Chapter 2

If \(y=a t^{2}+2 b t+c\) and \(t=a x^{2}+2 b x+c\), then \(\frac{d^{3} y}{d x^{3}}\) equals to (a) \(24 a^{2}(a t+b)\) (b) \(24 a(a x+b)^{2}\) (c) \(24 a(a t+b)^{2}\) (d) \(24 a^{2}(a x+b)\)

Differential coefficient of \(\left(x^{\frac{1+m}{m-n}}\right)^{\frac{1}{n-l}} \cdot\left(x^{\frac{m+n}{n-l}}\right)^{\frac{1}{l-m}} \cdot\left(x^{\frac{n+l}{l-m}}\right)^{\frac{1}{m-n}}\) w.r.t. \(x\), is (a) 1 (b) 0 (c) \(-1\) (d) \(x^{l m n}\) ots .$

If \(y=(A+B x) e^{m x}+(m-1)^{-2} e^{x}\), then \(\frac{d^{2} y}{d x^{2}}-2 m \frac{d y}{d x}+m^{2} y\) is equal to (a) \(e^{x}\) (b) \(e^{m x}\) (c) \(e^{-m x}\) (d) \(e^{(1-m) x}\)

If \(f\) and \(g\) are differentiable functions such that \(g^{\prime}(a)=2\) and \(g(a)=b\) and if \(f o g\) is an identity function, then \(f^{\prime}(b)\) has the value equal to (a) \(2 / 3\) (b) 1 (c) 0 (d) \(1 / 2\)

The derivative of the function, \(f(x)=\cos ^{-1}\left\\{\frac{1}{\sqrt{13}}(2 \cos x-3 \sin x)\right\\}+\sin ^{-1}\left\\{\frac{1}{\sqrt{13}}(2 \cos x+3 \sin x)\right\\}\) w.r.t. \(\sqrt{1+x^{2}}\) at \(x=\frac{3}{4}\) is (a) \(\frac{3}{2}\) (b) \(\frac{5}{2}\) (c) \(\frac{10}{3}\) (d) 0

If \(f(x)=\sqrt{x+2 \sqrt{2 x-4}}+\sqrt{x-2 \sqrt{2 x-4}}\), then the value of \(10 f^{\prime}\left(102^{+}\right)\), is (a) \(-1\) (b) 0 (c) \(\underline{1}\) (d) does not exist

Let \(y=\ln (1+\cos x)^{2}\), then the value of \(\frac{d^{2} y}{d x^{2}}+\frac{2}{e^{y / 2}}\) equals to (a) 0 (b) \(\frac{2}{1+\cos x}\) (c) \(\frac{4}{(1+\cos x)}\) (d) \(\frac{-4}{(1+\cos x)^{2}}\)

If \(f(x)=\frac{a+\sqrt{a^{2}-x^{2}}+x}{\sqrt{a^{2}-x^{2}}+a-x}\) where \(a>0\) and \(x

If \(f(x)\) is given by \(f(x)=(\cos x+i \sin x)(\cos 3 x+i \sin 3 x) \ldots\) \(\ldots(\cos (2 n-1) x+i \sin (2 n-1) x)\) then \(f^{\prime \prime}(x)\) is equal to (a) \(n^{3} f(x)\) (b) \(-n^{4} f(x)\) (c) \(-n^{2} f(x)\) (d) \(n^{4} f(x)\)

Let \(f(x)=x^{n}, n\) being a non-negative integer. The value of \(n\) for which the equality \(f^{\prime}(x+y)=f^{\prime}(x)+f^{\prime}(y)\) is valid for all \(x, y>0\), is (a) 0,1 (b) 1,2 (c) 2,4 (d) None of these

If \(y=\sin ^{-1}\left(\frac{\sin \alpha \sin x}{1-\cos \alpha \sin x}\right)\), then \(y^{\prime}(0)\) is (a) 1 (b) \(2 \tan \alpha\) (c) \(\frac{1}{2} \tan \alpha\) (d) \(\sin \alpha\)

If \(f(x)=\sin \left\\{\frac{\pi}{3}[x]-x^{2}\right\\}\) for \(2

The functions \(u=e^{x} \sin x, v=e^{x} \cos x\) satisfy the equation (a) \(v \frac{d u}{d x}-u \frac{d v}{d x}=u^{2}+v^{2}\) (b) \(\frac{d^{2} u}{d x^{2}}=2 v\) (c) \(\frac{d^{2} v}{d x^{2}}=-2 u\) (d) All of these

If \(f(x)=\log _{x}\\{\ln (x)\\}\), then \(f^{\prime}(x)\) at \(x=e\), is (a) \(e\) (b) \(-e\) (c) \(e^{2}\) (d) \(e^{-1}\)

Let \(f\) be a differentiable function satisfying \([f(x)]^{n}=f(n x)\) for all \(x \in R\). Then, \(f^{\prime}(x) f(n x)\) (a) \(f(x)\) (b) 0 (c) \(f(x) f^{\prime}(n x)\) (d) None of these

If \(y=f(x)\) is an odd differentiable function defined on \((-\infty, \infty)\) such that \(f^{\prime}(3)=-2\), then \(f^{\prime}(-3)\) equals (a) 4 (b) 2 \(\begin{array}{ll}\text { (c) }-2 & \text { (d) } 0\end{array}\)

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 (a) 1 (b) \(-1\) (c) 0 (d) None of these

If \(f(x)=|\cos x-\sin x|\) then \(f^{\prime}\left(\frac{\pi}{4}\right)\) is equal to (a) \(\sqrt{2}\) (b) \(-\sqrt{2}\) (c) 0 (d) None of these

If \(f(x)=x^{n}\), then the value of \(f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\frac{f^{(\omega)}(1)}{4 !}-\ldots\) \(\ldots+\frac{(-1)^{n} f^{n}(1)}{n !}\) is (a) 1 (b) \(2^{n}\) (c) \(2^{n-1}\) (d) 0

Which of the following functions are not derivable at \(x=0\) ? (a) \(f(x)=\sin ^{-1} 2 x \sqrt{1-x^{2}}\) (b) \(g(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^{x}}\right)\) (c) \(h(x)=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\) (d) \(k(x)=\sin ^{-1}(\cos x)\)

Let \(f(x)=\frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1} \cdot x\), then (a) \(f^{\prime}(10)=1\) (b) \(f^{\prime}(3 / 2)=-1\) (c) domain of \(f(x)\) is \(x \geq 1\) (d) None of these

if \(2^{x}+2^{y}=2^{x+y}\), then \(\frac{d y}{d x}\) has the value equal to (a) \(-\frac{2^{y}}{2^{x}}\) (b) \(\frac{1}{1-2^{x}}\) (c) \(1-2^{y}\) (d) \(\frac{2^{x}\left(1-2^{y}\right)}{2^{y}\left(2^{x}-1\right)}\)

for the function \(y=f(x)=\left(x^{2}+b x+c\right) e^{x}\), which of the following holds? (a) If \(f(x)>0\) for all real \(x \Rightarrow f^{\prime}(x)>0\) (b) If \(f(x)>0\) for all real \(x \Rightarrow f^{\prime}(x)>0\) (c) If \(f^{\prime}(x)>0\) for all real \(x \Rightarrow f(x)>0\) (d) If \(f^{\prime}(x)>0\) for all real \(x \neq f(x)>0\)

If \(\sqrt{y+x}+\sqrt{y-x}=c\) (where \(c \neq 0\) ), then \(\frac{d y}{d x}\) has the value equal to (a) \(\frac{2 x}{c^{2}}\) (b) \(\frac{x}{y+\sqrt{y^{2}-x^{2}}}\) (c) \(\frac{y-\sqrt{y^{2}-x^{2}}}{x}\) (d) \(\frac{c^{2}}{2 y}\)

If \(y=\tan x \tan 2 x \tan 3 x\), then \(\frac{d y}{d x}\) has the value equal to (a) \(3 \sec ^{2} 3 x \tan x \tan 2 x+\sec ^{2} x \tan 2 x \tan 3 x+2 \sec ^{2} 2 x \tan 3 x \tan x\) (b) \(2 y(\operatorname{cosec} 2 x+2 \operatorname{cosec} 4 x+3 \operatorname{cosec} 6 x)\) (c) \(3 \sec ^{2} 3 x-2 \sec ^{2} 2 x-\sec ^{2} x\) (d) \(\sec ^{2} x+2 \sec ^{2} 2 x+3 \sec ^{2} 3 x\)

If \(x-c\) is a factor of order \(m\) of the polynomial \(f(x)\) of degree \(n(1

If \(a_{1} x^{3}+b_{1} x^{2}+c_{1} x+d_{1}=0\) and \(a_{2} x^{3}+b_{2} x^{2}+c_{2} x+d_{2}=0\) have a pair of repeated common roots, then $$ \left|\begin{array}{ccc} 3 a_{1} & 2 b_{1} & c_{1} \\ 3 a_{2} & 2 b_{2} & c_{2} \\ a_{2} b_{1}-a_{1} b_{2} & c_{1} a_{2}-c_{2} a_{1} & d_{1} a_{2}-d_{2} a_{1} \end{array}\right| \text { is } $$ (a) 0 (b) \(c_{1} a_{2}-c_{2} a_{1}\) (c) \(a_{1} b_{2}-a_{2} b_{1}\) (d) \(d_{1} a_{2}-d_{2} a_{1}\)

If \(\alpha\) occurs \(p\) times and \(\beta\) occurs \(q\) times in polynomial equation \(f(x)=0\) of degree \(n(1

Match the following : \begin{tabular}{l|l} \hline Column I & Column II \\ \hline (A) \(y=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\), then \(\frac{d y}{d x}=-\frac{2}{1+x^{2}}\) & (p) for \(x<0\) \\ (B) \(y=\cos ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)\), then \(\frac{d y}{d x}=\frac{1}{1+x^{2}}\) & (q) for \(x>1\) \\ (C) \(y=\left|e^{|x|}-e\right|\), then \(\frac{d y}{d x}>0\) & (r) for \(x<-1\) \\ (D) \(u=\log |2 x|, v=\left|\tan ^{-1} x\right|\), then \(\frac{d u}{d v}>2\) & (s) for \(-1

Suppose, \(A=\frac{d y}{d x}\) of \(x^{2}+y^{2}=4\) at \((\sqrt{2}, \sqrt{2}), B=\frac{d y}{d x}\) of \(\sin y+\sin x=\sin x \cdot \sin y\) at \((\pi, \pi)\) and \(C=\frac{d y}{d x}\) of \(2 e^{x y}+e^{x} e^{y}-e^{x}-e^{y}=e^{x y+1}\) at \((1,1)\), then \((A-B-C)\) has the value equal to .........

A function is represented parametrically by the equations \(x=\frac{1+t}{t^{3}}\); \(y=\frac{3}{2 t^{2}}+\frac{2}{t}\), then \(\frac{d y}{d x}-x \cdot\left(\frac{d y}{d x}\right)^{3}\) has the absolute value equal to \(\ldots \ldots \ldots .\)

Suppose, the function \(f(x)-f(2 x)\) has the derivative 5 at \(x=1\) and derivative 7 at \(x=2\). The derivative of the function \(f(x)-f(4 x)\) at \(x=1\), has the value \(10+\lambda\), then the value of \(\lambda\) is equal to \(\ldots \ldots \ldots\)

If \(x+y=3 e^{2}\), then \(D\left(x^{y}\right)\) vanishes when \(x\) equals to \(\lambda e^{2}\), then the value of \(\lambda\) is equal to .........

Let \(h(x)\) be differentiable for all \(x\) and let \(f(x)=\left(k x+e^{x}\right) h(x)\) where \(k\) is some constant. If \(h(0)=5, h^{\prime}(0)=-2\) and \(f^{\prime}(0)=18\), then the value of \(k\) is equal to $\ldots \ld