Chapter 13

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

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) \(n(\log a)^{n}\)

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

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

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) \(\frac{1}{\sqrt{(a-x)(x-b)}}\)

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) \(\frac{y}{27}\)

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

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

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) \(f^{\prime}(3)=f^{\prime}(-3)\)

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) \(f(x)=e^{x}\)

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

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) \(b=-n\)

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 \(a\)

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

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

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) \(-\frac{1}{\sqrt{1+x^{2}}}\)

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) \(\sec ^{-1} x\)

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

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

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

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) \(\frac{2}{x^{2}-1}\)

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

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

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) \(-2 n x\)

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) \(-2 n^{2}\)

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

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

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: \(f(x)=\pm x^{n}+1\)

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: \(y=\frac{1-x^{2^{2-1}}}{1-x}\)

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: \(f(x)=\cos \frac{n(n-1)}{2} x+i \sin \frac{n(n-1)}{2} x\)

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) \(2 x^{2} y\)

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) \(\frac{\sin ^{2}(\alpha-y)}{\sin \alpha}\)

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) \(\frac{\log x}{(1+\log x)^{2}}\)

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

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

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) \(\frac{1+x}{x}\)

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

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

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) \((0, \infty)\)

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) \(-\log 2\)

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

\(\frac{d^{2} x}{d y^{2}}\) 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) \(\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\)

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

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) \(1+\\{g(x)\\}^{5}\)