Chapter 12

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

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 \(x=0\)

The function \(f: R \sim\\{0\\} \rightarrow R\) given by \(f(x)=\frac{1}{x}-\frac{2}{e^{2 x}-1}\) can be made continuous at \(x=0\) by defining \(f(0)\) as [2007] (A) 2 (B) \(-1\) (C) 0 (D) 1

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 \(f(x)=\left\\{\begin{array}{l}\frac{\sin (p+1) x+\sin x}{x}, x<0 \\ q, x=0 \\\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}}, x>0\end{array} \quad\right.\) 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) \(\quad p=\frac{1}{2}, q=\frac{3}{2}\) (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 [2012] (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 \(f(x)=|x-2|+|x-5|, x \in R\). Statement 1: \(f^{\prime}(4)=0\) Statement \(\mathbf{2}: f\) 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

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) \(g^{\prime}(0)=-\cos (\log 2)\)