Chapter 14

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) \(f(x)g(x) \forall x \in(-\infty, 0)\)

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

\((1+x)^{p} \leq 1+x^{p}\), 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 \(\left(0, \frac{1}{2}\right)\)

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) \(h\) is increasing whenever \(f\) is decreasing (C) \(h\) is 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) \(f(x)g(x) \forall x \in(-\infty, 0)\)

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 \(a+b+c=0\), then the equation \(3 a x^{2}+2 b x+c=0\) has, in the interval \((0,1)\) (A) at least one root (B) at most one root (C) no root (D) None of these

If \(a+b+c=0\), then the equation \(3 a x^{2}+2 b x+c=0\) has, in the interval \((0,1)\) (A) at least one root (B) at most one root (C) no root (D) None of these

The equation \(x \log x=3-x\) has, in the interval \((1,3)\) (A) exactly one root (B) at least one root (C) at most one root (D) no root

\(\frac{\sin \alpha-\sin \beta}{\cos \beta-\cos \alpha}=F(\alpha)\), where \(0<\alpha<\theta

$$ \begin{array}{ll} \text { Column-I } & \text { Column-II } \\ \hline \text { I. Let } f(x)=\left(1+b^{2}\right) x^{2}+2 b x+1 & \text { (A) }(0,1] \\ \text { and } m(b) \text { the minimum value of } \\ f(x) \text { for a given } b . \text { As } b \text { varies, the } \\ \text { range of } m(b) \text { is } \\ \text { II. The set of values of } x \text { for which } & \text { (B) }(0,1) \\\ \log (1+x)

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 \(y=x^{2}-x+1\)

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

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

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 \(x\)

Assertion: If \(f^{\prime}(x)=\frac{1}{1+x^{2}}\) for all \(x\) and \(f(0)=0\), then \(0.4

Assertion: If \(f(x)=\tan x, x \in\left[0, \frac{\pi}{7}\right]\), then \(\frac{\pi}{7}\) \(

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 \((a, b)\)

Assertion: \(\sin (\tan x) \geq x, \forall x \in\left[0, \frac{\pi}{4}\right]\) Reason: \(1-\cos x \leq \frac{x^{2}}{2}\)

Assertion: \(303^{202}<202^{303}\) Reason: The function \(f(x)=\frac{\ln x}{x}\) strictly increases in \((e, \infty)\)

The two curves \(x^{3}-3 x y^{2}+2=0\) and \(3 x^{2} y-y^{3}-2=0\) : \([2002]\) (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: [2002] (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 \(\quad\) [2003] (A) 3 (B) I (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 [2004] (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 \(\quad\) [2005] (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 \(\quad\) [2005] (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) \(\frac{5}{6 \pi} \mathrm{cm} / \mathrm{min}\)

The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minimum at (A) \(x=2\) (B) \(x=-2\) (C) \(x=0\) (D) \(x=1\)

Angle between the tangents to the curve \(y=x^{2}-\) \(5 x+6\) at the points \((2,0)\) and \((3,0)\) is [2006] (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 \(\quad[2007]\) (A) \(2 \log _{3} e\) (B) \(\frac{1}{2} \log _{e} 3\) (C) \(\log _{3} e\) (D) \(\log _{e^{e}}\)

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 \(|2007|\) (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? \(\quad\) [2008] (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) \(\frac{\sqrt{3}}{4}\)

The equation of the tangent to the curve \(y=x+\frac{4}{x^{2}}\), which is parallel to the \(x\)-axis, is (A) \(y=1\) (B) \(y=2\) (C) \(y=3\) (D) \(y=0\)

Let \(f: R \rightarrow R\) be defined by \(f(x)= \begin{cases}k-2 x, & \text { If } x \leq-1 \\ 2 x+3 & \text { if } x>-1\end{cases}\) If \(f\) has a local minimum at \(x=-1\), then apossible value of \(k\) is (A) 0 (B) \(-\frac{1}{2}\) (C) \(-1\) (D) 1

A spherical balloon is filled with \(4500 \pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72 \pi\) cubic meters per minute, then the rate (in meters per minute) at which theradius of the balloon decreases 49 minutes after the leakage began is (A) \(\frac{9}{7}\) (B) \(\frac{7}{9}\) (C) \(\frac{2}{9}\) (D) \(\frac{9}{2}\)

Let the real values \(a, b\) be such that the function \(\mathrm{f}\) given by \(f(x)=\ln |x|+b x^{2}+a x, x \neq 0\) has extreme values at \(x=-1\) and \(x=2 .\) Statement \(1: f\) has local maximum at \(x=-1\) and at \(x=2\) Statement 2: \(a=\frac{1}{2}\) and \(b=\frac{-1}{4}\) (A) Statement 1 is false, statement 2 is true (B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1 (C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1 (D) Statement 1 is true, statement 2 is false

If \(x=-1\) and \(x=2\) are extreme points of \(f(x)=\alpha \log |x|+\beta x^{2}+x\), then(A) \(\alpha=-6, \beta=\frac{1}{2}\) (B) \(\alpha=-6, \beta=-\frac{1}{2}\) (C) \(\alpha=2, \beta=-\frac{1}{2}\) (D) \(\alpha=2, \beta=\frac{1}{2}\)

The normal to the curve, \(x^{2}+2 x y-3 y^{2}=0\), at \((1,1):\) \([2015 \mid\) (A) meets the curve again in the second quadrant. (B) meets the curve again in the third quadrant. (C) meets the curve again in the fourth quadrant. (D) does not meet the curve again.

A wire of length 2 units is cur into two parts which are bent respectively to form a square of side \(=x\) units and a circle of radius \(=\mathrm{r}\) units. If the sum of the areas of the square and the circle so formed is minimum, then:(A) \(2 x=r\) (B) \(2 x=(\pi+4) r\) (C) \((4-\pi) x=\pi r\) (D) \(x=2 r\)

Consider \(f(x)=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), \quad x \in\left(0, \frac{\pi}{2}\right)\). A normal to \(y=f(x)\) at \(x=\frac{\pi}{6}\) also passes through the point: \([2016]\) (A) \(\left(\frac{\pi}{4}, 0\right)\) (B) \((0,0)\) (C) \(\left(0, \frac{2 \pi}{3}\right)\) (D) \(\left(\frac{\pi}{6}, 0\right)\)