Chapter 12

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 \(f(x)=\operatorname{sgn}(x)\) and \(g(x)=x\left(1-x^{2}\right)\), then \(f o g(x)\) and \(g o f(x)\) are continuous everywhere Reason: \(f o g(x)=\left\\{\begin{aligned}-1, & x \in(-1,0) \cup(1, \infty) \\\ 0, & x \in\\{-1,0,1\\} \\ 1, & x \in(-\infty,-1) \cup(0,1) \end{aligned}\right.\) and, \(g o f(x)=0, \forall x \in R\)

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: Let \(f\) be a function such that \(f(x y)=f(x)\). \(f(y), \forall y \in R\) and \(f(1+x)=1+x(1+g(x))\), where \(\lim _{x \rightarrow 0} g(x)=0\), then $$ \begin{aligned} &\int_{1}^{2} \frac{f(x)}{f^{\prime}(x)} \cdot \frac{1}{1+x^{2}} d x=\frac{1}{2} \log \left(\frac{5}{2}\right) \\ &\text { Reason: } f^{\prime}(x)=\frac{f(x)}{x} \end{aligned} $$

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: The function \(y=f(x)\), defined parametrically as \(y=t^{2}+t|t|, x=2 t-|t|, t \in R\), is continuous \mathrm{\\{} f o r ~ a l l ~ r e a l ~ \(x\) Reason: \(f(x)=\left\\{\begin{array}{r}2 x^{2}, x \geq 0 \\ 0, x<0\end{array}\right.\)

Let \(f(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right] .\) If \(f(x)\) is continuous in \(\left[0, \frac{\pi}{2}\right]\), then \(f\left[\frac{\pi}{4}\right]\) is \(\quad\) [2004] (A) 1 (B) \(\frac{1}{2}\) (C) \(-\frac{1}{2}\) (D) \(-1\)

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\min\) \(\\{x+1,|x|+1\\} .\) Then which of the following is true? [2007] (A) \(f(x) \geq 1\) for all \(x \in \mathrm{R}\) (B) \(f(x)\) is not differentiable at \(x=1\) (C) \(f(x)\) is differentiable everywhere (D) \(f(x)\) is not differentiable at \(x=0\)

Consider the following relations: \(R=\\{(x, y) \mid x, y\) are real numbers and \(x=w y\) for some rational number \(w\\} ;\) \(S=\left\\{\begin{array}{l}\left(\frac{m}{p}, \frac{p}{q}\right) m, n, p \text { and } q \in \mathbf{Z} \\ \text { such that } n, q \neq 0 \text { and } q m=p n\end{array}\right\\}\) Then [2010] (A) neither \(R\) nor \(S\) is an equivalence relation (B) \(S\) is an equivalence relation but \(R\) is not an equivalence relation (C) \(R\) and \(S\) both are equivalence relations (D) \(R\) is an equivalence relation but \(S\) is not an equivalence relation

The real values of \(p\) and \(q\) for which the function \(f(x)=\left\\{\begin{array}{l}\frac{\sin (p+1) x+\sin x}{x}, x<0 \\ q, x=0 \quad \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}}, x>0\end{array} \quad\right.\) is continuous for all \(x\) in \(R\), is (A) \(p=\frac{5}{2}, q=\frac{1}{2}\) (B) \(p=-\frac{3}{2}, q=\frac{1}{2}\) (C) \(p=\frac{1}{2}, q=\frac{3}{2}\) (D) \(p=\frac{1}{2}, q=-\frac{3}{2}\)

If \(f: R \rightarrow R\) is a function defined by \(f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([x]\) denotes the greatest integer function, then \(f\) is (A) continuous for every real \(x\) (B) discontinuous only at \(x=0\) (C) discontinuous only at non-zero integral values of \(x\) (D) continuous only at \(x=0\)

Consider the function \(f(x)=|x-2|+|x-5|, x \in R\). [2012] Statement 1: \(f^{\prime}(4)=0\) Statement 2: \(f\) is continuous in \([2,5]\), differentiable in \((2,5)\) and \(f(2)=f(5)\). (A) Statement 1 is false, statement 2 is true (B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1 (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 false

If the function \(g(x)=\left\\{\begin{array}{ll}k \sqrt{x+1}, & 0 \leq x \leq 3 \\\ m x+2, & 3

For \(x \in R, f(x)=|\log 2-\sin x|\) and \(g(x)=f(f(x))\), then: [2016] (A) \(g\) is differentiable at \(x=0\) and \(g^{\prime}(0)=-\sin (\log 2)\) (B) \(g\) is not differentiable at \(x=0\) (C) \(g^{\prime}(0)=\cos (\log 2)\) (D) \(g^{\prime}(0)=-\cos (\log 2)\)