on the basis of the information supplied
• 2)

Deepa rides her car at 25 km/hr, She has to spend Rs. 2 per km on diesel and if she rides it at a faster speed of 40 km/hr, the diesel cost increases to Rs. 5 per km. She has Rs. 100 to spend on diesel. Let she travels x kms with speed 25 km/hr and y kms with speed 40 km/hr. The feasible region for the LPP is shown below:
Based on the above information, answer the following questions

Based on the above information, answer the following questions.
(i) What is the point of intersection of line l1 and l2,

 $\text { (a) }\left(\frac{40}{3}, \frac{50}{3}\right)$ $\text { (b) }\left(\frac{50}{3}, \frac{40}{3}\right)$ $\text { (c) }\left(\frac{-50}{3}, \frac{40}{3}\right)$ $\text { (d) }\left(\frac{-50}{3}, \frac{-40}{3}\right)$

(ii) The corner points of the feasible region shown in above graph are

 $\text { (a) }(0,25),(20,0),\left(\frac{40}{3}, \frac{50}{3}\right)$ $\text { (b) }(0,0),(25,0),(0,20)$ $\text { (c) }(0,0),\left(\frac{40}{3}, \frac{50}{3}\right),(0,20)$ $\text { (d) }(0,0),(25,0),\left(\frac{50}{3}, \frac{40}{3}\right),(0,20)$

(iii) If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point

 $\text { (a) }\left(\frac{50}{3}, \frac{40}{3}\right)$ (b) (0, 0) (c) (25, 0) (d) (0, 20)

(iv) If Z = 6x - 9y be the objective function, then maximum value of Z is

 (a) -20 (b) 150 (c) 180 (d) 20

(v) If Z = 6x + 3y be the objective function, then what is the minimum value of Z?

 (a) 120 (b) 130 (c) 0 (d) 150
• 3)

Corner points of the feasible region for an LPP are (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the objective function.
Based on the above information, answer the following questions.
(i) The minimum value of Z occurs at

 (a) (6, 8) (b) (5, 0) (c) (0, 3) (d) (0, 8)

(ii) Maximum value of Z occurs at

 (a) (5, 0) (b) (0, 8) (c) (0, 3) (d) (6, 8)

(iii) Maximum of Z - Minimumof Z =

 (a) 58 (b) 68 (c) 78 (d) 88

(iv) The corner points of the feasible region determined by the system of linear inequalities are

 (a) (0, 0), (-3, 0), (3, 2), (2, 3) (b) (3, 0), (3, 2), (2, 3), (0, -3) (c) (0, 0), (3, 0), (3, 2), (2, 3), (0, 3) (d) None of these

(v) The feasible solution of LPP belongs to

• 4)

Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18.
Based on the above information, answer the following questions.

(i) Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased atleast one of the given machines, then

 (a) x + y ≥  0 (b) x + y < 0 (c) x + y > 0 (d) x + y ≤ 0

(ii) Let the constraints in the given problem is represented by the following inequalities
x + y ≤ 20
360x + 240y ≤ 5760
x, y ≥ 0
Then which of the following point lie in its feasible region.

 (a) (0, 24) (b) (8, 12) (c) (20, 2) (d) None of these

(iii) If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at

 (a) (0, 0) (b) (16, 0) (c) (8, 12) (d) (0, 20)

(iv) Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation. of given problem. Then which of the following represent the coordinates of one of its corner points.

 (a) (0, 24) (b) (12, 8) (c) (8, 12) (d) (6, 14)

(v) If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

 (a) the required optimal solution is at the midpoint of the line joining two points. (b) the optimal solution occurs at every point on the line joining these two points. (c) the LPP under consideration is not solvable. (d) the LPP under consideration must be reconstructed.
• 5)

Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to, constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
Based on the above information, answer the following questions.
(i) Objective function of a L.P.P. is

 (a) a constant (b) a function to be optimised (c) a relation between the variables (d) none of these

(ii) Which of the following statement is correct?

 (a) Every LPP has at least one optimal solution. (b) Every LPP has a unique optimal solution. (c) If an LPP has two optimal solutions, then it has infinitely many solutions (d) none of these

(iii) In solving the LPP : "minimize f = 6x + 10y subject to constraints x ≥  6, Y ≥ 2, 2x + y  ≥ 10,x  ≥ 0,y ≥ 0" redundant constraints are

 (a) x ≥ 6, y ≥ 2 (b) 2x + y ≥ 10, x ≥ 0, y ≥ 0 (c) x ≥ 6 (d) none of these

(iv) The feasible region for a LPP is shown shaded in the figure. Let Z = 3x - 4y be the objective function. Minimum of Z occurs at

 (a) (0, 0) (b) (0, 8) (c) (5, 0) (d) (4, 10)

(v) The feasible region for a LPP is shown shaded in the figure. Let F = 3x - 4y be the objective function. Maximum value of F is

 (a) 0 (b) 8 (c) 12 (d) -18

#### Class 12th Maths - Probability Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A = {1, 5},B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw.

Based on the above information, answer the following questions.
(i) P (A I B) =

 $(a) \ \frac{1}{6}$ $(b) \ \frac{1}{3}$ $(c) \ \frac{1}{2}$ $(d) \ \frac{2}{3}$

(ii) P (B I C) =

 $(a) \ \frac{2}{3}$ $(b) \ \frac{1}{12}$ $(c) \ \frac{1}{9}$ $(d) \ 0$

(iii) P (A ⋂ B I C) =

 $(a) \ \frac{1}{6}$ $(b) \ \frac{1}{2}$ $(c) \ \frac{1}{12}$ $(d) \ \frac{1}{3}$

(iv) P (A I C) =

 $(a) \ \frac{1}{4}$ $(b) \ 1$ $(c) \ \frac{2}{3}$ (d) None of these

(v) P (A ∪ B I C) =

 $(a) \ 0$ $(b) \ \frac{1}{2}$ $(c) \ \frac{2}{3}$ $(d) \ 1$
• 2)

In a play zone, Aastha is playing crane game. It has 12 blue balls, 8 red balls, 10 yellow balls and 5 green balls. If Aastha draws two balls one after the other without replacement, then answer the following questions.

(i) What is the probability that the first ball is blue and the second ball is green?

 $(a) \ \frac{5}{119}$ $(b) \ \frac{12}{119}$ $(c) \ \frac{6}{119}$ $(d) \ \frac{5}{119}$

(ii) What is the probability that the first ball is yellow and the second ball is red?

 $(a) \ \frac{6}{119}$ $(b) \ \frac{8}{119}$ $(c) \ \frac{24}{119}$ (d) None of these

(iii) What is the probability that both the balls are red?

 $(a) \ \frac{4}{85}$ $(b) \ \frac{24}{595}$ $(c) \ \frac{12}{119}$ $(c) \ \frac{64}{119}$

(iv) What is the probability that the first ball is green and the second ball is not yellow?

 $(a) \ \frac{10}{119}$ $(b) \ \frac{6}{85}$ $(c) \ \frac{12}{119}$ (d) None of these

(v) What is the probability that both the balls are not blue?

 $(a) \ \frac{6}{595}$ $(b) \ \frac{12}{85}$ $(c) \ \frac{15}{17}$ $(d) \ \frac{253}{595}$
• 3)

Ajay enrolled himself in an online practice test portal provided by his school for better practice. Out of 5 questions in a set-I, he was able to solve 4 of them and got stuck in the one which is as shown below.

If A and B are independent events, P(A) = 0.6 and P(B) = 0.8, then answer the following questions.
(i) P (A $\cap$ B) =

 (a) 0.2 (b) 0.9 (c) 0.48 (d) 0.6

(ii) P (A $\cup$ B) =

 (a) 0.92 (b) 0.08 (c) 0.48 (d) 0.64

(iii) P (B | A) =

 (a) 0.14 (b) 0.2 (c) 0.6 (d) 0.8

(iv) P (A | B) =

 (a) 0.6 (b) 0.9 (c) 0.19 (d) 0.11

(v) P ( not A and not B ) =

 (a) 0.01 (b) 0.48 (c) 0.08 (d) 0.91
• 4)

A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35 and 0.1 if he comes by cab, metro, bike and other means of transport respectively.

Based on the above information, answer the following questions.
(i) When the doctor arrives late, what is the probability that he comes by metro?

 $(a) \ \frac{5}{4}$ $(b) \ \frac{2}{7}$ $(c) \ \frac{5}{21}$ $(d) \ \frac{1}{6}$

(ii) When the doctor arrives late, what is the probability that he comes by cab?

 $(a) \ \frac{4}{21}$ $(b) \ \frac{1}{7}$ $(c) \ \frac{5}{14}$ $(d) \ \frac{2}{21}$

(iii) When the doctor arrives late, what is the probability that he comes by bike?

 $(a) \ \frac{5}{21}$ $(b) \ \frac{4}{7}$ $(c) \ \frac{5}{6}$ $(d) \ \frac{1}{6}$

(iv) When the doctor arrives late, what is the probability that he comes by other means of transport?

 $(a) \ \frac{6}{7}$ $(b) \ \frac{5}{14}$ $(c) \ \frac{4}{21}$ $(d) \ \frac{2}{7}$

(v) What is the probability that the doctor is late by any means?

 $(a) \ 1$ $(b) \ 0$ $(c) \ \frac{1}{2}$ $(d) \ \frac{1}{4}$
• 5)

Suman was doing a project on a school survey, on the average number of hours spent on study by students selected at random. At the end of-survey, Suman prepared the following report related to the data. Let X denotes the average number of hours spent on study by students. The probability that X can take the values x, has the following form, where k is some unknown constant.
$P(X=x)=\left\{\begin{array}{l} 0.2, \text { if } x=0 \\ k x, \text { if } x=1 \text { or } 2 \\ k(6-x), \text { if } x=3 \text { or } 4 \\ 0, \text { otherwise } \end{array}\right.$

Based on the above information, answer the following questions.
(i) Find the value of k.

 (a) 0.1 (b) 0.2 (c) 0.3 (d) 0.05

(ii) What is the probability that the average study time of students is not more than 1 hour?

 (a) 0.4 (b) 0.3 (c) 0.5 (d) 0.1

(iii) What is the probability that the average study time of students is at least 3 hours?

 (a) 0.5 (b) 0.9 (c) 0.8 (d) 0.1

(iv) What is the probability that the average study time of students is exactly 2 hours?

 (a) 0.4 (b) 0.5 (c) 0.7 (d) 0.2

(v) What is the probability that the average study time of students is at least 1 hour?

 (a) 0.2 (b) 0.4 (c) 0.8 (d) 0.6

#### Class 12th Maths - Vector Algebra Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Ritika starts walking from his house to shopping mall. Instead of going to the mall directly, she first goes to a ATM, from there to her daughter's school and then reaches the mall. In the diagram, A, B, C and D represent the coordinates of House, ATM, School and Mall respectively.

Based on the above information, answer the following questions.
(i) Distance between House (A) and ATM (B) is

 (a) 3 units (b) 3$\sqrt 2$ units (c) $\sqrt 2$units (d) 4$\sqrt 2$ units

(ii) Distance between ATM (B) and School (C) is

 (a) $\sqrt 2$ units (b) 2$\sqrt 2$ units (c) 3$\sqrt 2$ units (d) 4$\sqrt 2$ units

(iii) Distance. between School (C) and Shopping mall (D) is

 a) 3$\sqrt 2$ units (b) 5$\sqrt 2$ units (c) 7$\sqrt 2$ units (d) 10$\sqrt 2$ units

(iv) What is the total distance travelled by Ritika ?

 a) 4$\sqrt 2$ units (b) 6$\sqrt 2$ units (c) 8$\sqrt 2$ units (d) 9$\sqrt 2$ units

(v) What is the extra distance travelled by Ritika in reaching the shopping mall?

 a) 3$\sqrt 2$ units (b) 5$\sqrt 2$ units (c) 6$\sqrt 2$ units (d) 7$\sqrt 2$ units
• 2)

Ginni purchased an air plant holder which is in the shape of a tetrahedron.
Let A, B, C and D are the coordinates of the air plant holder where A $\equiv$ (1, 1, 1), B $\equiv$ (2, 1, 3), C $\equiv$ (3, 2, 2) and D $\equiv$(3, 3, 4).

Based on the above information, answer the following questions.
(i) Find the position vector of $\overrightarrow{A B}$.

 (a) $-\hat{i}-2 \hat{k}$ (b) $2 \hat{i}+\hat{k}$ (c) $\hat{i}+2 \hat{k}$ (d)$-2 \hat{i}-\hat{k}$

(ii) Find the position vector of $\overrightarrow{A C}$.

 (a) $2 \hat{i}-\hat{j}-\hat{k}$ (b) $2 \hat{i}+\hat{j}+\hat{k}$ (c) $-2 \hat{i}-\hat{j}+\hat{k}$ (d) $\hat{i}+2 \hat{j}+\hat{k}$

(iii) Find the position vector of $\overrightarrow{AD} .$.

 (a) $2 \hat{i}-2 \hat{j}-3 \hat{k}$ (b) $\hat{i}+\hat{j}-3 \hat{k}$ (c) $3 \hat{i}+2 \hat{j}+2 \hat{k}$ (d) $2\hat{i}+2 \hat{j}+3 \hat{k}$

(iv) Area of $\Delta A B C$

 (a) $\frac{\sqrt{11}}{2} \mathrm{sq .units}$ (b) $\frac{\sqrt{14}}{2} sq. units$ (c) $\frac{\sqrt{13}}{2}$ (d)$\frac{\sqrt{17}}{2} \mathrm{sq .units}$

(v) Find the unit vector along $\overrightarrow{AD} .$

 (a) $\frac{1}{\sqrt{17}}(2 \hat{i}+2 \hat{j}+3 \hat{k})$ (b)$\frac{1}{\sqrt{17}}(3 \hat{i}+3 \hat{j}+2 \hat{k})$ (c) $\frac{1}{\sqrt{11}}(2 \hat{i}+2 \hat{j}+3 \hat{k})$ (d) $(2 \hat{i}+2 \hat{j}+3 \hat{k})$
• 3)

Geetika's house is situated at Shalimar Bagh at point 0, for going to Aloks house she first travels 8 km by bus in the East. Here at point A, a hospital is situated. From Hospital, Geetika takes an auto and goes 6 km in the North, here at point B school is situated. From school, she travels by bus to reach Aloks house which is at 30° East, 6 km from point B.

Based on the above information, answer the following questions.
(i) What is the vector distance between Geetikas house and school ?

 (a) $8 \hat{i}-6 \hat{j}$ (b) $8 \hat{i}+6 \hat{j}$ (c) $8 \hat{i}$ (d) $6 \hat{j}$

(ii) How much distance Geetika travels to reach school?

 (a) 14 km (b) 15 km (c) 16 km (d) 17 km

(iii) What is the vector distance from school to Alok's house ?

 (a) $\sqrt{3} \hat{i}+\hat{j}$ (b) $3 \sqrt{3} \hat{i}+3 \hat{j}$ (c) $6 \hat{i}$ (d) $6 \hat{j}$

(iv) What is the vector distance from Geetikas house to Alok's house?

 (a) $(8+3 \sqrt{3}) \hat{i}+9 \hat{j}$ (b) $4\hat{i}+6 \hat{j}$ (c) $15 \hat{i}$ (d) $16 \hat{j}$

(v) What is the total distance travelled by Geetika from her house to Alok's house?

 (a) 19 km (b) 20 km (c) 21 km (d) 22 km
• 4)

Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below).
Team A pulls with force F1$\hat{4}+\hat{0} \hat{j}$ KN
Team B ⟶ F2$-2 \hat{i}+4 \hat{j}$ KN
Team C ⟶ F3 = $-3 \hat{i}-3 \hat{j}$ KN

Based on the above information, answer the following questions.
(i) Which team will win the game ?

 (a) Team B (b) Team A (c) Team C (d) No one

(ii) What is the magnitude of the teams combined force ?

 (a) 7 KN (b) 1.4 KN (c) 1.5 KN (d) 2 KN

(iii) In what direction is the ring getting pulled?

(iv) What is the magnitude of the force of Team B?

 (a) 2$\sqrt 5$ KN (b) 6 KN (c) 2 KN (d) $\sqrt 6$ KN

(v) How many KN force is applied by Team A?

 (a) 5 KN (b) 4 KN (c) 2 KN (d) 16 KN
• 5)

A plane started from airport situated at 0 with avelocity of 120 m/s towards east. Air is blowing at a velocity of 50 m/s towards the north as shown in the figure.
The plane travelled 1 hr in OP direction with the resultant velocity. From P to R the plane travelled 1 hr keeping velocity of 120 m/s and finally landed at R.

Based on the above information, answer the following questions.
(i) What is the resultant velocity from O to P?

 (a) 100 m/s (b) 130 m/s (c) 126 m/s (d) 180 m/s

(ii) What is the direction of travel of plane from O to P with East?

 (a) $\tan ^{-1}\left(\frac{5}{12}\right)$ (b) $\tan ^{-1}\left(\frac{12}{3}\right)$ (c) 50 (d) 80

(iii) What is the displacement from O to P?

 (a) 600 km (b) 468 km (c) 532 km (d) 500 km

(iv) What is the resultant velocity from P to R?

 (a) 120  m/s (b) 70 m/s (c) 170 m/s (d) 200 m/s

(v) What is the displacement from P to R?

 (a) 450 km (b) 532 km (c) 610 km (d) 612 km

#### Class 12th Maths - Differential Equations Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

In a college hostel accommodating 1000 students, one of the hostellers came in carrying Corona virus, and the hostel was isolated. The rate at which the virus spreads is assumed to be proportional to the product of the number of infected students and remaining students. There are 50 infected students after 4 days

Based on the above information, answer the following questions
(i) If n(t) denote the number of students infected by Corona virus at any time t, then maximum value of n(t) is

 (a) 50 (b) 100 (c) 500 (d) 1000

(ii) $\frac{d n}{d t}$ is proportional to

 (a) n(100-n) (b) n(100+ n) (c) n(100 - n) (d) n(100 + n)

(iii) The value of n( 4) is

 (a) 1 (b) 50 (c) 100 (d) 1000

(iv) The most general solution of differential equation formed in given situation is

 (a) $\frac{1}{1000} \log \left(\frac{1000-n}{n}\right)=\lambda t+c$ (b)  $\log \left(\frac{n}{100-n}\right)=\lambda t+c$ (c) $\frac{1}{1000} \log \left(\frac{n}{1000-n}\right)=\lambda t+c$ (d) None of these

(v) The value of n at any time is given by

 (a) $n(t)=\frac{1000}{1+999 e^{-0.9906 t}}$ (b) $n(t)=\frac{1000}{1-999 e^{-0.9906 t}}$ (c) $n(t)=\frac{100}{1-999 e^{-0.996 t}}$ (d) $n(t)=\frac{100}{999+e^{1000 t}}$
• 2)

A thermometer reading 800P is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50of At any time t the thermometer reading be TOP and the outside temperature be SoF.
Based on the above information, answer the following questions.
(i) If $\lambda$ is positive constant of proportionality, then $\frac{d T}{d t}$ is

 (a) $\lambda(T-S)$ (b) $\lambda(T+S)$ (c) $\lambda T S$ (d) $-\lambda(T-S)$

(ii) The value of T(S) is

 (a) 300F (b) 40oF (c) 50oF (d) 60oF

(iii) The value of T(10) is

 (a) 50oF (b) 40oF (c) 50oF (d) 60oF

(iv) Find the general solution of differential equation formed in given situation.

 (a) logT=St+c (b) $\log (T-S)=-\lambda t+c$ (c) log S = tT + c (d) $\log (T+S)=\lambda t+c$

(v) Find the valiie of constant of integration c in the solution of differential equation formed in given situation.

 (a) log (60 -S) (b) log (80 + S) (c) log (80 - S) (d) log (60 + S)
• 3)

It is known that, if the interest is compounded continuously, the principal changes at the rate equal to the produd' of the rate of bank interest per annum and the principal. Let P denotes the principal at any time t and rate of interest be r % per annum.

Based on the above information, answer the following questions.
(i) Find the value of $\frac{d P}{d t}$ .

 (a) $\frac{\operatorname{Pr}}{1000}$ (b) $\frac{P r}{100}$ (c) $\frac{\operatorname{Pr}}{10}$ (d) Pr

(ii) fPo be the initial principal, then find the solution of differential equation formed in given situation.

 (a) $\log \left(\frac{P}{P_{0}}\right)=\frac{r t}{100}$ (b) $\log \left(\frac{P}{P_{0}}\right)=\frac{r t}{10}$ (c) $\log \left(\frac{P}{P_{0}}\right)=r t$ (d) $\log \left(\frac{P}{P_{0}}\right)=100 r t$

(iii) If the interest is compounded continuously at 5% per annum, in how many years will Rs. 100 double itself?

 (a) 12.728 years (b) 14.789 years (c) 13.862 years (d) 15.872 years

(iv) At what interest rate will Rs.100 double itself in 10 years? (log e2 = 0.6931).

 (a) 9.66% (b) 8.239% (c) 7.341% (d) 6.931%

(v) How much will Rs. 1000 be worth at 5% interest after 10 years? (e0.5 = 1.648).

 (a) Rs. 1648 (b) Rs. 1500 (c) Rs. 1664 (d) Rs. 1572
• 4)

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective measured the body temperature and found it to be 70°F. Two hours later, the detective measured the body temperature again and found it to be 60°F, where the room temperature is 50°F. Also, it is given the body temperature at the time of death was normal, i.e., 98.6°F.
Let T be the temperature of the body at any time t and initial time is taken to be 8 p.m.

Based on the above information, answer the following questions.
(i) By Newton's law of cooling,$\frac{d T}{d t}$ is proportional to

 (a) T - 60 (b) T - 50 (c) T - 70 (d) T - 98.6

(ii) When t = 0, then body temperature is equal to

 (a) 50°F (b) 60°F (c) 70oF (d) 98.6°F

(iii) When t = 2, then body temperature is equal to

 (a) 50°F (b) 60°F (c) 70oF (d) 98.6°F

(iv) The value of T at any time tis

 (a) $50+20\left(\frac{1}{2}\right)^{t}$ (b) $50+20\left(\frac{1}{2}\right)^{t-1}$ (c) $50+20\left(\frac{1}{2}\right)^{t / 2}$ (d) None of these

(v) If it is given that loge(2.43) = 0.88789 and loge(0.5) = -0.69315, then the time at which the murder occur is

 (a) 7:30 p.m. (b) 5:30 p.m. (c) 6:00 p.m. (d) 5:00 p.m.
• 5)

A rumour on whatsapp spreads in a population of 5000 people at a rate proportional to the product of the number of people who have heard it and the number of people who have not. Also, it is given that 100 people initiate the rumour and a total of 500 people know the rumour after 2 days.

Based on the above information, answer the following questions
(i) If yet) denote the number of people who know the rumour at an instant t, then maximum value of yet) is

 (a) 500 (b) 100 (c) 5000 (d) none of these

(ii) $\frac{d y}{d t}$ is proptional to

 (a) (y - 5000) (b) y(y - 500) (c) y(500 - y) (d) y(5000 - y)

(iii) The value of y(0) is

 (a) 100 (b) 500 (c) 600 (d) 200

(iv) The value of y(2) is

 (a) 100 (b) 500 (c) 600 (d) 200

(v) The value of y at any time t is given by

 (a) $y=\frac{5000}{e^{-5000 k t}+1}$ (b) $y=\frac{5000}{1+e^{5000 k t}}$ (c) $y=\frac{5000}{49 e^{-5000 k t}+1}$ (d) $y=\frac{5000}{49\left(1+e^{-5000 k t}\right)}$

#### Class 12th Maths - Application of Integrals Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Consider the following equations of curves : x?- = y and y = x.
On the basis of above information, answer the following questions.
(i) The point(s) of intersection of both the curves is (are)

 (a) (0,0), (2, 2) (b) (0,0), (1, 1) (c) (0,0), (-1, -1) (d) (0,0), (-2, -2)

(ii) Area bounded by the curves is represented by which of the following graph?

(iii) The value of the integral $\int_{0}^{1} x d x$ is

 (a) 1/4 (b) 1/3 (c) 1/2 1

(iv) The value of the integral $\int_{0}^{1} x^{2} d x$

 (a) 1/4 (b) 1/3 (c) 1/2 1

(v) The value of area bounded by the curves x?- = y and x = y is

 (a) $\frac{1}{6} \mathrm{sq} . \text { unit }$ (b) $\frac{1}{3} \text { sq. unit }$ (c) $\frac{1}{2} \mathrm{sq} . \text { unit }$ (d) 1 sq. unit
• 2)

Consider the curve x2 +y2 = 16 and line y = x in the first quadrant. Based on the above information, answer the following questions.
(i) Point of intersection of both the given curves is

 (a) (0, 4) (b) $(0,2 \sqrt{2})$ (c) $(2 \sqrt{2}, 2 \sqrt{2})$ (d) $(2 \sqrt{2}, 4)$

(ii) Which of the following shaded portion represent the area bounded by given two curves?

(iii) The value of the integral $\int_{0}^{2 \sqrt{2}} x d x$ is

 (a) 0 (b) 1 (c) 2 (d).4

(iv) The value of the integral $\int_{2 \sqrt{2}}^{4} \sqrt{16-x^{2}} d x$ is

 (a) $2(\pi-2)$ (b) $2(\pi-8)$ (c) $4(\pi-2)$ (d) $4(\pi+2)$

(v) Area bounded by the two given curves is

 (a) $3 \pi \text { sq. units }$ (b) $\frac{\pi}{2} \text { sq. units }$ (c) $\pi \text { sq. units }$ (d) $2 \pi \text { sq. units }$
• 3)

A child cut a pizza with a knife. Pizza is circular in shape which is represented by knife represents a straight line given by $x=\sqrt{3} y$ .
Based on the above information, answer the following questions

(i) The point(s) of intersection of the edge of knife (line) and pizza shown in the figure is (are)

 (a) $(1, \sqrt{3}),(-1,-\sqrt{3})$ (b) $(\sqrt{3}, 1),(-\sqrt{3},-1)$ (c) $(\sqrt{2}, 0),(0, \sqrt{3})$ (d) $(-\sqrt{3}, 1),(1,-\sqrt{3})$

(ii) Which of the following shaded portion represent the smaller area bounded by pizza and edge of knife in first quadrant?

(iii) Value of area of the region bounded by circular pizza and edge of knife in first quadrant is

 (a) $\frac{\pi}{2} \text { sq. units }$ (b) $\frac{\pi}{3} \text { sq. units }$ (c) $\frac{\pi}{5} \text { sq. units }$ (d) $\pi \text { sq. units }$

(iv) Area of each slice of pizza when child cut the pizza into 4 equal pieces is

 (a) $\pi \text { sq. units }$ (b) $\frac{\pi}{2} \mathrm{sq} . \text { units }$(c) $3 \pi \text { sq. units }$ (d) $2 \pi \text { sq. units }$

(v) Area of whole pizza is

 (a) $3 \pi \text { sq. units }$ (b) $2 \pi \text { sq. units }$ (c) $5 \pi \text { sq. units }$ (d) $4 \pi \text { sq. units }$
• 4)

Consider the following equation of curve I' = 4x and straight line x + y = 3.
Based on the above information, answer the following questions.
(i) The line x + y = 3 cuts the x-axis and y-axis respectively at

 (a) (0, 2), (2, 0) (b) (3, 3), (0, 0) (c) (0, 3), (3, 0) (d) (3, 0), (0, 3)

(ii) Point(s) of intersection of two given curves is (are)

 (a) (1, -2), (-9, 6) (b) (2, 1), (-6, 9) (c) (1, 2), (9, -6) (d) None of these

(iii) Which of the following shaded portion represent the area bounded by given curves?

(iv) Value of the integral $\int_{-6}^{2}(3-y) d y$ is

 (a) 10 (b) 20 (c) 30 (d) 40

(v) Value of area bounded by given curves is

 (a) 56 sq. units (b) $\frac{63}{5} \text { sq; units }$ (c) $\frac{64}{3} \text { sq. units }$ (d) 31 sq. units
• 5)

In a classroom, teacher explains the properties of a particular curve by saying that this particular curve has
beautiful up and downs. It starts at 1 and heads down until rt radian, and then heads up again and closely related to sine function and both follow each other, exactly $\frac{\pi}{2}$ radians apart as shown in figure.

Based on the above information, answer the following questions
(i) Name the curve, about which teacher explained in the classroom

 (a) cosine (b) sine (c) tangent (d) cotangent

(ii) Area of curve explained in the passage from 0 to $\frac{\pi}{2}$ is

 (a) $\frac{1}{3} \mathrm{sq} . \text { unit }$ (b) $\frac{1}{2} \mathrm{sq} . \text { unit }$ (c) 1 sq. unit (d) 2 sq. units

(iii) Area of curve discussed in classroom from $\frac{\pi}{2} \text { to } \frac{3 \pi}{2}$ is

 (a) -2 sq. units (b) 2 sq. units (c) 3 sq. units (d) -3 sq. units

(iv) Area of curve discussed in classroom from $\frac{3 \pi}{2} \text { to } 2 \pi$ is

 (a) 1 sq. unit (b) 2 sq. units (c) 3 sq. units (d) 4 sq. units

(v) Area of explained curve from 0 to $2 \pi$ is

 (a) 1 sq. unit (b) 2 sq. units (c) 3 sq. units (d) 4 sq. units

#### Class 12th Maths - Application of Derivatives Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Megha wants to prepare a handmade gift box for her friend's birthday at home. For making lower part of box, she takes a square piece of cardboard of side 20 cm.

Based on the above information, answer the following questions.
(i) If x cm be the length of each side of the square cardboard which is to be cut off from corners of the square piece of side 20 cm, then possible value of x will be given by the interval

 (a) [0, 20] (b) (0, 10) (c) (0, 3) (d) None of these

(ii) Volume of the open box formed by folding up the cutting corner can be expressed as

 (a) V = x(20 - 2x)(20 - 2x) (b) $$$V=\frac{x}{2}(20+x)(20-x)$$$ (c) $$$V=\frac{x}{3}(20-2 x)(20+2 x)$$$ (d) V = x(20 - 2x)(20 - x)

(iii) The values of x for which $$$\frac{d V}{d x}=0$$$ ,are

 (a) 3, 4 (b)  $$$0, \frac{10}{3}$$$ (c) 0, 10 (d) $$$10, \frac{10}{3}$$$

(iv) Megha is interested in maximising the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum?

 (a) 12 cm (b) 8 cm (c) $$$\frac{10}{3} \mathrm{~cm}$$$ (d) 2 cm

(v) The maximum value of the volume is

 (a) $$$\frac{17000}{27} \mathrm{~cm}^{3}$$$ (b) $$$\frac{11000}{27} \mathrm{~cm}^{3}$$$ (c) $$$\frac{8000}{27} \mathrm{~cm}^{3}$$$ (d) $$$\frac{16000}{27} \mathrm{~cm}^{3}$$$
• 2)

Shobhit's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in' figure. He has 200 ft of wire fencing.

Based on the above information, answer the following questions.
(i) To construct a garden using 200 ft of fencing, we need to maximise its

 (a) volume (b) area (c) perimeter (d) length of the side

(ii) If x denote the length of side of garden perpendicular to brick wall and y denote the length, of side parallel to brick wall, then find the relation representing total amount of fencing wire.

 (a) x + 2y = 150 (b) x+2y=50 (c) y+2x=200 (d) y+2x=100

(iii) Area of the garden as a function of x, say A(x), can be represented as

 (a) 200 + 2x2 (b) x - 2x2 (c) 200x - 2x2 (d) 200-x2

(iv) Maximum value of A(x) occurs at x equals

 (a) 50 ft (b) 30 ft (c) 26ft (d) 31 ft

(v) Maximum area of garden will be

 (a) 2500 sq.ft (b) 4000 sq.ft (c) 5000 sq.ft (d) 6000 sq. ft
• 3)

The Government declare that farmers can get Rs.300 per quintal for their onions on 1st July and after that,the price will be dropped by Rs. 3 per quintal per extra day.
Shyams father has 80 quintal of onions in the field on 1st July and he estimates that crop is increasing at the rate of 1 quintal per day.

Based on the above information, answer the following questions.
(i) If x is the number of days after 1st July, then price and quantity ofonion respectively can be expressed as

 (a) Rs. (300 - 3x), (80 + x) quintals (b) Rs. (300 - 3x), (80 - x) quintals (c) Rs. (300 + x), 80 quintals (d) None of these

(ii) Revenue R as a function of x can be represented as

 (a) R(x) = 3x2 - 60x - 24000 (b) R(x) = -3x2 + 60x + 24000 (c) R(x) = 3x2 + 40x - 16000 (d) R(x) = 3x2- 60x - 14000

(iii) Find the number of days after 1stJuly, when Shyams father attain maximum revenue.

 (a) 10 (b) 20 (c) 12 (d) 22

(iv) On which day should Shyam's father harvest the onions to maximise his revenue?

 (a) 11thuly (b) 20th July (c) 12th July (d) 22nd July

(v) Maximum revenue is equal to

 (a) Rs. 20,000 (b) Rs. 24,000 (c) Rs. 24,300 (d) Rs. 24,700
• 4)

An owner of an electric bi~e rental company have determined that if they charge customers Rs. x per day to rent a bike, where 50 Rs. x Rs. 200, then number of bikes (n), they rent per day can be shown by linear function n(x) = 2000 - 10x. If they charge Rs. 50 per day or less, they will rent all their bikes. If they charge Rs. 200 or more per day, they will not rent any bike. Based on the above information, answer the following questions.

Based on the above information, answer the following questions
(i) Total revenue R as a function of x can be represented as

 (a) 2000x - 10x2 (b) 2000x + 10x2 (c) 2000 - 10x (d) 2000 - 5x2

(ii) If R(x) denote the revenue, then maximum value of R(x) occur when x equals

 (a) 10 (b) 100 (c) 1000 (d) 50

(iii) At x = 260, the revenue collected by the company is

 (a) Rs. 10 (b) Rs. 500 (c) Rs. 0 (d) Rs. 1000

(iv) The number of bikes rented per day, if x = 105 is

 (a) 850 (b) 900 (c) 950 (d) 1000

(v) Maximum revenue collected by company is

 (a) Rs. 40,000 (b) Rs. 50,000 (c) Rs. 75,000 (d) Rs. 1,00,000
• 5)

Mr. Sahil is the owner of a high rise residential society having 50 apartments. When he set rent at Rs. 10000/month, all apartments are rented. If he increases rent by Rs. 250/ month, one fewer apartment is rented. The maintenance cost for each occupied unit is Rs. 500/month. Based on the above information answer the following questions.

Based on the above information answer the following questions.
(i) If P is the rent price per apartment and N is the number of rented apartment, then profit is given by

 (a) NP (b) (N - 500)P (c) N(P - 500) (d) none of these

(ii) If x represent the number of apartments which are not rented, then the profit expressed as a function of x is

 (a) (50 - x) (38 + x) (b) (50 + x) (38 - x) (c) 250(50 - x) (38 + x) (d) 250(50 + x) (38 - x)

(iii) If P = 10500, then N =

 (a) 47 (b) 48 (c) 49 (d) 50

(iv) If P = 11,000, then the profit is

 (a) Rs. 11000 (b) Rs. 11500 (c) Rs. 15800 (d) Rs.16500

#### Class 12th Maths - Continuity and Differentiability Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

Let f(x) be a real valued function, then its
Left Hand Derivative (L.H.D.) : $$$\mathrm{L} f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}$$$
Right Hand Derivative (R.H.D.) : $$$\mathrm{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$$$
Also, a function jfx) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal
For the function $$$f(x)=\left\{\begin{array}{l} |x-3|, x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, x<1 \end{array}\right.$$$ answer the following questions
(i) R.H.D. of f(x) at x = 1is

 (a) 1 (b) -1 (c) 0 (d) 2

(ii) L.H.D. of f(x) at x = 1 is

 (a) 1 (b) -1 (c) 0 (d) 2

(iii) f(x) is non-differentiable at

 (a) x = 1 (b) x = 2 (c) x = 3 (d) x = 4

(iv) Find the value of f'(2).

 (a) 1 (b) 2 (c) 3 (d) -1

(v) The value of f'( -1) is

 (a) 2 (b) 1 (c) -2 (d) -1
• 2)

Let x = f(t) and y = get) be parametric forms with t as a parameter,
then $$$\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)}$$$ ,where $$$f^{\prime}(t) \neq 0$$$ $$$\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)}$$$ where $$$f^{\prime}(t) \neq 0$$$.
(i) The derivative off (tanx) w.r.t. $$$g(\sec x) \text { at } x=\frac{\pi}{4}$$$ ,where f'(1) and $$$g^{\prime}(\sqrt{2})=4$$$ is

 (a) $$$\frac{1}{\sqrt{2}}$$$ (b) $$$\sqrt{2}$$$ (c) 1 (d) 0

(ii) The derivate of $$$\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$$$ ,with respect to $$$\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$$$ is

 (a) -1 (b) 1 (c) 2 (d) 4

(iii) The derivative of $$$e^{x^{3}}$$$ with respect to log x is

 (a) $$$e^{x^{3}}$$$ (b) $$$3 x^{2} 2 e^{x^{3}}$$$ (c) $$$3 x^{3} e^{x^{3}}$$$ (d) $$$3 x^{2} e^{x^{3}}+3 x$$$

(iv) The derivative of $$$\cos ^{-1}\left(2 x^{2}-1\right)$$$ w.r.t. cos-1x is

 (a) 2 (b) $$$\frac{-1}{2 \sqrt{1-x^{2}}}$$$ (c) $$$\frac{2}{x}$$$ (d) 1 -x2

(v) If $$$y=\frac{1}{4} u^{4}$$$  and $$$u=\frac{2}{3} x^{3}+5$$$ then $$$\frac{d y}{d x}=$$$

 (a) $$$\frac{2}{27} x^{2}\left(2 x^{3}+15\right)^{3}$$$ (b) $$$\frac{2}{7} x^{2}\left(2 x^{3}+15\right)^{3}$$$ (c) $$$\frac{2}{27} x\left(2 x^{3}+5\right)^{3}$$$ (d) $$$\frac{2}{7}\left(2 x^{3}+15\right)^{3}$$$
• 3)

Let $$$f: A \rightarrow B$$$ and $$$g: B \rightarrow C$$$ be two functions defined on non-empty sets A, B, C,
then $$$\text { gof }: A \rightarrow C$$$  be is called the composition off and g defined as, $$$g o f(x)=g\{f(x)\} \forall x \in A$$$ .
Consider the functions $$$f(x)=\left\{\begin{array}{ll} \sin x, & x \geq 0 \\ 1-\cos x, & x \leq 0 \end{array}, g(x)=e^{x}\right.$$$ and
(i) The function gof(x) is defined as

 (a) $$$g o f(x)=\left\{\begin{array}{ll} e^{x} & , x \geq 0 \\ 1-e^{\cos x} & , x \leq 0 \end{array}\right.$$$ (b) $$$\operatorname{gof}(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \leq 0 \\ e^{1-\cos x} & , x \geq 0 \end{array}\right.$$$ (c) $$$g o f(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \leq 0 \\ 1-e^{\cos x} & , x \geq 0 \end{array}\right.$$$ (d)  $$$g o f(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \geq 0 \\ e^{1-\cos x} & , x \leq 0 \end{array}\right.$$$

(ii) $$$\frac{d}{d x}\{\operatorname{gof}(x)\}=$$$

 (a) $$$[g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ e^{1-\cos x} \cdot \sin x & , x \leq 0 \end{array}\right.$$$ (b) $$$[g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ -\sin x \cdot e^{1-\cos x} & , x \leq 0 \end{array}\right.$$$ (c) $$$[g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ \sin x \cdot(1-\cos x) & , x \leq 0 \end{array}\right.$$$ (d) $$$[g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ (1-\sin x) \cdot e^{1-\cos x} & , x \leq 0 \end{array}\right.$$$

(iii) R.H.D. of gof(x) at x = 0 is

 (a) 0 (b) 1 (c) -1 (d) 2

(iv) L.H.D. of gof(x) at x = 0 is

 (a) 0 (b) 1 (c) -1 (d) 2

(v) The value of $$$f^{\prime}(x) \text { at } x=\frac{\pi}{4}$$$ is

 (a) 1/9 (b) $$$1 / \sqrt{2}$$$ (c) 1/2 (d) not defined
• 4)

The function f(x) will be discontinuous at x = a if f(x) has
(a) Discontinuity of first kind $$$\lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h)$$$ both exist but are not equal. If is also known as irremovable discontinuity.
(b)  Discontinuity of second kind: If none of the limits $$$\lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h)$$$ exist.
(c) Removable discontinuity: $$$\lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h)$$$ both exist and equal but not equal to f(a).
Based on the above information, answer the following questions.
(i) If $$$f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3}, & \text { for } x \neq 3 \\ 4, & \text { for } x=3 \end{array}\right.$$$ ,then at x= 3

 (a) f has removable discontinuity (b) f is continuous (c) f has irremovable discontinuity (d) none of these

(ii) Let $$$f(x)=\left\{\begin{array}{ll} x+2, & \text { if } x \leq 4 \\ x+4, & \text { if } x>4 \end{array}\right.$$$ ,then at x = 4

 (a) f is continuous (b) f has removable discontinuity (c) f has irremovable discontinuity (d) none of these

(iii) Consider the function f(x) defined $$$f(x)=\left\{\begin{array}{l} \frac{x^{2}-4}{x-2} \\ 5 \end{array}\right.$$$, for $$$x \neq 2$$$

 (a) f has removable discontinuity (b) f has irremovable discontinuity (c) f is continuous (d) f is continuous if f(2) = 3

(iv) If $$$f(x)=\left\{\begin{array}{cc} \frac{x-|x|}{x}, & if\ x \neq 0 \\ 2, & if\ x=0 \end{array}\right.$$$ ,then x = 0

 a) f is continuous (b) f has removable discontinuity (c) f has irremovable discontinuity (d) none of these

(v) If $$$f^{\prime}(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log (1+2 x)}, & \text { if } x \neq 0 \\ 7, & \text { if } x=0 \end{array}\right.$$$, then at x = 0

 (a) f is continuous if f(0) = 2 (b) f is continuous (c) f has irremovable discontinuity (d) f has removable discontinuity
• 5)

If a real valued function f(x) is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
For example, every polynomial. constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
Based on the above information, answer the following questions.
(i) If $$$f(x)=\left\{\begin{array}{l} x, \text { for } x \leq 0 \\ 0, \text { for } x>0 \end{array}\right.$$$ , then at x = 0

 (a) f(x) is differentiable and continuous (b) j(x) is neither continuous nor differentiable (c) f(x) is continuous but not differentiable (d) none of these

(ii) If $$$f(x)=|x-1|, x \in R$$$ ,then at x= 1

 (a) f(x) is not continuous (b) f(x) is continuous but not differentiable (c) f(x) is continuous and differentiable (d) none of these

(iii) f(x) = x3 is

 (a) continuous but not differentiable at x = 3 (b) continuous and differentiable at x = 3 (c) neither continuous nor differentiable at x = 3 (d) none of these

(iv) f(x) = [sin x], then which of the following is true?

 (a) j(x) is continuous and differentiable at x = o. (b) j(x) is discontinuous at x = o. (c) j(x) is continuous at x = 0 but not differentiable (d) fix) is differentiable but not continuous at  $$$x=\pi / 2$$$

(v) If f(x) = sin-1x, $$$-1 \leq x \leq 1$$$, then

 (a) f(x) is both continuous and differentiable (b) f(x) is neither continuous nor differentiable. (c) f(x) is continuous but not differentiable (d) None of these

#### Class 12th Maths - Determinants Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

A company produces three products every day. Their production on certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product.

Using the concepts of matrices and determinants, answer the following questions.
(i) If x, y and z respectively denotes the quantity (in tons) of first, second and third product produced, then which of the following is true?

 (a)  x + y + z = 45 (b)  x + 8 = z (c)  -2y+z=0 (d) all of these

(ii) If $\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & -1 & 1 \end{array}\right)^{-1}=\frac{1}{6}\left(\begin{array}{ccc} 2 & 2 & 2 \\ 3 & 0 & -3 \\ 1 & -2 & 1 \end{array}\right)$ , then the inverse of $\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)$ is

 (a) $\left(\begin{array}{lll} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{2} & 0 & \frac{-1}{2} \\ \frac{1}{6} & \frac{-1}{3} & \frac{1}{6} \end{array}\right)$ (b) $\left(\begin{array}{ccc} \frac{1}{2} & 0 & -\frac{1}{2} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{6} & \frac{-1}{3} & \frac{1}{6} \end{array}\right)$ (c) $\left(\begin{array}{ccc} \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ \frac{1}{3} & 0 & \frac{-1}{3} \\ \frac{1}{3} & \frac{-1}{2} & \frac{1}{6} \end{array}\right)$ (d) none of these

(iii) x :y : z is equal to

 (a) 12: 13: 20 (b)  11:15:19 (c) 15: 19: 11 (d)  13: 12: 20

(iv) Which of the following is not true?

 (a) IAI = IA'I (b) (A'rl = (A-I), (c) A is skew symmetric-matrix of odd order, then IAI = 0 (d) IABI = IAI + IBI
• 2)

If there is a statement involving the natural number n such that
(i) The statement is true for n = 1
(ii) When the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1.
Then, the statement is true for all natural numbers n.
Also, if A is a square matrix of order n, then A 2 is defined as AA. In general, Am = AA .... A (m times), where m is any positive integer.
Based on the above information, answer the following questions.
(i) If $A=\left[\begin{array}{ll} 3 & -4 \\ 1 & -1 \end{array}\right]$,then for any positive integer n,

 (a) $A^{n}=\left[\begin{array}{cc} 3 n & -4 n \\ n & -n \end{array}\right]$ (b) $A^{n}=\left[\begin{array}{cc} 1+2 n & -4 n \\ n & 1-2 n \end{array}\right]$ (c) $A^{n}=\left[\begin{array}{cc} 3 n & -8 n \\ 1 & -n \end{array}\right]$ (d) $A^{n}=\left[\begin{array}{cc} 1+3 n & -4 n \\ n & 1-3 n \end{array}\right]$

(ii) If $A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]$,then |An| where $n \in N$, is equal to

 (a) 2n (b) 3n (c) n (d) 1

(iii) If $A=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right]$, and $I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$ then which of the following holds for all natural numbers $n \geq 1$ ?

 (a) An = nA-(n-1}I (b) An = 2n-1A-(n-1}I (c) An = nA+(n-1)I (d) An = 2n-1A+(n-1}I

(iv) Let $A=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{array}\right]$ , and $A^{n}=\left[a_{i j}\right]_{3 \times 3}$ for some positive integer n, then the cofactor of a13 is

 (a) an (b) -an (c) 2an (d) 0

(v) If A is a square matrix such that IAI = 2, then for any positive integer n, IAnl is equal to

 (a) 0 (b) 2n (c) 2n (d) n2
• 3)

Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 25 smaller equilateral triangles as shown in the figure.

Using the above information and concept of determinants, answer the following questions
(i) If the vertices of one of the smaller equilateral triangle are (0, 0), $$$(3, \sqrt{3})$$$ and $$$(3,-\sqrt{3})$$$ then the area of such triangle is

 (a) $$$\sqrt{3} \text { sq. units }$$$ (b) $$$2 \sqrt{3} \text { sq. units }$$$ (c) $$$3 \sqrt{3} \text { sq. units }$$$ (d) none of these

(ii) The area of a face of the Pyramid is

 (a)  $$$25 \sqrt{3} \text { sq. units }$$$ (b) $$$50 \sqrt{3} \text { sq. units }$$$ (c) $$$75 \sqrt{3} \text { sq. units }$$$ (d)  $$$35 \sqrt{3} \text { sq. units }$$$

(iii) The length of a altitude of a smaller equilateral triangle is

 (a) 2 units (b) 3 units (c) $$$\sqrt{3} \text { units }$$$ (d) 4 units

(iv) If (2, 4), (2, 6) are two vertices of a smaller equilateral triangle, then the third vertex will lie on the line represented by

 (a) x +y = 5 (b) $$$x=1+\sqrt{3}$$$ (c) $$$x=2+\sqrt{3}$$$ (d) 2x + y = 5

(v) Let A(a, 0), B(O, b) and C(1, 1) be three points. If $$$\frac{1}{a}+\frac{1}{b}=1$$$ ,then the three points are

 (a) vertices of an equilateral triangle (b) vertices of a right angled triangle (c)collinear (d) vertices of an isosceles triangle
• 4)

Let $$$A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right]$$$ , and U1 U2 are first and second columns respectively of a 2 x 2 matrix U.
Also, let the column matrices UI and U2 satisfying $$$A U_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right]$$$ and $$$A U_{2}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right]$$$
Based on the above information, answer the following questions
(i) The matrix U1 + U2 is equal to

 (a) $$$\left[\begin{array}{c} 1 \\ -1 \end{array}\right]$$$ (b) $$$\left[\begin{array}{c} 2 \\ -2 \end{array}\right]$$$ (c) $$$\left[\begin{array}{c} 3 \\ -3 \end{array}\right]$$$ (d) $$$\left[\begin{array}{c} 4 \\ -4 \end{array}\right]$$$

(ii) The value of IUI is

 (a) 2 (b) -2 (c) 3 (d) -3

(iii) If $$$X=\left[\begin{array}{ll} 3 & 2 \end{array}\right] U\left[\begin{array}{l} 3 \\ 2 \end{array}\right]$$$,then the value of IXI =

 (a) 3 (b) -3 (c) -5 (d) 5

(iv) The minor of element at the position a22 in U is

 (a) 1 (b) 2 (c) -2 (d) -1

(v) If $$$U=\left[a_{i j}\right]_{2 \times 2}$$$ , then the value of a11A11 + a12Al2 where Aij denotes the cofactor of aij is

 (a) 1 (b) 2 (c) -3 (d) 3
• 5)

The upward speed v(t) of a rocket at time t is approximated by $$$v(t)=a t^{2}+b t+c, 0 \leq t \leq 100$$$ ,where a, band c are constants. It has been found that the speed at times t = 3, t = 6 and t = 9 seconds are respectively 64, 133 and 208 miles per second.

If $$$\left(\begin{array}{ccc} 9 & 3 & 1 \\ 36 & 6 & 1 \\ 81 & 9 & 1 \end{array}\right)^{-1}=\frac{1}{18}\left(\begin{array}{ccc} 1 & -2 & 1 \\ -15 & 24 & -9 \\ 54 & -54 & 18 \end{array}\right)$$$ ,then answer the following questions
(i) The value of b + c is

 (a) 20 (b) 21 (c) 3/4 (d) 4/3

(ii) The value of a + c is

 (a) 1 (b) 20 (c) 4/3 (d) none of these

(iii) v(t) is given by

 (a) t2+20t+l (b) $$$\frac{1}{3} t^{2}+20 t+1$$$ (c) $$$t^{2}+\frac{1}{3} t+20$$$ (d) P + t + 1

(iv) The speed at time t = 15 seconds is

 (a) 346 miles/see (b) 356 miles/see (c) 366 miles/see (d) 376 miles/see

(v) The time at which the speed of rocket is 784 miles/see is

 (a) 20 seconds (b) 30 seconds (c) 25 seconds (d) 27 seconds

#### Class 12th Maths - Matrices Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III respectively are 80, 70'and 65 in factory A and 85, 65 and 72 are in factory B. For girls the number of units of types I, II and III respectively are 80, 75, 90 in factory A and 50, 55, 80 are in factory B.

Based on the above information, answer the following questions.
(i) If P represents the matrix of number of units of each type produced by factory A for both boys and girls, then P is given by

(ii) If Q represents the matrix of number of units of each type produced by factory B for both boys and girls, then Q is given by

(iii) The total- production of sports clothes of each type for boys is given by the matrix

(iv) The total production of sports clothes of each type for girls is given by the matrix

(v) Let R be a 3 x 2 matrix that represent the total production of sports clothes of each type for boys and girls, then transpose of R is(iv) The total production of sports clothes of each type for girls is given by the matrix

• 2)

To promote the making of toilets for women, an organisation tried to generate awareness through (i) house calls (ii) emails and (iii) announcements. The cost for each mode per attempt is given below:

(i) Rs.50 (ii) Rs.20 (iii) Rs.40
The number of attempts made in the villages X, Y and Z are given below:
$\begin{array}{llll} & (\mathrm{i}) & (\mathrm{ii}) & (\mathrm{iii}) \\ X & 400 & 300 & 100 \\ Y & 300 & 250 & 75 \\ Z & 500 & 400 & 150 \end{array}$
Also, the chance of making of toilets corresponding to one attempt of given modes is
(i) 2% (ii) 4% (iii) 20%
Based on the above information, answer the following questions.
(i) The cost incurred by the organisation on village X is

 (a) 10000 (b)  Rs.15000 (c) 30000 (d) Rs.20000

(ii) The cost incurred by the organisation on village Y is

 (a) Rs.25000 (b) Rs.18000 (c) Rs.23000 (d) Rs.28000

(iii) The cost incurred by the organisation on village Z is

 (a)  Rs.19000 (b)  Rs.39000 (c)  Rs.4500 (d)  Rs.5000

(iv) The total number of toilets that can be expected after the promotion in village X, is

 (a)  20 (b)  30 (c)  40 (d)  50

(v) The total number of toilets that can be expected after the promotion in village Z, is

 (a) 26 (b) 36 (c)  46 (d)  56
• 3)

Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan,S SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan,S SUV cars in 2020.

Based on the above information, answer the following questions.
(i) The matrix summarizing sales data of 2019 is

(ii) The matrix summarizing sales data of 2020 is

(iii) The total number of cars sold in two given years, by each dealer, is given by the matrix

(iv) The increase in sales from 2019 to 2020 is given by the matrix

(v) If each dealer receive profit of Rs. 50000 on sale of a Hatchback, Rs. 100000 on sale of a Sedan and Rs. 200000 on sale of a SUV (v) then amount of profit received in the year 2020 by each dealer is given by the matrix.

• 4)

Three schools A, Band C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100 and Rs.50 each. The number of articles sold by school A, B, C are given below.

 Artilcle\School A B C Fans 40 25 35 Mats 50 40 50 Plates 20 30 40

Based on above information, answer the following questions.
(i) If P be a 3 x 3 matrix represent the sale of handmade fans, mats and plates by three schools A, Band C, then

(ii) If Q be a 3 x 1 matrix represent the sale prices (in Rs) of given products per unit, then

(iii) The funds collected by school A by selling the given articles is

 (a) Rs. 7000 (b) Rs. 6125 (c) Rs. 7875 (d) Rs. 8000

(iv) The funds collected by school B by selling the given articles is

 (a) Rs. 5125 (b) Rs. 6125 (c) Rs. 7125 (d) Rs. 8125

(v) The total funds collected for the required purpose is

 (a) Rs. 20000 (b) Rs. 21000 (c) Rs. 30000 (d) Rs. 35000
• 5)

Two farmers Shyam and Balwan Singh cultivate only three varieties of pulses namely Urad, Masoor and Mung. The sale (in Rs.) of these varieties of pulses by both the farmers in the month of September and October are given by the following matrices A and B.

Using algebra of matrices, answer the following questions.
(i) The combined sales of Masoor in September and October, for farmer Balwan Singh, is

 (a) Rs. 80000 (b) Rs. 90000 (c) Rs. 40000 (d) Rs. 135000

(ii) The combined sales of Urad in September and October, for farmer Shyam is

 (a) Rs. 20000 (b) Rs. 30000 (c) Rs. 36000 (d) Rs. 15000

(iii) Find the decrease in sales of Mung from September to October, for the farmer Shyam.

 (a) Rs. 24000 (b) Rs. 10000 (c) Rs. 30000 (d) No change

(iv) If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October.

(v) Which variety of pulse has the highest selling value in the month of September for the farmer Balwan Singh?

 (a) Urad (b) Masoor (c) Mung (d) All of these have the same price

#### Class 12th Maths - Relations and Functions Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read

• 1)

A relation R on a set A is said to be an equivalence relation on A iff it is
(a) Reflexive i.e.., $(a, a) \in R \ \forall \ a \in A$
(b) Symmetric i.e., $(a, b) \in R \Rightarrow(b, a) \in R \ \forall \ a, b \in A$
(c) Transitive i.e., $(a, b) \in R$ and $(b, c) \in R \Rightarrow(a, c) \in R\ \forall\ a, b, c \in A$
Based on the above information, answer the following questions.
(i) If the relation R = {(1, 1), (1, 2), (1, 3), (2,2), (2, 3), (3,1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is

 (a) reflexive (b) symmetric (c) transitive (d) equivalence

(ii) If the relation R = {(1, 2), (2,1), (1, 3), (3, I)} defined on the setA = {1, 2, 3}, then R is

 (a) reflexive (b) symmetric (c) transitive (d) equivalence

(iii) If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}, then R is

 (a) reflexive (b) symmetric (c) transitive (d) equivalence

(iv) If the relation R on the set A = {1, 2, 3, , 13, 14}defined as R = {(x, y) : 3x - y = 0}, then R is

 (a) reflexive (b) symmetric (c) transitive (d) equivalence
• 2)

Consider the mapping $f: A \rightarrow B$ is defined by $f(x)=\frac{x-1}{x-2}$ such that f is a bijection.
Based on the above information, answer the following questions.
(i) Domain of f is

 (a) R - {2} (b) R (C) R-{1,2} (d) R-{0}

(ii) Range of f is

 (a) R (b) R -{1} (C) R-{0} (d) R-{1,2}

(iii) If g: $R-\{2\} \rightarrow R-\{1\}$ is defined by g(x) = 2f(x) - I, then g(x) in terms of x is

 (a) $\frac{x+2}{x}$ (b) $\frac{x+1}{x-2}$ (c) $\frac{x-2}{x}$ (d) $\frac{x}{x-2}$

(iv) The function g defined above, is

 (a) One-one (b) Many-one (c) into (d) None of these

(v) A function J(x) is said to be one-one iff

 (a) $f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow-x_{1}=x_{2}$ (b) $f\left(-x_{1}\right)=f\left(-x_{2}\right) \Rightarrow-x_{1}=x_{2}$ (c) $f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow x_{1}=x_{2}$ (d) None of these

#### Test - by QB365 School - View & Read

• 1)

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

• 2)

The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

• 3)

Which of the following functions are decreasing on 0, $\frac{\pi}{2}$?

• 4)

On which of the following intervals is the function f given by f (x) = x100 + sin x–1 decreasing?

• 5)

The interval in which y = x2 e–x is increasing is

#### 12th Standard CBSE Mathematics Public Model Question Paper IV 2019 - 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let R = { (P,Q) : OP = OQ , O being the origin} be an equivalence relation on A . The equivalence class [( 1,2)] is

• 2)

The angle between the curve y² = x and x² = y at (1, 1) is

• 3)

The area enclosed between the lines x = 2 and x = 7 is

• 4)

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

• 5)

If A is square matrix such that A2 = A, then write the value of (I + A)2-3A.

#### 12th Standard CBSE Mathematics Public Model Question Paper III 2019 - 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let A = {1,2,3,4} and B = {x,y,z}. Then R = {(1,x) , ( 2,z), (1,y), (3,x)} is

• 2)

The point on the curve x2 = 2y which is nearest to the point (0, 5) is

• 3)

Area of the shaded region in the given figure is:

• 4)

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

• 5)

Evaluate the following : $[a\quad b]\left[ \begin{matrix} c \\ d \end{matrix} \right] +[a\quad b\quad c\quad d]\left[ \begin{matrix} a \\ b \\ c \\ d \end{matrix} \right]$

#### 12th Standard CBSE Mathematics Public Model Question Paper II 2019 - 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Given set A ={1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be

• 2)

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

• 3)

Write the shaded region as an integral

• 4)

The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are

• 5)

if $\left[ \begin{matrix} a+b & 2 \\ 5 & ab \end{matrix} \right] =\left[ \begin{matrix} 6 & 2 \\ 5 & 8 \end{matrix} \right]$find the relation between a and b

#### 12th Standard CBSE Mathematics Public Model Question Paper I 2019 - 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is

• 2)

A stone is dropped into a quiet lake and waves move in circles at a speed of 2cm per second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area changing ?

• 3)

Area of the region bounded by the curve y2 = 2y – x and y-axis is:

• 4)

The equations of y-axis in space are

• 5)

If, A = |aij| = $\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right]$and B =|bij| =$\left[ \begin{matrix} 2 & -1 \\ -3 & 4 \\ 1 & 2 \end{matrix} \right]$ Write the value of
(i)a22 + b21
(ii) a11b11 +a22b22

#### 12th Standard CBSE Mathematics Public Model Question Paper V 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Given triangles with sides T1 : 3, 4, 5; T2 : 5, 12, 13; T3 : 6, 8, 10; T4 : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ1, Δ2) : Δ1 is similar to Δ2}. Which triangles belong to the same equivalence class?

• 2)

Using approximation find the value of $y=\sqrt{4.01}$

• 3)

Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is

• 4)

Distance between planes $\overrightarrow { r } .(2\widehat { i } +\widehat { j } -2\widehat { k } )+5=0$ and $\overrightarrow { r } .(6\widehat { i } +3\widehat { j } -6\widehat { k } )+2=0$ is

• 5)

If ${ X }_{ m\times 3 }{ Y }_{ p\times 4 }={ Z }_{ 2\times b }$, for three matrices X, Y and Z, find the values of m, p and b.

#### 12th Standard CBSE Mathematics Public Model Question Paper IV 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Given triangles with sides T1 : 3, 4, 5; T2 : 5, 12, 13; T3 : 6, 8, 10; T4 : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ1, Δ2) : Δ1 is similar to Δ2}. Which triangles belong to the same equivalence class?

• 2)

The area enclosed between the lines x = 2 and x = 7 is

• 3)

The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are

• 4)

If $\left[ \begin{matrix} xy & 4 \\ z+6 & x+y \end{matrix} \right] =\left[ \begin{matrix} 8 & w \\ 0 & 6 \end{matrix} \right]$, write the value of x + y + z.

#### 12th Standard CBSE Mathematics Public Model Question Paper III 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

In the set N x N the relation R is defined by (a, b) R (c, d) ⇔ ad = bc. Then R is

• 2)

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

• 3)

Area bounded by the curve y = x3, the x-axis and the ordinates x = – 2 and x = 1 is

• 4)

Find the direction cosines of a line which makes an angle with all three the coordinate axes.

• 5)

For what value of k, the matrix $\left[ \begin{matrix} 2k+3 & 4 & 5 \\ -4 & 0 & -6 \\ -5 & 6 & -2k-3 \end{matrix} \right]$ is a skew symmetric matrix?

#### 12th Standard CBSE Mathematics Public Model Question Paper II 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let R be a relation on N (set of natural numbers) such that (m, n) R (p, q)mq(n + p) = np(m + q). Then, R is

• 2)

The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

• 3)

The area enclosed between the lines x = 2 and x = 7 is

• 4)

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

• 5)

Show that all the elements on the main diagonal of a skew symmetric matrix are zero.

#### 12th Standard CBSE Mathematics Public Model Question Paper I 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let $f:[4, \infty) \rightarrow[4, \infty)$ be defined by $f(a)=5^{a(a-4)}$,Then,$f^{-1}(a)$is

• 2)

The total cost associated with the production of x units of a product is given by c(x) = 5x2 + 14x + 6. Find marginal cost when 5 units are produced

• 3)

Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is

• 4)

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

• 5)

If $A=\left[ \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} \right]$, find ${ A }^{ 2 }$. Hence find ${ A }^{ 6 }$.

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper IV 2019 -2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let C = {(a, b): a2 + b2 = 1; a, b ∈ R} a relation on R, set of real numbers. Then C is

• 2)

If sec-1 x + sec-1 y = $\frac{\pi}{2}$ the value of cosec-1x + cosec-1y is

• 3)

$\left[ \begin{matrix} 2 & 3 & 1 \\ 1 & 2 & 4 \end{matrix}\begin{matrix} 5 & 1 \\ 2 & 2 \end{matrix} \right]$ is a matrix of order

• 4)

If a, b, c, are in A.P, then the determinant
$\left| \begin{matrix} x+2 & x+3 & x+2b \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{matrix} \right|$ is

• 5)

If y = Ae5x,+ Be-5x x then $\frac { { d }^{ 2 }y }{ dx^{ 2 } }$ is equal to

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper III 2019 -2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let f : R ➝ R be defined as f (x) = 3x. Choose the correct answer

• 2)

Principal value of the expression cos-1[cos(-680°)] is

• 3)

What is the element in the 2nd row and 1st column of a 2 x 2 Matrix A= [ aij], such that a = (i + 3) (j – 1)

• 4)

If a, b, c, are in A.P, then the determinant
$\left| \begin{matrix} x+2 & x+3 & x+2b \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{matrix} \right|$ is

• 5)

A function f is said to be continuous for x ∈ R, if

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper II 2019 -2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

• 2)

If sin-1x + sin-1y + sin-1z = then the value of x + y² + z3 is

• 3)

If a matrix A is both symmetric and skew symmetric then matrix A is

• 4)

If Δ = $\left| \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right|$ and Aij is Cofactors of aij, then value of Δ is given by

• 5)

A function $f(x)=\begin{cases} \frac { sinx }{ x } +cosx,x\neq 0 \\ 2k\quad \quad \quad \quad ,x=0 \end{cases}$ is continuous at x = 0 for

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper I 2019 -2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

• 2)

tan–1 $\sqrt3$ sec-1(-2) is equal to

• 3)

If the matrix A is both symmetric and skew symmetric, then

• 4)

Let x, yeR, then the determinant $\triangle =$ $\left| \begin{matrix} cosx & -sinx & 1 \\ sinx & cosx & 1 \\ cos(x+y) & -sin(x+y) & 0 \end{matrix} \right|$, lies in the interval

• 5)

If y = xx-∞, then x(l -y log x)$\frac { dy }{ dx }$ is equal to

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper V 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let A = {1,2,3,4} and B = {x,y,z}. Then R = {(1,x) , ( 2,z), (1,y), (3,x)} is

• 2)

sec{tan-1 (-$\frac y3$)} is equal to

• 3)

$\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$ is example of

• 4)

If A is an invertible matrix of order 2, then det (A–1) is equal to

• 5)

Derivative of cot x° with respect to x is

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper IV 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

In the set N x N the relation R is defined by (a, b) R (c, d) ⇔ ad = bc. Then R is

• 2)

If sec-1 x + sec-1 y = $\frac{\pi}{2}$ the value of cosec-1x + cosec-1y is

• 3)

If a matrix A is both symmetric and skew symmetric then matrix A is

• 4)

Which of the following is correct

• 5)

If y = xx-∞, then x(l -y log x)$\frac { dy }{ dx }$ is equal to

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper III 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let f : R ⟶ R be defined as f(x) = x4. Choose the correct answer

• 2)

If sin-1x + sin-1y + sin-1z = then the value of x + y² + z3 is

• 3)

If A = $\begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}$ and A = A’ then

• 4)

If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4). Then k is

• 5)

Derivative of cot x° with respect to x is

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper II 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

If A = {1, 2, 3, 4} and B = {1, 3, 5} and R is a relation from A to B defined by (a, b) ∈ element of R ⇔ a < b. Then, R = ?

• 2)

Value of ${ cot }^{ -1 }\left( sin\left( -\frac { \pi }{ 2 } \right) \right)$

• 3)

Consider the following information regarding the number of men and women workers in three BPOs I, II and III

 Men Women I 35 20 II 20 23 III 25 25

What does the entry in the second row and first column represent if the information is represented as a 3 x 2 matrix?

• 4)

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

• 5)

If y = tan-1 $\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right)$, then $\frac { dy }{ dx }$ is equal to

#### 12th Standard CBSE Mathematics Board Exam Model Question Paper I 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Let R be a relation on a finite set A having n elements. Then, the number of relations on A is

• 2)

If sin–1 x = y, then

• 3)

If A, B are symmetric matrices of same order, then AB – BA is a

• 4)

If $\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right]$ and Aij is cofactor of aij, then the value of Δ is given by

• 5)

What is the point of discontinuity for signum function?

#### 12th CBSE Mathematics - Public Model Question Paper 2019 - 2020 - by Shalini Sharma - Udaipur - View & Read

• 1)

Given set A ={1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be

• 2)

If the curves ay + x2 = 7 and x3 = y cut orthogonally at (1,1), then the value of a is

• 3)

Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

• 4)

The area bounded by the curve y = x |x| , x-axis and the ordinates x = – 1 and x = 1 is given by

#### CBSE 12th Mathematics - Linear Programming Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

A furniture firm manufactures chairs and tables, each requiring the use of three machines 'A', 'B' and 1 hour on machine 'C'. Each table requires 1 hour each on machines 'A' and 'B' and 3 hours on machine 'C'. The profit realized by selling one chair is RS. 30 while for a table is Rs. 60. The total time available per week o machine 'A' is 70 hours, on machine 'B' is 40 hours and on machine 'C' is 90 hours. Find the mathematical formulation so as to find the number of chairs and tables that should be made per week so as to maximize the profit.

• 2)

Solve the following Linear Programming Problems graphically:
Maximise Z = 5x + 3y
subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 3)

If a man rides his motor cycle at 25 km/hr., he has to spend Rs. 2 per km on petrol, if he rides at a faster speed of 40 km/hr., the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find maximum distance he can travel within one hour. Express this as a linear programming problem and then solve it graphically.

• 4)

A furniture firm manufactures chairs and tables, each requiring the use of three machines -A, B and C. Production of one chair requires 2 hours on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as LLP and solve it graphically.

• 5)

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchantshould stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an LPP and solve it graphically.

#### CBSE 12th Mathematics - Three Dimensional Geometry Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the distance of the plane 3x - 4y + 12z = 3 from the origin.

• 2)

Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line $\frac { x+3 }{ 3 } =\frac { 4-y }{ 5 } =\frac { z+8 }{ 6 }$

• 3)

Find the distance between the planes 2x + 3y + 4z = 10 and 4x + 6y + 8z = 18.

• 4)

Find the distance of point $2\check { i } +\check { j } -\check { k }$ from the plane $\overrightarrow{r}.(\hat{i}-\hat{2}j+4\hat{k})=9.$

#### CBSE 12th Mathematics - Probability Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Given P(A) = $1\over2$, P(B) = $1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

Bayes’ Theorem If E1 , E2 ,..., En are n non empty events which constitute a partition of sample space S, i.e. E1 , E2 ,..., En are pairwise disjoint and E1∪ E2∪ ... ∪ En = S and A is any event of nonzero probability, then
$\mathrm{P}\left(\mathrm{E}_i \mid \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_i\right) \mathrm{P}\left(\mathrm{A}_{\mid} \mathrm{E}_i\right)}{\sum_{j=1}^n \mathrm{P}\left(\mathrm{E}_j\right) \mathrm{P}\left({\left.\mathrm{A} \mid E_j\right)}_1\right.} \text { for any } i=1,2,3, \ldots, n$

• 3)

Given P(A) = 0.2, P(B) = 0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 4)

Events E and F are given to be independent. Find P(F) if it is given that P(E) = 0.60 and P(E$\cap$F) = 0.35

• 5)

Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.

#### CBSE 12th Mathematics - Vector Algebra Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

For what value of $\lambda$ are the vectors $\overrightarrow { a } =2\overrightarrow { i } +\lambda \overrightarrow { j } +\overrightarrow { k }$ and $\overrightarrow { b } =\overrightarrow { i } -2\overrightarrow { j } +3\overrightarrow { k }$  perpendicular to each other?

• 2)

If $\overrightarrow { a } =\overrightarrow { i } +2\overrightarrow { j } -\overrightarrow { k } \ and\ \overrightarrow { b } =3\overrightarrow { i } +\overrightarrow { j } -5\overrightarrow { k }$ find a unit vector in the direction of $\overrightarrow { a } -\overrightarrow { b }$

• 3)

Find $\lambda$, if $(2\hat { i } +6\hat { j } +14\hat { k } )\times (\hat { i } -\lambda \hat { j } +7\hat { k } )=\overrightarrow { 0 }$

• 4)

Find the projection of the vector $\overrightarrow { a } =2\hat { i } +3\hat { j } +2\hat { k }$ on the vector $\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k }$

• 5)

Write the position vector of the point which divides the join of points with position vectors $3\overset\rightarrow a-2\overset\rightarrow b$ and $2\overset\rightarrow a+3\overset\rightarrow b$ in the ratio 2:1

#### CBSE 12th Mathematics - Differential Equations Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Write the degree of the differential equation: $5x{ \left( \frac { dy }{ dx } \right) }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0.$

• 2)

Write the degree of the differential equation ${ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+x\left( \frac { dy }{ dx } \right) ^{ 4 }=0$

• 3)

Find the differential equation of the family of lines passing through the origin.

• 4)

Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.

• 5)

Find the solution of the differential equation $\frac { dy }{ dx } ={ x }^{ 3 }{ e }^{ -2y }$

#### CBSE 12th Mathematics - Application of Integrals Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the area of the region by the curve $y=\frac { 1 }{ x }$ , x-axis and between x = 1, x = 4.

• 2)

On sketching the graph of  $y=\left| x-2 \right|$  and evaluating $\int _{ -1 }^{ 3 }{ \left| x-2 \right| } dx$ , what does $\int _{ -1 }^{ 3 }{ \left| x-2 \right| } dx$ represent on the graph ?

• 3)

The area bounded by the curve $y=x\left| x \right|$, x - axis and the ordinates x = -1 and x = 1 given by :
(A) 0
(B) $\frac { 1 }{ 3 }$
(C) $\frac { 2 }{ 3 }$
(D) $\frac { 4 }{ 3 }$

• 4)

The area bounded by the y = axis y = cos x and y = sin x, where $0\le x\le \frac { \pi }{ 2 }$ is:
(A) $2(\sqrt { 2 } -1)$
(B) $\sqrt { 2 } -1$
(C) $\sqrt { 2 } +1$
(D) $\sqrt { 2 }$

• 5)

Find the area of the region bounded by:
y2 = 4x, x = 1, x = 4 and x - axis in the first quadrant.

#### CBSE 12th Mathematics - Integrals Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral: $\int { x^2\ +\ 4x\over x^3\ +\ 6x^2\ +\ 5 } dx$

• 2)

Evaluate the integral: $\int {dx\over x^2\ +\ 16}$

• 3)

If $\int {(ax\ +\ b)^2dx\ =\ f\ (x)\ +\ c}$, find f (x).

• 4)

$\int tan^{-1}(cot\ x)dx.$

• 5)

$\int \sqrt{tan\ x}(1+tan^2\ x)\ dx.$

#### CBSE 12th Mathematics - Application of Derivatives Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue(marginal revenue).If the total revenue(in rupees)received from the sale of x units of a product is given by R9x)=3x2+36+5,find the marginal revenue,when=5,and write which value does the question indicate?

• 2)

The total revenue received from the sale of x souvenirs in connection with 'PEACE DAY'is given by R(x)=3x2+40x+10.
Find the marginal revenue when 100souvenirs were sold.What is the importance of celebrating Peace Day in our life?

• 3)

Find the points on the curve ${x^2\over4}+{y^2\over 25}=1$at which the  tangents are
(i) parallel to the x-axis.
(ii)parallel to the y-axis.

• 4)

For the function y=x3, if x=5 and $\Delta$x=0.01,find $\Delta$y

• 5)

f(x)=-|x+1|+

#### CBSE 12th Mathematics - Continuity and Differentiability Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Examine the continuity of the function f (x) = $\frac { 1 }{ x+3 } , x\ \in \ R$.

• 2)

Differentiate cos x, with respect to ex.

• 3)

Verify MVT for the following : f (x) = | x | in [-1, 1].

• 4)

Given an example of a function which is continous but not differtiable

• 5)

Find the derivative of sin ($cos^{ 2 }\left( \sqrt { x } \right)$).

#### CBSE 12th Mathematics - Determinants Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If $\begin{vmatrix} 2x+5 & 3 \\ 5x+2 & 9 \end{vmatrix}=0$ find x.

• 2)

If A=$\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ write A-1 in terms of A .

• 3)

Find the value of X, such that the points (0, 2), (1, x) and (3, 1) are collinear.

• 4)

If A is a $3\times 3$ matrix, $\left| A \right| \neq 0$ and $\left| 3A \right| =k\left| A \right|$, then write the value of k.

• 5)

Evaluate x if: $\left| \begin{matrix} 2 & 4 \\ 5 & 1 \end{matrix} \right| =\left| \begin{matrix} 2x & 4 \\ 6 & x \end{matrix} \right|$

#### CBSE 12th Mathematics - Matrices Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If matrix A = $[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 2)

Show that all the elements on the main diagonal of a skew symmetric matrix are zero.

• 3)

Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

• 4)

Let A = [aij]be a matric of order 2 x 3 and aij = $\frac { i-j }{ i+j }$, write the value of a23

#### CBSE 12th Mathematics - Inverse Trigonometric Functions Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

if (a < 0) and x $\varepsilon$ (-a, a), simplify tan-1 $\left( \frac { x }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } \right)$

#### CBSE 12th Mathematics - Relations and Functions Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2 * 4.

• 2)

If f is an invertible function defined as f(x) = ${3X-4}\over5$, write f-1(x).

• 3)

Let f:$R\rightarrow R$ is defined by f(x) = x2. Is f one-one?

• 4)

Let the function f : R$\rightarrow$R to be defined by f(x) = cos x $\forall$ x $\in$R. Show that  is neither one-one nor onto.

#### CBSE 12th Mathematics - Full Syllabus Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Show that the function $f:R[x\in R:1 defined by \(f(x)=\frac { x }{ 1+|x|^{ ' } } ,x\in R$ is one-one and onto function Hence find $f^{ -1 }(x)$

• 2)

If $A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right)$, prove that A3 - 6A2 + 7A + 2I = 0

• 3)

If  $A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right]$ Prove that , A =$\left[ \begin{matrix} { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \end{matrix} \right]$ for every positive integer n.

• 4)

A school wants to award its students for the value of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for hard work added to that given for honesty amounts to Rs. 11,000.The award money given for honesty and hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.

#### CBSE 12th Mathematics - Full Syllabus Four Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Define a binary operation '*' on the set A = {0, 1, 2, 3, 4, 5}, given by a*b=(ab) mod 6. Show that for *,1 and 5 are only invertible elements with $1^{ -1 }=1$ and $5^{ -1 }=5$.
[Here (a, b) mod 6, we mean the remainder after dividing ab by 6 ]

• 2)

Write the value of  $tan^{-1}\left[2sin\left(2cos^{-1}{\sqrt{3}\over2}\right)\right]$

• 3)

Using elementary transformations, find the inverse of the matrix

$\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right]$.

• 4)

If $A=\begin{bmatrix} 4 & 1 \\ 5 & 8 \end{bmatrix}$, show that A + AT is a symmetric matrix, where AT denotes the transpose of matrix A.

#### CBSE 12th Mathematics - Full Syllabus Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a*b $=\frac { a }{ 2 } +\frac { b }{ 3 }$then value of 2*3 is.......

• 2)

Let P be set of all subsets of given set X. Show that $\cup :P\times P\rightarrow P$ given by $(A,B)\rightarrow A\cup B$ and $\cap :P\times P\rightarrow P$ given by $(A,B)\rightarrow A\cap B$ are binary operations on the set P.

• 3)

If the mappings f and g are given by:
f = {(1, 2), (3, 5) (4, 1) and g = {(2, 3), (5, 1), (1, 3)}, write fog.

• 4)

Find te principal values of the following:
${ cos }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right)$

• 5)

If $A=\begin{bmatrix} cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$, then $A+A\prime =I$, if the value of $\alpha$is:

(A)  $\frac { \pi }{ 6 }$
(B) $\frac { \pi }{ 3 }$
(C) $\pi \quad$
(D) $\frac { 3\pi }{ 2 }$.

#### CBSE 12th Mathematics - Full Syllabus Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Define Transitive Relation. Give one example.

• 2)

Write in the simplest form : $sin\left[ 2{ tan }^{ -1 }\sqrt { \frac { 1-x }{ 1+x } } \right]$

• 3)

If is $A=\left[ \begin{matrix} 0 & b & -2 \\ 3 & 1 & 3 \\ 2a & 3 & -1 \end{matrix} \right]$skew symmetric matrix, find the values of a and b.

• 4)

The side of an equilateral triangle is increasing at the rate of 5 cm/sec. At what rate its area increasing when the side of the triangle is 10 cm.

• 5)

$\int { { e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ x^{ 2 } } \right) } dx$

#### 12th CBSE Mathematics - Probability Five Marks Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the binomial distribution for which mean is 4 and variance 3.

• 2)

There are 2000 scooter drivers, 4000 car drivers and 6000 truck drivers all insured. The probabilities of an accident involving a scooter, a car, a truck are 0.01, 0.03, 0.15 respectively. One of the insured drivers meets with an accident. What is the probability that he is a scooter driver?

• 3)

12 cards, numbered 1 to 12, are placed in box mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

• 4)

A and B throw a pair of die turn by turn. The first to throw 9 is awarded a prize. If A starts the game, show that the probability of A getting the prize is $9\over17$

• 5)

A pair of dice is thrown 4 times. If getting a doublet is considered a success find the mean and variance of the number of successes.

#### 12th CBSE Mathematics - Linear Programming Six Marks Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

A manufacture produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours for day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine, Each unit of product A is sold at Rs. 7 profit and  that of B at a profit of Rs. 4. Find the production level per day for maximum profit graphically.

• 2)

A manufacturer produces nuts and bolts. It takes 2 hours work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 2 hours on machine B to produce a package of bolts. He earns a profit of Rs. 24 per package on nuts and Rs. 18 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines both for at the most 10 hours a days. Make an LPP from above and solve it graphically?

• 3)

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below: 2x + 4y $\le$ 8 $\Rightarrow$ x + 2y $\le$ 4
3x + y $\le$ 6
x + y $\le$ 4
x $\\ \ge$ 0, y  $\\ \ge$ 0

• 4)

A company manufactures two types of sweaters, type A and B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make at most 300 sweaters and can spend Rs. 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs 200 on each unit of type A. The company charging a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve for max. profit.

• 5)

A decorative item dealer deals in two items A and B. He has Rs. 15,000 to invest and a space to store at the most 80 pieces. Item A cost him Rs. 300 and item B costs him Rs 150. He can sell items A and B at respective profits of Rs. 50 and Rs. 28. Assuming he can sell all he buys, formulate the linear programming problem in order to maximize his profit and solve it graphically.

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Six Marks Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

If lines $\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 }$ and $\frac { x-3 }{ 1 } =\frac { y-k }{ 32 } =\frac { z }{ 1 }$intersect, then find the value of k and hence find the equation of plane containing these lines.

• 2)

Find the distance of the point $3\hat { i } -2\hat { j } +\hat { k }$ from the plane 3x + y - z + 2 = 0 measured parallel to the line $\frac { x-1 }{ 2 } =\frac { y+2 }{ -3 } =\frac { z-1 }{ 1 }$ . Also, find the foot of the . Also, find the foot of the perpendicular from the given point upon the given plane.

• 3)

Find the equation of plane passing through the line of intersection of the planes$\vec { r } .\left( 2\hat { i } +3\hat { j } -\hat { k } \right) =-1$ and $\vec { r } .\left( \hat { i } +\hat { j } -2\hat { k } \right) =0$and passing through the point (3,- 2, -1). Also, find the angle between the two given planes

• 4)

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, - 3) B(- 2, - 3, 5) and C(5, 3, - 3).

• 5)

Find the distance of the point (2, 12, 5) from the point of intersection of the lines $\vec { r } =\left( 2\hat { i } -4\hat { j } +2\hat { k } \right) +\lambda \left( 3\hat { i } +4\hat { j } +2\hat { k } \right)$ and the plane.

#### CBSE 12th Mathematics - Probability Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of success.

• 2)

12 cards, numbered 1 to 12, are placed in box mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

• 3)

In a bulb factory machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. Find the probability that this bulb was produced by the machine A.

• 4)

The probabilities of A, B, C solving a problem are $\frac { 1 }{ 3 } ,\frac { 2 }{ 7 } and\frac { 3 }{ 8 }$ respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it.

• 5)

A die is throewn again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the vector equation of the plane through the intersection of the planes $\vec { r } .(\hat { i } +\hat { j } +\hat { k } )=6$ and $\vec { r } .(2\hat { i } +3\hat { j } +4\hat { k } )=-5$ and the point (1, 1, 1)

• 2)

Find the value of $\lambda$ so that the lines $\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2\lambda } =\frac { 5z-10 }{ 11 }$ and $\frac { 7-7x }{ 3\lambda } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 }$ are perpendicular to each other.

• 3)

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0

• 4)

Find the shortest distance between the following two lines:
$\overrightarrow { r } =(1+\lambda )\acute { i } +(2-\lambda )\acute { j } +(\lambda +1)\acute { k } \\ \vec { r } =(2\acute { i } -\acute { j } -\acute { k } )+\mu (2\acute { i } +\acute { j } +2\acute { k } )$

#### 12th Standard CBSE Mathematics - Vector Algebra Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find a unit vector perpendicular to each of the vectors $\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b }$  where $\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$

• 2)

If two vectors $\overrightarrow { a }\ and \ \overrightarrow { b }$ are such that $\left| \overrightarrow { a } \right| =2,\left| \overrightarrow { b } \right| =1\ and \ \overrightarrow { a } .\overrightarrow { b } =1$ then find the value of ($3\overrightarrow { a } -5\overrightarrow { b }$).($2\overrightarrow { a } +7\overrightarrow { b }$).

• 3)

Using vectors find the area of the triangle with vertices A(1,1,2),B(2,3,5) and C(1,5,5).

• 4)

If $\overrightarrow { \alpha } =3\hat { i } +4\hat { j } +5\hat { k } \ and\ \overrightarrow { \beta } =2\hat { i } +\hat { j } -4\hat { k }$ the express $\overrightarrow { \beta }$ in the form $\overrightarrow { \beta } ={ \overrightarrow { \beta } }_{ 1 }+{ \overrightarrow { \beta } }_{ 2 }$  where ${ \overrightarrow { \beta } }_{ 1 }$ is parallel to $\overrightarrow { \alpha }$  and ${ \overrightarrow { \beta } }_{ 2 }$ is perpendicular to $\overrightarrow { \alpha }$

• 5)

If $\overrightarrow { a } \ and\ \overrightarrow { b }$ are two vectors such that $\left| \overrightarrow { a } +\overrightarrow { b } \right| =\left| \overrightarrow { a } \right|$ then prove that vector $2\overrightarrow { a } +\overrightarrow { b }$ is perpendicular to vector $\overrightarrow { b }$

#### CBSE 12th Mathematics - Linear Programming Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solve graphically.

• 2)

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

• 3)

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchantshould stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an LPP and solve it graphically.

• 4)

A diet for a sick person must contain at least 4,000 units of vitamins, 50 units of minerals and 1,400 calories. Two foods X and Y are available at a cost of Rs. 4 and Rs. 3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements?

• 5)

A firm deals with two kinds of fruit juices- pineapple and orange juice. These are mixed and two mixtures are sold as soft drinks A and B. One tin of A requires 4 litres of pineapple and 1 litre of orange juice. One tin of B requires 2 litres of pineapple and 3 litres of orange juice. The firm has only 46 litres of pineapple juice and 24 litres of orange juice. Each tin of A and B are sold at a profit of Rs. 4 and Rs. 3 respectively. How many tins of each type should the firm produce to maximise the profit? Solve the problem graphically.

#### CBSE 12th Mathematics - Differential Equations Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Solve the differential equation:$\frac { dy }{ dx } +1={ e }^{ x+y }$

• 2)

Solve the differential equation: y - x $\frac { dy }{ dx } =a\left( { y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } \right) ,$ where x = a, y = a.

• 3)

Solve the differential equation: sec2 y (1+x2) dy 2x tan y dx = 0, given that $y=\frac {\pi}{4}$, when x = 1.

• 4)

Solve the differential equation: $\frac {dy}{dx}$ + y cot x = 2 cos x .

• 5)

Solve the differential equation: $\frac {dy}{dx} = tan (x + y)$

#### 12th Standard CBSE Mathematics - Application of Integrals Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Draw the rough sketch of y2 = x + 1 and y2 = x + 1 and determine the area enclosed by the two curves.

• 2)

Using integration, find the area of the quadrant of the circle x2 + y2 = 4

• 3)

Find the area bounded by y = x, the x - axis and the lines x = -1 and x = 2.

• 4)

Calculate the area under the curve: $y=\sqrt { 2 } x$ between the ordinates x =0 and x = 1.

• 5)

Find the area lying above the x - axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.

#### 12th Standard CBSE Mathematics - Integrals Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral $\int {x^2cot^{-1}x}\ dx$

• 2)

Evaluate the integral $\int {1\over a^2 sin^2\ x+b^2 cos^2 x}dx$

• 3)

Evaluate the integral: $\int {x^2+4\over x^4+16}dx.$

• 4)

Evaluate the integral: $\int {(x-4)e^x\over(x-2)^3}dx$

• 5)

Evaluate the integral: $\int{dx\over\sqrt{5-4x-2x^2}}$

#### 12th Standard CBSE Mathematics - Vector Algebra Four Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the projection of $\overrightarrow { b } +\overrightarrow { c } \ on\ \overrightarrow { a }$ where $\overrightarrow { a } =2\hat { i } -2\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$ and $\overrightarrow { c } =2\hat { i } -\hat { j } +4\hat { k }$

• 2)

If $\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =4\hat { i } -2\hat { j } +3\hat { k }$  and $\overrightarrow { c } =\hat { i } -2\hat { j } +\hat { k }$ find a vector of a magnitude 6 units which is parallel to the vector $2\overrightarrow { a } -\overrightarrow { b } +3\overrightarrow { c }$

• 3)

Using vectors find the area of the triangle with vertices A(1,1,2),B(2,3,5) and C(1,5,5).

• 4)

If $\overrightarrow { a } \times \overrightarrow { b } =\overrightarrow { a } \times \overrightarrow { c } \ and\ \overrightarrow { a } \times \overrightarrow { c } =\overrightarrow { b } \times \overrightarrow { d }$ prove that $\overrightarrow { a } -\overrightarrow { d }$ is parallel to $\overrightarrow { b } -\overrightarrow { c }$ provided $\overrightarrow { a } \neq \overrightarrow { d } \ and\ \overrightarrow { b } \neq \overrightarrow { c }$

#### 12th CBSE Mathematics - Differential Equations Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Obtain the differential equation of all the circles of radius r.

• 2)

Find the particular solution of the differential equation (tan-1 y - x) dy = (1 + y2) dx, given that when x = 0, y = 0.

• 3)

Find the particular solution of the differential equation $\frac { dy }{ dx } =\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } }$ given that y = 1, when x = 0.

• 4)

Find the particular solution satisfying the given condition : ${ x }^{ 2 }dy+\left( xy+{ y }^{ 2 } \right) dx=0$; y = 1, when x = 1.

• 5)

Find the particular solution of the following differential equation given that : y = 0, when x = 1 (x2 + xy) dy = (x2 + y2) dx.

#### CBSE 12th Mathematics - Application of Derivatives Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Radius of a variable circle is changing at the rate of 5cm/sec. What is the radius of the circle at the times when area is changing at the rate of 100cm2 /sec?

• 2)

The bottom of a rectangular swimming tank is 50cm $\times$20cm.Water is pumped into the tank at the rate of 500c.c/min.Find the rate at which the level of water in the tank rising

• 3)

A water tank has a shape of an inverted right circular cone with its axis vertical and vertex lowermost. It semi-vertical angle is tan-1(0.5). Water is poured into it at a constant rate of 5cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m.

• 4)

A woman is moving away from a tower 41.5m high at the rate of 2m/sec. Find the rate at which the angle of elevation of the top of the tower is changing, when she is at a distance of 30m from the foot of the tower. Assume that eye level is 1.5m from the ground.

• 5)

Find the intervals in which the function f given by f(x) =sin x+cos x,0$\le$x$\le$2$\pi$,is strightlly increasing or strightly decreasing.

#### CBSE 12th Mathematics - Continuity and Differentiability Four Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If y = x2, prove that $\frac { d^{ 2 }y }{ dx^{ 2 } } \frac { 1 }{ y } \left( \frac { dy }{ dx } \right) ^{ 2 }-\frac { y }{ x } =0$

• 2)

If xmyn = (x + y)m+n, prove that $\frac { dy }{ dx } =\frac { y }{ x }$

• 3)

Differentiate log $\left( x+\sqrt { 1+{ x }^{ 2 } } \right)$

• 4)

Differentiate $\frac { \sqrt { a+x } +\sqrt { a-x } }{ \sqrt { a+x } -\sqrt { a-x } }$

• 5)

Find $\frac { dy }{ dx }$ for sin (xy) + $\frac { x }{ y }$ = x2 - y

#### 12th Standard CBSE Mathematics - Determinants Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Using properties of determinants solve for $x\begin{vmatrix} a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{vmatrix}=0$

• 2)

Using properties of determinants solve the following for X:
$\begin{vmatrix} x-a & x & x \\ x & x+a & x \\ x & x & x+a \end{vmatrix}=0,a\neq 0$

• 3)

Use product $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}$ to solve the system of equations.

x - y + 2z = 1;
2y - 3z = 1;
3x - 2y + 4z = 2

• 4)

solve for x,y,z
${2\over x}+{3\over y}+{10\over z}=4$;
${4\over x}-{6\over y}+{5\over z}=1$;
${6\over x}+{9\over y}-{20\over z}=2$

• 5)

A school wants to award its student or the values of Honesty, Regularity and Hard Work with a total cash award of Rs.6,000. Three times the award money for Hard work added to that given for Honesty amounts to Rs.11,000. The award money given for Honesty and Hard work together is double the one given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

#### 12th CBSE Maths - Application of Integrals Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Using integration, find the area of the triangle formed by positive x-axis and tangent and normal to the circle x + y = 4 at (1, $\sqrt3$).

• 2)

Using integration, find the area of the $\triangle$PQR co-ordinates whose vertices are P(2, 0), Q(4, 5) and R(6, 3).

• 3)

Using integration, find the area of the region enclosed between the two circles x2 + y2 = 9 and (x32)2 + y2 = 9.

• 4)

Find the area of the region bounded by the two parabolas y2 = 4ax and x2 = 4ay, when a > 0.

• 5)

Using integration find the area of the region given by {(x,y) : (x2$\le$y$\le$ |x|)}.

#### 12th CBSE Maths - Integrals Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find : $\int { \frac { sinx }{ sin^{ 3 }x+cos^{ 3 }x } } dx$

• 2)

Evaluate : $\int { \frac { 8 }{ (x+2){ (x }^{ 2 }+4) } } dx$

• 3)

Find : $\int { \frac { sin^{ -1 }\sqrt { x } -cos^{ -1 }\sqrt { x } }{ sin^{ -1 }\sqrt { x } +cos^{ -1 }\sqrt { x } } } dx,x\epsilon \left[ 0,1 \right]$

• 4)

Evaluate : $\int { \frac { { x }^{ 2 }+x+1 }{ (x+2)({ x }^{ 2 }+1) } } dx$

• 5)

Evaluate: $\int _{ 2 }^{ 5 }{ \left( { x }^{ 2 }+3 \right) } dx$

#### CBSE 12th Mathematics - Matrices Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Using elementary transformations, find the inverse of the matrix

$\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right]$.

• 2)

If $A=\left[ \begin{matrix} \cos { \alpha } & \sin { \alpha } \\ -\sin { \alpha } & \cos { \alpha } \end{matrix} \right] ,$ then show that ${ A }^{ 2 }=\begin{bmatrix} \cos { 2\alpha } & \sin { 2\alpha } \\ -\sin { 2\alpha } & \cos { 2\alpha } \end{bmatrix}$

• 3)

If $A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $l=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, prove that $(al+bA)^{ 3 }={ a }^{ 3 }l+{ 3a }^{ 2 }bA.$

• 4)

Let f(x) = x2 - 5x+6 find f(A) If, A =$\left[ \begin{matrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{matrix} \right]$

• 5)

If, A = $\left[ \begin{matrix} a & 0 \\ 1 & 1 \end{matrix} \right]$and B = $\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right]$ find all those values of a for which A = B

#### CBSE 12th Mathematics - Inverse Trigonometric Functions Four Marks and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Solve for X, 2tan-1(sin x) = tan-1(2sec x), $x\neq \frac { \pi }{ 2 }$

#### 12th Standard CBSE Mathematics - Relations and Functions Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Consider $f:R_+\rightarrow[-5,\infty)$ given by f(x) = 9x+ 6x - 5. Show that f is invertible with $f^{-1}(y)={(\sqrt{(y+6)}-1)\over3}$

• 2)

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a*b = min{a, b}. Write the operation table of the operation *.

• 3)

Find $fof^{ -1 }$ and $f^{ -1 }$ of for the function:
$f(x)=\frac { 1 }{ x } ,x\neq 0$. Also prove that $fof^{ -1 }$$f^{ -1 }$ of .

• 4)

Let R be a relation defined on the set of natural numbers Nas follow:
R = {(x, y) : x $\in$ N, y $\in$N and 2x + y = 24}
Find the domain and range of the relation R. Also, find if R is an equivalence relation or not.

• 5)

Let X be a non-empty set, Let * be a binary operation on the power set P(X) defined by A * B = A n B. What is the identify element for the operation * ? Given X is a set of people in a locality, A is a set of children and B is a set of citizens aged above 75 years in the same locality. Is * an invertible binary Feration for these sets as defined above?
What qualities would you suggest that elements of A should have towards elements of B?

#### CBSE 12th Standard Mathematics - Application of Derivatives Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the equation of the tangent line to the curve $y=x^2-2x+7$ which is
(i) parallel to the line $2x-y+9=0$
(ii) perpendicular to the line $5y-15x=13$

• 2)

Find the intervals in which $f(x)-\sin3x-\cos3x,0 is strictly increasing or strictly decreasing. • 3) A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 mand volume is 75 cm3. If building of tank costs Rs. 100 per square metre for the base and Rs. 50 per square metre for the sides, find the cost of least expensive tank. • 4) A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs. 5 per cm2 and the material for the sides cost Rs. 2.50/cm2. Find the least cost of the box. • 5) Show that semi-vertical angle of a cone of maximum volume and given slant height is \(\cos^{-1}(\frac{1}{\sqrt3})$

#### 12th CBSE Mathematics - Continuity and Differentiability Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the value of k, so that the function
$f(x)=\begin{cases} { kx }^{ 2 },\quad if\quad x\ge 1 \\ 4\quad ,\quad if\quad x<1 \end{cases}$ is continuous at x=1.

• 2)

For what value of λ is the function defined by
$f(x)=\begin{cases} \lambda ({ x }^{ 2 }-2x)\quad ,\ if\ x\le 0 \\ 4x+1\quad \quad \ ,\quad if\ x>0 \end{cases}$  continuous at x = 0?
What about continuity at x = 1?

• 3)

For what value of k is the function $f(x)=\begin{cases} \frac { { e }^{ x }+{ e }^{ -x }-2 }{ x^{ 2 } }\ \ ,if\quad x\neq 0 \\ \quad 4k \ \ \ \ \ \ \ \ \ , if\quad x=0 \end{cases}$is continuous at x = 0?

• 4)

For what value of k, is the following function continuous at x=0?
$f(x)=\begin{cases} \frac { 1-cos\quad 4x }{ 8x^{ 2 } } \ , \ \ if\quad x\neq 0 \\ \quad \quad \quad k \ \ \ , \ \ if\quad x=0 \end{cases}$

• 5)

Find the derivative of each of the following function w.r.t. x, or find $\frac { dy }{ dx }$:
$y=\sqrt { \frac { sec \ \ x-1 }{ sec \ \ \ x+1 } }$

#### 12th CBSE Mathematics - Determinants Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Using properties of determinants, show that triangle ABC is isosceles if:
$\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0$

• 2)

Using properties of determinants, prove that:
$\left| \begin{matrix} { (y+z) }^{ 2 } & xy & zx \\ xy & { (x+z) }^{ 2 } & yz \\ xz & yz & { (x+y) }^{ 2 } \end{matrix} \right| =2xyz{ (x+y+z) }^{ 3 }.$

• 3)

If $A=\left[ \begin{matrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{matrix} \right]$find A-1
Hence solve the following system of equations:
x+2y-3z=-4,
2x+3y+2z=2,
3x-3y-4z=11.

• 4)

Determine the product $\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] ,$  and use it to solve the system of equations: x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.

• 5)

If $A=\left[ \begin{matrix} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{matrix} \right] ,$ find A-1. Hence solve the following system of equations:
x + 2y + 5z = 10, x - y - z = -2, 2x + 3y - z = -11.

#### CBSE 12th Standard Mathematics - Matrices Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

If $A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right)$, prove that A3 - 6A2 + 7A + 2I = 0

• 2)

If $A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right)$ and A3 - 6A2 + 7A + kI3 = 0, find k.

• 3)

Using elementary column operations, find the inverse of the following matrix :
$\left[ \begin{matrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right]$

• 4)

If A = $\left[ \begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix} \right]$ find x, y such that A2 +xI = yA Hence find A-1

#### 12th Standard CBSE Mathematics - Probability Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

• 2)

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

• 3)

Six balls are drawn successively from an urn containing 7 red and 9 black balls.Tell whether or not the trials of drawing black balls are Bernoulli trials when after each draws the ball drawn is:
(i) replaced
(ii) not replaced in the urn.

• 4)

If a machine is correctly set up, it produces 90%  acceptable item.If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the setups are correctly done. If after a certain setup, the machine produces 2 acceptable items, find the probability that the machine is correctly set up.

• 5)

An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a line has direction -ratios<2,-1,-2>,determine its direction cosines.

• 2)

Find the direction cosines of the line passing through the two points (-2, 4, -5) and (1, 2, 3)

• 3)

Find the direction-cosines of x, y and z-axis.

• 4)

Find the direction cosines of the unit vector perpendicular to the plane $\vec { r } .\left( 6\hat { i } -3\hat { j } -2\hat { k } \right) +1=0$ through the origin.

• 5)

Find the distance of a point (2, 5, - 3) from the plane:
$\hat { r } .(6\hat { i } -3\hat { j } +2\hat { k }) =4$

#### 12th Standard CBSE Mathematics - Vector Algebra Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write two different having same magnitude.

• 2)

Find the values of x and y so that the vector $2 \hat{i}+3 \hat{j} \text { and } x \hat{i}+y \hat{j}$ are equal.

• 3)

Find out the vector in the direction of vector $\overset { \rightarrow }{ PQ }$ where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.

• 4)

Find a vecto in a direction of vector $5 \hat{i}-\hat{j}+2 \hat{k}$  which has magnitude 8 units.

• 5)

Find the direction-consines of the vector $\hat{i}+2 \hat{j}+3 \hat{k}$ .

#### 12th Standard CBSE Mathematics - Differential Equations Three Marks - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the general solution of differential equation $\log { \left( \frac { dy }{ dx } \right) } =x+1$

• 2)

Find the general solution of differential equation
$\frac { dy }{ dx } +y={ e }^{ -x }$

• 3)

Find the general solution of the differential equation: $(x-y){dy\over dx}={x+2y}$

• 4)

Form the differential equation of the family of parabolas having vertex at the orgin and axis along positive y-axis.

• 5)

Show that the differential equation 2y ex/y dx + (y-2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y= 1.

#### 12th Standard CBSE Mathematics - Application of Integrals Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x - axis in the first quadrant.

• 2)

Find the area of the region in the first quadrant enclosed by x - axis and $x=\sqrt { 3 } y$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }=4$.

• 3)

Find the area of the region bounded by the parabola y = x2 and $y=\left| x \right|$

• 4)

Find the area bounded by the curve x2 = 4y and the line x = 4y - 2.

• 5)

Find the area under the given curves and given lines :
y = x2, x = 1, x = 2 and x - axis

#### CBSE 12th Mathematics - Linear Programming Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following linear programming problem graphically: Maximise Z = 4x + y subject to the constraints:
$x+y\le 50,$
$3x+y\le 90,$
$x\ge 0,y\ge 0.$

• 3)

(Manufacturing Problem) A manufacturer has Three machines I,II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for at least 5 hours a day. She produces only two items M and N each on the three machines are given in the following table:

 Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25

She makes a profit of RS.600 and Rs 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

• 4)

Maximise Z = 3x + 4y subject to the constraints:
$x+y\le 4,x\ge 0,y\ge 0.$

• 5)

Solve the following Linear Programming Problems graphically:
Minimise Z = – 3x + 4 y
subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

#### CBSE 12th Mathematics - Integrals Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write an antiderivative for each of the followings functions, using method of inspection :
(i) $\cos { 2x }$
(ii)${ 3x }^{ 2 }+{ 4x }^{ 3 }$
(iii)  $\frac { 1 }{ x } ,x\neq 0$

• 2)

Find the following integrals:
$(i)\int { \frac { { x }^{ 3 }-1 }{ { x }^{ 2 } } } dx$
$(ii) \int { { (x }^{ 2/3 }+1) } dx$
$(iii) \int { { (x }^{ 3/2 }+{ 2e }^{ x }-\frac { 1 }{ x } ) } dx$

• 3)

Find: $\int { \frac { { x }^{ 2 }+1 }{ { x }^{ 2 }-5x+6 } dx }$

• 4)

Find: $\int { \frac { { x }^{ 2 } }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+4) } } dx$

• 5)

Find:$\int { \sqrt { 3-2x-{ x }^{ 2 } } } dx.$

#### CBSE 12th Mathematics - Application of Derivatives Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The volume of a cube is increasing at the rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of an edge is 10 centimetres?

• 2)

Show that the function  given by:
$f(x)=7x-3$ is strictly increasing on R.

• 3)

The total cost C(x) in Rupees associated with the production of x units of an item is given by
$C(x)=0.007x^{ 3 }-0.003x^{ 2 }+15x+4000$
Find the marginal cost when 17 units are produced.

• 4)

The total revenue in Rupees received from the sale of x units of a product is given by
$R(x)=13x^{ 2 }+26x+15$
Find the marginal revenue when x = 7

• 5)

Prove that the function given by
$f(x)=x^{ 3 }-3x^{ 2 }+3x-100\quad$

#### 12th Standard CBSE Mathematics - Relations and Functions Five Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Let f : $R\rightarrow R$  be defined as f(x) = 10x + 7. Find the function g : $R\rightarrow R$ such that gof = fog = IR

• 2)

Let A = R- 5(3), B = R-[1] Let $f:A\rightarrow B$ be defined by $f(x)=\left( \frac { x-2 }{ x-3 } \right) \forall x\in A$. Then show that f is bijective Hence find  $f^{ -1 }(x)$

• 3)

Let x be a non-empty set and * be a binary operation on P(X) (the power set of set X) defined by
$A*B=A\cup B\ for\ all\ A,B\in P(x)$
Prove that '*' is both commutative and associative on P(X). Find the identity element with respect to on P(X). Also, show that <1>E P(X) is the only invertible element of P(X).

#### 12th CBSE Mathematics - Inverse Trigonometric Functions Five Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Show that :
${ sin }^{ -1 }(2x\sqrt { 1-{ x }^{ 2 } } )={ 2sin }^{ -1 }x,\frac { 1 }{ \sqrt { 2 } } \le x\le \frac { 1 }{ \sqrt { 2 } }$

• 2)

Find the value of cos $({ sec }^{ -1 }x+{ cosec }^{ -1 }x),\left| x \right| \ge 1$

• 3)

Find te principal values of the following: ${ sec }^{ -1 }\left( \frac { 2 }{ \sqrt { 3 } } \right)$

#### 12th Standard CBSE Mathematics - Continuity and Differentiability Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Prove that the identity function on real numbers given by: f(x) = x is continuous at every real number.

• 2)

Discuss the continuity of the function f defined by:
$f(x)={ x }^{ 3 }+{ x }^{ 2 }-1$

• 3)

Find the derivative of tan(2x+3).

• 4)

Differentiate $sin(cos({ x }^{ 2 }))\quad with\quad respect\quad to\quad x.$

• 5)

$Find\ \frac { dy }{ dx } \ if\ x-y=\pi$

#### 12th Standard CBSE Mathematics - Determinants Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Prove that $\left| \begin{matrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{matrix} \right| =4!$

• 2)

If $f(x)=\left| \begin{matrix} a & -1 & 0 \\ ax & a & -1 \\ { a }x^{ 2 } & ax & a \end{matrix} \right|$, using properties of determinants, find the value of f(2x)-f(x).

• 3)

Prove that : $\left| \begin{matrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2ab \end{matrix} \right| =2(a+b+c{ ) }^{ 3 }$

• 4)

Using the properties of determinants, solve the following for 'x' $\left| \begin{matrix} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{matrix} \right| =0$

• 5)

Prove that $\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & c^{ 2 } \\ b+c & c+a & a+b \end{matrix} \right| =(a+b+c)(a-b)(b-c)(c-a)$

#### CBSE 12th Mathematics - Matrices Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

• 2)

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

• 3)

If $x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$ , find the values of x and y.

• 4)

Given:$3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},$ find the values of x, y, z and w.

• 5)

Given: $3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},$find the values of x,y,z and w.

#### CBSE 12th Standard Mathematics - Inverse Trigonometric Functions Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the principal value of ${ cot }^{ -1 }\left( -\frac { 1 }{ \sqrt { 3 } } \right)$

• 2)

Show that :
${ sin }^{ -1 }(2x\sqrt { 1-{ x }^{ 2 } } )={ 2sin }^{ -1 }x,\frac { 1 }{ \sqrt { 2 } } \le x\le \frac { 1 }{ \sqrt { 2 } }$

• 3)

Express $({ tan }^{ -1 }\left( \frac { cosx }{ 1-sinx } \right) ,-\frac { 3\pi }{ 2 }$ in the simplest form :

• 4)

Write ${ cot }^{ -1 }\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ,x>1$ in the simplest form.

• 5)

Prove that :
${ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ { 1-x }^{ 2 } } ={ tan }^{ -1 }\left( \frac { { 3x-x }^{ 3 } }{ { 1-3x }^{ 2 } } \right) ,\left| x \right| <\frac { 1 }{ \sqrt { 3 } }$

#### 12th Standard CBSE Mathematics Unit 1 Relations and Functions Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Show that the function $f\ :\ N\rightarrow N$ given by f (x) = 2x  is one-one but not onto.

• 2)

Show that the function $f:N\rightarrow N$ given by $f(2)=1$ and $f(x)=x-1$ , for every $x>2$ is onto but not one-one.

• 3)

Show that if $f$ :$A\rightarrow B$ and $B\rightarrow C$ are one-one, then $gof:A\rightarrow C$ is also one-one.

• 4)

Show that the relation R in the set {1, 2, 3} given by:
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

• 5)

Show that $\vee :R\times R\rightarrow R$ given by $(a,b)\rightarrow$ max. {a,b} and $\wedge :R\times R\rightarrow R$ given by $(a,b)\rightarrow$ min. (a, b} are binary operations.

#### CBSE 12th Mathematics - Probability Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

One card is drawn is drawn from a pack of 52 cards. Find the probability of getting :
(a) a red card
(b) a jack of hearts
(c) a black face card
(d) a king.

• 2)

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

• 3)

If E and F are two events such that $P(E)=\frac { 1 }{ 4 } ,$ $P(E)=\frac { 1 }{ 2 }$ and $P(E\cap F)=\frac { 1 }{ 8 }$, find
(a) P(E or F)
(b) P(not E and not F).

• 4)

If P(E) = $\frac { 6 }{ 11 }$, P(F) = $\frac { 5 }{ 11 }$ and P(E $\cup$F) = $\frac { 7 }{ 11 }$ then find (a) P(E/F), (b) P(F/E)

• 5)

If P(E) = $\frac { 7 }{ 13 }$ , P(F) = $\frac { 9 }{ 13 }$ and P(E$\cap$F) = $\frac { 4 }{ 13 }$,then evaluate :
(a) $P(\overline { E } /F)$
(b) $P(\overline { E } /F)$

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a lines makes angle 60° and 45° with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the y-axis.

• 2)

If a line makes angle $\alpha ,\beta ,\gamma$ with the coordinates axis ,then find the value of cos $cos2\alpha +cos2\beta +cos2\gamma$

• 3)

Let $I_{ e }m_{ i }n_{ i }i=1,2,3$ be the direction cosines of three mutually perpendicular vector ion space
$\left[ \begin{matrix} { l }_{ 4 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right]$
Show that AA'= l3, where A =

• 4)

If the equation of a line $\frac { x-2 }{ 2 } =\frac { 2y-5 }{ -3 } ,z=-1$ then find the ratio of the line and a point on the line.

• 5)

If the equation of a line is x = ay + b z = cy + d, then find direction ratios of the line and a point on the line.

#### 12th Standard CBSE Mathematics - Vector Algebra Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Classify the following on scalar and vector quantities:
(i) Work
(ii) Force
(iii) Velocity
(iv) Displacement.

• 2)

In a triangle ABC, Show that $\overset\rightarrow {AB}+\overset\rightarrow {BC}+\overset\rightarrow {CA}=0$

• 3)

Find the unit vector in the direction of $\overset\rightarrow a+\overset\rightarrow b$if $\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k$, and $\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k$

• 4)

Find the position vector of c which divides the line segment joining A & B whose position vectors are $3\overset\rightarrow a+\overset\rightarrow b$and $\overset\rightarrow a-3\overset\rightarrow b$ internally in the ratio 2:3.

• 5)

Find the direction cosines of the vector joining the points A(1, 2, - 3) and B(- 1, - 2, 1) directed from B to A.

#### 12th Standard CBSE Mathematics - Differential Equations Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Show that the solution of differential equation :
$y=\left( { 2x }^{ 2 }-1 \right) +c{ e }^{ { -x }^{ 2 } }$ is $\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0$

• 2)

From the differential equation of equation y = a cos2x + b sin2x, where a and b are constant.

• 3)

Find the sum of the order and degree of the following differential equations :
$\frac { { d }^{ 2 }y }{ dx^{ 2 } } +\sqrt [ 3 ]{ \frac { dy }{ dx } } +\left( 1+x \right) =0$

• 4)

Obtain the differential equation of the family of circles passing through the points (a,0) and (- a, 0).

• 5)

Solve the differential equation $\frac { dy }{ dx } ={ e }^{ x-y }+x{ e }^{ -y }$.

#### 12th Standard CBSE Mathematics - Application of Integrals Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

$\int { sin2xcos3xdx }$

• 2)

$\int { \frac { dx }{ 1+sinx } }$

• 3)

$\int { tan^{ -1 } } \sqrt { \frac { 1-cos2x }{ 1+cos2x } } dx$

• 4)

$\int { { cos }^{ 3 } } xdx$

• 5)

$\int { \frac { dx }{ 1+{ e }^{ x } } }$

#### 12th Standard CBSE Mathematics Integrals Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

$\int { sin2xcos3xdx }$

• 2)

$\int { \frac { dx }{ 1+sinx } }$

• 3)

$\int { sin^{ -1 }(cosx)dx }$

• 4)

$\int { tan^{ -1 } } \sqrt { \frac { 1-cos2x }{ 1+cos2x } } dx$

• 5)

$\int { \frac { a }{ b+ce^{ x } } } dx$

#### 12th Standard CBSE Mathematics Unit 6 Application of Derivatives Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The sides of an equilateral triangle are increasing at the rate of 2 cm/see, Find the rate at which its area increases, when side is 10 cm long.

• 2)

If x changes from 4 to 4.01, then find the approximate change in log, x.

• 3)

The side of an equilateral triangle is increasing at the rate of 5 cm/sec. At what rate its area increasing when the side of the triangle is 10 cm.

• 4)

The length x of a rectangle in decreasing at the rate of 5 cm/min and the width y increasing at the rate of 4 cm/min. find the rate of change its area when x = 5 cm and y = 8 cm.

• 5)

The volume of a cube increasing at the rate of 9 cm3/sec. How fast is the surface area increasing when the length of an edge is 10 cm.

#### 12th CBSE Mathematics Continuity and Differentiability Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

if y = $f({ e }^{ { { sin }^{ -1 } } }2x)$, find dy/dx.

• 2)

If y = log(sin x), find $\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } }$

• 3)

If x = $\theta$sin$\theta$, y = $\theta$cos$\theta$ find dy/dx at $\theta$ = $\pi/4$

• 4)

If y = tan-1$\sqrt { \frac { 1-x }{ 1+x } } find\frac { dy }{ dx }$

• 5)

If y = tan-1$\sqrt { \frac { sinx }{ 1+cosx } , } find\frac { dy }{ dx }$

#### CBSE 12th Mathematics - Matrices Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the value of x, y, z if
$\left[ \begin{matrix} 2x+y & x-y \\ x-z & x+y+z \end{matrix} \right] =\left[ \begin{matrix} 10 & -1 \\ 2 & 8 \end{matrix} \right]$

• 2)

If $A=\left[ \begin{matrix} 1 & 4 \\ 3 & 2 \\ 2 & 1 \end{matrix} \right] B=\left[ \begin{matrix} 5 & 2 \\ -1 & 0 \\ 1 & 1 \end{matrix} \right]$, then find the matrix X for which A + B - X = 0.

• 3)

Solve the matrix equation $\left[ \begin{matrix} { x }^{ 2 } \\ { y }^{ 2 } \end{matrix} \right] -3\left[ \begin{matrix} x \\ 2y \end{matrix} \right] =\left[ \begin{matrix} -2 \\ -9 \end{matrix} \right]$

• 4)

If $A=\left[ \begin{matrix} 1 & 2 & 3 \end{matrix} \right]$ and $B=\left[ \begin{matrix} -2 \\ 3 \\ 1 \end{matrix} \right]$, find AB and BA.

• 5)

Find the value of x and y in each if AB exist
(i) ${ A }_{ 3\times x },{ B }_{ 4\times y }$ and ${ AB }_{ 3\times 3 }$
(ii) ${ A }_{ x\times 2 },{ B }_{ y\times 4 }$ and ${ AB }_{ 3\times 4 }$

#### CBSE 12th Mathematics Unit 2 Inverse Trigonometric Functions Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write in the simplest form : ${ tan }^{ -1 }\left[ \frac { cos\quad x }{ 1+sin\quad x } \right] ,x\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right]$

• 2)

Show that :  ${ tan }^{ -1 }\frac { x }{ y } -{ tan }^{ -1 }\frac { x-y }{ x+y } =\frac { \pi }{ 4 }$

• 3)

Show that : ${ tan }^{ -1 }\frac { 2 }{ 3 } =\frac { 1 }{ 2 } { tan }^{ -1 }\frac { 12 }{ 5 }$

• 4)

Show that : ${ tan }^{ -1 }\left( \frac { 3a^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3a{ x }^{ 2 } } \right) =3tan^{ -1 }\left( \frac { x }{ a } \right)$

• 5)

show that : ${ tan }^{ -1 }\frac { 1 }{ 4 } +{ tan }^{ -1 }\frac { 2 }{ 9 } =\frac { 1 }{ 2 } { tan }^{ -1 }\frac { 4 }{ 3 }$

#### CBSE 12th Mathematics Unit 1 Relations and Functions Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Define Reflexive. Give one example.

• 2)

Define symmetric Relation. Give one example

• 3)

Define Transitive Relation. Give one example.

• 4)

Consider the relation perpendicular on a set  of lines in a plane. Show that this relation is symmetric and neither reflexive and nor transitive.

• 5)

What is meant by one-one function?

#### CBSE Class 12th Mathematics Unit 13 Probability Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Given P(A) = $1\over2$, P(B) = $1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

Bayes’ Theorem If E1 , E2 ,..., En are n non empty events which constitute a partition of sample space S, i.e. E1 , E2 ,..., En are pairwise disjoint and E1∪ E2∪ ... ∪ En = S and A is any event of nonzero probability, then
$\mathrm{P}\left(\mathrm{E}_i \mid \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_i\right) \mathrm{P}\left(\mathrm{A}_{\mid} \mathrm{E}_i\right)}{\sum_{j=1}^n \mathrm{P}\left(\mathrm{E}_j\right) \mathrm{P}\left({\left.\mathrm{A} \mid E_j\right)}_1\right.} \text { for any } i=1,2,3, \ldots, n$

• 3)

Given P(A) = 0.2, P(B) = 0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 4)

Given P(A) = 0.4, P(B) = 0.7 and P(B/A) = 0.6, Find $P(A\cup B)$

• 5)

Events E and F are given to be independent. Find P(F) if it is given that P(E) = 0.60 and P(E$\cap$F) = 0.35

#### CBSE Class 12th Mathematics Unit 12 Linear Programming Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following linear programming problem graphically: Minimise Z = 200 x + 500 y subject to the constraints
$x+2y\ge 10,$
$3x+4y\le 24,$
$x\ge 0,y\ge 0.$

• 3)

Maximise Z = 3x + 4y subject to the constraints:
$x+y\le 4,x\ge 0,y\ge 0.$

• 4)

Solve the following Linear Programming Problems graphically:
Minimise Z = 3x + 5y
such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.

• 5)

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts while It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Formulate this mathematically and then solve it.

### CBSE Education Study Materials

#### 12th CBSE Mathematics 2019 - 2020 Academic Syllabus - by Shalini Sharma - Udaipur Aug 21, 2019 Aug 21, 2019

Mathematics 2019 - 2020 Academic Syllabus

#### CBSEStudy Material - Sample Question Papers with Solutions for Class 12 Session 2020 - 2021

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