Chapter 2

If the period of the function \(f(x)=\sin (\sqrt{[n]} x)\), where \([n]\) denotes the greatest integer less than or equal to \(n\), is \(2 \pi\), then (A) \(1 \leq n<2\) (B) \(1

The period of the function \(f(x)=3 x+3-[3 x+3]+\) \(\sin \frac{\pi x}{2}\), where \([x]\) denotes the greatest integer \(\leq x\), is (A) 4 (B) 1 (C) 2 (D) None of these

\(\pi\) is the period of the function (A) \(|\sin x|+|\cos x|\) (B) \(\sin ^{4} x+\cos ^{4} x\) (C) \(\sin (\sin x)+\sin (\cos x)\) (D) \(\frac{1+2 \cos x}{\sin x(2+\sec x)}\)

The period of the function \(f(x)=\sin 5 x+\cos \sqrt{3} x\) is (A) \(\sqrt{3} \pi\) (B) \(\pi\) (C) non-periodic (D) None of these

If \(e^{x}+e^{f(x)}=e\), then range of the function \(f\) is (A) \((-\infty, 1]\) (B) \((-\infty, 1)\) (C) \((1, \infty)\) (D) \([1, \infty)\)

The range of the function \(f(x)=\frac{\sin \left(\pi\left[x^{2}+1\right]\right)}{x^{4}+1}\), where \([\cdot]\) denotes the greatest integer function, is (A) \([0,1]\) (B) \([-1,1]\) (C) \(\\{0\\}\) (D) None of these

Range of values of \(f(x)=1+\sin x+\sin ^{3} x+\sin ^{5} x, x \in\) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) is (A) \((0,1)\) (B) \((-\infty, \infty)\) (C) \((-2,2)\) (D) None of these

If \([x]\) denotes the integral part of \(x\), then the domain of the function \(f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left[\log _{1 / 2}\left(x^{2}-5 x\right.\right.\) \(+5\) ) \(]\) is (A) \(\left(-\sqrt{\frac{5}{2}},-1\right]\) (B) \(\left[1, \sqrt{\frac{5}{2}}\right)\) (C) \(\left(-\sqrt{\frac{5}{2}},-1\right] \cup\left[1, \sqrt{\frac{5}{2}}\right)\) (D) None of these

Column-I I. Domain of the function \(f(x)=\frac{1}{\sqrt{|\sin x|+\sin x}}\) II. Domain of the function \(f(x)=\) \(\log _{3}\left[-\log _{1 / 2}\left(1+\frac{1}{x^{1 / 5}}\right)-1\right]\) III. Domain of the function \(f(x)=\log _{3}\left[-\left(\log _{3} x\right)^{2}\right.\) \(\left.+5 \log _{3} x-6\right]\) IV. Domain of the function \(f(x)=\cot ^{-1}\left(\frac{x}{\sqrt{x^{2}-\left[x^{2}\right]}}\right)\) \(x \in R\) Column-II (A) \(R-\\{\sqrt{n}\), \(n \geq 0, n \in I\\}\) (B) \((2 n \pi,\), \((2 n+1) \pi)\) (C) \((0,1)\) (D) \((9,27)\)

Column-I I. Range of the function \(f(x)=\sqrt{3 x^{2}-4 x+5}\) II. Range of the function \(f(x)=\log _{\varepsilon}\left(3 x^{2}-4 x+5\right)\) III. The value of the function \(f(x)=\frac{x^{2}-3 x+2}{x^{2}+x-6}\) lies in the interval IV. The range of the function \(f(x)=\) \(\sin \left[\log \left(\frac{\sqrt{4-x^{2}}}{1-x}\right)\right]\) Column-II (A) \(\left[\log _{\epsilon} \frac{11}{3}, \infty\right)\) (B) \((-\infty, \infty)\) (C) \(\left[\sqrt{\frac{11}{3}}, \infty\right)\) (D) \([-1,1]\)

Assertion: If \(f(x)=\frac{a^{x}}{a^{x}+\sqrt{a}}(a>0)\), then \(\sum_{r=1}^{2 n-1} 2 f\left(\frac{r}{2 n}\right)=2 n-1\) Reason: \(f(x)+f(1-x)=1 \forall x\)

Assertion: Suppose, \(f(x)=(x+1)^{2}\) for \(x \geq-1\). If \(g(x)\) is the function whose graph is the reflection of the graph of \(f(x)\) with respect to the line \(y=x\), then \(g(x)=\) \(\sqrt{x}-1, x \geq 0 .\) Reason: \(g(x)\) is the inverse of \(f(x)\)

The period of \(\sin ^{2} \theta\) is : (A) \(\pi^{2}\) (B) \(\pi\) (C) \(2 \pi\) (D) \(\pi 2\)

The domain of \(\sin ^{-1}\left[\log _{3}(x / 3)\right]\) is: (A) \([1,9]\) (B) \([-1,9]\) (C) \([-9,1]\) (D) \([-9,-1]\)

The period of the function \(f(x)=\sin ^{4} x+\cos ^{4} x\) is : (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(2 \pi\) (D) None of these

A function \(f\) from the set of natural numbers to integers defined by \(f(n)=\left\\{\begin{array}{ll}\frac{n-1}{2}, & \text { when is odd } \\ & \text { is } \\ -\frac{n}{2}, & \text { when } n \text { is even }\end{array}\right.\) (A) one-one but not onto (B) onto but not one-one (C) one-one and onto both (D) neither one-one nor onto

Domain of definition of the function \(f(x)=\frac{3}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)\), is (A) \((1,2)\) (B) \((-1,0) \cup(1,2)\) (C) \((1,2) \cup(2, \infty)\) (D) \((-1,0) \cup(1,2) \cup(2, \infty)\)

If \(f: R \rightarrow S\), defined by \(f(x)=\sin x-\sqrt{3} \cos x+1\), is onto, then the interval of \(S\) is (A) \([0,3]\) (B) \([-1,1]\) (C) \([0,1]\) (D) \([-1,3]\)

The graph of the function \(y=f(x)\) is symmetrical about the line \(x=2\), then (A) \(f(x+2)=f(x-2)\) (B) \(f(2+x)=f(2-x)\) (C) \(f(x)=f(-x)\) (D) \(f(x)=-f(-x)\)

The domain of the function \(f(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^{2}}}\) is (A) \([2,3]\) (B) \([2,3)\) (C) \([1,2]\) (D) \([1,2)\)

Let \(f:(-1,1) \rightarrow B\), be a function defined by \(f(x)=\tan ^{-1} \frac{2 x}{1-x^{2}}\), then \(f\) is both one-one and onto when \(B\) is the interval (A) \(\left(0, \frac{\pi}{2}\right)\) (B) \(\left[0, \frac{\pi}{2}\right)\) (C) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) (D) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

A real valued function \(f(x)\) satisfies the functional equation \(f(x-y)=f(x) f(y)-f(a-x) f(a+y)\) where \(a\) is a given constant and \(f(0)=1, f(2 a-x)\) is equal to (A) \(-f(x)\) (B) \(f(x)\) (C) \(f(A)+f(a-x)\) (D) \(f(x)\)

The largest interval lying in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for which the function \(f(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\) is defined, is (A) \([0, \pi]\) (B) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (C) \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)\) (D) \(\left[0, \frac{\pi}{2}\right)\)

For real \(x\), let \(f(x)=x^{3}+5 x+1\), then \(\quad\) (A) \(f\) is one-one but not onto \(R\) (B) \(f\) is onto \(R\) but not one-one (C) \(f\) is one-one and onto \(R\) (D) \(f\) is neither one-one nor onto \(R\)

The domain of the function \(f(x)=\frac{1}{\sqrt{|x|-x}}\) is (A) \((0, \infty)\) (B) \((-\infty, 0)\) (C) \((-\infty, \infty)-\\{0\\}\) (D) \((-\infty, \infty)\)

If \(f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0\), and \(S=\\{x \in R: f(x)=f(-x)\\} ;\) then \(\mathrm{S}:\) (A) contains more than two elements. (B) is an empty set. (C) contains exactly one element (D) contains exactly two elements