Question Set B

CBSE 11th Mathematics Sets Model Questions Papers Study Material - QB365

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Questions:

Generic 2

7 x 2 = 14

Solve the inequality 5x - 1> 3x + 7 and show the graph of the solution on number line.

Let f = {(1,1), (2,3), (0, –1), (–1, –3)} be a linear function from Z into Z. Find f(x).

Find the modulus and argument of the following complex number \(\frac { -16 }{ 1+\sqrt { 3 } i } \)

Find the domain and range of the following functions:
f(x) = x²- 3x +\({13\over4}\)

How many different 4-digit numbers can be formed from the digits 2, 3, 4 and 6, if each digit is used only once in a number? Further, how many of these numbers
(ii) end at 3?

If a cos\(\theta\)+b sin\(\theta\)=c, then \(\theta\)=2n\(\pi\)+\(\alpha \pm \beta \), where cos \(\alpha\)=\(\frac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \),cos \(\beta \)=\(\frac { c }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)and sin \(\alpha\)=\(\frac { b }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)

List all the proper subsets of the set A={a,b}.

Generic 3

8 x 3 = 24

Let f = \(\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in R\right\}\) be a functions R into R determine the range of R.

Find the value of sin 15°.

10.

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed?
(ii) repetition of the digits is not allowed?

11.

How many natural numbers less than 1000 can be formed with the digits 1, 2, 3, 4, and 5, if repetition of digits is allowed?

12.

Solve graphically \(4x+3y\ge 12\quad and\quad 4x-5y\ge -20\)

13.

Find the value of x and y, if \(\frac { \left( 1+i \right) x-2i }{ 3+i } +\frac { \left( 2-3i \right) y+i }{ 3-i } =i\)

14.

The Moon's distance from the Earth is 360000 km and its diameter subtend an angle of 31' at the eye of observer. Find the diameter of the Moon.

15.

If X = {a,b,c,d} and Y = {f,b,d,g}, then find
Y - X

Generic 5

6 x 5 = 30

16.

In any triangle ABC, prove that \(\frac{a^2sin(B-C)}{sinB+sinC}+\frac{b^2sin(C-A)}{sinC+sinA}+\frac{c^2sin(A-B)}{sinA+sinB}=0\) .

17.

Find the values of x and y if (x+iy) (4+5i) =6-2i.

18.

Draw the Venn diagrams to illustrate the following relationship among sets, E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.
There is no student who studies both Mathematics and English.

19.

Let A = {1, 2, 3, 4} and B = {5, 6, 7}. If R = {(a, b) ; a \(\in\) A, b \(\in\) B} and a - b is even} then find R.

20.

There are 4 routes between Delhi and Patna. In how many different ways can a man o from Delhi to Patna and return , if for returning
(i) any of the routes is taken ?
(ii) the same routes is taken ?
(iii) the same routes is not taken ?

21.

Solve the following system of linear inqualities \(-2-\frac { x }{ 4 } \ge \frac { 1+x }{ 3 } \) and \(3-x<4\left( x-3 \right) \)

Multiple Choice Question

20 x 1 = 20

22.

If P(x) is a n degree polynomial, then number of roots are _______.

(a)

(b)

(c)

(d)

23.

Representation of z = x + iy in terms of rand e is called _______.

(a)

polar form of the complex number

(b)

cartesian form of the complex number

(c)

complex form of the complex number

(d)

real form of the complex number

24.

If a + ib = c + id, then _______.

(a)

\(a^{2}+c^{2}=0\)

(b)

\(b^{2}+c^{2}=0\)

(c)

\(b^{2}+d^{2}=0\)

(d)

\(a^{2}+b^{2}=c^{2}+d^{2}\)

25.

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. The number of ways in which the seating arrangement can be made, is _______.

(a)

\(\frac{11 !}{6 ! 5 !} \times 9 ! \times 9 !\)

(b)

\(9 ! \times 9 !\)

(c)

\(\frac{11 !}{6 ! 5 !} \times 5 ! 6 !\)

(d)

None of these

26.

If (n + 1)! = 12 x (n - 1)! then the value of n is equal to

(a)

(b)

(c)

(d)

27.

Sabnam has 2 school bags, 3 tiffin boxes and 2 water bottles. In how many ways can she carry these items ( choosing one each)?

(a)

(b)

(c)

(d)

28.

In an experiment, a solution of hydrochloric acid is to be kept between 30°C and 35°C.The range of temperature in degree Fahrenheit, if conversion formula is given byC \(=\frac{5}{9}(F-32)\) where Cand F represent temperature in degree Celsius and degree Fahrenheit respectively, is between _______.

(a)

86°F and 95°F

(b)

54°F and 63°F

(c)

54°F and 95°F

(d)

63°F and 86°F

29.

Two real numbers or two algebraic expressions related by the symbol '<', '>', '\(\le\)' or '\(\ge\)'forms an _______.

(a)

equation

(b)

inequality

(c)

set

(d)

None of the above

30.

Which of the following is the solution set of the inequality \(\frac{x}{4}<\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5} ?\)

(a)

\((4, \infty)\)

(b)

\((-\infty, 4)\)

(c)

\({[4, \infty)}\)

(d)

\((-\infty, 4]\)

31.

Which of the following solution set represent the following figure?

(a)

\(x+2 y \leq 8,2 x+y \leq 8, x \geq 0, y \geq 0 \)

(b)

\(x-2 y \leq 7, x+y \leq 8, x \geq 0, y \geq 0 \)

(c)

\(x+y<82 x+y<8, r>0 y>0\)

(d)

None of the above

32.

The value ofcos1°cos2°cos3° ... cos179° is ______.

(a)

\(\frac{1}{\sqrt{2}}\)

(b)

(c)

(d)

-1

33.

Angles of a triangle are in the ratio 4 : 1 : 1. The ratio between its greatest side and perimeter is ______.

(a)

\(\frac{3}{2+\sqrt{3}}\)

(b)

\(\frac{1}{2+\sqrt{3}}\)

(c)

\(\frac{\sqrt{3}}{\sqrt{3}+2}\)

(d)

\(\frac{2}{2+\sqrt{3}}\)

34.

The graph of the function defined by \(f(x)=\left\{\begin{array}{cl} 1-x, & x<0 \\ 1 & , x=0 \text { is } \\ 1+x, & x>0 \end{array}\right.\)______.

(a)

(b)

(c)

(d)

35.

Let R be a relation in N defined by \(R=\left\{\left(1+x, 1+x^{2}\right): x \leq 5, x \in N\right\}\). Which of the following is false?

(a)

R = {(2, 2), (3, 5), (4, 10), (5, 17), (6, 25)}

(b)

Domain of R = {2, 3, 4, 5, 6}

(c)

Range of R = {2,5, 10, 17, 26}

(d)

None of the above

36.

Domain of \(\sqrt{a^{2}-x^{2}}(a>0)\) is ______.

(a)

(- a, a)

(b)

[- a, a]

(c)

[0, a]

(d)

(- a, 0]

37.

The minute hand of a watch is 1.5 cm long. The distance travelled by the minute hand in 40 minutes is equal to ______.

(a)

3.28 cm

(b)

4.28 cm

(c)

5.28 cm

(d)

6.28 cm

38.

Radian measure of 40°20' is equal to ______.

(a)

\(\frac{120 \pi}{504}\) radian

(b)

\(\frac{121 \pi}{540}\) radian

(c)

\(\frac{121 \pi}{3}\) radian

(d)

None of these

39.

Let A and B be two sets such that n (A) = 0.16, n(B) = 0.14 and n(A \(\cup\) B) = 0.25. Then, n(A \(\cap\) B) is equal to_____.

(a)

0.3

(b)

0.5

(c)

0.05

(d)

None of these

40.

The set of all letters in the word 'BETTER' in roster form is _____.

(a)

{B, E, T, R}

(b)

{B, T, R}

(c)

{B,E,R}

(d)

{B, R}

41.

Which of the following is not a set?

(a)

(a) The collection of days of week.

(b)

(b) The collection of prime number less than 10.

(c)

(d)

(d) All of the above.

Case Study Questions

3 x 4 = 12

42.

The water acidity in a pool is considered normal, when the average pH reading of three daily measurements is between 7.2 and 7.8. The first two pH reading are 7.48 and 7.85, and pH reading of 3rd day is x.

On the basis of above information, answer the following questions.
(i) The average pH of three days is

(a) 5.11+x

(b) \(5.11+\frac{x}{3}\)

(c) 15.33+x

(d) None of these

(ii) The system of linear inequality, which shows the given information is

(a) \(7.2 \leq 5.11+\frac{x}{3} \leq 7.8\)

(b) \(7.2<5.11+\frac{x}{3} \leq 7.8\)

(c) \(7.2 \leq 5.11+\frac{x}{3}<7.8\)

(d) \(7.2<5.11+\frac{x}{3}<7.8\)

(iii) The solution of linear inequality \(7.2 \leq 5.11+\frac{x}{3}\) is

(a) x≥6.27

(b) x>6.27

(c) x≤6.27

(d) x<6.27

(iv) The solution of linear inequality \(5.11+\frac{x}{3} \leq 78\) is

(a) x≤8.07

(b) x<8.07

(c) x≥8.07

(d) x>8.07

(v) The value of pH on third day is

(a) [6.27,8.07]

(b) (627,8.07)

(c) (627,8.07]

(d) [6.27,8.07)

43.

A complex number z is purely real if and only if \(\bar{z}=z\) and is purely imaginary if and only if \(\bar{z}=-z\). Based on the above information, anwer the following qustions.
(i) If \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is purely real, then \(\theta\) is equal to

(a) \(n \pi\)

(b) \(\frac{n \pi}{2}\)

(c) \(n \pi \pm \frac{\pi}{3}\)

(d) \(2 n \pi \pm \frac{\pi}{4}\)

(ii) If x and y are rol numbers and the complex number \(\frac{(2+6) x-1}{4+1}+\frac{(1-6) y+21}{4}\) is purely teal, the relation between x and y is

(a) 8x+7 y=15

(b) 8x-17 y=16

(c) 17x-8 y=15

(d) 17x+8 y=15

(iii) If z be a complex number satisfying \(|\operatorname{Re}(z)|+|m(z)|=4\), then |z| cannot be

(a) \(\sqrt{7}\)

(b) \(\sqrt{\frac{17}{2}}\)

(c) \(\sqrt{10}\)

(d) \(\sqrt{8}\)

(iv) The smallest positive integer ( n ) for which \((1+i)^{2 n}=(1-i)^{2 n}\) is

(a) 2

(b) 3

(c) 1

(d) 4

(v) If \(z_1\) and z₂ are complex numbers such that \(\left|\frac{z_1-z_2}{z_1+z_2}\right|=1\), then

(a) \(\frac{z_1}{z_2}\) is purely real

(b) \(\frac{z_1}{z_2}\) is purely imaginary

(c) z₁ is purely real

(d) z₁ and z₂ are purely imaginary

44.

Consider thc information given below Let P (a, b) be any point on the unit circle given below, which has its centre at origin O. It is given that ∠AOP = x radian.
Now, answer the questions based on the figure given below.

(i) If \(a = \frac{ \sqrt{3}}{2}\) and \(b = \frac{-1}{2}\) then the value of x in radian is

(a) \(\frac{\pi}{6}\)

(b) \(\frac{7\pi}{6}\)

(c) \(\frac{5\pi}{6}\)

(d) \(\frac{11\pi}{6}\)

(ii) If x = 75^o, then the value of tan x is equal to

(a) \(2 + \sqrt3\)

(b) \(2-\sqrt3\)

(c) \(\sqrt3 + 3\)

(d) \(\sqrt2 + 1\)

(iii) If x = 25°, then the value of x in radian is

(a) \(\frac{5\pi}{18}\)

(b) \(\frac{\pi}{18}\)

(c) \(\frac{5\pi}{36}\)

(d) \(\frac{\pi}{36}\)

(iv) Which of the following is incorrect?

(a) sin x is positive for 0 < x < π

(b) sin x isnegative for π < x < 2π

(c) cos x is positive for 0 < x < π/2 and 3π/2 < x < 2π

(d) cos x is positive for π/2 < x < 3π/2

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Answers Key:

Generic 2

7 x 2 = 14

Subtracting (3x -1) from both sides of5x -1 > 3x + 7.
Ans : (4, \(\infty\))

Since f is a linear function, f (x) = mx + c. Also, since (1, 1), (0, – 1) \(\in\) R, f (1) = m + c = 1 and f (0) = c = –1. This gives m = 2 and f(x) = 2x – 1.

\(8,\frac { 2\pi }{ 3 } \)

Domain = R, Range = [1,\(\infty\))

The number of 4-digit numbers formed by the digits 2, 3, 4 and 6
= P(4,4) = \(\frac { 4! }{ 4-4! } \)= \(\frac { 4! }{ 0! } \) [\(\because \) ⁿP_{r =}\(\frac { n! }{ (n-r)! } ]\)
\(=\frac { 4! }{ 1 } = 4\times 3\times 2\times 1= 24\) \([\because 0!=1]\)
In this case, '3' is fixed at the unit's place. Therefore, remaining 3-digits can be 2, 4 and 6. Hence, the required number of 4- digits number = P (3, 3)
\(\frac { 3! }{ 0! } =6\) ... (ii)

Given, a cos\(\theta\)+ b sin \(\theta\)=c....(i)
On dividing Eq.(i) throughout by \(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \), we get
\(\frac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } cos\theta +\frac { b }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } sin\theta =\frac { c }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)
\(\Rightarrow\)cos\(\theta\)cos\(\alpha\)+sin\(\theta\)sin\(\alpha\)=cos\(\beta\)
where, cos\(\alpha\) =\(\frac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \) ,sin\(\alpha\) =\(\frac { b }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)and cos \(\beta\)=\(\frac { c }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)
\(\Rightarrow\)cos(\(\theta-\alpha\)) = cos\(\beta\)\(\Rightarrow\)\(\theta-\alpha\) = \(2n\pi \pm \beta \)
\(\therefore \theta =2n\pi +\alpha \pm \beta \)

ϕ,{a},{b}

Generic 3

8 x 3 = 24

f(x) is defined for all x \(\epsilon\) R. As x²+1 for any x \(\epsilon\) R.
Now, let \(y=f(x)=\frac{x^{2}}{r^{2}+1}\)
\(\Rightarrow x^{2}=\frac{y}{1-y} \Rightarrow x=\pm \sqrt{\frac{y}{1-y}}\)
Clearly, x will be real, if
\(\frac{y}{1-y} \geq 0 \text { and } y \neq 1\)
\(\begin{array}{ll} \Rightarrow \frac{y}{y-1} \leq 0 \text { and } y \neq 1 \Rightarrow 0 \leq y<1 \\ \therefore \text { Range }(f)=[0,1) . \end{array}\)

We have sin 15° = sin (45° – 30°)
= sin 45° cos 30° – cos 45° sin 30°
\(=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \times \frac{1}{2}=\frac{\sqrt{3}-1}{2 \sqrt{2}}\)

10.

(i) There will be as many ways as there are ways of filling 3 vacant places in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filled in by any of the given five digits. Similarly, tens and hundreds digits can be filled in by any of the given five digits.Thus, by the multiplication principle, the number of ways in which three-digit numbers can be formed from the given digits is 5 × 5 × 5 = 125
(ii) In this case, repetition of digits is not allowed. Here, if units place is filled in first, then it can be filled by any of the given five digits. Therefore, the number of ways of filling the units place of the three-digit number is 5. Then, the tens place can be filled with any of the remaining four digits and the hundreds place can be filled with any of the remaining three digits. Thus, by the multiplication principle, the number of ways in which three-digit numbers can be formed without repeating the given digits is 5 × 4 × 3 = 60

11.

Required numbers will be of 1-digit, 2-digit or 3-digit.
Ans. 155

12.

13.

Given \(\frac { \left( 1+i \right) x-2i }{ 3+i } +\frac { \left( 2-3i \right) y+i }{ 3-i } =i\)
\(\Rightarrow \frac { x+\left( x-2 \right) i }{ 3+i } +\frac { 2y+\left( 1-3y \right) i }{ 3-i } =i\)
\(\Rightarrow \frac { \left[ x+\left( x-2 \right) i \right] \left( 3-i \right) +\left[ 2y+\left( 1-3y \right) i \right] \left( 3+i \right) }{ \left( 3+i \right) \left( 3-i \right) } =i\)
\(\Rightarrow \left( 4x+9y-3 \right) +i\left( 2x-7y-3 \right) =10i\)
\(\Rightarrow 4x+9y-3=0\quad and\quad 2x-7y-3=10\)
\(Ans.\ x=3\ and \ y=-1\)

14.

\(\theta =31'=\left( \frac { 31 }{ 60 } \times \frac { \pi }{ 180 } \right) rad\quad and\quad r=360000km\)
\(\because \quad \theta =\frac { l }{ r } \therefore \frac { 31 }{ 60 } \times \frac { \pi }{ 180 } =\frac { 1 }{ 360000 } \)
Ans. 3247.62 km

15.

\(Y-X=\{ f,g\} \)

Generic 5

6 x 5 = 30

16.

From sine formula
\(\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=k(say)\)
\(\therefore\) a=ksinA,b=ksinB and c=ksinC
L.H.S \(\frac{a^2sin(B-C)}{sinB+sinC}+\frac{b^2sin(C-A)}{sinC+sinA}+\frac{c^2sin(A-B)}{sinA+sinB}\)
\(=\frac{k^2sin^2A.sin(B-C)}{sinB+sinC}+\frac{k^2sin^2B.sin(C-A)}{sinC+sinA}+\frac{k^2sin^2C.sin(A-B)}{sinA+sinB}\)
\(=k^2[\frac{sinA.sin(\pi-\overline {B+C}).sin(B-C)}{sinB+sinC}+\frac{sinB.sin(\pi-\overline {C+A}).sin(C-A)}{sinC+sinA}+\frac{sinC.sin(\pi-\overline {A+B}).sin(A-B)}{sinA+sinB}]\)
\(=k^2[\frac{sinA.sin(B+C).sin(B-C)}{sinB+sinC}+\frac{sinB.sin(C+A).sin(C-A)}{sinC+sinA}+\frac{sinC.sin(A+B).sin(A-B)}{sinA+sinB}]\)
\(=k^2\frac{sinA(sin^2B-sin^2C)}{sinB+sinC}+\frac{sinB(sin^2C-sin^2A)}{sinC+sinA}+\frac{sinC(sin^2A-sin^2B)}{sinA+sinB}\)
= k² [sin A (sin B - sin C) + sin B (sin C - sin A) + sin C (sin A - sin B)]
= k² [sin A sin B - sin A sin C + sin B sin C - sin A sin B + sin A sin C - sin B sin C]
=k²[0]
=0 R.H.S.

17.

(x + iy)(4 + 5i) = 6-2i
⇒ 4x + 5xi + 4yi + 5yi² = 6-2i
⇒ (4x - 5y) + (5x + 4y)i = 6-2i
Comparing real and imaginary parts on both sides,we have
4x - 5y = 6 and 5x + 4y = -2
Solving these two equations for x and y, we get
x=\(\frac { 14 }{ 41 } \) and y=\(-\frac { 38 }{ 41 } \)

18.

19.

Here A = {1, 2, 3, 4} and B = {5, 6, 7}, a \(\in\) A, b \(\in\) B.
\(\therefore\) a - b = 1 - 5, 1 - 6, 1 - 7, 2 - 5, 2 - 6, 2 - 7, 3 - 5, 3 - 6, 3 - 7, 4 - 5, 4-6,4-7
= -4, -5, -6, -3, -4, -5, -2, -3, -4, -1, -2,-3
R = {(1, 5), (1, 7), (2, 6), (3,5), (3, 7), (4,6)}.

20.

(i) Total number of different routes = 4 x 4
(ii) Total number of different routes = 4
(iii) Total number of different routes = 4 x 3

21.

No solution exists

Multiple Choice Question

20 x 1 = 20

22.

(d)
n

23.

(a)
polar form of the complex number

24.

(d)
\(a^{2}+b^{2}=c^{2}+d^{2}\)

25.

(a)
\(\frac{11 !}{6 ! 5 !} \times 9 ! \times 9 !\)

26.

(b)
3

27.

(b)
12

28.

(a)
86°F and 95°F

29.

(a)
equation

30.

(a)
\((4, \infty)\)

31.

32.

(b)
0

33.

34.

35.

(a)
R = {(2, 2), (3, 5), (4, 10), (5, 17), (6, 25)}

36.

(b)
[- a, a]

37.

(d)
6.28 cm

38.

(b)
\(\frac{121 \pi}{540}\) radian

39.

40.

(a)
{B, E, T, R}

41.

Case Study Questions

3 x 4 = 12

42.

(i) (b)
(ii) (d)
(iii) (a)
(iv) (a)
(v) (b)

43.

(i) (a)
(ii) (b)
(iii) (a)
(iv) (a)
(v) (b)

44.

(i) (d) Hint \(a=\sqrt{3}\) and b=-2
\(\therefore x\) lies in 4 th quadrant.
\( \cot x=\frac{a}{b}=-\sqrt{3}=\cot \left(2 \pi-\frac{\pi}{6}\right) \)
\( \cot x=\cot \frac{11 \pi}{6} \Rightarrow x=\frac{11 \pi}{6}\)
(ii) (a) Hint \(\tan 75^{\circ} =\tan \left(45^{\circ}+30^{\circ}\right)=\frac{\tan 45^{\circ}+\tan 30^{\circ}}{1-\tan 45^{\circ} \tan 30^{\circ}} \)
\(=\frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}=\frac{\sqrt{3}+1}{\sqrt{3}-1}=2+\sqrt{3}\)
(iii) (c) Hint \(x=25^{\circ}\)
\(25^{\circ}=\left(25 \times \frac{\pi}{180^{\circ}}\right) \text { radian }=\frac{5 \pi}{36}\)
(iv) (d) Hint cos x is negative in \(x \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\).