12th Standard CBSE Maths Application Of Derivatives Important Quetions
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12th Standard CBSE Maths Application Of Derivatives Important Quetions
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Application Of Derivatives
12th Standard CBSE

Reg.No. :
Maths
PartA
use only black and blue ink to write and pencil to draw

If P(A) = \(\frac12\), P(B) = 0, then P(AB) is
(a)0
(b)\(\frac12\)
(c)not defined
(d)1

Let f : R ➝ R be defined as f (x) = 3x. Choose the correct answer
(a)f is oneone onto
(b)f is manyone onto
(c)f is oneone but not onto
(d)f is neither oneone nor onto

Consider a binary operation * on N defined as a * b = a^{3} + b^{3}. Choose the correct answer
(a)Is * both associative and commutative?
(b)Is * commutative but not associative?
(c)Is * associative but not commutative?
(d)Is * neither commutative nor associative?

If A = {1,3,5,7} and define a relation, such that R = { (a,b) a,b ∈ A : a+b = 8}. Then how many elements are there in the relation R
(a)8
(b)16
(c)1
(d)4

If A = diag(3, 1), then matrix A is
(a)\(\begin{bmatrix} 0 & 3 \\ 0 & 1 \end{bmatrix}\)
(b)\(\begin{bmatrix} 1 & 0 \\ 3 & 0 \end{bmatrix}\)
(c)\(\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\)
(d)\(\begin{bmatrix} 3 & 1 \\ 0 & 0 \end{bmatrix}\)

The number of all possible matrices of order 3 × 3 with each entry
(a)27
(b)18
(c)81
(d)512

If A, B are symmetric matrices of same order, then AB – BA is a
(a)Skew symmetric matrix
(b)Symmetric matrix
(c)Zero matrix
(d)Identity matrix

For what real value of y will matrix A be equal to matrix B, where
\(A=\begin{bmatrix} 3x4 & 5y \\ 8 & { y }^{ 2 }4y \end{bmatrix};B=\begin{bmatrix} x+1 & 6{ y }^{ 2 }+1 \\ 8 & 3 \end{bmatrix}\)(a)1, 3
(b)No real value
(c)1/3, 1/2
(d)2 and 3

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
(a)k = 1
(b)k = 2
(c)K = \(\frac12\)
(d)k = \(\frac32\)

If y = tan^{1} \(\left( \frac { 1{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) \), then \(\frac { dy }{ dx } \) is equal to
(a)\(\frac { 1 }{ 1+{ x }^{ 4 } } \)
(b)\(\frac { 2x }{ 1+{ x }^{ 4 } } \)
(c)\(\frac { 1 }{ 1+{ x }^{ 4 } } \)
(d)\(\frac { { x }^{ 2 } }{ 1+{ x }^{ 4 } } \)

The equation of the normal to the curve y = sin x at (0, 0) is
(a)x = 0
(b)y = 0
(c)x + y = 0
(d)x – y = 0

The absolute maximum value of y = x^{3} – 3x + 2 in 0 ≤ x ≤ 2 is
(a)4
(b)6
(c)2
(d)0

The interval in which y = x^{2} e^{–x} is increasing is
(a)(– ∞, ∞)
(b)(– 2, 0)
(c)(2, ∞)
(d)(0, 2)

For all real values of x, the minimum value of \(\frac { 1x+{ x }^{ 2 } }{ 1+x+{ x }^{ 2 } } \) is
(a)0
(b)1
(c)3
(d)\(\frac { 1 }{ 3 } \)

Let \({ I }_{ 1 }=\int _{ 1 }^{ 2 }{ \frac { dx }{ \sqrt { 1+{ x }^{ 2 } } } } \) and \({ I }_{ 2 }=\int _{ 1 }^{ 2 }{ \frac { dx }{ x } } \), then
(a)I_{1} > I_{2}
(b)I_{2} > I_{1}
(c)I_{1} = I_{2}
(d)I_{1} > 2I_{2}

\(\int { \sqrt { { x }^{ 2 }8x+7 } } dx\) is equal to
(a)\(\frac { 1 }{ 2 } (x4)\sqrt { { x }^{ 2 }8x+7 } + 9logx4+\sqrt { { x }^{ 2 }8x+7 } +\)C
(b)\(\frac { 1 }{ 2 } (x+4)\sqrt { { x }^{ 2 }8x+7 } + 9logx+4+\sqrt { { x }^{ 2 }8x+7 } +\)C
(c)\(\frac { 1 }{ 2 } (x4)\sqrt { { x }^{ 2 }8x+7 }  3\sqrt2logx4+\sqrt { { x }^{ 2 }8x+7 } +\)C
(d)\(\frac { 1 }{ 2 } (x4)\sqrt { { x }^{ 2 }8x+7 }  \frac92logx4+\sqrt { { x }^{ 2 }8x+7 } +\)C

Area of a rectangle having vertices A, B, C and D with position vectors \(\widehat { i } +\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), \(\widehat { i } +\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), \(\widehat { i } \frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \) and \(\widehat { i } \frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), respectively is
(a)\(\frac12\)
(b)1
(c)2
(d)4

If a, b, c and d are the position vectors of the points A, B, C and D such that a + c = b + d, then ABCD is a
(a)Trapezium
(b)Rectangle
(c)Square
(d)Parallelogram

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(a)0
(b)\(\frac13\)
(c)\(\frac{1}{12}\)
(d)\(\frac{1}{36}\)

Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is
(a)\(\frac{37}{221}\)
(b)\(\frac{5}{13}\)
(c)\(\frac{1}{13}\)
(d)\(\frac{2}{13}\)

If f(x)=27x^{3} and g(x)=x^{1/3} find gof(x).
(a)\(gof(x)=g\left( f\left( x \right) \right) =g\left( 27{ x }^{ 3 } \right) ={ \left( 27{ x }^{ 3 } \right) }^{ 1/3 }=3x\)

In the group (z, *) of all integers , where a * b = a + b + 1 for a, b \(\epsilon \)Z, then what is the inverse of 2?
(a)0

Find whether the relation R in the set {1,2,3} given by R = {(1,1), (2,2), (3,3), (1,2),(1,3)} is reflexive , symmetric or transitive
(a) 
If \(2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix}+\begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\), then write the value of (x+y).
(a)\(\begin{bmatrix} 2 & 6 \\ 0 & 2x \end{bmatrix}+\begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\)
\(\Rightarrow \begin{bmatrix} 2+y & 6 \\ 1 & 2x+2 \end{bmatrix}=\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\Rightarrow 2+y=5,2x+2=8\Rightarrow x=3,y=3\)
\(\therefore \quad x+y=3+3=6\) 
If the following matrix is skew symmetric, find the values of a, b, c
\(A=\left[ \begin{matrix} 0 & a & 3 \\ 2 & b & 1 \\ c & 1 & 0 \end{matrix} \right] \)(a)a = 2, b = 0, c = 3

Show that the function f (x)= \(\begin{cases} { x }^{ 3 }+3\quad ,\quad if\quad x\neq 0 \\ 1\quad \quad \quad ,\quad if\quad x=0 \end{cases}\) is not continuous at x=0.
(a)3\(\neq \)f (0), discontinuous.

Discuss the continuity of the function f(x) = x  x1
(a){continuous}

The total cost C(x)of producing x units of an article is given by C(x)=0.005x^{2}0.02x^{2}+30x+5000.Find the marginal cost when 3 units are produced.
(a)Rs.30.015

Evaluate the integral: \(({\int{\sqrt x+{1\over\sqrt x}}})^2dx.\)
(a)Consider : \(\int({\sqrt x+{1\over \sqrt x}})^2dx=\int({x+{1\over x+2}+2})dx={x^2\over2}+log x + 2x + c\)

\(\int {1+tan\ x\over 1tan\ x}dx\).
(a)Consider, = log t + c = log cos x  sin x+ c

\(\int tan^{1}(cot\ x)dx.\)
(a)\(={\pi \over2}\times{x^2\over2}+c\)

What is the cosine of the angle which the vector \(\sqrt { 2 } \hat { i } +\hat { j } +\hat { k } \) makes with yaxis?
(a)\(cos\beta =\frac { 1 }{ 2 } \)

Find the scalar components of the vector \(\overset\rightarrow {AB} \) with initial point A(2,1) and terminal point B(5,7).
(a)\(\overset\rightarrow {AB}\) = Position vector of BPosition vector of A
\(=(5\overset\wedge i+7\overset\wedge j)(2\overset\wedge i+1\overset\wedge j)\)
\(=(52)\overset\wedge i+(71)\overset\wedge j\)
\(=7\overset\wedge i+6\overset\wedge j\)
\(\therefore\) The scalar components are (7,6,0). 
Find \(\lambda\) and \(\mu\) if \((\overset\wedge i+3\overset\wedge j+9\overset\wedge k)\times(3\overset\wedge i\lambda \overset\wedge j+\mu \overset\wedge k)=0\)
(a)Getting \(\lambda =9\)
and \(\mu=27\)
Alternative Method :
\((\overset\wedge i+3\overset\wedge j+9\overset\wedge k)\times(3\overset\wedge i\lambda \overset\wedge j+\mu\overset\wedge k)=0\)
\(\Rightarrow \left \begin{matrix} \overset { \wedge }{ i } & \overset { \wedge }{ j } & \overset { \wedge }{ k } \\ 1 & 3 & 9 \\ 3 & \lambda & \mu \end{matrix} \right \)
\(\Rightarrow \overset\wedge i(3\mu+9\lambda)\overset\wedge j(\mu27)+\overset\wedge k(\lambda9)=0\)
\(\Rightarrow 3\mu+9\lambda=0 ....(i)\)
\(\Rightarrow \mu27=0 ...(ii)\)
\(\Rightarrow \lambda9=0...(iii)\)
from eqn.(ii) and (iii),
\(\mu=27\)
and \(\lambda=9\) 
Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find \(P(A\cup B)\)
(a)0.86

Check whether functions in the following diagrams are onto:
(i) Since, every element in B has preimage in A. so \(f_{ 1 }\) is onto function.
(ii) Since, element in B has preimage of A. So \(f_{ 2 }\) is onto function.
(ill) Since 3 E B does not have preimage A. So \(f_{ 3 }\) is not onto function. Yz
(iv) Since, 6 E B does not have preimage of A. So \(f_{ 4 }\) is not onto function.(a) 
Let f and g be real function be \(f(x)=\sqrt { x+4 } ,x\ge 4\) find the function fg, \(\frac { f }{ g } \)
(a) 
Find the value of X and Y if
\(X+Y=\left[ \begin{matrix} 2 & 3 \\ 5 & 1 \end{matrix} \right] ,XY=\left[ \begin{matrix} 6 & 5 \\ 7 & 3 \end{matrix} \right] \)(a) 
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 \({ AB }_{ 3\times 4 }\)(a) 
If y= tan^{1}\(\sqrt { \frac { 1x }{ 1+x } } find\frac { dy }{ dx } \)
(a) 
If x = \(\frac { at }{ 1+{ t }^{ 2 } } ,y=\frac { a{ t }^{ 2 } }{ 1+{ t }^{ 2 } } ,find\frac { dy }{ dx } at\) t = 2
(a) 
Find the point on the curve \(y^2 =8x + 3\) for which the ycoordinate changes 4 times more than coordinate of x.
(a) 
\(\int { { sin }^{ 2 }x{ \quad cos }^{ 2 }x } dx\)
(a) 
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 \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j\overset\wedge k\)
(a) 
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) \(P(\overline { E } /F)\)(a) 
Show that the relation R in the set Z of integers given by:
R={(a,b):2divides ab} is an equivalence relation(a) 
(Diet Problem) A dietician has to develop a special diet using two foods P and Q. Each packet (containing 50g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet. What is the minimum amount of vitamin A?
(a) 
In each of the following cases, state whether the function is oneone, onto or bijective. Justify your answer.
(i) \(f:R\rightarrow R\) defined by f(x)=34x
(ii) \(f:R\rightarrow R\) defined by \(f(x)=1+x^{ 2 }\) .(a) 
Find X, if \(Y=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\) and \(2X+Y=\begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix}\) .
(a) 
Discuss the continuity of the function f given by:
\(f(x)=\left x \right at\quad x=0.\)(a) 
Discuss the continuity of the function f defined by:
\(f(x)={ x }^{ 3 }+{ x }^{ 2 }1\)(a) 
Differentiate \({ tan }^{ 1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } 1 }{ x } \right) \) w.r.t.x.
(a) 
Prove that the function given by:
\(f(x)cosxis:\)
(a) strictly decreasing in \((0,\pi )\)
(b) strictly increasing in \((\pi ,2\pi )\)
(c) neither increasing nor decreasing in \((0,2\pi )\)(a) 
Find the absolute maximum and absolute value of the function given by:
\(f(x)sin^{ 2 }xcosx,x\in [0,\pi ]\\ \)(a) 
Let \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } \) and \(\overset { \rightarrow }{ b } =2\overset { \wedge }{ i } +\overset { \wedge }{ j } \quad \) Is \(\overset { \rightarrow }{ a } =\overset { \rightarrow }{ b } \)?
Are the vector \(\overset { \rightarrow }{ a } \) is \(\overset { \rightarrow }{ b } \) equal?
(a) 
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.(a) 
Two numbers are selected at random (without replacement)from first six positive integers.Let X denote the larger of the two numbers obtained.Find the probability distribution of X.Find the mean and variance of this distribution
(a) 
Show that the relation S in the set R of real numbers, defined as
S={(a,b): a,b \(\in\)R and a\(\le\)b^{3}} is neither reflexive, nor symmetric nor transitive.
(a) 
Check the injectivity and surjectivity of the following
(i) f: N \(\rightarrow\) N given by f(x) = x^{2}
(ii) f : R \(\rightarrow\) R given by f(x) = x^{2}(a) 
Using elementary transformations, find the inverse of the matrix
\(\left[ \begin{matrix} 1 & 3 & 2 \\ 3 & 0 & 1 \\ 2 & 1 & 0 \end{matrix} \right] \).
(a) 
Find the value of a for which the function f defined by
\(f(x)=\begin{cases} a\quad sin\frac { \pi }{ 2 } (x+1)\quad \quad ,\quad x\le 0 \\ \frac { tan\quad x\quad \quad sin\quad x }{ { x }^{ 3 } } \ \ \ \ ,\quad x>0 \end{cases}\) is continuous at x=0.
(a) 
A leaded 5m long is leaning against a wall.The bottom of the ladder is pulled along the ground,away from the wall,at the rate of 2cm/s.how fast is its height on the wall decreasing when the foot of ladder is 4m away from the wall?
(a) 

Check whether the function:
\(f(x)=x^{ 100 }+sinx1\)
is strictly increasing or strictly decreasing or none of both on(1,1)
Should the nature of a man be like this function? Justify your answer.(a) 
Evaluate the integral: \(\int sin\ x.\ sin2x.\ sin3x\ dx\)
(a)



Evaluate: \(\int _{ 0 }^{ \pi /2 }{ \frac { x\sin { x\cos { x } } }{ \sin ^{ 4 }{ x } +\cos ^{ 4 }{ x } } } dx.\)
(a) 
Points L,M,N divide the sides BC,CA and AB of a triangle ABC in the ratio 1:4,3:2 and 3:7 respectively. Prove that \(\overrightarrow { AL } +\overrightarrow { BM } +\overrightarrow { CN } \) is a vector parallel to \(\overrightarrow { CK } \) where k divides AB in the ratio 1:3
(a)



In a factory which manufactures boltsm machines A,B and C, manufacture respectively 25%, 35% and 40% of the bolts. Of their output 5,4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the total production and is found to be defective. Find the probability that it is manufactured by the machine B.
(a) 
Show that the function \(f:R[x\in R:1
defined by \(f(x)=\frac { x }{ 1+x^{ ' } } ,x\in R\) is oneone and onto function Hence find \(f^{ 1 }(x)\) (a)



if f(x) = \(\frac { \left( 4x+3 \right) }{ \left( 6x4 \right) } \), show that fof (x) = x for all x \(\neq \) \(\frac { 2 }{ 3 } \) , what is the inverse of f(x)?
(a) 
Using elementary transformation, find the inverse of the matrix \(A=\left[ \begin{matrix} 8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \) and use it to solve the following system of lines equations :
8x + 4y + 3z = 19
2x + y + z = 5
x + 2y + 2z = 7(a)



The sum of three numbers is 1. If we multiply the second number by 2 , third number by 3 and add them we get 5. If we subtract the third number from the sum of first and second numbers we get 1. Represent it by a system of equations . Find the three numbers using inverse of a matrix
(a) 
Find the point P on the curve \(y^2= 4ax\) which is nearest to the point (11a, 0).
(a)



Find : \(\int { \frac { \sqrt { x^{ 2 }+1 } \left\{ log({ x }^{ 2 }+1)2logx \right\} }{ x^{ 4 } } } dx\)
(a) 
A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is 1. Find the probability that it is actually 1.
(a)
