A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 13

Differentiation - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

If $f(x)=\sqrt{x^{2}-10 x+25}$, then the derivative of $f(x)$ on the interval $[0,7]$ is
(A) 1
(B) $-1$
(C) 0
(D) Does not exist

Aman Gupta

Aman Gupta

Numerade Educator

Problem 2

If the capital letters denote the cofactors of the corresponding small letters in the determinant $\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|$ then the value of $\Delta^{\prime}=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|$ is
(A) 0
(B) $2 \Delta$
(C) $\Delta^{2}$
(D) $\Delta$

Aayush Gupta

Numerade Educator

Problem 3

If $f(x)=\cos ^{2} x+\cos ^{2}\left(x+\frac{\pi}{3}\right)+\sin x \sin \left(x+\frac{\pi}{3}\right)$ and
$g\left(\frac{5}{4}\right)=3$ then $(g o f)(x)$ is equal to
(A) 1
(B) 2
(C) 3
(D) None of these

Aman Gupta

Numerade Educator

Problem 4

If $f(x)=\cos x \cos 2 x \cos 4 x \cos 8 x$, then $f^{\prime}\left(\frac{\pi}{4}\right)$ is
(A) $-1$
(B) 2
(C) $\sqrt{2}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 5

If $y=e^{n x}$, then $\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d^{2} x}{d y^{2}}\right)$ is equal to
(A) $n e^{n x}$
(B) $n^{2} e^{n x}$
(C) $-n e^{n x}$
(D) $-n e^{-n x}$

Urvashi Arora

Numerade Educator

Problem 6

If the parametric equation of a curve is given by $x=\cos \theta+\log \tan \frac{\theta}{2}$ and $y=\sin \theta$, then the points for which $\frac{d^{2} y}{d x^{2}}=0$ are given by
(A) $\theta=n \pi, n \in Z$
(B) $\theta=(2 n+1) \pi / 2, n \in Z$
(C) $\theta=(2 n+1) \pi, n \in Z$
(D) $\theta=2 n \pi, n \in Z$.

Aman Gupta

Numerade Educator

Problem 7

Let $f(x)=\sqrt{x-1}+\sqrt{x+24-10 \sqrt{x-1}} ; 1<x<26$ be real valued function. Then $f^{\prime}(x)$, for $1<x<26$ is
(A) 0
(B) $\frac{1}{\sqrt{x-1}}$
(C) $2 \sqrt{x-1}-5$
(d) None of these

Aman Gupta

Numerade Educator

Problem 8

If $y=\tan ^{-1}\left(\frac{\log \left(e / x^{3}\right)}{\log \left(e x^{3}\right)}\right)+\tan ^{-1}\left(\frac{\log \left(e^{4} x^{3}\right)}{\log \left(e / x^{12}\right)}\right)$, then $\frac{d^{2} y}{d x^{2}}$ is equal to
(A) 1
(B) 0
(C) $-1$
(D) None of these

Aman Gupta

Numerade Educator

Problem 9

Let $f(x)=\left|\begin{array}{ccc}x^{3} & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^{2} & p^{3}\end{array}\right|$, where $p$ is a constant. Then $\frac{d^{3}}{d x^{3}}[f(x)]$ at $x=0$ is
(A) $p$
(B) $p+p^{2}$
(C) $p+p^{3}$
(D) independent of $p$

Aman Gupta

Numerade Educator

Problem 10

The function $y$ defined by the equation $x y-\log y=1$ satisfies $x\left(y y^{\prime \prime}+y^{\prime 2}\right)-y^{\prime \prime}+k y y^{\prime}=0 .$ The value $k$ is
(A) $-3$
(B) 3
(C) 1
(D) None of these

Aman Gupta

Numerade Educator

Problem 11

If the function $y(x)$ represented by $x=\sin t$ $y=a e^{l \sqrt{2}}+b e^{l \sqrt{2}}, t \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfies the equation
$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}=k y$, then $k$ is equal to
(A) 1
(B) $-2$
(C) 2
(D) None of these

Aman Gupta

Numerade Educator

Problem 12

If $f(x)=\frac{x^{2}-x}{x^{2}+2 x}$, then $\frac{d f^{-1}(x)}{d x}$ is equal to
(A) $-\frac{3}{(1-x)^{2}}$
(B) $\frac{3}{(1-x)^{2}}$
(C) $\frac{1}{(1-x)^{2}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 13

Let $F(x)=f(x) g(x) h(x)$ for all real $x$, where $f(x), g$ $(x)$ and $h(x)$ are differentiable functions. At some point $x_{0}$, if $F^{\prime}\left(x_{0}\right)=21 F\left(x_{0}\right), f^{\prime}\left(x_{0}\right)=4 f\left(x_{0}\right), g^{\prime}\left(x_{0}\right)=-7$ $g\left(x_{0}\right)$ and $h^{\prime}\left(x_{0}\right)=k h\left(x_{0}\right)$ then $k$ is equal to
(A) 24
(B) 12
(C) $-12$
(D) $-24$

Aman Gupta

Numerade Educator

Problem 14

If $f(x)$ is a polynomial of degree $n(>2)$ and $f(x)=$ $f(k-x)$, (where $k$ is a fixed real number), then degree of $f^{\prime}(x)$ is
(A) $n$
(B) $n-1$
(C) $n-2$
(D) None of these

Aman Gupta

Numerade Educator

Problem 15

If $f(x)=|x-1|$ and $g(x)=f\{f[f(x)]\}$, then for $x>2$, $g^{\prime}(x)$ is equal to
(A) $-1$ if $2 \leq x<3$
(B) 1 if $2 \leq x<3$
(C) 1 for all $x>2$
(D) None of these

Aman Gupta

Numerade Educator

Problem 16

If $f(x)=|(x-4)(x-5)|$, then $f^{\prime}(x)$ is equal to
(A) $-2 x+9$, for all $x \in R$
(B) $2 x-9$ if $4<x<5$
(C) $-2 x+9$ if $4<x<5$
(D) None of these

Aman Gupta

Numerade Educator

Problem 17

If $2 f(\sin x)+f(\cos x)=x$, then $\frac{d}{d x} f(x)$ is
(A) $\sin x+\cos x$
(B) 2
(C) $\frac{1}{\sqrt{1-x^{2}}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 18

Let $f(x)=|x-a| ;(a>0)$ and $g(x)=f\{f[f(x)]\}$. Then $g^{\prime}(\alpha) ;(\alpha>3 a)$
(A) does not exist
(B) equal to 3
(C) equal to 1
(D) None of these

Aman Gupta

Numerade Educator

Problem 19

Let $\phi(x)$ be the inverse of the function $f(x)$ and $f^{\prime}(x)=\frac{1}{1+x^{5}}$ then $\frac{d}{d x} \phi(x)$ is
(A) $\frac{1}{1+[\phi(x)]^{5}}$
(B) $\frac{1}{1+[f(x)]^{5}}$
(C) $1+[\phi(x)]^{5}$
(D) $1+[f(x)]^{5}$

Aman Gupta

Numerade Educator

Problem 20

If $y=\frac{1}{x}$ then $\frac{d y}{\sqrt{1+y^{4}}}+\frac{d x}{\sqrt{1+x^{4}}}=$
(A) 0
(B) 1
(C) $\frac{x}{y}$
(D) $\frac{y}{x}$

Urvashi Arora

Numerade Educator

Problem 21

If $y=\frac{f(x)}{\phi(x)}$ and $z=\frac{f^{\prime}(x)}{\phi^{\prime}(x)}$, then $\frac{f^{\prime \prime}}{f}-\frac{\phi^{\prime \prime}}{\phi}+\frac{2(y-z)}{f \phi}\left(\phi^{\prime}\right)^{2}=$
(A) $\frac{d^{2} y}{d x^{2}}$
(B) $\frac{1}{y} \frac{d^{2} y}{d x^{2}}$
(C) $y \frac{d^{2} y}{d x^{2}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 22

Let $g(x)$ be the inverse of an invertible function $f(x)$ which is differentiable for all real $x$, then $g^{\prime \prime}(f(x))$ equals
(A) $-\frac{f^{\prime \prime}(x)}{\left[f^{\prime}(x)\right]^{3}}$
(B) $\frac{f^{\prime}(x) f^{\prime \prime}(x)-\left[f^{\prime}(x)\right]^{2}}{f^{\prime}(x)}$
(C) $\frac{f^{\prime}(x) f^{\prime \prime}(x)-\left[f^{\prime}(x)\right]^{2}}{\left[f^{\prime}(x)\right]^{2}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 23

Let $f(x)=x^{n}, n$ being a non-negative integer. The value of $n$ for which equality $f^{\prime}(a+b)=f^{\prime}(a)+f^{\prime}(b)$ is valid for all $a, b>0$ is
(A) 5
(B) 1
(C) 2
(D) 4

Aman Gupta

Numerade Educator

Problem 24

The solution set of $f^{\prime}(x)>g^{\prime}(x)$ where $f(x)=(1 / 2) 5^{2 x+1}$ and $g(x)=5^{x}+4 x \log 5$ is
(A) $(1, \infty)$
(B) $(0,1)$
(C) $(0, \infty)$
(D) $[0, \infty)$

Aman Gupta

Numerade Educator

Problem 25

If $I_{n}=\frac{d^{n}}{d x^{n}}\left(x^{n} \log x\right)$, then $I_{n}-n I_{n-1}=$
(A) $n$
(B) $n-1$
(C) $n !$
(D) $(n-1) !$

Urvashi Arora

Numerade Educator

Problem 26

If $u=f\left(x^{3}\right), v=g\left(x^{2}\right), f^{\prime}(x)=\cos x$ and $g^{\prime}(x)=\sin x$
then $\frac{d u}{d v}$ is
(A) $\frac{3}{2} x \cdot \cos x^{3} \cdot \operatorname{cosec} x^{2}$
(B) $\frac{2}{3} \sin x^{3} \cdot \operatorname{cosec} x^{2}$
(C) $\tan x$
(D) None of these

Aman Gupta

Numerade Educator

Problem 27

If $f(x)$ be a differentiable function such that $f(x y)=f(x)$ $+f(y)$ for all $x$ and $y$, then $f(e)+f(1 / e)=$
(A) 1
(B) 0
(C) $-1$
(D) None of these

Aman Gupta

Numerade Educator

Problem 28

If for all $x, y$ the function $f$ is defined by $f(x)+f(y)+$ $f(x) \cdot f(y)=1$ and $f(x)>0$ then
(A) $f^{\prime}(x)$ does not exist
(B) $f^{\prime}(x)=0$ for all $x$
(C) $f^{\prime}(0)<f^{\prime}(1)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 29

Let $3 f(x)-2 f(1 / x)=x$, then $f^{\prime}(2)$ is equal to
(A) $\frac{2}{7}$
(B) $\frac{1}{2}$
(C) 2
(D) $\frac{7}{2}$

Aman Gupta

Numerade Educator

Problem 30

If $\sqrt{x+y}+\sqrt{y-x}=c$ then $\frac{d^{2} y}{d x^{2}}$ equals
(A) $\frac{2}{c^{2}}$
(B) $\frac{-2}{c^{2}}$
(C) $\frac{2}{c}$
(D) $\frac{-2}{c}$

Aman Gupta

Numerade Educator

Problem 31

$\frac{d^{2} x}{d y^{2}}$ equals
(A) $-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}$
(B) $\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}$
(C) $-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}$
(D) $\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-2}$

Aman Gupta

Numerade Educator

Problem 32

Let $y$ be an implicit function of $x$ defined by, $x^{2 x}-2 x^{x} \cot y-1=0$
Then $y^{\prime}(1)$ equals
(A) $-1$
(B) 1
(C) $\log 2$
(D) $-\log 2$

Aman Gupta

Numerade Educator

Problem 33

If $y$ and $z$ are the functions of $x$ and if $y^{2}+z^{2}=\lambda^{2}$, then $y \frac{d}{d x}\left(\frac{y}{\lambda}\right)+\frac{d}{d x}\left(\frac{z^{2}}{\lambda}\right)$ is equal to
(A) $\frac{z}{\lambda} \frac{d z}{d x}$
(B) $\frac{z}{\lambda} \frac{d x}{d z}$
(C) $\frac{\lambda}{z} \frac{d z}{d x}$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 34

If $S_{n}$ denotes the sum of $n$ terms of a G.P. whose common ratio is $r$, then $(r-1) \frac{d S_{n}}{d r}$ is equal to
(A) $(n-1) S_{n}+n S_{n-1}$
(B) $(n-1) S_{n}-n S_{n-1}$
(C) $(n-1) S_{n}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 35

Let $f\left(\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)=$ $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots+f\left(x_{n}\right)}{n}$ where all $x_{i} \in R$ are independent to each other and $n \in N$. If $f(x)$ is differentiable and $f^{\prime}(0)=a, f(0)=b$ then $f^{\prime}(x)$ is equal to
(A) $a$
(B) 0
(C) $b$
(D) None of these

Aman Gupta

Numerade Educator

Problem 36

If $y^{2}=P(x)$, a polynomial of degree $n \geq 3$, then $2 \frac{d}{d x}\left(y^{3} \frac{d^{2} y}{d x^{2}}\right)=$
(A) $-P(x) \times P^{\prime \prime \prime}(x)$
(B) $P(x) \times P^{\prime \prime \prime}(x)$
(C) $P(x) \times P^{\prime \prime}(x)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 37

A function $f(x)$ is so defined that for all $x,[f(x)]^{n}=$ $f(n x)$. If $f^{\prime}(x)$ denotes derivative of $f(x)$ with respect to $x$, then $f^{\prime}(x) \times f(n x)=$
(A) $f(x)$
(B) 0
(C) $f(x) \times f^{\prime}(n x)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 38

If $f, g, h$ are differentiable functions of $x$ and $\Delta=\left|\begin{array}{ccc}f & g & h \\ (x f)^{\prime} & (x g)^{\prime} & (x h)^{\prime} \\ \left(x^{2} f\right)^{\prime \prime} & \left(x^{2} g\right)^{\prime \prime} & \left(x^{2} h\right)^{\prime \prime}\end{array}\right|$ then $\Delta^{\prime}$ (the derivative of $\Delta$ with respect to $x$ ) is given by (A) $\left|\begin{array}{ccc}f^{\prime} & g^{\prime} & h^{\prime} \\ f & g & h \\ \left(x^{3} f^{\prime \prime}\right)^{\prime} & \left(x^{3} g^{\prime \prime}\right)^{\prime} & \left(x^{3} h^{\prime \prime}\right)^{\prime}\end{array}\right|$
(B) $\left|\begin{array}{ccc}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ \left(x^{2} f^{\prime \prime}\right)^{\prime} & \left(x^{2} g^{\prime \prime}\right)^{\prime} & \left(x^{2} h^{\prime \prime}\right)^{\prime}\end{array}\right|$
(C) $\left|\begin{array}{ccc}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ \left(x^{3} f^{\prime \prime}\right)^{\prime} & \left(x^{3} g^{\prime \prime}\right)^{\prime} & \left(x^{3} h^{\prime \prime}\right)^{\prime}\end{array}\right|$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 39

If $\alpha$ is a repeated root of a quadratic equation $f(x)=0$ and $A(x), B(x), C(x)$ be polynomials of degree $>2$, then the determinant $\left|\begin{array}{ccc}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|$ is divisible by
(A) $A(x)$
(B) $B(x)$
(C) $C(x)$
(D) $f(x)$

Aman Gupta

Numerade Educator

Problem 40

If the capital letters denote the cofactors of the corresponding small letters in the determinant $\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|$, then the value of $\Delta^{\prime}=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|$ is
(A) 0
(B) $2 \Delta$
(C) $\Delta^{2}$
(D) $\Delta$

Aayush Gupta

Numerade Educator

Problem 41

If $f(x)=\frac{x^{2}-x}{x^{2}+2 x}$, then $\frac{d f^{-1}(x)}{d x}$ is equal to
(A) $-\frac{3}{(1-x)^{2}}$
(B) $\frac{3}{(1-x)^{2}}$
(C) $\frac{1}{(1-x)^{2}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 42

If $f(x)$ is a polynomial of degree $n(>2)$ and $f(x)=f(k-x)$, (where $k$ is a fixed real number), then degree of $f^{\prime}(x)$ is
(A) $n$
(B) $n-1$
(C) $n-2$
(D) None of these

Aman Gupta

Numerade Educator

Problem 43

If $y=\frac{f(x)}{\phi(x)}$ and $z=\frac{f^{\prime}(x)}{\phi^{\prime}(x)}$, then
$\frac{f^{\prime \prime}}{f}-\frac{\phi^{\prime \prime}}{\phi}+\frac{2(y-z)}{f \phi}\left(\phi^{\prime}\right)^{2}=$
(A) $\frac{d^{2} y}{d x^{2}}$
(B) $\frac{1}{y} \frac{d^{2} y}{d x^{2}}$
(C) $y \frac{d^{2} y}{d x^{2}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 44

The solution set of $f^{\prime}(x)>g^{\prime}(x)$ where $f(x)=(1 / 2) 5^{2 x+1}$ and $g(x)=5^{x}+4 x \log 5$ is
(A) $(1, \infty)$
(B) $(0,1)$
(C) $(0, \infty)$
(D) $[0, \infty)$

Aman Gupta

Numerade Educator

Problem 45

If for all $x, y$ the function $f$ is defined by $f(x)+f(y)+$ $f(x) \cdot f(y)=1$ and $f(x)>0$, then
(A) $f^{\prime}(x)$ does not exist
(B) $f^{\prime}(x)=0$ for all $x$
(C) $f^{\prime}(0)<f^{\prime}(1)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 46

If $\sqrt{x+y}+\sqrt{y-x}=c$ then $\frac{d^{2} y}{d x^{2}}$ equals
(A) $\frac{2}{c^{2}}$
(B) $\frac{-2}{c^{2}}$
(C) $\frac{2}{c}$
(D) $\frac{-2}{c}$

Aman Gupta

Numerade Educator

Problem 47

Let $f(x)=\prod_{k=1}^{n}(\cos (2 k-1) x+i \sin (2 k-1) x)$, then $(\operatorname{Re} f(x))^{\prime \prime}+i(\operatorname{Im} f(x))^{\prime \prime}$ is equal to
(A) $n^{2} f(x)$
(B) $-n^{4} f(x)$
(C) $-n^{2} f(x)$
(D) $n^{4} f(x)$

Aayush Gupta

Numerade Educator

Problem 48

If $2 f(\sin x)+f(\cos x)=x$, then $\frac{d}{d x} f(x)$ is
(A) $\sin x+\cos x$
(B) 2
(C) $\frac{1}{\sqrt{1-x^{2}}}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 49

Let $f(x)=\sqrt{x-1}+\sqrt{x+24-10 \sqrt{x-1}} ; 1<x<26$ be a real valued function. Then, $f^{\prime}(x)$, for $1<x<26$ is
(A) 0
(B) $\frac{1}{\sqrt{x-1}}$
(C) $2 \sqrt{x-1}-5$
(D) None of these

Aman Gupta

Numerade Educator

Problem 50

If $f(x)=|x-2|$ and $g(x)=f\{f(x)\}$, then $g^{\prime}(x)$ for $x>2$ is
(A) $-1$
(B) 1
(C) 0
(D) does not exist

Aman Gupta

Numerade Educator

Problem 51

The derivative of the function represented parametrically as $x=2 t-|t|, y=t^{3}+t^{2}|t|$ at $t=0$ is
(A) 0
(B) 1
(C) $-1$
(D) does not exist

Aman Gupta

Numerade Educator

Problem 52

A polynomial $f(x)$ leaves remainder 15 when divided by $(x-3)$ and $(2 x+1)$ when divided by $(x-1)^{2} .$ When $f$ is divided by $(x-3)(x-1)^{2}$, the remainder is
(A) $2 x^{2}+2 x+3$
(B) $2 x^{2}-2 x-3$
(C) $2 x^{2}-2 x+3$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 53

If for a non-zero $x$, the function $f(x)$ satisfies the equation $a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5(a \neq b)$
then $f^{\prime}(x)$ is equal to
(A) $\frac{1}{b^{2}-a^{2}}\left(\frac{a}{x^{2}}+b\right)$
(B) $\frac{1}{a^{2}-b^{2}}\left(\frac{a}{x^{2}}+b\right)$
(C) $\frac{1}{a^{2}-b^{2}}\left(\frac{a}{x^{2}}-b\right)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 54

If $x=\cos ^{7} \theta$ and $y=\sin \theta$, then $\frac{d^{3} x}{d y^{3}}=$
(A) $\frac{105}{4} \sin 4 \theta$
(B) $\frac{105}{2} \sin 2 \theta$
(C) $\frac{105}{4} \cos 4 \theta$
(D) None of these

Aman Gupta

Numerade Educator

Problem 55

If $y=\sec ^{-1}\left(\frac{x+1}{x-1}\right)+\sin ^{-1}\left(\frac{x-1}{x+1}\right)$, then $\frac{d y}{d x}=$
(A) $0 \forall x \in R$
(B) $0 \forall x \in(0, \infty)$
(A) $0 \forall x \in R-\{0\}$
(D) None of these

Aman Gupta

Numerade Educator

Problem 56

If $f(x)=\cos ^{-1}\left(\frac{x^{-1}-x}{x^{-1}+x}\right)$, then $f^{\prime}(x)$ is
(A) odd
(B) even
(C) periodic
(D) None of these

Aman Gupta

Numerade Educator

Problem 57

If $f(x)=(1-x)^{n}$, then the value of $f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^{n}(0)}{n !}$ is (A) $n$
(B) 0
(C) $2^{n}$
(D) $2^{n}-1$

Aman Gupta

Numerade Educator

Problem 58

If $y=\cot ^{-1}\left(\frac{x^{x}-x^{-x}}{2}\right)$, then $\frac{d y}{d x}$ at $x=1$, equals
(A) 0
(B) 1
(C) $-1$
(D) None of these

Aman Gupta

Numerade Educator

Problem 59

If $y=\sqrt{\frac{1+\cos 2 \theta}{1-\cos 2 \theta}}$, then
(A) $y^{\prime}\left(\frac{\pi}{4}\right)=y^{\prime}\left(\frac{3 \pi}{4}\right)$
(B) $y^{\prime}\left(\frac{\pi}{4}\right) \cdot y^{\prime}\left(\frac{3 \pi}{4}\right)=-4$
(C) $y^{\prime}\left(\frac{\pi}{4}\right)$ and $y^{\prime}\left(\frac{3 \pi}{4}\right)$ do not exist
(D) None of these

Urvashi Arora

Numerade Educator

Problem 60

Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and $a_{1}, a_{2}, a_{3}$ are in A. P., then $f^{\prime}\left(a_{1}\right)$ $f^{\prime}\left(a_{2}\right), f^{\prime}\left(a_{3}\right)$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) None of these

Aman Gupta

Numerade Educator

Problem 61

If $5 f(x)+3 f\left(\frac{1}{x}\right)=x+2$ and $y=x f(x)$ then $\frac{d y}{d x}$ at $x=1$ is equal to
(A) 1
(B) $-1$
(C) $\frac{7}{8}$
(D) $-\frac{7}{8}$

Aman Gupta

Numerade Educator

Problem 62

Let $f(x)$ be a polynomial function of degree 2 and $f(x)$ $>0$ for all $x \in R$. If $g(x)=f(x)+f^{\prime}(x)+f^{\prime \prime}(x)$, then for any $x$
(A) $g(x)>0$
(B) $g(x)<0$
(C) $g(x)=0$
(D) $g(x) \leq 0$

Aman Gupta

Numerade Educator

Problem 63

Let $f(x+y)=f(x)+f(y)+2 x y-1 \forall x, y \in R$. If $f(x)$ is differentiable and $f^{\prime}(0)=\sin \theta$, then
(A) $f(x)>0 \forall x \in R$
(B) $f(x)<0 \forall x \in R$
(C) $f(x)=\sin \theta \forall x \in R$
(D) None of these

Aman Gupta

Numerade Educator

Problem 64

If $f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$ for all $x \in R$,
then which of the following is false?
(A) $f(0)+f(2)=f(1)$
(B) $f(0)+f(3)=0$
(C) $f(1)+f(3)=f(2)$
(D) $f(1)+f(3)=f(0)$

Aayush Gupta

Numerade Educator

Problem 65

Let $f\left(\frac{x+y}{2}\right)=\frac{1}{2}[f(x)+f(y)]$ for real $x$ and $y .$ If $f^{\prime}(0)$ exists and equals $-1$ and $f(0)=1$ then the value of $f(2)$ is
(A) 1
(B) $-1$
(C) 0
(D) None of these

Aayush Gupta

Numerade Educator

Problem 66

Let the function $f$ satisfy the equation $f(x+y)=f(x) f(y)$ for all $x$ and $y$ and $f(x)=1+x g(x)$ where $\lim _{x \rightarrow 0} g(x)=\log a$. If $f^{n}(x)=k f(x)$, then $k=$
(A) $\log a$
(B) $n \log a$
(C) $(\log a)^{n}$
(D) $n(\log a)^{n}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 67

Let $f$ be a differentiable function satisfying $f(x+y)=$ $f(x)+f(y)+x y .$ If $\lim _{h \rightarrow 0} \frac{1}{h} f(h)=3$, then
(A) $f(x)=3 x$
(B) $f(x)=3 x+x^{2}$
(C) $f(x)=3 x+\frac{x^{2}}{2}$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 68

If $f(x)=x+\tan x$ and $f$ is the inverse of $g$, then $g^{\prime}(x)$ is equal to
(A) $\frac{1}{1+[g(x)-x]^{2}}$
(B) $\frac{1}{2-[g(x)-x]^{2}}$
(C) $\frac{1}{2+[g(x)-x]^{2}}$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 69

If $y=\sqrt{(a-x)(x-b)}-(a-b) \tan ^{-1} \sqrt{\frac{a-x}{x-b}}$, then $\frac{d y}{d x}=$
(A) 1
(B) $\sqrt{\frac{a-x}{x-b}}$
(C) $\sqrt{(a-x)(x-b)}$
(D) $\frac{1}{\sqrt{(a-x)(x-b)}}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 70

If $y^{3}-y=2 x$, then $\left(x^{2}-\frac{1}{27}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=$
(A) $y$
(B) $\frac{y}{3}$
(C) $\frac{y}{9}$
(D) $\frac{y}{27}$

Patha Sharma

Numerade Educator

Problem 71

If $x<1$, then $\frac{1-2 x}{1-x+x^{2}}+\frac{2 x-4 x^{3}}{1-x^{2}+x^{4}}+\frac{4 x^{3}-8 x^{7}}{1-x^{4}+x^{8}}$.
$+\ldots \infty=$
(A) $\frac{1}{1+x+x^{2}}$
(B) $\frac{1+2 x}{1+x+x^{2}}$
(C) $\frac{1-x+x^{2}}{1+x+x^{2}}$
(D) 1

Aayush Gupta

Numerade Educator

Problem 72

If $f(x)=x^{m}, m$ being a non-negative integer, then the value of $m$ for which $f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)$, for all $\alpha, \beta>0$, is
(A) 1
(B) 2
(C) 0
(D) None of these

Aman Gupta

Numerade Educator

Problem 73

If $f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$ for all $x \in \mathbb{R}$
where $f(x)$ is a polynomial of degree 3 , then
(A) $f(0)+f(2)=f(1)$
(B) $f(0)+f(3)=0$
(C) $f(1)+f(3)=f(2)$
(D) None of these

Aman Gupta

Numerade Educator

Problem 74

If $f(x-y), f(x) \cdot f(y)$ and $f(x+y)$ are in A.P. for all $x, y$ and $f(0) \neq 0$, then
(A) $f(2)=f(-2)$
(B) $f(3)+f(-3)=0$
(C) $f^{\prime}(2)+f^{\prime}(-2)=0$
(D) $f^{\prime}(3)=f^{\prime}(-3)$

Aman Gupta

Numerade Educator

Problem 75

If $f(x)+f(y)+f(z)+f(x) \cdot f(y) \cdot f(z)=14$ for all $x, y, z \in R$, then
(A) $f(0)=2$
(B) $f^{\prime}(x)=0$, for all $x \in R$
(C) $f^{\prime}(x)>0$, for all $x \in R$
(D) None of these

Aman Gupta

Numerade Educator

Problem 76

If $f(x-y), f(x) \cdot f(y)$ and $f(x+y)$ are in A. P. for all $x, y$ and $f(0) \neq 0$, then
(A) $f^{\prime}(3)+f^{\prime}(-3)=0$
(B) $f(3)+f(-3)=0$
(C) $f^{\prime}(2)+f^{\prime}(-2)=0$
(D) $f^{\prime}(3)=f^{\prime}(-3)$

Aman Gupta

Numerade Educator

Problem 77

A function $f:(0, \infty) \rightarrow R$ satisfies the equation $f(x y)=2 f(x)-f\left(\frac{x}{y}\right)$
If $f$ is differentiable on $R$ and $f(1)=0, f^{\prime}(1)=1$, then
(A) $f(y)=-f\left(\frac{1}{y}\right)$
(B) $f^{\prime}(x)=\frac{1}{x}$
(C) $f(x)=\ln x$
(D) $f(x)=e^{x}$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 78

If $f: R \rightarrow R$ be a function such that $f(x)=x^{3}+x^{2} f^{\prime}(1)+$ $x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$ is differentiable for every $x \in R$, then
(A) $f^{\prime}(1)=-5$
(B) $f^{\prime \prime}(2)=2$
(C) $f^{\prime \prime \prime}(3)=6$
(D) None of these

Aman Gupta

Numerade Educator

Problem 79

If $f(x-y)+f(x+y)=2 f(x) f(y) \forall x, y \in R$,then
(A) $f$ is even
(B) $f$ is odd
(C) $f^{\prime}$ is even
(D) $f^{\prime}$ is odd

Aman Gupta

Numerade Educator

Problem 80

If $f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2)$ and $g(x)=f(1) \cdot x^{2}+x f^{\prime}(x)$
$+f^{\prime \prime}(x)$, then
(A) $f(x)=x^{2}-3 x$
(B) $f(x)=x^{2}+3 x$
(C) $g(x)=3 x+2$
(D) $g(x)=-3 x+2$

Aman Gupta

Numerade Educator

Problem 81

If $\sum_{r=1}^{n} r x^{r-1}=\frac{1}{(1-x)^{2}} \cdot\left\{1+a x^{n}+b x^{n+1}\right\}$, then
(A) $a=(n+1)$
(B) $b=n$
(C) $a=-(n+1)$
(D) $b=-n$

Aman Gupta

Numerade Educator

Problem 82

If $y=f(x)=\min \phi(t) ;-3 \leq t \leq x$ where $\phi(x)=\| x-1$ $|-| x+1 \|$, then
(A) $f(x)$ is non-differentiable at $x=0,-1$
(B) $f(x)$ is non-differentiable at $x=1,-1$
(C) $f^{\prime \prime}(100)=0$
(D) $\int_{-3}^{10} f(x) d x=5$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 83

If $f(x)=\left|\begin{array}{ccc}x^{n} & \sin x & -\cos x \\ n ! & \sin (n \pi / 2) & \cos (n \pi / 2) \\ a & a^{2} & a^{3}\end{array}\right|$, then $f^{n}(0)$ for $n=2 m+1$ is
(A) 1
(B) $-1$
(C) 0
(D) independent of $a$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 84

Let $f(x)=x^{3}+3 x^{2}-33 x-33$ for $x>0$ and $g$ be its inverse, then the value of $k$ such that $k g^{\prime}(2)=1$ is equal to
(A) $-36$
(B) 51
(C) 72
(D) 36

Aman Gupta

Numerade Educator

Problem 85

If $F(x)=f(x) g(x)$ and $f^{\prime}(x) g^{\prime}(x)=c$, then
(A) $F^{\prime}=c\left(\frac{f}{f^{\prime}}+\frac{g}{g^{\prime}}\right)$
(B) $\frac{F^{\prime \prime}}{F}=\frac{f^{\prime \prime}}{f}+\frac{g^{\prime \prime}}{g}+\frac{2 c}{f g}$
(C) $\frac{F^{\prime \prime \prime}}{F}=\frac{f^{\prime \prime \prime}}{f}+\frac{g^{\prime \prime \prime}}{g}$
(D) $\frac{F^{\prime \prime \prime}}{F^{\prime \prime}}=\frac{f^{\prime \prime \prime}}{f^{\prime \prime}}+\frac{g^{\prime \prime \prime}}{g^{\prime \prime}}$

Aayush Gupta

Numerade Educator

Problem 86

Let $f$ be a function such that $f:(-1,1) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Let $f$
satisfy the equation $f(x)+f(y)=f\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)$
The function $f(x)$ is
(A) even
(B) odd
(C) constant
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 87

Let $f$ be a function such that $f:(-1,1) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Let $f$
satisfy the equation $f(x)+f(y)=f\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)$
If $f(x)$ is differentiable on $(-1,1)$ and $f^{\prime}(0)=1$, then $f^{\prime}(x)$ is equal to
(A) $\frac{1}{\sqrt{1-x^{2}}}$
(B) $-\frac{1}{\sqrt{1-x^{2}}}$
(C) $\frac{1}{\sqrt{1+x^{2}}}$
(D) $-\frac{1}{\sqrt{1+x^{2}}}$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 88

Let $f$ be a function such that $f:(-1,1) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Let $f$
satisfy the equation $f(x)+f(y)=f\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)$
The function $f(x)$ is equal to
(A) $\cos ^{-1} x$
(B) $\sin ^{-1} x$
(C) $\tan ^{-1} x$
(D) $\sec ^{-1} x$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 89

Let $f: R \rightarrow R$ be a function satisfying the condition $f\left(\frac{x+y}{k}\right)=\frac{f(x)+f(y)}{k}$, where $k \neq 0,2$. The function $f(x)$ is differentiable on $R$ and $f^{\prime}(0)=m$.
$f^{\prime}(x)$ is equal to
(A) $m$
(B) $2 m$
(C) $m+1$
(D) 0

Aayush Gupta

Numerade Educator

Problem 90

Let $f: R \rightarrow R$ be a function satisfying the condition $f\left(\frac{x+y}{k}\right)=\frac{f(x)+f(y)}{k}$, where $k \neq 0,2$. The function $f(x)$ is differentiable on $R$ and $f^{\prime}(0)=m$.
The function $f(x)$ is equal is
(A) $m x$
(B) $m x+1$
(C) $-2 m x$
(D) None of these

Aayush Gupta

Numerade Educator

Problem 91

A function $f: R \rightarrow[1, \infty)$ satisfies the equation $f(x y)=f(x)$ $f(y)-f(x)-f(y)+2 .$ The function $f$ is differentiable on $R$ and $f(2)=5$
$f^{\prime}(x)$ is equal to
(A) $\frac{f(x)-1}{x} f^{\prime}(1)$
(B) $\frac{f(x)+1}{x} f^{\prime}(1)$
(C) $\frac{1-f(x)}{x} f^{\prime}(1)$
(D) None of these

Aayush Gupta

Numerade Educator

Problem 92

A function $f: R \rightarrow[1, \infty)$ satisfies the equation $f(x y)=f(x)$ $f(y)-f(x)-f(y)+2 .$ The function $f$ is differentiable on $R$ and $f(2)=5$
$f(x)$ is equal to
(A) $x^{2}-1$
(B) $1-x^{2}$
(C) $x^{2}+1$
(D) None of these

Aayush Gupta

Numerade Educator

Problem 93

A function $f: R \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfies the equation $f(x)+f(y)=f\left(\frac{x+y}{1-x y}\right)$. The function $f$ is differentiable on $R$ and $f^{\prime}(0)=2$.
The function $f$ is
(A) an even function
(B) an odd function
(C) a constant function
(D) None of these

Aman Gupta

Numerade Educator

Problem 94

A function $f: R \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfies the equation $f(x)+f(y)=f\left(\frac{x+y}{1-x y}\right)$. The function $f$ is differentiable on $R$ and $f^{\prime}(0)=2$.
$f^{\prime}(x)$ is equal to
(A) $\frac{1}{1+x^{2}}$
(B) $\frac{1}{1-x^{2}}$
(C) $\frac{2}{1+x^{2}}$
(D) $\frac{2}{x^{2}-1}$

Aayush Gupta

Numerade Educator

Problem 95

A function $f: R \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfies the equation $f(x)+f(y)=f\left(\frac{x+y}{1-x y}\right)$. The function $f$ is differentiable on $R$ and $f^{\prime}(0)=2$.
$f(x)$ is equal to
(A) $\tan ^{-1} x$
(B) $2 \tan ^{-1} x$
(C) $4 \tan ^{-1} x$
(D) None of these

Aayush Gupta

Numerade Educator

Problem 96

Let $z=f(x, y)$ be a function of two variables $x$ and $y$. The partial derivative with respect to $x$ of the function $z=$ $f(x, y)$ at $(x, y)$ is defined as $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}$, provided the limit exists and is finite. It is denoted by $\frac{\partial z}{\partial x}$ or $\frac{\partial f}{\partial x}$ or $f_{x^{-}}$Clearly, $\frac{\partial z}{\partial x}$ is the derivative of $z=f(x, y)$ with respect to $x$, regarding $y$ as a constant. Similarly, we can define $\frac{\partial z}{\partial y} \cdot \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial x}\right)$, denoted by $\frac{\partial^{2} z}{\partial x^{2}}$ or $f_{x x}, \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)$, denoted by $\frac{\partial^{2} z}{\partial y \partial x}$ or $f_{y x}, \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)$, denoted by $\frac{\partial^{2} z}{\partial x \partial y}$ or $f_{x y}, \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial y}\right)$, denoted by $\frac{\partial^{2} z}{\partial y^{2}}$ or $f_{y y}$ are called second order partial derivatives of $z=f(x, y)$.
If $u=\sin ^{-1} \frac{x}{y}+\tan ^{-1} \frac{y}{x}$, then $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=$
(A) 0
(B) 1
(C) $-1$
(D) None of these

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 97

If $U$ and $V$ are two functions of $x$ having derivatives of the $n$th order, then $(U V)_{n}=U_{n} V+{ }^{n} C_{1} U_{n-1} V_{1}+{ }^{n} C_{2} U_{n-2} V_{2}+\ldots$ $+{ }^{n} C_{r} U_{n-r} V_{r}+\ldots+{ }^{n} C_{n} U V_{n}$
If $y=x^{2} \sin x$, then $\frac{d^{n} y}{d x^{n}}=\left(x^{2}-n^{2}+n\right) \sin$ $\left(x+\frac{n \pi}{2}\right)+k \cos \left(x+\frac{n \pi}{2}\right)$, where $k=$
(A) $n x$
(B) $2 n x$
(C) $-\overline{n x}$
(D) $-2 n x$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 98

If $U$ and $V$ are two functions of $x$ having derivatives of the $n$th order, then $(U V)_{n}=U_{n} V+{ }^{n} C_{1} U_{n-1} V_{1}+{ }^{n} C_{2} U_{n-2} V_{2}+\ldots$ $+{ }^{n} C_{r} U_{n-r} V_{r}+\ldots+{ }^{n} C_{n} U V_{n}$
If $\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}$, then $x^{2} y_{n+2}+(2 n+1) x y_{n+1}$
$+k y_{n}=0$ where $k=$
(A) $n^{2}$
(B) $2 n^{2}$
(C) $-n^{2}$
(D) $-2 n^{2}$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 99

If $U$ and $V$ are two functions of $x$ having derivatives of the $n$th order, then $(U V)_{n}=U_{n} V+{ }^{n} C_{1} U_{n-1} V_{1}+{ }^{n} C_{2} U_{n-2} V_{2}+\ldots$ $+{ }^{n} C_{r} U_{n-r} V_{r}+\ldots+{ }^{n} C_{n} U V_{n}$
If $f(x)=\tan x$, then
$f^{n}(0)-{ }^{n} C_{2} f^{n-2}(0)+{ }^{n} C_{4} f^{n-4}(0)-\ldots=$
(A) $\sin \frac{n \pi}{2}$
(B) $\cos \frac{n \pi}{2}$
(C) $\tan \frac{n \pi}{2}$
(D) None of these

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 100

If $U$ and $V$ are two functions of $x$ having derivatives of the $n$th order, then $(U V)_{n}=U_{n} V+{ }^{n} C_{1} U_{n-1} V_{1}+{ }^{n} C_{2} U_{n-2} V_{2}+\ldots$ $+{ }^{n} C_{r} U_{n-r} V_{r}+\ldots+{ }^{n} C_{n} U V_{n}$
If $I_{n}=\frac{d^{n}}{d x^{n}}\left(x^{n} \log x\right)$, then $I_{n}=n I_{n-1}+k$, where $k=$
(A) $n !$
(B) $(n-1) !$
(C) $(n-2) !$
(D) None of these

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 101

Column-I Column-II
The derivative of $f(\tan x)$ with
(A) 3 respect to $g(\sec x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(1)=2$ and $g^{\prime}(\sqrt{2})=4$,
is
II. Let $y=x^{3}-8 x+7$ and $x=f(t)$.
(B) $-4$ If $\frac{d y}{d t}=2$ and $x=3$ at $t=0$, then $\frac{d x}{d t}$ at $t=0$ is
(C) $\frac{2}{19}$
III. Let $f(x)=\sin x, g(x)=2 x$ and $h(x)$ (C
$=\cos x$. If $\phi(x)=\left[g_{0}(f h)\right](x)$
then $\phi^{\prime \prime}\left(\frac{\pi}{4}\right)$ is equal to
IV. If $f(x)=\cos ^{2} x+\cos ^{2}\left(x+\frac{\pi}{3}\right)$
(D) $\frac{1}{\sqrt{2}}$
$+\sin x \sin \left(x+\frac{\pi}{3}\right)$ and $g\left(\frac{5}{4}\right)$
$=3$, then $(g o f)(x)$ is equal to

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 102

Column-I Column-II
I. The function $y$ defined by the equa-
(A) 24 tion $x y-\log y=1$ satisfies $x\left(y y^{\prime \prime}+\right.$
$\left.y^{\prime 2}\right)-y^{\prime \prime}+k y y^{\prime}=0 .$ The value of $k$ is
II. If the function $y(x)$
(B) 2 represented by $x=\sin t, y=$ $a e^{t \sqrt{2}}+b e^{t \sqrt{2}}, t \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
satisfies the equation $\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}$ $=k y$, then $k$ is equal to
III. Let $F(x)=f(x) g(x) h(x)$ for all real
(C) 4
$x$, where $f(x), g(x)$ and $h(x)$ are differentiable functions. At some point $x_{0}$, if $F^{\prime}\left(x_{0}\right)=21 F\left(x_{0}\right), f^{\prime}\left(x_{0}\right)=4$
$f\left(x_{0}\right), g^{\prime}\left(x_{0}\right)=-7 g\left(x_{0}\right)$ and $h^{\prime}\left(x_{0}\right)=$
$k h\left(x_{0}\right)$ then $k$ is equal to
IV. Let $f(x)=x^{n}, n$ being a non-negative
(D) 3 integer. The number of values of $n$ for which the equality $f^{\prime}(a+b)$ $=f^{\prime}(a)+f^{\prime}(b)$ is valid for all $a, b$
$>0$, is

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 103

Instructions: In the following questions an Assertion (A) is given followed by a Reason $(R) .$ Mark your responses from the following options:
(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A)
(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A)
(C) Assertion(A) is True, Reason(R) is False
(D) Assertion(A) is False, Reason(R) is TrueAssertion: Let $f(x)$ be a polynomial function satisfying $f(x) . f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right) .$ If $f(4)=65$ and $l_{1}, l_{2}$,
$l_{3}$ are in G.P., then $f^{\prime}\left(l_{1}\right), f^{\prime}\left(l_{2}\right), f^{\prime}\left(l_{3}\right)$, are also in G.P.
Reason: $f(x)=\pm x^{n}+1$

Aayush Gupta

Numerade Educator

Problem 104

Instructions: In the following questions an Assertion (A) is given followed by a Reason $(R) .$ Mark your responses from the following options:
(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A)
(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A)
(C) Assertion(A) is True, Reason(R) is False
(D) Assertion(A) is False, Reason(R) is True
Assertion: If $y=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2 n}\right)$,
then $\frac{d y}{d x}$ at $x=0$ is 1 .
Reason: $y=\frac{1-x^{2^{2-1}}}{1-x}$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 105

Instructions: In the following questions an Assertion (A) is given followed by a Reason $(R) .$ Mark your responses from the following options:
(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A)
(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A)
(C) Assertion(A) is True, Reason(R) is False
(D) Assertion(A) is False, Reason(R) is True
Assertion: If $f(x)=(\cos x+i \sin x)(\cos 2 x+i \sin 2 x)$ $(\cos 3 x+i \sin 3 x) \ldots(\cos n x+i \sin n x)$ and $f(1)=1$
then $f^{\prime \prime}(1)$ is equal to $-\left(\frac{n(n+1)}{2}\right)^{2}$.
Reason: $f(x)=\cos \frac{n(n-1)}{2} x+i \sin \frac{n(n-1)}{2} x$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 106

If $y=\left(\mathrm{x}+\sqrt{1+x^{2}}\right)^{\mathrm{n}}$, then$\left(1+\mathrm{x}^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}$ is :
(A) $n^{2} y$
(B) $-n^{2} y$
(C) $-y$
(D) $2 x^{2} y$

Aman Gupta

Numerade Educator

Problem 107

If $\sin y=x \sin (\alpha+y)$, then $\frac{d y}{d x}$ is:
(A) $\frac{\sin \alpha}{\sin ^{2}(\alpha+y)}$
(B) $\frac{\sin ^{2}(\alpha+y)}{\sin \alpha}$
(C) $\sin \alpha \sin ^{2}(\alpha+y)$
(D) $\frac{\sin ^{2}(\alpha-y)}{\sin \alpha}$

Aman Gupta

Numerade Educator

Problem 108

If $x^{y}=e^{x-y}$, then $\frac{d y}{d x}$ is:
(A) $\frac{1+x}{1+\log x}$
(B) $\frac{1-\log x}{1+\log x}$
(C) not defined
(D) $\frac{\log x}{(1+\log x)^{2}}$

Urvashi Arora

Numerade Educator

Problem 109

Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and $a, b, c$ are in A. P., then $f^{\prime}(a)$, $f^{\prime}(B)$ and $f^{\prime}(C)$ are in
(A) A.P.
(B) G.P.
(C) H. P.
(D) arithmetic-geometric progression

Aman Gupta

Numerade Educator

Problem 110

If $f(x)=x$, then the value of $f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\ldots+\frac{(-1)^{n} f^{n}(1)}{n !}$ is
(A) $2^{n}$
(B) $2^{n-1}$
(C) 0
(D) 1

Aman Gupta

Numerade Educator

Problem 111

If $x=e^{y+e^{\prime \prime}}, x>0$, then $\frac{d y}{d x}$ is
(A) $\frac{x}{1+x}$
(B) $\frac{1}{x}$
(C) $\frac{1-x}{x}$
(D) $\frac{1+x}{x}$

Aman Gupta

Numerade Educator

Problem 112

Suppose $f(x)$ is differentiable $x=1$ and $\lim _{h \rightarrow 0} \frac{1}{h} f(1+h)=5$, then $f^{\prime}(1)$ equals
(A) 3
(B) 4
(C) 5
(D) 6

Aman Gupta

Numerade Educator

Problem 113

If $f$ is a real-valued differentiable function satisfying $|f(x)-f(y)| \leq(x-y)^{2}, x, y \in R$ and $f(0)=0$, then $f(1)$
equals
(A) $-1$
(B) 0
(C) 2
(D) 1

Aman Gupta

Numerade Educator

Problem 114

The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is
(A) $(-\infty, 0) \cup(0, \infty)$
(B) $(-\infty,-1) \cup(-1, \infty)$
(C) $(-\infty, \infty)$
(D) $(0, \infty)$

Aman Gupta

Numerade Educator

Problem 115

If $x^{m} \cdot y^{m}=(x+y)^{m+n}$, then $\frac{d y}{d x}$ is
(A) $\frac{y}{x}$
(B) $\frac{x+y}{x y}$
(C) $x y$
(D) $\frac{x}{y}$

Aman Gupta

Numerade Educator

Problem 116

Let $y$ be an implicit function of $x$ defined by $x^{2 x}-2 x^{x}$ $\cot y-1=0 .$ Then $y^{\prime}(1)$ equals
(A) $-1$
(B) 1
(C) $\log 2$
(D) $-\log 2$

Aman Gupta

Numerade Educator

Problem 117

Let $f:(-1,1) \rightarrow R$ be a differentiable function such that $f(0)=-1$ and $f^{\prime}(0)=1$. Let $g(x)=[f(2 f(x)+2)]^{2}$.
Then $g^{\prime}(0)=$
(A) $-4$
(B) 0
(C) $-2$
(D) 4

Aman Gupta

Numerade Educator

Problem 118

$\frac{d^{2} x}{d y^{2}}$ is equal to
(A) $-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}$
(B) $\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-2}$
(C) $-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}$
(D) $\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}$

Aman Gupta

Numerade Educator

Problem 119

If $y=\sec \left(\tan ^{-1} x\right)$, then $\frac{d y}{d x}$ at $x=1$ is equal to
(A) $\frac{1}{2}$
(B) 1
(C) $\sqrt{2}$
(D) $\frac{1}{\sqrt{2}}$

Urvashi Arora

Numerade Educator

Problem 120

If $g$ is the inverse of a function $f$ and $f^{\prime}(x)=\frac{1}{1+x^{5}}$, then $g^{\prime}(x)$ is equal to
(A) $1+x^{5}$
(B) $5 x^{4}$
(C) $\frac{1}{1+\{g(x)\}^{5}}$
(D) $1+\{g(x)\}^{5}$

Aman Gupta

Numerade Educator