Chapter 2

Let \(f(x)=x^{3}+x^{2}+100 x+7 \sin x\), then the equation \(\frac{1}{y-f(1)}+\frac{2}{y-f(2)}+\frac{3}{y-f(3)}=0\) has (A) one real root (B) two real roots (C) more than two real roots (D) no real root

If \(b^{2}-4 a c=0\) and \(a>0\), then the domain of the function \(f(x)=\log \left(a x^{3}+(2 a+b) x^{2}+(2 b+c) x+2 c\right)\) is (A) \((-2, \infty) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (B) \([-2, \infty) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (C) \((-\infty,-2) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (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) \([1, \infty)\)

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

The graph of the function \(\cos x \cos (x+2)-\cos ^{2}(x+1)\) is (A) a straight line passing through \(\left(0,-\sin ^{2} 1\right)\) with slope 2 (B) a straight line passing through \((0,0)\) (C) a parabola with vertex \(\left(1,-\sin ^{2} 1\right)\) (D) a straight line parallel to \(x\)-axis passing through the point \(\left(\frac{\pi}{2},-\sin ^{2} 1\right)\)

Let \(f: R \rightarrow R\) be a function defined by, \(f(x)=\frac{x^{2}-8}{x^{2}+2}\), then \(f\) is (A) one-one but not onto (B) one-one and onto (C) onto but not one-one (D) neither one-one nor onto

If \(f(x)=64 x^{3}+\frac{1}{x^{3}}\) and \(a, b\) are the roots of \(4 x+\frac{1}{x}=3\), (A) \(f(a)=12\) (B) \(f(b)=11\) (C) \(f(a)=f(b)\) (D) None of these

If the functions \(f, g, h\) are defined from the set of real numbers \(R\) to \(R\) such that \(f(x)=x^{2}-1, g(x)=\sqrt{x^{2}+1}, h(x)=\left\\{\begin{array}{l}0, \text { if } x \leq 0 \\ x, \text { if } x \geq 0\end{array}\right.\) then the composite function (hofog) \((x)=\) (A) \(\begin{cases}0, & x=0 \\ x^{2}, & x>0 \\ -x^{2}, & x<0\end{cases}\) (B) \(\begin{cases}0, & x=0 \\ x^{2}, & x \neq 0\end{cases}\) (C) \(\begin{cases}0, & x \leq 0 \\ x^{2}, & x>0\end{cases}\) (D) None of these

If \(S\) is the set of all real \(x\) and such that \(\frac{2 x-1}{2 x^{3}+3 x^{2}+x}\) is positive, then \(S\) contains (A) \(\left(-\infty,-\frac{3}{2}\right)\) (B) \(\left(-\frac{3}{2},-\frac{1}{4}\right)\) (C) \(\left(-\frac{1}{4}, \frac{1}{2}\right)\) (D) \(\left(\frac{1}{2}, 3\right)\)

The number of values of \(x\), where the function \(f(x)=\) \(\cos x+\cos (\sqrt{2} x)\) attains its maximum, is (A) 0 (B) 1 (C) 2 (D) infinite

The distinct linear function (s) which map (s) \([-1,1]\) onto \([0,2]\) is (are) (A) \(x+1,-x+1\) (B) \(x-1, x+1\) (C) \(-x+1\) (D) None of these

Let \(f(x)=\max .\\{(1-x),(1+x), 2\\}, \forall x \in \mathrm{R}\). Then (A) \(f(x)=\left\\{\begin{array}{lc}1+x, & x \leq-1 \\ 2, & -1

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

The image of the interval \([1,3]\) under the mapping \(f: R \rightarrow R\), given by \(f(x)=2 x^{3}-24 x+107\) is (A) \([0,89]\) (B) \([75,89]\) (C) \([0,75]\) (D) None of these

If \(2 f(x)-3 f\left(\frac{1}{x}\right)=x^{2}, x\) is not equal to zero, then \(f(2)\) is equal to (A) \(-\frac{7}{4}\) (B) \(\frac{5}{2}\) (C) \(-1\) (D) None of these

Let \(f(x)=\left(1+b^{2}\right) x^{2}+2 b x+1\) and \(m(b)\) the minimum value of \(f(x)\) for a given \(b\). As \(b\) varies, the range of \(m(b)\) is \(\left.\begin{array}{ll}\text { (A) }[0,1] & \text { (B) }\left(0, \frac{1}{2}\right]\end{array}\right]\) (C) \(\left[\frac{1}{2}, 1\right]\) (D) \((0,1]\)

Let \(f: R \rightarrow R, g: R \rightarrow R\) be two functions given by \(f(x)=2 x-3, g(x)=x^{3}+5\). Then \((f o g)^{-1}(x)\) is equal to (A) \(\left(\frac{x-7}{2}\right)^{1}\) (B) \(\left(\frac{x+7}{2}\right)^{1}\) (C) \(\left(x-\frac{7}{2}\right)^{1 / 3}\) (D) \(\left(\frac{x-2}{7}\right)^{1 / 3}\)

The functions \(f(x)=\log (x-1)-\log (x-2)\) and \(g(x)=\) \(\log \left(\frac{x-1}{x-2}\right)\) are identical when \(x\) lies in the interval (A) \([1,2]\) (B) \([2, \infty]\) (C) \((2, \infty)\) (D) \((-\infty, \infty)\)

The domain of the function \(y=\sqrt{\log \frac{1}{|\sin x|}}\) (A) \(R \backslash\\{n \pi: n \in Z\\}\) (B) \(R^{\prime}(-\pi, \pi)\) (C) \(R \backslash\\{2 n \pi: n \in Z\\}\) (D) \((-\infty, \infty)\)

If \(x\) is real, then the expression \(\frac{x^{2}+34 x-71}{x^{2}+2 x-7}\) (A) cannot lie between 5 and 9 (B) always lies between 5 and 9 (C) is not real (D) None of these

If \(f(x)\) is an odd periodic function with period 2 , then \(f(4)\) equals (A) \(-4\) (B) 4 (C) 2 (D) 0

The function \(f(x)=\cot ^{-1}[\sqrt{(x+3) x}]+\cos ^{-1}\left(\sqrt{x^{2}+3 x+1}\right)\) is defined on the set \(S\), where \(S\) is equal to (A) \(\\{-3,0\\}\) (B) \([-3,0]\) (C) \([0,3]\) (D) \((-3,0)\)

If \(f(x)=a^{\cos x}\) and \(g(x)=(\sin x)^{a}, a \in \mathrm{N}\), then (A) \(f(x)>g(x), \forall x\) (B) \(f(x)

Let \(f\) be a function satisfying \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\). If \(f(1)=3\), then \(\sum f(r)\) is equal to (A) \(\frac{3}{2}\left(3^{n}-1\right)\) (B) \(\frac{3}{2} n(n+1)\) (C) \(3^{n+1}-3\) (D) None of these

Let \(f(x)\) be defined for all \(x>0\) and be continuous. Let \(f(x)\) satisfy \(f\left(\frac{x}{y}\right)=f(x)-f(y)\) for all \(x, y\) and \(f(e)=1 .\) Then \(\begin{array}{ll}\text { (A) } f(x) \text { is bounded } & \text { (B) } f\left(\frac{1}{x}\right) \rightarrow 0 \text { as } x \rightarrow 0\end{array}\) (C) \(x f(x) \rightarrow 1\) as \(x \rightarrow 0\) (D) \(f(x)=\log x\)

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

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

The function \(f:(-\infty,-1] \rightarrow\left(0, e^{5}\right]\) defined by, \(f(x)=e^{x^{3}-3 x+2}\) is (A) Many one and onto (B) Many one and into (C) One-one and onto (D) One-one and into

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

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

If \(a f(x)+b f\left(\frac{1}{x}\right)=x+\frac{5}{x},(a \neq b)\), then \(f(x)\) is equal to (A) \(\frac{1}{a^{2}-b^{2}}\left(x+\frac{1}{x}\right)\) (B) \(\frac{1}{a^{2}-b^{2}}\left[x(5 a-b)+\frac{1}{x}(5 b-a)\right]\) (C) \(\frac{1}{a^{2}-b^{2}}\left[x(a-5 b)+\frac{1}{x}(5 a-b)\right]\) (D) None of the above

Let \(f: R \rightarrow R\) defined by, \(f(x)=x^{3}+x^{2}+100 x+5 \sin x\), then \(f\) is (A) many-one onto (B) many-one into (C) one-one onto (D) one-one into

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) \(\vec{a}\) (B) \(2 a\) (C) non-periodic (D) None of these

Let \(f(x)=\sin x+\cos x, g(x)=x^{2}-1\). Then \(g(f(x))\) is invertible for \(x \in\) (A) \(\left[-\frac{\pi}{2}, 0\right]\) (B) \(\left[-\frac{\pi}{2}, \pi\right]\) (C) \(\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]\) (D) \(\left[0, \frac{\pi}{2}\right]\)

If \(f\left(2 x+\frac{y}{8}, 2 x-\frac{y}{8}\right)=x y\), then \(f(m, n)+f(n, m)=0\) (A) only when \(m=n\) (B) only when \(m \neq n\) (C) only when \(m=-n\) (D) for all \(m\) and \(n\).

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

The value of \(\left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{100}\right]+\left[\frac{1}{2}+\frac{2}{100}\right]+\cdots+\) \(\left[\frac{1}{2}+\frac{99}{100}\right]\) is (A) 49 (B) 50 (C) 51 (D) 98

The domain of definition of $$ f(x)=\sqrt{\frac{\log _{0.3}|x-2|}{|x|}} $$ (A) \([1,2) \cup(2,3]\) (B) \([1,3]\) (C) \(\mathbb{R}-(1,3]\) (D) None of these

Let \(f: R \rightarrow R\) be a function defined by, \(f(x)=\) \(-\frac{|x|^{3}+|x|}{1+x^{2}}\), then the graph of \(f(x)\) lies in which quadrant \((s) ?\) (A) I and II (B) I and III (C) II and III (D) III and IV

If \(f: R \rightarrow R\) and \(g: R \rightarrow R\) are given by \(f(x)=|x|\) and \(g(x)=[x]\) for each \(x \in R\), then \(\\{x \in R: g[f(x)] \leq f\) \([g(x)]\\}=\) (A) \(Z \cup(-\infty, 0)\) (B) \((-\infty, 0)\) (C) \(Z\) (D) \(R\)

The function $$ f(x)=\sin ^{-1}\left(x-x^{2}\right)+\sqrt{1-\frac{1}{|x|}}+\frac{1}{\left[x^{2}-1\right]} $$ is defined in the interval (where [-] is the greatest integer) (A) \(x \in\left(\sqrt{2}, \frac{1+\sqrt{5}}{2}\right)\) (B) \(x \in\left(1, \frac{1+\sqrt{5}}{2}\right)\) (C) \(x \in\left[\frac{1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right]\) (D) \(x \in\left(-\sqrt{2}, \frac{1+\sqrt{5}}{2}\right)\)

Let \(f: \mathbb{R} \rightarrow A=\left\\{y: 0 \leq y<\frac{\pi}{2}\right\\}\) be a function such that \(f(x)=\tan ^{-1}\left(x^{2}+x+k\right)\), where \(k\) is a constant. The minimum value of \(k\) for which \(f\) is an onto function, is (A) 1 (B) 0 (C) \(\frac{1}{4}\) (D) None of these

Suppose \(f:[2,2] \rightarrow R\) is defined by, $$ f(x)= \begin{cases}-1 & \text { for }-2 \leq x \leq 0 \\ x-1 & \text { for } 0 \leq x \leq 2\end{cases} $$ then \(\\{x \in(-2,2): x \leq 0\) and \(f(x \mid)=x\\}=\) (A) \(\\{-1\\}\) (B) \(\\{0\\}\) (C) \(\\{-1 / 2\\}\) (D) \(\phi\)

Let \(f(x)=[x]^{2}+[x+1]-3\), where \([x]\) is greatest integer less than or equal to \(x\), then (A) \(f(x)\) is a many one and into function (B) \(f(x)=0\) for infinite number of values of \(x\) (C) \(f(x)=0\) for only two real values (D) None of these

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 \([x]\) denotes the integral part of \(x\), then the domain of the function \(f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left[\log _{12}\left(x^{2}-5 x+5\right)\right]\) 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

If \(f(n+1)=\frac{2 f(n)+1}{2}, n=1,2, \ldots\) and \(f(1)=2\), then \(f(101)\) equals (A) 52 (B) 49 (C) 48 (D) 51

If \(f(x)=\frac{\cos ^{2} x+\sin ^{4} x}{\sin ^{2} x+\cos ^{4} x}\) for \(x \in R\), then \(f(2002)=\) (A) 1 (B) 2 (C) 3 (D) \(\underline{4}\)

Let \(f(x)=x+1\) and \(\phi(x)=x-2\), then the values of \(x\) satisfying| \(f(x)+\phi(x)|=| f(x)|+| \phi(x) \mid\) are (A) \((-\infty, 1)\) (B) \((2, \infty)\) (C) \((-\infty,-2)\) (D) \((1, \infty)\)

A function \(f\) from the set of natural numbers to integers defined by, $$ f(n)= \begin{cases}\frac{n-1}{2}, & \text { when } n \text { is odd } \\\ -\frac{n}{2}, & \text { when } n \text { is even }\end{cases} $$ (A) neighter one-one nor onto (B) one-one but not onto (C) onto but not one-one (D) one-one and onto both