Chapter 20

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) \((0, \infty)\)

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

If \(f(x)=\left\\{\begin{array}{ll}x e^{-\left(\frac{1}{|x|} \mid \frac{1}{x}\right)}, x \neq 0 \\ 0 & , x=0\end{array}\right.\) then \(\mathrm{f}(\mathrm{x})\) is (a) discontinuous every where (b) continuous as well as differentiable for all \(x\) (c) continuous for all \(x\) but not differentiable at \(x=0\) (d) neither differentiable nor continuous at \(x=0\)

The derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)\) with respect to \(\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)\) at \(x=\frac{1}{2}\) is : (a) \(\frac{2 \sqrt{3}}{5}\) (b) \(\frac{\sqrt{3}}{12}\) (c) \(\frac{2 \sqrt{3}}{3}\) (d) \(\frac{\sqrt{3}}{10}\)

If \(y=\sum_{k=1}^{6} k \cos ^{-1}\left\\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\\}\), then \(\frac{d y}{d x}\) at \(x=0\) is

If \(x=2 \sin \theta-\sin 2 \theta\) and \(y=2 \cos \theta-\cos 2 \theta, \theta \in[0,2 \pi]\), then \(\frac{d^{2} y}{d x^{2}}\) at \(\theta=\pi\) is : \(\quad\) (a) \(\frac{3}{4}\) (b) \(-\frac{3}{8}\) (c) \(\frac{3}{2}\) (d) \(-\frac{3}{4}\)

If \(y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)\), then \(\begin{array}{ll}\frac{d y}{d \alpha} & \text { at } \alpha=\frac{5 \pi}{6} \text { is: } & \text { [Jan. } 7,2020 \text { (I)] }\end{array}\) (a) 4 (b) \(\frac{4}{3}\) (c) \(-4\) (d) \(-\frac{1}{4}\)

Let \(y=y(x)\) be \(a\) function of \(x\) satisfying \(y \sqrt{1-x^{2}}=k-x \sqrt{1-y^{2}}\) where \(k\) is \(a\) constant and \(y\left(\frac{1}{2}\right)=-\frac{1}{4}\). Then \(\frac{d y}{d x}\) at \(x=\frac{1}{2}\), is equal to: \mathrm{\\{} (a) \(-\frac{\sqrt{5}}{4}\) (b) \(-\frac{\sqrt{5}}{2}\) (c) \(\frac{2}{\sqrt{5}}\) (d) \(\frac{\sqrt{5}}{2}\)

If \(\mathrm{e}^{y}+x y=e\), the ordered pair \(\left(\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}\right)\) at \(x=0\) is equal to: (a) \(\left(\frac{1}{e},-\frac{1}{e^{2}}\right)\) (b) \(\left(-\frac{1}{e}, \frac{1}{e^{2}}\right)\) (c) \(\left(\frac{1}{e}, \frac{1}{e^{2}}\right)\) (d) \(\left(-\frac{1}{e},-\frac{1}{e^{2}}\right)\)

The derivative of \(\tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)\), with respect to \(\frac{x}{2}\), where \(\left(x \in\left(0, \frac{\pi}{2}\right)\right)\) is: \(\quad\) (a) 1 (b) \(\frac{2}{3}\) (c) \(\frac{1}{2}\) (d) 2

If \(2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^{2}, x \in\left(0, \frac{\pi}{2}\right)\) then \(\frac{d y}{d x}\) is equal to : \(\quad\) \(\begin{array}{llll}\text { (a) } \frac{\pi}{6}-x & \text { (b) } x-\frac{\pi}{6} & \text { (c) } \frac{\pi}{3}-x & \text { (d) } 2 x-\frac{\pi}{3}\end{array}\)

Let \(\mathrm{S}\) be the set of all points in \((-\pi, \pi)\) at which the function \(f(x)=\min \\{\sin x, \cos x\\}\) is not differentiable. Then \(S\) is a subset of which of the following? (a) \(\left\\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\\}\) (b) \(\left\\{-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}\right\\}\) (c) \(\left\\{-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\\}\) (d) \(\left\\{-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}\right\\}\)

For \(x>1\), if \((2 x)^{2 y}=4 e^{2 x-2 y}\), then \(\left(1+\log _{e} 2 x\right)^{2} \frac{d y}{d x}\) is equal to (a) \(\frac{x \log _{e} 2 x-\log _{e} 2}{x}\) (b) \(\log _{e} 2 x\) (c) \(\frac{x \log _{e} 2 x+\log _{e} 2}{x}\) (d) \(x \log _{e} 2 x\)

If \(f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^{2} & 2 x \\\ \tan x & x & 1\end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{f^{\prime}(\mathrm{x})}{x}\) (a) Exists and is equal to-2 (b) Does not exist (c) Exist and is equal to 0 (d) Exists and is equal to 2

If \(f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)\), then \(f^{\prime}\left(-\frac{1}{2}\right)\) equals. (a) \(\sqrt{3} \log _{e} \sqrt{3}\) (b) \(-\sqrt{3} \log _{e} \sqrt{3}\) (c) \(-\sqrt{3} \log _{e} 3\) (d) \(\sqrt{3} \log _{e} 3\)

If \(x^{2}+y^{2}+\sin y=4\), then the value of \(\frac{d^{2} y}{d x^{2}}\) at the point \((-2,0)\) is \(\quad\) [Online April 15, 2018] \(\begin{array}{llll}\text { (a) }-34 & \text { (b) }-32 & \text { (c) }-2 & \text { (d) } 4\end{array}\)

If for \(\mathrm{x} \in\left(0, \frac{1}{4}\right)\), the derivative of \(\tan ^{-1}\left(\frac{6 \mathrm{x} \sqrt{\mathrm{x}}}{1-9 \mathrm{x}^{3}}\right)\) is \(\sqrt{x} \cdot g(x)\), then \(g(x)\) equals: (a) \(\frac{3}{1+9 x^{3}}\) (b) \(\frac{9}{1+9 x^{3}}\) (c) \(\frac{3 x \sqrt{x}}{1-9 x^{3}}\) (d) \(\frac{3 x}{1-9 x^{3}}\)

For \(x \in R, f(x)=|\log 2-\sin x|\) and \(g(x)=f(f(x))\), then : (a) \(g^{\prime}(0)=-\cos (\log 2)\) (b) \(g\) is differentiable at \(x=0\) and \(g^{\prime}(0)=-\sin (\log 2)\) (c) g is not differentiable at \(x=0\) (d) \(g^{\prime}(0)=\cos (\log 2)\)

If \(f(x)=x^{2}-x+5, x>\frac{1}{2}\), and \(g(x)\) is its inverse function, then \(g^{\prime}(7)\) equals: (a) \(-\frac{1}{3}\) (b) \(\frac{1}{13}\) (c) \(\frac{1}{3}\) (d) \(-\frac{1}{13}\)

If \(y=\sec \left(\tan ^{-1} x\right)\), then \(\frac{d y}{d x}\) at \(x=1\) is equal to: \([\mathbf{2 0 1 3}]\) (a) \(\frac{1}{\sqrt{2}} \quad\) (b) \(\frac{1}{2}\) (c) 1 (d) \(\sqrt{2}\)

If the curves \(\frac{x^{2}}{\alpha}+\frac{y^{2}}{4}=1\) and \(y^{3}=16 x\) intersect at right angles, then a value of \(\alpha\) is : \(\quad\) (a) 2 (b) \(\frac{4}{3}\) (c) \(\frac{1}{2}\) (d) \(\frac{3}{4}\)

For \(a>0, t \in\left(0, \frac{\pi}{2}\right)\), let \(x=\sqrt{a^{\sin ^{-1} t}}\) and \(y=\sqrt{a^{\cos ^{-1} t}}\), Then, \(1+\left(\frac{d y}{d x}\right)^{2}\) equals: (a) \(\frac{x^{2}}{y^{2}}\) (b) \(\frac{y^{2}}{x^{2}}\) (c) \(\frac{x^{2}+y^{2}}{y^{2}}\) (d) \(\frac{x^{2}+y^{2}}{x^{2}}\)

Let \(f(x)=\frac{x^{2}-x}{x^{2}+2 x} x \neq 0,-2\). Then \(\frac{d}{d x}\left[f^{-1}(x)\right]\) (wherever it is defined) is equal to : \(\quad\) (a) \(\frac{-1}{(1-x)^{2}}\) (b) \(\frac{3}{(1-x)^{2}}\) (c) \(\frac{1}{(1-x)^{2}}\) (d) \(\frac{-3}{(1-x)^{2}}\)

If \(f^{\prime}(x)=\sin (\log x)\) and \(y=f\left(\frac{2 x+3}{3-2 x}\right)\), then \(\frac{d y}{d x}\) equals (a) \(\sin \left[\log \left(\frac{2 x+3}{3-2 x}\right)\right]\) (b) \(\frac{12}{(3-2 x)^{2}}\) (c) \(\frac{12}{(3-2 x)^{2}} \sin \left[\log \left(\frac{2 x+3}{3-2 x}\right)\right]\) (d) \(\frac{12}{(3-2 x)^{2}} \cos \left[\log \left(\frac{2 x+3}{3-2 x}\right)\right]\)

Let \(f:(-1,1) \rightarrow \mathbb{R}\) be a differentiable function with \(f(0)=-\) 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

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 \(\quad\) (a) (b) \(\log 2 \quad\) (c) \(-\log 2 \quad\) (d) \(-1\)

If \(x^{m} \cdot y^{n}=(x+y)^{m+n}\), then \(\frac{d y}{d x}\) is (a) \(\frac{y}{x}\) (b) \(\frac{x+y}{x y}\) (c) \(x y\) (d) \(\frac{x}{y}\)

For all twice differentiable functios \(f: \mathrm{R} \rightarrow \mathrm{R}\), with \(f(0)=f(1)=f^{\prime}(0)=0\) (a) \(f^{\prime \prime}(x) \neq 0\) at every point \(x \in(0,1)\) (b) \(f^{\prime \prime}(x)=0\), for some \(x \in(0,1)\) (c) \(f^{\prime \prime}(0)=0\) (d) \(f^{\prime \prime}(x)=0\), at every point \(x \in(0,1)\)

Let \(x^{k}+y^{k}=a^{k},(a, k>0)\) and \(\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0\), then \(k\) is: (a) \(\frac{3}{2}\) (b) \(\frac{4}{3}\) (c) \(\frac{2}{3}\) (d) \(\frac{1}{3}\)

The value of \(c\) in the Lagrange's mean value theorem for the function \(f(x)=x^{3}-4 x^{2}+8 x+11\), when \(x \in[0,1]\) is: (a) \(\frac{4-\sqrt{5}}{3}\) (b) \(\frac{4-\sqrt{7}}{3}\) (c) \(\frac{2}{3}\) (d) \(\frac{\sqrt{7}-2}{3}\)

If \(y=\left[x+\sqrt{x^{2}-1}\right]^{15}+\left[x-\sqrt{x^{2}-1}\right]^{15}\), then \(\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) is equal to (a) \(12 y\) \(\begin{array}{lll}\text { (b) } 224 y^{2} & \text { (c) } 225 y^{2} & \text { (d) } 225 y\end{array}\)

If \(f\) and \(g\) are differentiable functions in \([0,1]\) satisfying \(f(0)=2=g(1), g(0)=0\) and \(f(1)=6\), then for some \(c \in] 0,1[\) (a) \(f^{\prime}(c)=g^{\prime}(c)\) (b) \(f^{\prime}(c)=2 g^{\prime}(c)\) (c) \(2 f^{\prime}(c)=g^{\prime}(c)\) (d) \(2 f^{\prime}(c)=3 g^{\prime}(c)\)

Let \(f(x)=x|x|, g(x)=\sin x\) and \(h(x)=(g \circ f)(x)\). Then (a) \(\mathrm{h}(\mathrm{x})\) is not differentiable at \(\mathrm{x}=0\). (b) \(\mathrm{h}(\mathrm{x})\) is differentiable at \(\mathrm{x}=0\), but \(\mathrm{h}^{\prime}(\mathrm{x})\) is not continuous at \(\mathrm{x}=0\) (c) \(\mathrm{h}^{\prime}(\mathrm{x})\) is continuous at \(\mathrm{x}=0 \mathrm{but}\) it is not differentiable at \(\mathrm{x}=0\) (d) \(\mathrm{h}^{\prime}(\mathrm{x})\) is differentiable at \(\mathrm{x}=0\)

Let for \(\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})\) be a polynomial of degree \(2 \mathrm{in} \mathrm{x}, \mathrm{p}_{\mathrm{i}}^{\prime}(\mathrm{x})\) and \(\mathrm{p}^{\prime \prime} \cdot(\mathrm{x})\) be the first and second order derivatives of \(\mathrm{p}_{\mathrm{i}}(\mathrm{x})\) respectively. Let, \(\mathrm{A}(\mathrm{x})=\left[\begin{array}{lll}\mathrm{p}_{1}(\mathrm{x}) & \mathrm{p}_{1}^{\prime}(\mathrm{x}) & \mathrm{p}_{1}^{\prime \prime}(\mathrm{x}) \\ \mathrm{p}_{2}(\mathrm{x}) & \mathrm{p}_{2}^{\prime}(\mathrm{x}) & \mathrm{p}_{2}^{\prime}(\mathrm{x}) \\\ \mathrm{p}_{3}(\mathrm{x}) & \mathrm{p}_{3}^{\prime}(\mathrm{x}) & \mathrm{p}_{3}^{\prime \prime}(\mathrm{x})\end{array}\right]\) and \(\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{\mathrm{T}} \mathrm{A}(\mathrm{x})\). Then determinant of \(\mathrm{B}(\mathrm{x}):\) (a) is a polynomial of degree 6 in \(x\). (b) is a polynomial of degree 3 in \(x\). (c) is a polynomial of degree 2 in \(x\). (d) does not depend on \(\mathrm{x}\).

If the Rolle's theorem holds for the function \(f(x)=2 x^{3}+a x^{2}+b x\) in the interval \([-1,1]\) for the point \(\mathrm{c}=\frac{1}{2}\), then the value of \(2 \mathrm{a}+\) bis: (a) 1 (b) \(-1\) (c) 2 (d) \(-2\)

If \(f(x)=\sin (\sin x)\) and \(f^{\prime \prime}(x)+\tan x f^{\prime}(x)+g(x)=0\), then \(g(x)\) is: (a) \(\cos ^{2} x \cos (\sin x)\) (b) \(\sin ^{2} x \cos (\cos x)\) (c) \(\sin ^{2} x \sin (\cos x)\) (d) \(\cos ^{2} x \sin (\sin x)\)

Consider a quadratic equation \(a x^{2}+b x+c=0\), where \(2 a+3 b+6 c=0\) and let \(g(x)=a \frac{x^{3}}{3}+b \frac{x^{2}}{2}+c x\) |Online May 19, 2012] Statement 1: The quadratic equation has at least one root in the interval \((0,1)\). Statement 2: The Rolle's theorem is applicable to function \(g(x)\) on the interval \([0,1]\). (a) Statement 1 is false, Statement 2 is true. (b) Statement 1 is true, Statement 2 is false. (c) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 . (d) Statement 1 is true, Statement 2 is true,, Statement 2 is a correct explanation for Statement 1 .

Let \(f(x)=x|x|\) and \(g(x)=\sin x\). Statement-1 : gof is differentiable at \(x=0\) and its derivative is continuous at that point. Statement-2: gof is twice differentiable at \(x=0\). (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.

A value of \(c\) for which conclusion of Mean Value Theorem holds for the function \(f(x)=\log _{e} x\) on the interval \([1,3]\) is (a) \(\log _{3} e\) (b) \(\log _{e} 3\) (c) \(2 \log _{3} e\) (d) \(\frac{1}{2} \log _{3} e\)

Let \(f\) be differentiable for all \(x\). If \(f(1)=-2\) and \(f^{\prime}(x) \geq 2\) for \(x \in[1,6]\), then (a) \(f(6) \geq 8(\) b \() f(6)<8\) (c) \(f(6)<5\) (d) \(f(6)=5\)

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots \ldots . .+a_{1} x=0\) \(a_{1} \neq 0, n \geq 2\), has a positive root \(x=\alpha\), then the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots \ldots .+a_{1}=0\) has a positive root, which is (a) greater than \(\alpha\) (b) smaller than \(\alpha\) (c) greater than or equal to \(\alpha\) (d) equal to \(\alpha\)

If \(2 a+3 b+6 c=0\), then at least one root of the equation \(a x^{2}+b x+c=0\) lies in the interval (a) \((1,3)\) (b) \((1,2)\) (c) \((2,3)\) (d) \((0,1)\)

Let \(f(a)=g(a)=k\) and their nth derivatives \(f^{n}(a), g^{n}(a)\) exist and are not equal for some \(n\). Further if \(\lim _{x \rightarrow a} \frac{f(a) g(x)-f(a)-g(a) f(x)+f(a)}{g(x)-f(x)}=4\) then the value of \(k\) is (a) 0 (b) 4 (c) 2 (d) 1

If \(y=\left(x+\sqrt{1+x^{2}}\right)^{n}\), then \(\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) is \([\mathbf]\) \(\begin{array}{lll}\text { (a) } n^{2} y & \text { (b) }-n^{2} y & \text { (c) }-y\end{array}\) (d) \(2 x^{2} y\)

If \(2 a+3 b+6 c=0,(a, b, c \in R)\) then the quadratic equation \(a x^{2}+b x+c=0\) has (a) at least one root in \([0,1]\) (b) at least one root in \([2,3]\) (c) at least one root in \([4,5]\) (d) None of these