Chapter 3

The sum of the maximum and minimum values of the function \(f(x)=\frac{1}{1+(2 \cos x-4 \sin x)^{2}}\) is (a) \(\frac{22}{21}\) (b) \(\frac{21}{20}\) (c) \(\frac{22}{20}\) (d) \(\frac{21}{11}\)

Let \(f: X \rightarrow Y, f(x)=\sin x+\cos x+2 \sqrt{2}\) is invertible, then \(X \rightarrow Y\) is/are (a) \(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right] \rightarrow[\sqrt{2}, 3 \sqrt{2}]\) (b) \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right] \rightarrow[\sqrt{2}, 3 \sqrt{2}]\) (c) \(\left[-\frac{3 \pi}{4}, \frac{3 \pi}{4}\right] \rightarrow[\sqrt{2},-3 \sqrt{2}]\) (d) \(\left[-\frac{3 \pi}{4},-\frac{\pi}{4}\right] \rightarrow[\sqrt{2}, 3 \sqrt{2}]\)

The range of values of \(a\) so that all the roots of the equation \(2 x^{3}-3 x^{2}-12 x+a=0\) are real and distinct belongs to (a) \((7,20)\) (b) \((-7,20)\) (c) \((-20,7)\) (d) \((-7,7)\)

If \(f(x)\) is continuous such that \(|f(x)| \leq 1, \forall x \in R\) and \(g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}}\), then range of \(g(x)\) is (a) \([0,1]\) (b) \(\left[0, \frac{e^{2}+1}{e^{2}-1}\right]\) (c) \(\left[0, \frac{e^{2}-1}{e^{2}+1}\right]\) (d) \(\left[\frac{1-e^{2}}{1+e^{2}}, 0\right]\)

Let \(f(x)=\sqrt{|x|-\\{x\\}}\) (where \\{\\} denotes the fractional part of \(x\) and \(X, Y\) and its domain and range respectively, then (a) \(f: X \rightarrow Y: y=f(x)\) is one-one function (b) \(X \in\left(-\infty,-\frac{1}{2}\right] \cup[0, \infty)\) and \(Y \in\left[\frac{1}{2}, \infty\right)\) (c) \(X \in\left(-\infty,-\frac{1}{2}\right] \cup[0, \infty)\) and \(Y \in[0, \infty)\) (d) None of the above

If the graphs of the functions \(y=\ln x\) and \(y=a x\) intersect at exactly two points, then \(a\) must be (a) \((0, e)\) (b) \(\left(\frac{1}{e}, 0\right)\) (c) \(\left(0, \frac{1}{e}\right)\) (d) None of these

A quadratic polynomial maps from \([-2,3]\) onto \([0,3]\) and touches \(x\)-axis at \(x=3\), then the polynomial is (a) \(\frac{3}{16}\left(x^{2}-6 x+16\right)\) (b) \(\frac{3}{25}\left(x^{2}-6 x+9\right)\) (c) \(\frac{3}{25}\left(x^{2}-6 x+16\right)\) (d) \(\frac{3}{16}\left(x^{2}-6 x+9\right)\)

The range of the function \(y=\sqrt{2\\{x\\}-\\{x\\}^{2}-\frac{3}{4}}\) is (where \\{\\} denotes the fractional part) (a) \(\left[-\frac{1}{4}, \frac{1}{4}\right]\) (b) \(\left[0, \frac{1}{2}\right]\) (c) \(\left[0, \frac{1}{4}\right]\) (d) \(\left[\frac{1}{4}, \frac{1}{2}\right]\)

Let \(f(x)\) be a fourth differentiable function such that \(f\left(2 x^{2}-1\right)=2 x f(x), \forall x \in R\), then \(f^{i v}(0)\) is equal to (where \(f^{i v}(0)\) represents fourth derivative of \(f(x)\) at \(x=0\) ) (a) 0 (b) 1 (c) \(-1\) (d) Data insufficient

Number of solutions of the equation \([y+[y]]=2 \cos x\) is (where \(y=\frac{1}{3}[\sin x+[\sin x+[\sin x]]]\) and [] denotes the greatest integer function) (a) 1 (b) 2 (c) 3 (d) None of these

If a function satisfies \(f(x+1)+f(x-1)=\sqrt{2} f(x)\), then period of \(f(x)\) can be (a) 2 (b) 4 (c) 6 (d) 8

If \(x\) and \(\alpha\) are real, then the inequation \(\log _{2} x+\log _{x} 2+2 \cos \alpha \leq 0\) (a) has no solution (b) has exactly two solutions (c) is satisfied for any real \(\alpha\) and any real \(x\) in \((0,1)\) (d) is satisfied for any real \(\alpha\) and any real \(x\) in \((1, \infty)\)

The range of values of ' \(a\) ' such that \(\left(\frac{1}{2}\right)^{|x|}=x^{2}-a\) is satisfied for maximum number of values of ' \(x\) ' (a) \((-\infty,-1)\) (b) \((-\infty, \infty)\) (c) \((-1,1)\) (d) \((-1, \infty)\)

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\\{|\cos x|\\}\), where \(\\{x\\}\) represents fractional part of \(x\). Let \(S\) be the set containing all real values \(x\) lying in the interval \([0,2 \pi]\) for which \(f(x) \neq|\cos x|\). Then, number of elements in the set \(S\) is (a) 0 (b) 1 (c) 3 (d) infinite

The domain of the function \(f(x)=\sqrt{\log _{\sin x+\cos x}(|\cos x|+\cos x)}, 0 \leq x \leq \pi\) is (a) \((0, \pi)\) (b) \(\left(0, \frac{\pi}{2}\right)\) (c) \(\left(0, \frac{\pi}{3}\right)\) (d) None of these

If \(f(x)=\left(x^{2}+2 \alpha x+\alpha^{2}-1\right)^{1 / 4}\) has its domain and range such that their union is set of real numbers, then \(\alpha\) satisfies (a) \(-1<\alpha<1\) (b) \(\alpha \leq-1\) (c) \(\alpha \geq 1\) (d) \(\alpha \leq 1\)

Let \(f:(e, \infty) \rightarrow R\) be a function defined by \(f(x)=\log (\log (\log x))\), the base of the logarithm being \(e\). Then, (a) \(f\) is one-one and onto (b) \(f\) is one-one but not onto (c) \(f\) is onto but not one-one (d) the range of \(f\) is equal to its domain

The period of \(\sin \frac{\pi[x]}{12}+\cos \frac{\pi[x]}{4}+\tan \frac{\pi[x]}{3}\) where \([x]\) represents the greatest integer less than or equal to \(x\) is (a) 12 (b) 4 (c) 3 (d) 24

If \(f(2 x+3 y, 2 x-7 y)=20 x\), then \(f(x, y)\) equals to (a) \(7 x-3 y\) (b) \(7 x+3 y\) (c) \(3 x-7 y\) (d) \(x-y\)

The range of the function \(f(x)=\sqrt{x-1}+2 \sqrt{3-x}\) is (a) \([\sqrt{2}, 2 \sqrt{2}\) ] (b) \([\sqrt{2}, \sqrt{10}]\) (c) \([2 \sqrt{2}, \sqrt{10}]\) (d) \([1,3]\)

The domain of the function \(f(x)=\cos ^{-1}\left(\sec \left(\cos ^{-1} x\right)\right)+\sin ^{-1}\left(\operatorname{cosec}\left(\sin ^{-1} x\right)\right)\) is (a) \(x \in R\) (b) \(x=1,-1\) (c) \(-1 \leq x \leq 1\) (d) \(x \in \phi\)

Let \(f(x)\) be a polynomial with real coefficients such that \(f(x)=f^{\prime}(x) \times f^{\prime \prime}(x)\). If \(f(x)=0\) is satisfied \(x=1,2,3\) only, then the value of \(f^{\prime}(1) f^{\prime}(2) f^{\prime}(3)\) is (a) positive (b) negative (c) 0 (d) Inadequate data

Let \(A=\\{1,2,3,4,5\\}\) and \(f: A \rightarrow A\) be an into function such that \(f(i) \neq i, \forall i \in A\), then number of such functions \(f\) are (a) 1024 (b) 904 (c) 980 (d) None of these

The range of \(y=\sin ^{3} x-6 \sin ^{2} x+11 \sin x-6\) is (a) \([-24,2]\) (b) \([-24,0]\) (c) \([0,24]\) (d) None of these

Let \(f(x)=x^{2}-2 x\) and \(g(x)=f(f(x)-1)+f(5-f(x))\), then (a) \(g(x)<0, \forall x \in R\) (b) \(g(x)<0\) for some \(x \in R\) (c) \(g(x) \geq 0\) for some \(x \in R\) (d) \(g(x) \geq 0, \forall x \in R\)

If \(f(x)\) and \(g(x)\) are non-periodic functions, then \(h(x)=f(g(x))\) is (a) non-periodic (b) periodic (c) may be periodic (d) always periodic, if domain of \(h(x)\) is a proper subset of real numbers

If \(f(x)\) is a real-valued function discontinuous at all integral points lying in \([0, n]\) and if \((f(x))^{2}=1 \forall x \in[0, n]\), then number of functions \(f(x)\) are (a) \(2^{n+1}\) (b) \(6 \times 3^{n}\) (c) \(2 \times 3^{n-1}\) (d) \(3^{n+1}\)

A function \(f\) from integers to integers is defined as \(f(x)=\left\\{\begin{array}{cc}n+3, & n \in \text { odd } \\ n / 2, & n \in \text { even }\end{array}\right.\) suppose \(k \in\) odd and \(f(f(f(k)))=27\), then the sum of digits of \(k\) is (a) 3 (b) 6 (c) 9 (d) 12

\(f: R \rightarrow R, f(x)=\frac{\sin (\pi\\{x\\})}{x^{4}+3 x^{2}+7}\) where \\{\\} is a fractional function, then (a) \(f\) is injective (b) \(f\) is not one-one and non-constant (c) \(f\) is a surjective (d) \(f\) is a zero function

Let \(f: R \rightarrow R\) and \(g: R \rightarrow R\) be two one-one and onto functions, such that they are the mirror images of each other about the line \(y=a\). If \(h(x)=f(x)+g(x)\), then \(h(x)\) is (a) one-one and onto (b) only one-one and not onto (c) only onto but not one-one (d) None of these

Domain of the function \(f(x)\), if \(3^{x}+3^{f(x)}=\) minimum of \(\phi(t)\) where \(\phi(t)=\) minimum of \(\left\\{2 t^{3}-15 t^{2}+36 t-25,2+|\sin t| ; 2 \leq t \leq 4\right\\}\) is (a) \((-\infty, 1)\) (b) \(\left(-\infty, \log _{3} e\right)\) (c) \(\left(0, \log _{3} 2\right)\) (d) \(\left(-\infty, \log _{3} 2\right)\)

Let \(f(x)=1-x-x^{3} .\) Then, the real values of \(x\) satisfying the inequality, \(1-f(x)-f^{3}(x)>f(1-5 x)\), are (a) \((-2,0)\) (b) \((0,2)\) (c) \((2, \infty)\) (d) \((-\infty,-2)\)

If a function satisfies \((x-y) f(x+y)-(x+y) f(x-y)=2\left(x^{2} y-y^{3}\right), \forall x, y \in R\) and \(f(1)=2\), then (a) \(f(x)\) must be polynomial function (b) \(f(3)=12\) (c) \(f(0)=0\) (d) \(f(x)\) may not be differentiable

If the fundamental period of function \(f(x)=\sin x+\cos \left(\sqrt{4-a^{2}}\right) x\) is \(4 \pi\), then the value of \(a\) is/are \(\begin{array}{ll}\text { (a) } \frac{\sqrt{15}}{2} & \text { (b) }-\frac{\sqrt{15}}{2}\end{array}\) (c) \(\frac{\sqrt{7}}{2}\) (d) \(-\frac{\sqrt{7}}{2}\)

Let \(f(x)\) be a real-valued function such that \(f(0)=\frac{1}{2}\) and \(f(x+y)=f(x) f(a-y)+f(y) f(a-x) \forall x, y \in R\), then for some real \(a\) (a) \(f(x)\) is a periodic function (b) \(f(x)\) is a constant function (c) \(f(x)=\frac{1}{2}\) (d) \(f(x)=\frac{\cos x}{2}\)

If \(f(g(x))\) is one-one function, then (a) \(g(x)\) must be one-one (b) \(f(x)\) must be one-one (c) \(f(x)\) may not be one-one (d) \(g(x)\) may not be one-one

Which of the following functions have their range equal to \(R\) (the set of real numbers)? (a) \(x \sin x\) (b) \(\frac{[x]}{\tan 2 x} \cdot x \in\left(-\frac{\pi}{4} \cdot \frac{\pi}{4}\right)-\\{0\\},[.]\) denotes the greatest integer function (c) \(\frac{x}{\sin x}\) (d) \([x]+\sqrt{\\{x\\}},[\cdot]\) and \(\\{\cdot\\}\) respectively denote the greatest integer and fractional part functions.

Which of the following pairs of function are identical? (a) \(f(x)=e^{\ln \sec ^{-1} x}\) and \(g(x)=\sec ^{-1} x\) (b) \(f(x)=\tan \left(\tan ^{-1} x\right)\) and \(g(x)=\cot \left(\cot ^{-1} x\right)\) (c) \(f(x)=\operatorname{sgn}(x)\) and \(g(x)=\operatorname{sgn}(\operatorname{sgn}(x))\) (d) \(f(x)=\cot ^{2} x \cdot \cos ^{2} x\) and \(g(x)=\cot ^{2} x-\cos ^{2} x\)

Statement I The function \(f(x)=x \sin x\) and \(f^{\prime}(x)=x \cos x+\sin x\) are both non-periodic. Statement II The derivative of differentiable function (non-periodic) is non-periodic function.

Statement I The maximum value of \(\sin \sqrt{2} x+\sin a x\) cannot be 2 . (where \(a\) is positive rational number) Statement II \(\frac{\sqrt{2}}{\alpha}\) is irrational.

Let \(f: R \rightarrow R\) be a function such that \(f(x)=\frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}\). Statement I \(f(x)\) is into function. Statement II \(f(x)\) is many-one function, and the many-one function is not onto.

Statement I The range of \(f(x)=\sin \left(\frac{\pi}{5}+x\right)-\sin \left(\frac{\pi}{5}-x\right)-\sin \left(\frac{2 \pi}{5}+x\right)\) \(+\sin \left(\frac{2 \pi}{5}-x\right)\) is \([-1,1]\) Statement II \(\cos \frac{\pi}{5}-\cos \frac{2 \pi}{5}=\frac{1}{2}\).

Statement I The period of \(f(x)=2 \cos \frac{1}{3}(x-\pi)+4 \sin \frac{1}{3}(x-\pi)\) is \(3 \pi\). Statement II If \(T\) is the period of \(f(x)\), then the period of \(f(a x+b)\) is \(\frac{T}{|a|}\).

Statement I The equation \(f(x)=4 x^{5}+20 x-9=0\) has only one real root. Statement II \(f^{\prime}(x)=20 x^{4}+20=0\) has no real root.

Statement I The range of \(\log \left(\frac{1}{1+x^{2}}\right)\) is \((-\infty, \infty)\). Statement II When \(0

\(\operatorname{Let} f(x)=\sin x\) Statement I \(f\) is not a polynomial function. Statement II \(n\)th derivative of \(f(x)\), w.r.t. \(x\), is not a zero function for any positive integer \(n\).

Statement I The function \(f: R \rightarrow R\), given \(f(x)=\log _{a}\left(x+\sqrt{x^{2}+1}\right), a>0\), \(a \neq 1\) is invertible. Statement II \(f\) is many to one and into.

\(f: R \rightarrow R \quad\) and \(\quad f(x)=a_{1} x+a_{3} x^{3}+a_{5} x^{5}+\ldots+a_{2 n+1} x^{2 n+1}-\cot ^{-1} x\) where \(0

\(f: R \rightarrow R\) and \(f(x)=\frac{x\left(x^{4}+1\right)(x+1)+x^{4}+2}{x^{2}+x+1}\), then \(f(x)\) is (a) one-one into (b) many-one onto (c) one-one onto (d) many-one into

\(f: R \rightarrow R\) and \(f(x)=2 a x+\sin 2 x\), then the set of values of \(a\) for which \(f(x)\) is one-one and onto is (a) \(a \in\left(-\frac{1}{2}, \frac{1}{2}\right)\) (b) \(a \in(-1,1)\) (c) \(a \in R-\left(-\frac{1}{2}, \frac{1}{2}\right)\) (d) \(a \in R-(-1,1)\)