Chapter 2

If \(\alpha \in\left(0, \frac{\pi}{2}\right)\) then \(\sqrt{x^{2}+x}+\frac{\tan ^{2} \alpha}{\sqrt{x^{2}+x}}\) is always greater than or equal to (A) \(2 \tan \alpha\) (B) 1 (C) 2 (D) \(\sec ^{2} \alpha\)

The function \(f(x)\) is defined in \([0,1]\), then the domain of definition of the function \(f\left[\log \left(1-x^{2}\right)\right]\), is (A) \(x \in\\{0\\}\) (B) \(x \in[-\sqrt{1+e},-1] \cup[1,+\sqrt{1+e}]\) (C) \(x \in(-\infty, \infty)\) (D) None of these

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) \([-1,3]\) (B) \([-1,1]\) (C) \([0,1]\) (D) \([0,3]\)

If \(a\) and \(b\) are natural numbers and \(f(x)=\sin \left(\sqrt{a^{2}-3}\right) x+\cos \left(\sqrt{b^{2}+7}\right) x\) is periodic with finite fundamental period, then period of \(f(x)\) is (A) \(\pi\) (B) \(2 \pi\) (C) \(2 \pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\) (D) \(\pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\)

If \(2 f(x)+3 f\left(\frac{1}{x}\right)=x^{2}-1\), then \(f(x)\) is (A) a periodic function (B) an even function (C) an odd function (D) None of these

If \(f\left(x_{1}\right)-f\left(x_{2}\right)=f\left(\frac{x_{1}-x_{2}}{1-x_{1} x_{2}}\right)\) for \(x_{1}, x_{2} \in[-1,1]\) then \(\begin{aligned}&f(x) \text { is } \\\&\text { (A) } \log \left(\frac{1-x}{1+x}\right) & \text { (B) } \tan ^{-1}\left(\frac{1-x}{1+x}\right)\end{aligned}\) (C) \(\log \left(\frac{1+x}{1-x}\right)\) (D) \(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\)

Let \(f: R \rightarrow R\) be a periodic function such that \(f(T+x)=1+\left\\{1-3 f(x)+3[f(x)]^{2}-[f(x)]^{3}\right\\}^{1 / 3}\) where \(T\) is a fixed positive number, then period of \(f(x)\) is (A) \(T\) (B) \(2 T\) (C) \(3 T\) (D) None of these

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 \(g(x)\) equals (A) \(-\sqrt{x}-1, x \geq 0\) (B) \(\frac{1}{(x+1)^{2}}, x>-1\) (C) \(\sqrt{x+1}, x \geq-1\) (D) \(\sqrt{x}-1, x \geq 0\)

The function \(f(x)=\frac{\sin ^{101} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}\), where \([x]\) denotes the integral part of \(x\) is (A) an odd function (B) an even function (C) neither odd nor even function (D) both odd and even function

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

For real \(x\), let \(f(x)=x^{3}+5 x+1\), then (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 \(R\)

Let \(f(x)=(x+1)^{2}-1, x \geq-1\) Statement 1: The set \(\left\\{x: f(x)=f^{-1}(x)\right\\}=\\{0,-1\\}\) Statement \(2: f\) is a bijection. (A) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement (B) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is false (D) Statement 1 is false, Statement 2 is true

The period of the function \(f(x)= \begin{cases}1, & \text { when } x \text { is a rational } \\ 0, & \text { when } x \text { is irrationa }\end{cases}\) (A) 1 (B) 2 (C) non-periodic (D) None of these

Let \(f_{1}(n)=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\), then \(f_{1}(1)+f_{1}(2)+f_{1}(3)\) \(+\ldots+f_{1}(n)\) is equal to (A) \(n f_{1}(n)-1\) (B) \((n+1) f_{1}(n)+n\) (C) \((n+1) f_{1}(n)-n\) (D) \(n f_{1}(n)+n\)

Range of the function \(f\) defined by \(f(x)=\left[\frac{1}{\sin \\{x\\}}\right]\) (where \([\cdot]\) and \(\\{\cdot\\}\) respectively denote the greatest integer and the fractional part functions is:) (A) \(\mathrm{Z}\), the set of integers (B) \(\mathrm{N}\), the set of natural numbers (C) \(\mathrm{W}\), the set of whole numbers (D) \(\\{2,3,4, \ldots\\}\)

Let \(f(x)\) be a function defined on \([0,1]\) such that $$ f(x)= \begin{cases}x & x \in Q \\ 1-x & x \notin Q\end{cases} $$ Then, for all \(x \in[0,1], f o f(x)\) is (A) a constant (B) \(1+x\) (C) \(x\) (D) None of these

Which of the following functions is (are) injective \(\operatorname{map}(s) ?\) (A) \(f(x)=x^{2}+2, x \in(-\infty, \infty)\) (B) \(f(x)=|x+2|, x \in[-2, \infty)\) (C) \(f(x)=(x-4)(x-5), x \in(-\infty, \infty)\) (D) \(f(x)=\frac{4 x^{2}+3 x-5}{4+3 x-5 x^{2}}, x \in(-\infty, \infty)\)

If \(g(x)=1+\sqrt{x}\) and \(f[g(x)]=3+2 \sqrt{x}+x\), then \(f(x)=\) \(\begin{array}{llll}\text { (A) } 1+2 x^{2} & \text { (B) } 2+x^{2} & \text { (C) } 1+x & \text { (D) } 2+x\end{array}\)

The domain of the function \(f(x)=\) $$ \log _{2}\left(-\log _{1 / 2}\left(1+\frac{1}{\sqrt[4]{x}}\right)-1\right) $$ (A) \(01\)

Let \(f\) be a real valued function with domain \(R\) satisfying \(0 \leq f(x) \leq \frac{1}{2}\) and for some fixed \(a>0\) $$ f(x+a)=\frac{1}{2}-\sqrt{f(x)-(f(x))^{2}} \forall x \in R $$ then the period of the function \(f(x)\) is (A) \(a\) (B) \(2 a\) (C) non-periodic (D) None of these

If \(f(x)\) is defined on \((0,1)\), then the domain of definition of \(f\left(e^{x}\right)+f(\log |x|)\) is (A) \((-e,-1)\) (B) \((-e,-1) \cup(1, e)\) (C) \((-\infty,-1)\) (D) \((-e, e)\)

If \(q^{2}-4 p r=0, p>0\), then the domain of the function \(f(x)=\log \left\\{p x^{3}+(p+q) x^{2}+(q+r) x+r\right\\}\) is (A) \(R-\left\\{-\frac{q}{2 p}\right\\}\) (B) \(R-\left[(-\infty,-1] \cup\left\\{-\frac{q}{2 p}\right\\}\right]\) (C) \(R-\left[(-\infty,-1) \cap\left\\{-\frac{q}{2 p}\right\\}\right]\) (D) None of these

If \(a\) and \(b\) are natural numbers and \(f(x)=\sin \left(\sqrt{a^{2}-3}\right) x+\cos \left(\sqrt{b^{2}+7}\right) x\) is periodic with finite fundamental period, then period of \(f(x)\) is: (A) \(\pi\) (B) \(2 \pi\) (C) \(2 \pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\) (D) \(\pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\)

The function \(f(x)=\frac{\sin ^{101} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}\), where \([x]\) denotes the integral part of \(x\), is (A) an odd function (B) an even function (C) neither odd nor even (D) both odd and even functon

The domain of definition of the function \(f(x)=\) \(\ln \\{x\\}+\sqrt{x-2\\{x\\}}\), where \\{\\} denotes the fractional part, is (A) \(\\{0\\} \cup[1, \infty)\) (B) \((1, \infty)\) (C) \((1, \infty)-I^{+}\) (D) None of these

The domain of the function \(f(x)=\ln \left\\{\operatorname{sgn}\left(9-x^{2}\right)\right\\}+\sqrt{[x]^{3}-4[x]}\), where \([\cdot]\) denotes integral part, is (A) \((-2,1) \cup(2,3)\) (B) \([-2,1) \cup[2,3)\) (C) \([-2,1] \cup[2,3)\) (D) \([-2,1) \cup[2,3]\)

The range of the function \(y=\left[x^{2}\right]-[x]^{2}, x \in[0,2]\) where [-] denotes the integral part, is (A) \(\\{0\\}\) (B) \(\\{0,1\\}\) (C) \(\\{1,2\\}\) (D) \(\\{0,1,2\\}\)

The range of the function \(y=\sin ^{-1}\left[x^{2}+\frac{1}{2}\right]+\cos ^{-1}\left[x^{2}-\frac{1}{2}\right]\), where \([\cdot]\) denotes the integral part, is (A) \((0, \pi)\) (B) \([0, \pi]\) (C) \(\\{\pi\\}\) (D) \(\\{0, \pi\\}\)

Let \(f(x)=\frac{a x^{2}+2 x+1}{2 x^{2}-2 x+1}\). If \(f: R \rightarrow[-1,2]\) is onto, then the values of \(a\) are (A) \((-\infty, 2)\) (B) \([2, \infty)\) (C) \((-\infty,-7] \cup[-2, \infty)\) (D) None of these

Let \(f: N \rightarrow N\), where \(f(x)=x+(-1)^{x-1}\). Then, (A) \(f^{-1}(x)=x+(-1)^{x-1}\) (B) \(f^{-1}(x)=x\) (C) \(f^{-1}(x)=x-(-1)^{x-1}\) (D) None of these

Domain of definition of the function \(f(x)=\sqrt{\sin x}+\sin ^{-1}\left(\frac{2|x|}{1+x^{2}}\right)\) is (A) \((2 n \pi,(2 n+1) \pi), n \in I\) (B) \([2 n \pi,(2 n+1) \pi], n \in I\) (C) \(R\) (D) None of these

The domain of the function \(f(x)=[\sin x] \cos \left(\frac{\pi}{[x-1]}\right)\) (A) \((1,2)\) (B) \(R-[1,2)\) (C) \(R-(1,2)\) (D) None of these

The range of the function \(y=\frac{x-[x]}{1-[x]+x}\) (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left[0, \frac{1}{2}\right]\) (C) \(\left[0, \frac{1}{2}\right)\) (D) \(\left(0, \frac{1}{2}\right]\)

If \(\\{x\\}\) and \([x]\) represent fractional and integral part of \(x\), then the value of \([x]+\sum_{r=1}^{2000} \frac{\\{x+r\\}}{2000}\) is (A) \(x\) (B) \(2000 x\) (C) 0 (D) None of these

Consider a function \(f(n)\) defined for all \(n \in N\). The function satisfies the following two conditions (i) \(f(1)+f(2)+f(3)+\ldots\) to \(\infty=1\) (ii) \(f(n)=\left\\{(1-p) p^{-1}\right\\}\\{f(n+1)+f(n+2)+\ldots\) to \(\infty\\}\) where \(0

If the function \(f\) satisfies the relation \(f(x+y)+\) \(f(x-y)=2 f(x) f(y) \forall x, y \in R\) and \(f(0) \neq 0\), then \(f(x)\) is (A) an even function (B) an odd function (C) odd if \(f(x)>0\) (D) neither even nor odd

Let \(f: R-\\{2\\} \rightarrow R\) be a function satisfying \(2 f(x)+\) \(3 f\left(\frac{2 x+29}{x-2}\right)=100 x+80 \forall x \in R-\\{2\\}\), then \(f(x)=\) (A) \(16-40 x-\frac{60(2 x+29)}{x-2}\) (B) \(100 x+80-\frac{3(2 x+29)}{x-2}\) (C) \(40-16 x+\frac{30(2 x+29)}{x-2}\) (D) None of these

Let \(g: R \rightarrow R\) be given by \(g(x)=3+4 x\). If \(g^{n}(x)=\) gogo ... \(\operatorname{og}(x)\), then \(g^{\circ}(x)\) (where \(g^{-n}(x)\) denotes inverse of \(g^{n}(x)\) ) is equal to (A) \(\left(4^{n}-1\right)+4^{u} x\) (B) \((x+1) 4^{-n}-1\) (C) \((x+1) 4^{n}-1\) (D) \(\left(4^{-u}-1\right) x+4^{n}\)

Let \(f(x+p)=1+\left[2-3 f(x)+3(f(x))^{2}-(f(x))^{3}\right]^{1 / 3}\), \(\forall x \in R\), where \(p>0\). Then, \(f(x)\) is periodic with period. (A) \(p\) (B) \(2 p\) (C) \(4 p\) (D) None of these

If \(y=\log _{3} x\) and \(S=(3,27)\), the set onto which the set \(S\) is mapped is (A) \((0,3)\) (B) \((1,4)\) (C) \((1,3)\) (D) \((0,2)\)

The values of \(x\) for which the functions \(f(x)=x-3\) and \(\phi(x)=4-x\) satisfy the inequality \(|f(x)+\phi(x)|<\) \(|f(x)|+|\phi(x)|\) are (A) \([3,4]\) (B) \((-\infty, \infty)\) (C) \((-\infty, \infty)-[3,4]\) (D) None of these

If \(f\) is an even function defined on the interval \([-5,5]\), then the real values of \(x\) satisfying the equation $$ f(x)=f\left(\frac{x+1}{x+2}\right) \text { are } $$ (A) \(\frac{-1 \pm \sqrt{5}}{2}\) (B) \(\frac{-3 \pm \sqrt{5}}{2}\) (C) \(\frac{-2 \pm \sqrt{5}}{2}\) (D) None of these

The distinct linear function which maps \([-1,1]\) onto \([0,2]\) is (A) \(x-1\) (B) \(x+1\) (C) \(-x+1\) (D) \(-x-1\)

Which of the following functions have period \(2 ?\) (A) \(\\{x\\}+\cos \pi x\) (B) \(\tan \left(\frac{\pi}{2}[x]\right)\) (C) \(\sin x+\\{x\\}\) (D) \(\sin (\cos x)\)

The values of \(x\) for which the domain of definition of the function, \(f(x)=\frac{1}{[|x-1|]+|7-x|-6}\), where [.] denotes the greatest integer part, is not defined are (A) \((0,1]\) (B) \([7,8)\) (C) \(\\{2,3,4,5,6\\}\) (D) \([0,1] \cup[7,8]\)

If \(f: R \rightarrow R\) be defined by \(f(x)=\frac{e^{x}-e^{-x}}{2}\), then (A) \(f\) is one-one (B) \(f\) is onto (C) \(f^{-1}(x)=\log \left(x-\sqrt{x^{2}+1}\right)\) (D) \(f^{-1}(x)=\log \left(x+\sqrt{x^{2}+1}\right)\)

Let \(f(x)=\sin ^{-1}(\log [x])+\log \left(\sin ^{-1}[x]\right)\), where [] denotes the greatest integer function. Then, (A) domain of \(f\) is \([1,2)\) (B) domain of \(f\) is \([1,3)\) (C) range of \(f\) is \(\left\\{\log \frac{\pi}{2}\right\\}\) (D) range of \(f\) is \(\\{0\\}\)

If the function \(f:[1, \infty) \rightarrow[1, \infty)\) is defined by \(f(x)=\) \(2^{x(x-1)}\), then (A) \(f\) is one-one (B) \(f\) is onto (C) \(f^{-1}(x)=\frac{1+\sqrt{1+4 \log _{2} x}}{2}\) (D) \(f^{-1}(x)=\frac{1-\sqrt{1+4 \log _{2} x}}{2}\)

Let \(f(x)=\frac{9^{x}}{9^{x}+3}\). Then, (A) \(f(x)+f(1-x)=1\) (B) \(f(x)+f(1-x)=-1\) (C) \(f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)\) \(+f\left(\frac{3}{1996}\right)+\ldots+f\left(\frac{1995}{1996}\right)=998\) (D) \(f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)\) \(+f\left(\frac{3}{1996}\right)+\ldots+f\left(\frac{1995}{1996}\right)=997 \frac{1}{2}\)

Let \(n\) be a positive integer with \(f(n)=1 !+2 !+3 !+\) \(\ldots+n !\) and \(P(x)\) and \(Q(x)\) be polynomials in \(x\) such that \(f(n+2)=P(n) f(n+1)+Q(n) f(n)\) for all \(n \geq 1\), then (A) \(P(x)=x+3\) (B) \(Q(x)=-x-2\) (C) \(P(x)=-x-2\) (D) \(Q(x)=x+3\)