Chapter 14

The function \(f(x)=|x+2|+|x-1|\) is (A) increasing in \((1, \infty)\) (B) increasing in \([1, \infty)\) (C) decreasing in \((-\infty,-2]\) (D) decreasing in \((-\infty,-2)\)

If \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) for \(0 \leq x \leq 1\), then (A) \(g(x)\) increases in \(\left(-\infty, \frac{1}{2}\right)\) (B) \(g(x)\) increases in \(\left(0, \frac{1}{2}\right)\) (C) \(g(x)\) decreases in \(\left(\frac{1}{2}, 1\right)\) (D) \(g(x)\) decreases in \(\left(\frac{1}{2}, \infty\right)\)

The function \(f(x)=\frac{|x-1|}{x^{2}}\) (A) increases in \((-\infty, 0) \cup(1,2)\) (B) increases in \((0,1) \cup(2, \infty)\) (C) decreases in \((0,1) \cup(2, \infty)\) (D) decreases in \((-\infty, \infty) \cup(1,2)\)

Let \(h(x)=f(x)-[f(x)]^{2}+[f(x)]^{3}\) for every real number \(x\). Then (A) \(h\) is increasing whenever \(f\) is increasing (B) his increasing whenever \(f\) is decreasing (C) his decreasing whenever \(f\) is decreasing (D) nothing can be said in general

Given that \(f^{\prime}(x)>g^{\prime}(x)\) for all real \(x\) and \(f(0)=g(0)\), then (A) \(f(x)>g(x) \forall x \in(0, \infty)\) (B) \(f(x)g(x) \forall x \in(-\infty, 0)\)

For the function \(f(x)=\int_{2}^{x} e^{-t^{4} / 4}\left(4-t^{2}\right) d t\), (A) maximum occurs at \(x=2\) (B) minimum occurs at \(x=-2\) (C) maximum occurs at \(x=-2\) (D) minimum occurs at \(x=2\)

Let \(f(x)=\sin x+\frac{1}{2} \cos 2 x\). Then (A) \(\min\) \(x \in\left[0, \frac{\pi}{2}\right] f(x)<\frac{4}{3}\) (B) \(\min _{x \in\left[0, \frac{\pi}{2}\right] f(x)>\frac{3}{4}}\) (C) \(\min _{x \in\left[0, \frac{\pi}{2}\right] f(x)>\frac{2}{3}}\) (D) \(\min _{x \in\left[0, \frac{\pi}{2}\right]} f(x)<\frac{3}{2}\)

If \((x-a)^{2 n}(x-b)^{2 m+1}\), where \(m\) and \(n\) are positive integers and \(a>b\), is the derivative of a function \(f\), then (A) \(x=a\) gives neither a maximum nor a minimum (B) \(x=a\) gives a maximum (C) \(x=b\) gives a minimum (D) \(x=b\) gives neither a maximum nor a minimum

If \(f(x)=|x|+|x-1|+|x-2|\), then (A) \(f(x)\) has minima at \(x=1\) (B) \(f(x)\) has maxima at \(x=0\) (C) \(f(x)\) has neither maxima nor minima at \(x=0\) (D) \(f(x)\) has neither maxima nor minima at \(x=2\)

\((1+x)^{p} \leq 1+x^{p}\), where (A) \(p>1\) (B) \(0 \leq p \leq 1\) (C) \(x>0\) (D) \(x<0\)

If \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) for \(0 \leq x \leq 1\), then (A) \(g(x)\) increases in \(\left(-\infty, \frac{1}{2}\right)\) (B) \(g(x)\) increases in \(\left(0, \frac{1}{2}\right)\) (C) \(g(x)\) decreases in \(\left(\frac{1}{2}, 1\right)\) (D) \(g(x)\) decreases in \(\left(\frac{1}{2}, \infty\right)\)

The function \(f(x)=\frac{|x-1|}{x^{2}}\) (A) increases in \((-\infty, 0) \cup(1,2)\) (B) increases in \((0,1) \cup(2, \infty)\) (C) decreases in \((0,1) \cup(2, \infty)\) (D) decreases in \((-\infty, \infty) \cup(1,2)\)

Given that \(f^{\prime}(x)>g^{\prime}(x)\) for all real \(x\) and \(f(0)=g(0)\), then (A) \(f(x)>g(x) \forall x \in(0, \infty)\) (B) \(f(x)g(x) \forall x \in(-\infty, 0)\)

The points on the curve \(a y^{2}=x^{3}\) where the normal line makes equal intercepts on the axes are (A) \(\left(\frac{2 a}{9}, \frac{8 a}{27}\right)\) (B) \(\left(\frac{4 a}{9}, \frac{8 a}{27}\right)\) (C) \(\left(\frac{4 a}{9}, \frac{-8 a}{27}\right)\) (D) \(\left(\frac{4 a}{9}, \frac{4 a}{27}\right)\)

The equation of the straight line which is tangent at one point and normal at another point to the curve \(y=8 t^{3}-1, x=4 t^{2}+3\), is (A) \(\sqrt{2} x-y=\frac{89 \sqrt{2}}{27}-1\) (B) \(\sqrt{2} x-y=\frac{89 \sqrt{2}}{27}+1\) (C) \(\sqrt{2} x+y=\frac{89 \sqrt{2}}{27}-1\) (D) \(\sqrt{2} x+y=\frac{89 \sqrt{2}}{27}+1\)

Let \(f(x)=\left\\{\begin{array}{l}x+2,-1 \leq x<0 \\ 1, x=0 \\ \frac{x}{2}, 0

If \(f(x)\) be a function of \(x\), where \(f(x)\) is continuous in the closed interval \([a, b]\) and differentiable in the open interval \((a, b)\). Also, \(f(a)=f(b)\), i.e., the values at the end points \(a\) and \(b\) are equal. Then, there exists at least one point \(c\) between \(a\) and \(\mathrm{b}\) (i.e., \(a

If \(f(x)\) be a function of \(x\), where \(f(x)\) is continuous in the closed interval \([a, b]\) and differentiable in the open interval \((a, b)\). Also, \(f(a)=f(b)\), i.e., the values at the end points \(a\) and \(b\) are equal. Then, there exists at least one point \(c\) between \(a\) and \(\mathrm{b}\) (i.e., \(a

If \(f(x)\) be a function of \(x\), where \(f(x)\) is continuous in the closed interval \([a, b]\) and differentiable in the open interval \((a, b)\). Also, \(f(a)=f(b)\), i.e., the values at the end points \(a\) and \(b\) are equal. Then, there exists at least one point \(c\) between \(a\) and \(\mathrm{b}\) (i.e., \(a

Column-I Column-II I. If \(f(x)=\) (A) \(R-[-\sqrt{3}, \sqrt{3}]\) \(\left\\{\begin{array}{c}4 x-x^{3}+\ln \left(a^{2}-3 a+3\right), 0 \leq x<3 \\\ x-18, & x \geq 3\end{array}\right.\) \(=\) has a local minima at \(x=3\), than \(a\) belongs to II. If the function \(f(x)=\) (B) \((0, \infty)\) \(\left(\frac{\sqrt{a+1}}{a-1}-1\right) x^{3}-x+\ln (a-1)\) is strictly decreasing \(\forall x \in R\) then \(a\) belongs to III. If the function \(f(x)=x^{3}+\) (C) \([1,2]\) \(a x^{2}+a^{2} x+2 \sin ^{2} x\) is strictly increasing \(\forall x \in R\), then \(a\) belongs to IV. The function \(f(x)=\left|e^{a x}-e^{-u x}\right|\), (D) \((3, \infty)\) \(a>0\) is strictly increasing in the interval

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If a quadratic curve touches the line \(y=x\) at the point \((1,1)\), then the values of \(x\) for which the curve has a negative gradient are \(x<\frac{1}{2}\) Reason: The equation of the curve is \(y=x^{2}-x+1\)

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The function \(f(x)=\frac{\sin x}{x}\) is decreasing in the interval \(\left(0, \frac{\pi}{2}\right)\) Reason: \(\tan x>x\) for \(0

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(0

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(0\cos x>\sin\) \((\cos x)\) Reason: \(\sin x

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(0<\alpha<\beta<, \frac{\pi}{2}\) then \(\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}\) Reason: \(x \tan x\) is increasing for \(0

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: Let \(f\) and \(g\) be increasing and decreasing functions respectively from \([0, \infty]\) to \([0, \infty] .\) Let \(h(x)=f(g(x))\). If \(h(0)=0\), then \(h(x)\) is always zero Reason: \(h(x)\) is an increasing function of \(x\)

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True \mathrm{\\{} A s s e r t i o n : ~ I f ~ \(f^{\prime}(x)=\frac{1}{1+x^{2}}\) for all \(x\) and \(f(0)=0\), then \(0.4

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: For \(b>a>1, \frac{1}{b \ln b}<\frac{f(b)-f(a)}{b-a}<\) \(\frac{1}{a \ln a}\), where \(f(x)=\ln (\ln x), x>1\) Reason: \(\frac{1}{x \ln x}\) is strictly decreasing in \((a, b)\)

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\sin (\tan x) \geq x, \forall x \in\left[0, \frac{\pi}{4}\right]\) Reason: \(1-\cos x \leq \frac{x^{2}}{2}\)

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \((a+b)^{1 / n} \leq a^{1 / n}+b^{1 / n}\), where \(a, b \geq 0\) and \(n \geq 1\) Reason: The function \(f(x)=(1+x)^{p}-x^{p}-1, x \geq 0\) and \(0

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(303^{202}<202^{303}\) Reason: The function \(f(x)=\frac{\ln x}{x}\) strictly increases in \((e, \infty)\).

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\ln (\cos \theta)<\cos (\ln \theta)\) where \(e^{-\theta 2}<\theta<\frac{\pi}{2}\) Reason: \(\ln x0\)

The two curves \(x^{3}-3 x y^{2}+2=0\) and \(3 x^{2} y-y^{3}-2=0\) : (A) cut at right angle (B) touch each other (C) cut at an angle \(\frac{\pi}{3}\) (D) cut at an angle \(\frac{\pi}{4}\)

The function \(f(x)=\cot ^{-1} x+x\) increases in the interval: (A) \((1, \infty)\) (B) \((-1, \infty)\) (C) \((-\infty, \infty)\) (D) \((0, \infty)\)

The greatest value of \(f(x)=(x+1)^{1 / 3}-(x-1)^{1 / 3}\) on \([0,1]\) is: (A) 1 (B) 2 (C) 3 (D) \(1 / 3\)

If the function \(f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1\), where \(a>0\), attains its maximum and minimum at \(p\) and \(q\) respectively such that \(p^{2}=q\), then \(a\) equals (A) 3 (B) 1 (C) 2 (D) \(\frac{1}{2}\)

A function \(y=f(x)\) has a second order derivative \(f^{\prime \prime}(x)=6(x-1)\). If its graph passes through the point \((2,1)\) and at that point the tangent to the graph is \(y=\) \(3 x-5\), then the function is (A) \((x-1)^{2}\) (B) \((x-1)^{3}\) (C) \((x+1)^{3}\) (D) \((x+1)^{2}\)

The normal to the curve \(x=a(1+\cos \theta), y=a \sin \theta\) at \(\theta\) always passes through the fixed point (A) \((a, 0)\) (B) \((0, a)\) (C) \((0,0)\) (D) \((a, a)\)

The normal to the curve \(x=a(\cos \theta+\theta \sin \theta), y=\) \(a(\sin \theta-\theta \cos \theta)\) at any point \(\theta\) is such that (A) It passes through the origin (B) It makes angle \(\frac{\pi}{2}+\theta\) with the \(x\)-axis (C) It passes through \(\left(a \frac{\pi}{2},-a\right)\) (D) It is at a constant distance from the origin

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval Function (A) \((-\infty, \infty)\) \(x^{3}-3 x^{2}+3 x+3\) (B) \([2, \infty)\) \(2 x^{3} 3 x^{2}-12 x+6\) (C) \(\left(-\infty, \frac{1}{3}\right]\) \(3 x^{2}-2 x+1\) (D) \((-\infty,-4]\) \(x^{3}+6 x^{2}+6\)

Let \(f\) be differentiable for all \(x\). If \(f(1)=-2\) and \(f^{\prime}(x) \geq 2\) for \(x \in[1,6]\), then (A) \(f(6) \geq 8\) (B) \(f(6)<8\) (C) \(f(6)<5\) (D) \(f(6)=5\)

A spherical iron ball \(10 \mathrm{~cm}\) in radius is coated with a layer of ice of uniform thickness than melts at a rate of \(50 \mathrm{~cm}^{3} / \mathrm{min}\). When the thickness of ice is \(5 \mathrm{~cm}\), then the rate at which the thickness of ice decreases, is (A) \(\frac{1}{36 \pi} \mathrm{cm} / \mathrm{min}\) (B) \(\frac{1}{18 \pi} \mathrm{cm} / \mathrm{min}\) (C) \(\frac{1}{54 \pi} \mathrm{cm} / \mathrm{min}\) (D) \(\frac{5}{6 \pi} \mathrm{cm} / \mathrm{min}\)

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+-+a_{1} x=0, a_{1} \neq 0\), \(n \geq 2\), has a positive root \(x=\alpha\), then the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots+a_{1}=0\) has a positive root, which is (A) greater than a (B) smaller than a (C) greater than or equal to a (D) equal to a

Angle between the tangents to the curve \(y=x^{2}-\) \(5 x+6\) at the points \((2,0)\) and \((3,0)\) is (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{6}\) (D) \(\frac{\pi}{4}\)

A value of \(C\) for which the conclusion of Mean Value Theorem holds for the function \(f(x)=\log _{e} x\) on the interval \([1,3]\) is (A) \(2 \log _{3} e\) (B) \(\frac{1}{2} \log _{e} 3\) (C) \(\log _{3} e\) (D) \(\log _{e} 3\)

The equation of a tangent to the parabola \(y^{2}=8 x\) is \(y=x+2 .\) The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is (A) \((-1,1)\) (B) \((0,2)\) (C) \((2,4)\) (D) \((-2,0)\)

Suppose the cube \(x^{3}-p x+q\) has three distinct real roots where \(p>0\) and \(q>0 .\) Then which one of the following holds? (A) The cubic has minima at \(\sqrt{\frac{p}{3}}\) and maxima at \(-\sqrt{\frac{p}{3}}\) (B) The cubic has minima at \(-\sqrt{\frac{p}{3}}\) and maxima at \(\sqrt{\frac{p}{3}}\) (C) The cubic has minima at both \(\sqrt{\frac{p}{3}}\) and \(-\sqrt{\frac{p}{3}}\) (D) The cubic has maxima at both \(\sqrt{\frac{p}{3}}\) and \(-\sqrt{\frac{p}{3}}\)

How many real solutions does the equation \(x^{7}+14 x^{5}\) \(+16 x^{3}+30 x-560=0\) have? (A) 7 (B) 1 (C) 3 (D) 5

Given \(P(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) such that \(x=0\) is the only real root of \(P^{\prime}(x)=0\). If \(P(-1)

The shortest distance between the line \(y-x=1\) and the curve \(x=y^{2}\) is (A) \(\frac{3 \sqrt{2}}{8}\) (B) \(\frac{2 \sqrt{3}}{8}\) (C) \(\frac{3 \sqrt{2}}{5}\) (D) \(\frac{\sqrt{3}}{4}\)