Chapter 13

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

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) \(-\frac{7}{8}\)

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

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

$$ \begin{aligned} &\text { If } y=\sqrt{(a-x)(x-b)}-(a-b) \tan ^{-1} \sqrt{\frac{a-x}{x-b}}, \text { then } \\ &\frac{d y}{d x}= \end{aligned} $$(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) \(f^{\prime}(3)=f^{\prime}(-3)\)

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

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

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) \(b=-n\)

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

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 \(\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}\), 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 \(f(x)=\tan x\), 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

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

\begin{tabular}{l} \hline \multicolumn{1}{c} { Column-I } \\ \hline I. The function \(y\) defined by the equa- \\ 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 \(y(x)\) \\ 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 \\ \(x\), where \(f(x), g(x)\) (C) 4 \\ ferentiable 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 \\ 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 \end{tabular}

If \(y=\left(x+\sqrt{1+x^{2}}\right)^{\mathrm{n}}\), then \(\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) is : [2002] (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 : \(\quad\) [2002] (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: [2002] (A) \(\frac{1+x}{1+\log x}\) (B) \(\frac{1-\log x}{1+\log x}\) (C) not defined (D) \(\frac{\log x}{(1+\log x)^{2}}\)

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 [2003] (A) \(2^{n}\) (B) \(2^{n-1}\) (C) 0 (D) 1

Suppose \(f(x)\) is differentiable \(x=1\) and \(\lim _{h \rightarrow 0} \frac{1}{h} f(1+h)=5\), then \(f^{\prime}(1)\) equals \([2005]\) (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 [2005] (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 \(\quad\) [2009] (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 \(g^{\prime}(0)=\) [2010] (A) \(-4\) (B) 0 (C) \(-2\) (D) 4 2

\(\frac{d^{2} x}{d y^{2}}\) is equal to [2011] (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) \(1+\\{g(x)\\}^{5}\)