Differential Calculus for JEE Main and Advanced

Vinay Kumar

Chapter 7

MAXIMA AND MINIMA - all with Video Answers

Educators

Chapter Questions

Problem 1

$f(x)=\left\{\begin{array}{cc}2-\left|x^{2}+5 x+6\right|, & x \neq-2 \\ a^{2}+1 & , x=-2\end{array}\right.$, then the
range of a so that $f(x)$ has maxima at $x=-2$ is $($ A) $|a| \geq 1$
(B) $\mid \mathrm{a}<1$
(C) $\mathrm{a}>1$
(D) $\mathrm{a}<1$

Hast Aggarwal

Hast Aggarwal

Numerade Educator

Problem 2

A triangle has one vertex at $(0,0)$ and the other two on the graph of $y=-2 x^{2}+54$ at $(x, y)$ and $(-\mathrm{x}, \mathrm{y})$ where $0<\mathrm{x}<\sqrt{27}$. The value of $\mathrm{x}$ so that the corresponding triangle has maximum area is
(A) $\frac{\sqrt{27}}{2}$
(B) 3
(C) $2 \sqrt{3}$
(D) None

Hast Aggarwal

Numerade Educator

Problem 3

The set of all values of the parameters a for which the points of minimum of the function $y=1+a^{2} x-x^{3}$ satisfy the inequality $\frac{x^{2}+x+2}{x^{2}+5 x+6} \leq 0$ is
(A) an empty set
(B) $(-3 \sqrt{3},-2 \sqrt{3})$
(C) $(2 \sqrt{3}, 3 \sqrt{3})$
(D) $(-3 \sqrt{3},-2 \sqrt{3}) \cup(2 \sqrt{3}, 3 \sqrt{3})$

Hast Aggarwal

Numerade Educator

Problem 4

If $f(x)=\frac{1+x}{[x]} x \in[1,3]$ then which of following statements about $\mathrm{f}(\mathrm{x})$ is correct
(A) $f(x)$ has local maxima at $x=2$
(B) $f(x)$ has no point of local maxima
(C) $\mathrm{f}(\mathrm{x})$ hasneither maxima nor minima at $\mathrm{x}=2$
(D) $f(x)$ has 1 point of local minimum and 2 points of local maxima

Hast Aggarwal

Numerade Educator

Problem 5

Which of the following statement is true for the function function $f(x)=\left[\begin{array}{ll}\sqrt{x} & x \geq 1 \\ x^{3} & 0 \leq x \leq 1 \\ \frac{x^{3}}{3}-4 x & x<0\end{array}\right.$
(A) It is monotonic increasing $\forall \mathrm{x} \in \mathrm{R}$
(B) $\mathrm{f}(\mathrm{x})$ fails to exist for 3 distinct real values of $\mathrm{x}$
(C) $\mathrm{f}^{\prime}(\mathrm{x})$ changes its sign twice as $\mathrm{x}$ varies from $(-\infty, \infty)$
(D) function attains its extreme values at $x_{1} \& x_{2}$, such that $\mathrm{x}_{1} \mathrm{x}_{2}>0$

Hast Aggarwal

Numerade Educator

Problem 6

Consider the function for $\mathrm{x} \in[-2,3]$, $f(x)=\left[\begin{array}{ll}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1 \\ \lfloor-6 & \text { if } x=1\end{array}\right.$ then
(A) $\mathrm{f}$ is discontinuous at $\mathrm{x}=1 \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
(B) $f(-2) \neq f(3) \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
(C) $\mathrm{f}$ is not derivable in $(-2,3) \Rightarrow$ Rolle's theorem is not applicable
(D) Rolle's theorem is applicable as f satisfies all the conditions and c of Rolle's theorem is $1 / 2$

Hast Aggarwal

Numerade Educator

Problem 7

The total number of values of $x$, where $f(x)=2^{-x}$ (cos $x$ $+\cos \sqrt{3} \mathrm{x}$ ) attains its maximum value is
(A) $\underline{1}$
(B) $\underline{2}$
(C) 4
(D) None

Hast Aggarwal

Numerade Educator

Problem 8

Let $f(x)=\left\{\begin{array}{r}x^{3}+x^{2}+3 x+\sin x \mid(3+\sin 1 / x), x \neq 0 \\ 0 \quad, x=0\end{array}\right.$
then number of points (where $f(x)$ attains its minimum value) is
(A) 1
(B) 2
(C) 3
(D) infinite many

Hast Aggarwal

Numerade Educator

Problem 9

In which of the following functions Rolle's theorem is applicable -
(A) $f(x)= \begin{cases}x & , 0 \leq x<1 \\ 0 & , x=1\end{cases}$
(B) $f(x)= \begin{cases}\frac{\sin x}{x},-\pi \leq x<0 \\ 0, & x=0\end{cases}$
(C) $f(x)=\frac{x^{2}-x-6}{x-1}$ on $[-2,3]$
(D) $f(x)= \begin{cases}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1, x \in[-2,3] \\ -6 & \text { if } x=1\end{cases}$

Hast Aggarwal

Numerade Educator

Problem 10

The function in which the conditions of Rolle's theorem are satisfied, is
(A) $f(x)=2 x^{3}+x^{2}-4 x-2,[-\operatorname{root} 2$, root 2$]$
(B) tanx in $[0, \pi]$
(C) $f(x)=\left\{\begin{array}{l}x^{2}+1 ; 0 \leq x \leq 1 \\ 3-x ; 1 \leq x \leq 2\end{array}\right.$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 11

If $a>b>0$ and $f(\theta)=\frac{\left(a^{2}-b^{2}\right) \cos \theta}{a-b \sin \theta}$, then the maximum
value of $f(\theta)$ is
(A) $\sqrt{a^{2}+b^{2}}$
(B) $\sqrt{a^{2}-b^{2}}$
(C) $a^{2}-b^{2}$
(D) $\frac{a-b}{a+b}$

Hast Aggarwal

Numerade Educator

Problem 12

Let $f(x)=\sin \left(x^{2}-3 x\right)$, if $x \leq 0 ;$ and $6 x+5 x^{2}$, if $x>0$, then at $x=0, f(x)$
(A) has a local maximum
(B) has a local minimum
(C) is discontinuous
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 13

The greatest value of $f(x)=\cos \left(x e^{|x|}+7 x^{2}-3 x\right)$, $x \in[-1, \infty)$ is
(A) $-1$
(B) 1
(C) 0
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 14

The function $f(x)=\left(4 \sin ^{2} x-1\right)^{n}\left(e^{x}-x+1\right)$
$\mathrm{n} \in \mathrm{N}$, has a local minimum at $\mathrm{x}=\frac{\pi}{6}$, then
(A) n can be any even natural number
(B) $\mathrm{n}$ can be an odd natural number
(C) $\mathrm{n}$ can be odd prime number
(D) n can be any natural number

Hast Aggarwal

Numerade Educator

Problem 15

Let $f(x)=\cos \pi x+10 x+3 x^{2}+x^{3},-2 \leq x \leq 3$. The absolute minimum value of $f(x)$ is
(A) 0
(B) $-15$
(C) $3-2 \pi$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 16

The global minimum value of $\mathrm{e}^{\left(2 \mathrm{x}^{2}-2 x+1\right) \sin ^{\prime} \mathrm{x}}$ is
(A) $\mathrm{c}$
(B) $1 / 3$
(C) 1
(D) 0

Hast Aggarwal

Numerade Educator

Problem 17

The global maximum value of $f(x)=\log _{10}\left(4 x^{3}-12 x^{2}+11 x-3\right), x \in[2,3]$ is
(A) $-\frac{3}{2} \log _{10} 3$
(B) $1+\log _{10} 3$
(C) $\log _{10} 3$
(D) $\frac{3}{2} \log _{\mathrm{t} 0} 3$

Hast Aggarwal

Numerade Educator

Problem 18

The least natural number a for which $x+a x^{-2}>2, \forall x \in$ $(0, \infty)$ is
(A) 1
(B) 2
(C) 5
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 19

Let $f(x)=\left\{\begin{array}{lr}x^{3}+x^{2}-10 x, & -1 \leq x<0 \\ \cos x, & 0 \leq x<\pi / 2 \\ 1+\sin x, & \pi / 2 \leq x \leq \pi\end{array}\right.$ Then $\mathrm{f}(\mathrm{x})$ has
(A) a local minimum at $x=\pi / 2$
(B) a global maximum at $\mathrm{x}=-1$
(C) an absolute minimum at $\mathrm{x}=-1$
(D) an absolute maximum at $\mathrm{x}=\pi$

Hast Aggarwal

Numerade Educator

Problem 20

If $a>b>0$ thenthe maximum value of $\frac{a b\left(a^{2}-b^{2}\right) \sin x \cos x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$ in $\left(0, \frac{\pi}{2}\right)$ is
(A) $a^{2}-b^{2}$
(B) $\frac{a^{2}-b^{2}}{2}$
(C) $\frac{a^{2}+b^{2}}{2}$
(D) none of these

Hast Aggarwal

Numerade Educator

Problem 21

Let $\mathrm{f}:[2,7] \rightarrow[0, \infty)$ be a continuous and differentiable function. Then the value of $(f(7)-f(2)) \frac{(f(7))^{2}+(f(2))^{2}+f(2) \cdot(7)}{3}$, is (where $c \in(2,7))$ ).
(A) $3 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(B) $4 \mathrm{f}^{(}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(C) $5 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 22

Let $f(x)=a x^{5}+b x^{4}+c x^{3}+d x^{2}+e x$, where $a, b_{1} c, d, e \in R$
and $f(x)=0$ has a positive root $\alpha$, then
(A) $\mathrm{f}(\mathrm{x})=0$ has root al such that $0<\alpha_{1}<\alpha$
(B) $\mathrm{f}^{\prime}(\mathrm{x})=0$ has at least one real root
(C) $\mathrm{f}^{\prime}(\mathrm{x})=0$ has at least two real roots
(D) All of the above

Hast Aggarwal

Numerade Educator

Problem 23

Bctween any two real roots of the equation $\mathrm{e}^{\pi} \sin \mathrm{x}-1=0$, the equation $e^{x} \cos x+1=0$ has
(A) Atleast one root
(B) Atmost one root
(C) Exactly one root
(D) No root

Hast Aggarwal

Numerade Educator

Problem 24

A differentiable function $\mathrm{f}(\mathrm{x})$ has a relative minimum at $x=0$, then the function $y=f(x)+a x+b$ has a relative minimum at $x=0$ for
(A) all a and all $b$
(B) all b if $\mathbf{a}=0$
(C) all $b>0$
(D) all $\mathrm{a}>0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 25

Suppose that $\mathrm{f}$ is a polynomial of degree 4 and that $f^{\prime}(x) \neq 0$, then
(A) $\mathrm{f}$ has exactly one stationary point
(B) $\mathrm{f}$ must have no stationary point
(C) $\mathrm{f}$ must have exactly 3 stationary points
(D) f has either 1 or 3 stationary points

Hast Aggarwal

Numerade Educator

Problem 26

If $f: \mathbb{R} \rightarrow R$ and $g: R \rightarrow R$ are two functions such that $f(x)+f^{\prime \prime}(x)=-x g(x) f(x)$ and $g(x)>0 \forall x \in R$, then the
function $\mathrm{f}^{\prime}(\mathrm{x})+\left(\mathrm{f}^{\prime}(\mathrm{x})\right)^{2}$ has
(A) a maxima at $x=0$
(B) a minima at $x=0$
(C) a point of inflection at $\mathrm{x}=0$
(D) none of these

Hast Aggarwal

Numerade Educator

Problem 27

For any real $\theta$, the maximum value of $\cos ^{2}(\cos \theta)+$ $\sin ^{2}(\sin \theta)$ is
(A) 1
(B) $1+\sin ^{2} 1$
(C) $\left.1+\cos ^{2}\right]$
(D) Does not exist

Hast Aggarwal

Numerade Educator

Problem 28

If the function $f(x)=\frac{t+3 x-x^{-}}{x-4}$, where $t$ is a parameter, has a minimum and maximum, then the range of values of $t$ is
(A) $(0,4)$
(B) $(0, \infty)$
(C) $(-\infty, 4)$
(D) $(4, \infty)$

Hast Aggarwal

Numerade Educator

Problem 29

Let $\mathrm{f}(\mathrm{x})=a x^{3}+\mathrm{bx}^{2}+\mathrm{cx}+1$ have extrema at $\mathrm{x}=\boldsymbol{\alpha}, \beta$ such
that $\alpha \beta<0$ and $f(\alpha) f(\beta)<0 .$ Then the equation $f(x)=0$ has
(A) three equal real roots
(B) one negative root if $f(\alpha)<0$ and $f(\beta)>0$
(C) one positive root if $f(\alpha)>0$ and $f(\beta)<0$
(D) none of these

Hast Aggarwal

Numerade Educator

Problem 30

A bell tent consists of a conical portion above a cylindrical portion placed on the ground. For a given volume and a circular base of a given radius, the amount of the canvasused is a minimum when the semi-vertical angle of the cone is
(A) $\cos ^{-1} 2 / 3$
(B) $\sin ^{-1} 2 / 3$
(C) $\cos ^{-1} 1 / 3$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 31

A cylindrical gas container is closed at the top and open at the botton. If the iron plate of the top is $5 / 4$ times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is
(A) $3 ; 4$
(B) $5 ; 6$
(C) $4: 5$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 32

A given right circular cone has a volume $p$, and the largest right circular cylinder that can be inscribed in the cone has a volume $\mathrm{q}$. Then $\mathrm{p}: \mathrm{q}$ is
(A) $9 ; 4$
(B) $8: 3$
(C) $7: 2$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 33

A wire of length $a$ is cut into two parts which are bent, respectively, in the form of a square and a circle. The least value of the sum of the areas so formed is
(A) $\frac{a^{2}}{\pi+4}$
(B) $\frac{\mathrm{a}}{\pi+4}$
(C) $\frac{a}{4(\pi+4)}$
(D) $\frac{a^{2}}{4(\pi+4)}$

Hast Aggarwal

Numerade Educator

Problem 34

The largest term of the sequance $\left\langle t_{n}\right\rangle$ where $t_{n}=\frac{n^{2}}{n^{4}+300}$ is
(A) $t_{3}$
(B) $\mathrm{t}_{4}$
(C) $t$.
(D) $t_{\text {. }}$

Hast Aggarwal

Numerade Educator

Problem 35

The set of value of $\mathrm{c}$ for which $\sin \{\ln (\cos \mathrm{x}+\mathrm{c})\}=1$ has
at most one solution in $[0, \pi]$ is
(A) $(2 \pi, \infty)$
(B) $\left(\mathrm{e}^{2 n}, \infty\right)$
(C) $\left(\frac{\mathrm{e}^{2 \pi}+1}{\mathrm{e}^{2 \pi}-1}, \infty\right)$
(D) null set

Hast Aggarwal

Numerade Educator

Problem 36

For $\alpha \neq \beta, \mathrm{n} \in \mathrm{N}$ which of the following is not always true
(A) $|\sin n \alpha-\sin n \beta|<\mathbf{n}|\alpha-\beta|$
(B) $|\cos n \alpha-\cos n \beta|>n|\alpha-\beta|$
(C) $|\sin n \alpha+\sin n \beta|<n|\alpha+\beta|$
(D) $|\cos n \alpha-\cos n \beta|<\mathbf{n} \mid \alpha-\beta$

Hast Aggarwal

Numerade Educator

Problem 37

The maximum value of the expression $\left|\sqrt{\sin ^{2} x+2 a^{2}}-\sqrt{2 a^{2}-1-\cos ^{2} x}\right|$, where a and $x$ are real numbers, is
(A) $\sqrt{2}$
(B) 1$]$
(C) $\sqrt{3}$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 38

Which of the following statements are true?
(i) If $\mathrm{p}(\mathrm{x})=\mathrm{Ax}^{2}+\mathrm{Bx}+\mathrm{C}, \mathrm{x} \in[\mathrm{a}, \mathrm{b}]$, then a value of ${ }^{\prime} \mathrm{c}^{7}$
given by Lagrange's Mean Value Theorem is the mid point of the interval.
(ii) A differentiable function fhas one critical point at $\mathrm{x}=5$. If $f(4)=-2$ and $f(6)=3$, then $x=5$ is a point of local minimum.
(iii) If $\mathrm{f}(\mathrm{x})$ is differentiable on $\mathrm{R}$ snd $\mathrm{f}^{\prime}(\mathrm{x})<1 \forall \mathrm{x} \in \mathbb{R}$,
then atmost one value of $x$ (say $x=c$ ) exists such that $\mathrm{f}(\mathrm{c})=\mathrm{c}$.
(A) TFT
(B) FTF
(C) TTT
(D) TTF

Hast Aggarwal

Numerade Educator

Problem 39

If $a, b, c d \in R$ such that $\frac{a+2 c}{b+3 d}+\frac{4}{3}=0$, then the equation $a x^{2}+c x+d=0$ has
(A) atleast one root in $(-1,0)$
(B) atleast one root in $(0,1)$
(C) no root in $(-1,1)$
(D) no root in $(0,2)$

Hast Aggarwal

Numerade Educator

Problem 40

A box, constructed from a rectangular metal sheet, is $21 \mathrm{~cm}$ by $16 \mathrm{~cm}$ by cutting equal squares of sides $\mathrm{x}$ from the corners of the sheet and then tuming up the projected portions. The value of $\mathrm{x}$ so that volume of the box is maximum is
(A) $\underline{1}$
(B) 2
(C) 3
(D) 4

Hast Aggarwal

Numerade Educator

Problem 41

If $\mathrm{A}\left(\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{B}\left(-\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{C}\left(-\frac{3}{\sqrt{2}},-\sqrt{2}\right)$ and $\mathrm{D}(3 \cos \theta, 2 \sin \theta)$ are four points, then the value of $\theta$ for which the area of quadrilateral $\mathrm{ABCD}$ is maximum, $\left(\frac{3 \pi}{2} \leq 0 \leq 2 \pi\right)$ is
(A) $2 \pi-\sin ^{-1} \frac{1}{3}$
(B) $\frac{7 \pi}{4}$
(C) $2 \pi-\cos ^{-1} \frac{3}{\sqrt{85}}$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 42

A rod of fixed length $\mathrm{k}$ slides along the coordinate axes. If it meets the axes at $\mathrm{A}(\mathrm{a}, 0)$ and $\mathrm{B}(0, \mathrm{~b})$, then the minimum value of $\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}$ is
(A) 0
(B) 8
(C) $\mathrm{k}^{2}-4+\frac{4}{\mathrm{k}^{2}}$
(D) $\mathrm{k}^{2}+4+\frac{4}{\mathrm{k}^{2}}$

Hast Aggarwal

Numerade Educator

Problem 43

Let $f(x)=\left\{\begin{array}{cl}x^{3}+x^{2}+3 x+\sin x \mid(3+\sin 1 / x), & x \neq 0 \\ 0 & , x=0\end{array}\right.$
then number of points (where $f(x)$ attains its minimum value) is
(A) $]$
(B) $\underline{2}$
(C) 3
(D) infinite many

Hast Aggarwal

Numerade Educator

Problem 44

The function $\mathrm{f}(\mathrm{x})=\left|\frac{\mathrm{x}^{2}-2}{\mathrm{x}^{2}-4}\right|$ has
(A) no point of local minima
(B) no point of local maxima
(C) exactly one point of local minima
(D) exactly one point of local maxima

Hast Aggarwal

Numerade Educator

Problem 45

Which of the follwoing is always correct?
(A) If $f^{\prime}(x)>0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one
(B) If $f^{\prime}(x)<0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one
(C) If $\mid f(x)$ be continuous at $x=a$, then $f(x)$ is also continuous at $\mathrm{x}=\mathrm{a}$
(D) If $f(x)$ is continuous at $x=a, f(A)=2$ and $x=a$ is the point of local minimum of $f(x)$, then $[f(x)]$, where [] denotes gresatest integer function, is also continuous

Hast Aggarwal

Numerade Educator

Problem 46

$f(x)$ is a continuous function having 4 critical points $a, b$, c, $d$ where $a<b<c<d$. If $a$ and $d$ are points of maxima, then $f(x)$
(A) two points of minima
(B) has two points of inflexion
(C) can have no point of inflection
(D) has exactly one point of inflection

Hast Aggarwal

Numerade Educator

Problem 47

Given $f(x)=4-\left(\frac{1}{2}-x\right)^{2 / 3} ; g(x)=\{x\}$
$h(x)=\left\{\begin{array}{ll}\frac{\tan [x]}{x}, & x \neq 0 \\ 1 \quad, \quad x=0\end{array}\right.$ and $p(x)=5^{\ln x+1}$ defined in $[0,1]$, the functions on which LMVT is applicable is $\begin{array}{ll}\text { (A) } \mathrm{f} & \text { (B) } \mathrm{g}\end{array}$
(C) $\mathrm{p}$
(D) $h$

Hast Aggarwal

Numerade Educator

Problem 48

The set of critical points of the function $f(x)=x-\log x+\int_{2}^{\pi}(1 / z-2-2 \cos 4 z) d z$ is
(A) $\left\{\frac{\pi}{6}, \frac{n \pi}{2}+\frac{\pi}{6}\right\}, n \in N$
(B) $\{n \pi\}, n \in N$
(C) $\left\{\frac{\pi}{2}, \mathrm{n} \pi+\frac{\pi}{6}\right\}, \mathrm{n} \in \mathrm{N}$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 49

If $f(\theta)=\frac{\sin ^{4} \frac{\theta}{n}-\sin ^{2} \frac{\theta}{n}+1}{\sin ^{4} \frac{\theta}{n}+\sin ^{2} \frac{\theta}{n}+1}$, where $\theta \neq \mathrm{k} \mathrm{n} \pi .$ $\mathrm{k} \in \mathrm{I}$, then range of $f(\theta)$ is
(A) $(1 / 3,1]$
(B) $[1 / 3,1]$
(C) $[1 / 3,1)$
(D) $(1 / 3,1)$

Hast Aggarwal

Numerade Educator

Problem 50

Consider $\mathrm{f}(\mathrm{x})=|1-\mathrm{x}| 1<\mathrm{x}<2 \quad 1 \leq \mathrm{x} \leq 2$
and $g(x)=f(x)+b \sin \pi / 2 x, \quad 1 \leq x \leq 2$
then which of the following is correct?
(A) Rolles Theorem is applicable to both $\mathrm{f}, \mathrm{g}$ and $\mathrm{b}=3 / 2$
(B) LMVT is not applicable to $\mathrm{f}$ and Rolle's Theorem if applicable to $g$ with $b=1 / 2$
(C) LMVT is applicable to $\mathrm{f}$ and Rolle's Theorem is applicable to $g$ with $b=1$
(D) Rolle's Theorem is not applicable to both $\mathrm{f}, \mathrm{g}$ for any real $\underline{b}$

Hast Aggarwal

Numerade Educator

Problem 51

If $f(x)=\left\{\begin{array}{ll}3 x^{2}+12 x-1 ;-1 \leq x \leq 2 \\ 37-x & ; 2<x \leq 3\end{array}\right.$ then
(A) $\mathrm{f}(\mathrm{x})$ is increasing in $[-1,2]$
(B) $\mathrm{f}(\mathbf{x})$ is continuous in $[-1,3]$
(C) $f^{\prime}(2)$ does not exist
(D) $f(x)$ has the maximum value at $x=2$

Hast Aggarwal

Numerade Educator

Problem 52

Which of the statements are necessarily true?
(A) If $\mathrm{f}$ is differentiable and $\mathrm{f}(-1)=\mathrm{f}(1)$, then there is a number $\mathrm{c}$ such that $|\mathrm{c}|<\mathrm{l}$ and $\mathrm{f}^{\prime}(\mathrm{C})=0$.
(B) If $f^{\prime \prime}(2)=0$, then $(2, f(2))$ is an inflection point of the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$.
(C) There exists a function $f$ such that $f(x)>0, f(x)<0$, and $f^{\prime \prime}(x)>0$ for all $x$.
(D) If $f^{\prime}(x)$ exists and is nonzero for all $x$, then $f(1) \neq f(0) .$

Hast Aggarwal

Numerade Educator

Problem 53

Let $f(x)=(x-1)^{4}(x-2)^{2}, n \in N$. Then $f(x)$ has
(A) a maximum at $\mathrm{x}=1$ if $\mathrm{n}$ is odd
(B) a maximum at $x=1$ if $n$ is even
(C) a minimum at $\mathrm{x}=1$ if $\mathrm{n}$ is even
(D) a minima at $\mathrm{x}=2$ if $\mathrm{n}$ is even

Hast Aggarwal

Numerade Educator

Problem 54

Identify the correct statements:
(A) If $f(x)=a x^{3}+b$ and $f$ is strictly increasing on $(-1,1)$ then $\mathrm{a}>0$.
(B) An $n$ th-degree polynomial has atmost ( $n-1)$ critical points.
(C) If $\mathrm{f}^{\prime}(\mathrm{x})>0$ for all real numbers $\mathrm{x}$, then $\mathrm{f}$ increases without bound.
(D) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Hast Aggarwal

Numerade Educator

Problem 55

Which of the given equation has exactly one solution in the indicated interval ?
(A) $x^{3}+2 x-3=0 ;[0,1]$
(B) $\mathrm{e}^{-x}=\mathrm{x}-1 ;[1,2]$
(C) $x \ln x=3 ;[2,4]$
(D) $\sin x=3 x-1 ;[-1,1]$

Hast Aggarwal

Numerade Educator

Problem 56

If $\mathrm{f}(\mathrm{x})$ is a differentiable function and $\phi(\mathrm{x})$ is twice differentiable function and $\alpha$ and $\beta$ are roots of the equation $f(x)=0$ and $\phi^{\prime}(x)=0$ respectively, then which of the following statement is true ? $(\alpha<\beta)$
(A) there exists exactly one root of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi^{\prime \prime}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$
(B) there exists atleast one root of the equation $\phi^{\prime}(x)$. $f(x)+\phi^{\prime \prime}(x) \cdot f(x)=0$ on $(\alpha, \beta)$
(C) there exists odd number of roots of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi "(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 57

An even polynomal function $f(x)$ satisfies a relation
$f(2 x)\left(1-f\left(\frac{1}{2 x}\right)\right)+f\left(16 x^{2} y\right)=f(-2)-f(4 x y) \forall x, y \in$
$\mathrm{R}-\{0\}$ and $\mathrm{f}(4)=-255, \mathrm{f}(0)=1$
Which of the following holds good?
(A) $\mathrm{f}(\mathrm{x})$ has local maximum at $\mathrm{x}=1$.
(B) $f(x) f\left(\frac{1}{x}\right) \leq 0$
(C) Range of values of $\mathrm{k}$ for which $|\mathrm{f}(\mathrm{x})|=\mathrm{k}-2$ has exactly four distinct solutions is $(2,3)$.
(D) $\int_{ }^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{3}{4}$.

Hast Aggarwal

Numerade Educator

Problem 58

Let $f(x)=a x^{3}+b x^{2}+c x+d(a, b, c, d, \alpha, \beta \in R)$ and
$\mathrm{f}^{\prime}(\alpha)=\mathrm{f}^{\prime}(\beta)=0,(\alpha \neq \beta)$, then
(A) if $f(\alpha), f(\beta)<0$ then all the roots of $f(x)=0$ are real
(B) if $f(\alpha), f(\beta)>0$ then all the roots of $f(x)=0$ is real
(C) if $\alpha$ is a point of local maxima and $\beta$ is point of local minima for the function $f(x)$ then $\alpha<\beta$
(D) If $a>0$ then $f(x)$ is decreasing on $(\alpha, \beta)$, where $\beta>\alpha$

Hast Aggarwal

Numerade Educator

Problem 59

The diagram shows the graph of the derivative of a function $f(x)$ for $0 \leq x \leq 4$ with $f(0)=0$. Which of the following could be correct statements for $\mathrm{y}=\mathrm{f}(\mathrm{x})$ ?
(A) Tangent line to $\mathrm{y}=\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=0$ makes an angle of $\sec ^{-1} \sqrt{5}$ with the $x$-axis.
(B) $\mathrm{f}$ is strictly increasing in $(0,3)$.
(C) $\mathrm{x}=1$ is both an inflection point as well as point of local extremum.
(D) Number of critical point on $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is two.

Hast Aggarwal

Numerade Educator

Problem 60

Which of the functions $f$ satisfies neither the hypotheses nor the conclusions of LMVT
(A) $f(x)=|x-2| ;[1,4]$
(B) $f(x)=1+|x-1| ;[0,3]$
(C) $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$ (the greatest integer function); $[-1,1]$
(D) $f(x)=3 x^{2 / 3} ;[-1,1]$

Hast Aggarwal

Numerade Educator

Problem 61

If $x+y=60, x>0, y>0$, then the expression $x^{2}(30-y)^{2}$ has
(A) least value $=0$
(B) greatest value $=15^{4}$
(C) two extrema
(D) no greatest value

Hast Aggarwal

Numerade Educator

Problem 62

Let $f(x)=\left\{\begin{array}{cl}x^{a} \sin ^{2} \frac{1}{n x}, & x \neq 0 \\ 0 & , x=0\end{array}\right.$ where $n \in I, n \neq 0$.
If Rolle's Theorem is applicable to $\mathrm{f}(\mathrm{x})$ in the interval $[0,1]$, then
(A) $\alpha>0$, greatest value of $\mathrm{n}$ is $\frac{1}{\pi}$
(B) $\alpha>2$, greatest value of $n$ is $\frac{1}{\pi}$
(C) $\alpha>0$, least value of $\mathrm{n}$ is $-\frac{1}{\pi}$
(D) $\alpha$ can not be $<1$

Hast Aggarwal

Numerade Educator

Problem 63

The function $f(x)=3+2(a+1) x+\left(a^{2}+1\right) x^{2}-x^{3}$ has a local minimum at $x=x_{1}$ and local maximum at $x=x_{2}$ such that $\mathrm{x}_{1}<2<\mathrm{x}_{2}$ then a belongs to the interval(s)
(A) $\left(-\infty,-\frac{3}{2}\right)$
(B) $\left(-\frac{3}{2}, 1\right)$
(C) $(0, \infty)$
(D) $(1, \infty)$

Hast Aggarwal

Numerade Educator

Problem 64

It $\mathrm{y}=\mathrm{g}(\mathrm{x})$ is a curve which is obtained by the reflecton of $y=f(x)=\frac{e^{x}-e^{-x}}{2}$ by the line $y=x$ then
(A) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has exactly one tangent parallel to $\mathrm{x}$-axis
(B) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has no tangent parallel to $\mathrm{y}$-axis
(C) The tangent to $y=g(x)$ at $(0,0)$ is $y=x$
(D) $g(x)$ has no extremum

Hast Aggarwal

Numerade Educator

Problem 65

An extreme value of $4 \sin ^{2} x+3 \cos ^{2} x-24 \sin \frac{x}{2}$
$-24 \cos \frac{\mathrm{x}}{2}$, where $0 \leq \mathrm{x} \leq \frac{\pi}{2}$, is
(A) $4+\sqrt{2}$
(B) $4(1-6 \sqrt{2})$
(C) $-21$
(D) 4

Hast Aggarwal

Numerade Educator

Problem 66

If $f(x)=(x-3)^{9}+\left(x-3^{2}\right)^{4}+\ldots .+\left(x-3^{4}\right)^{5}$, then
(A) $f(x)$ is always increasing
(B) $f(x)=0$ has one real $\&$ eight imaginary roots
(C) $x=3,3^{2} \quad 3^{9}$ are the roots of $f(x)=0$
(D) $f(x)=0$ has a negative real root

Hast Aggarwal

Numerade Educator

Problem 67

If $f(x)=-\frac{1}{3} x^{3}+t^{2} x$, where $t$ is a real parameter. Let $m(t)$ denote the minimum of $\mathrm{f}(\mathrm{x})$ over $[0,1]$ then
(A) $m(t)=0$ if $t^{2} \geq \frac{1}{3}$
(B) $\mathrm{m}(\mathrm{t})=0$ for all $\mathrm{t}$
(C) $m(t)=t^{2}-\frac{1}{3}$ if $t^{2}<\frac{1}{3}$
(D) $\mathrm{m}(\mathrm{t})=\frac{1}{3}-\mathrm{t}^{2}$ for all $\mathrm{t}$.

Hast Aggarwal

Numerade Educator

Problem 68

If $\mathrm{f}:[-1,1] \rightarrow \mathrm{R}$ is a continuously differetiable function such that $f(1)>f(-1)$ and $|P(y)| \leq 1$ for all $y \in[-1,1]$ then
(A) there exists an $\mathrm{x} \in[-1,1]$ such that $\mathrm{f}(\mathrm{x})>0$
(B) there exists an $x \in[-1,1]$ such that $f(x)<0$
(C) $f(1) \leq f(-1)+2$
(D) $f(-1), f(1)<0$

Hast Aggarwal

Numerade Educator

Problem 69

The figure shows the graph of $f(x)$. Then the correct statement(s) is/are
(A) $\mathrm{f}$ is decreasing in $(-1,1)$
(B) $\mathrm{f}$ has a local minimum at $\mathrm{x}=2$
(C) $\mathrm{f}$ is an odd function
(D) $\mathrm{f}$ has atmost 4 zeros.

Hast Aggarwal

Numerade Educator

Problem 70

Let $g(x)=-\frac{f(-1)}{2} x^{2}(x-1)-f(0)\left(x^{2}-1\right)$
$+\frac{f(1)}{2} x^{2}(x+1)-f^{\prime}(0) x(x-1)(x+1)$ where $f$ is a thrice
differentiable function. Then the correct statements are
(A) there exists $x \in(-1,0)$ such that $f^{\prime}(x)=g^{\prime}(x)$
(B) there exists $x \in(0,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime}(x)$
(C) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime \prime}(x)$
(D) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=3 f(1)-3 f(-1)$ $-6 \mathrm{f}^{\prime}(0)$

Hast Aggarwal

Numerade Educator

Problem 71

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathrm{A}): \mathrm{A}$ tangent parallel to $\mathrm{x}$-axis can be drawn for $f(x)=(x-1)(x-2)(x-3)$ in the interval $[1,3]$ Reason (R): A horizontal tangent can be drawn to any cubic function.

Hast Aggarwal

Numerade Educator

Problem 72

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathrm{A}):$ The least value of the function $f(x)=-x^{2}$ $+4 x+1+\sin ^{-1}\left(\frac{x}{2}\right)$ on the interval $[-1,1]$ is $-4-\frac{\pi}{6}$
Reason $(\mathrm{R}):$ The least value of $\mathrm{f}(\mathrm{x})$ in $[-1,1]=\min (\mathrm{f}(-1)$
$f(1)\}=\min \left\{-4-4 \frac{\pi}{6}, 4+\frac{\pi}{6}\right\}=-4-\frac{\pi}{6}$

Hast Aggarwal

Numerade Educator

Problem 73

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Let $\mathrm{f}(\mathrm{x})$ be a twice differentiable function. Assertion $(\mathrm{A}):$ If $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}$ and $\mathrm{f}(\mathrm{a})=0, \mathrm{f}(\mathrm{b})=1$,
$\mathrm{f}(\mathrm{c})=-1, \mathrm{f}(\mathrm{d})=0$, then the minimum number of zeroes $g(x)=\left(f^{\prime}(x)\right)^{2}+f(x) f^{\prime}(x)$ in $[a, d]$ is $4 .$
Reason (R) : If $f(\alpha) f(\beta)<0$ then $f(\gamma)=0$ for some $\alpha<\gamma<\beta$ and if $f(\alpha)=f(\beta)=0$
then $f^{\prime}(\gamma)=0$ for some $\alpha<\gamma<\beta$

Hast Aggarwal

Numerade Educator

Problem 74

Assertion $(\mathbf{A}):$ Let $f(x)=5-4(x-2)^{21}$, then at $x=2$ the function $f(x)$ attains neither the least value nor the greatest value. Reason $(\mathbf{R}):$ At $x=2$, the first derivative does not exist.

Hast Aggarwal

Numerade Educator

Problem 75

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(A): f(x)=\frac{x}{3}+\frac{a x^{2}}{2}+x+5$ has positive point of maxima for $\mathrm{a}<-2$. Reason $(\mathrm{R}): \mathrm{x}^{2}+a \mathrm{x}+1=0$ has both roots positive for $\mathrm{a}<-2$

Hast Aggarwal

Numerade Educator

Problem 76

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion (A) : For all a, b $\in \mathrm{R}$, the function $f(x)=3 x^{4}-$ $4 x^{3}+6 x^{2}+a x+b$ has exactly one extremum. Reason $(\mathbf{R}):$ If a cubic function is monotonic, then its graph cuts $\mathrm{x}$-axis only once.

Hast Aggarwal

Numerade Educator

Problem 77

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion (A) : Conditions of Lagrange's Mean Value Theorem are not satisfied when $f(x)=|x-1|(x-1)$ in $[0,2]$.
Reason $(\mathbf{R}):|\mathrm{x}-1|$ is not differentiable at $\mathrm{x}=1$.

Hast Aggarwal

Numerade Educator

Problem 78

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathbf{A}):$ If $27 a+9 b+3 c+d=0$, then the equation $f(x)=4 a x^{3}+3 b x^{2}+2 c x+d=0$ has at least one real root Iying between $(0,3)$. Reason $(\mathrm{R}):$ If $f(x)$ is continuous in $[a, b]$, derivable in
b) such that $f(a)=f(b)$, then ther exists atleast one point (a, $c \in(a, b)$ such that $f^{\prime}(c)=0$

Hast Aggarwal

Numerade Educator

Problem 79

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion (A) : The maximum value of $\left(\sqrt{-3+4 x-x^{2}}+4\right)^{2}+(x-5)^{2}($ where $1 \leq x \leq 3)$ is 36
Reason (R) : The maximum distance between the point $(5,-4)$ and the point on the circle $(\mathrm{x}-2)^{2}+\mathrm{y}^{2}=1$ is 6 .

Hast Aggarwal

Numerade Educator

Problem 80

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(A):$ If $g(x)$ is a differentiable function $g(2) \neq 0$, $g(-2) \neq 0$ and Rolle's Theorem is not applicable to $f(x)=\frac{x^{2}-4}{g(x)}$ in $[-2,2]$, then $g(x)$ has atleast one root in $(-2,2)$
Reason ( $(\mathrm{R})$ : If a function $\mathrm{f}$ is differentiable in $(\mathrm{a}, \mathrm{b})$ and $f(a)=f(b)$, then Rolle's Theorem is applicable to $f(x)$ for $x \in(a, b)$

Hast Aggarwal

Numerade Educator

Problem 81

Consider $f, g$ and $h$ be three real valued dilierentiable functions defined on $\mathrm{R}$.
Let $g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1)$, $f(x)=x g(x)-12 x+1$ and $f(x)=(h(x))^{2}$ where $h(0)=1$.
The function $y=f(x)$ has
(A) Exactly one local minima and no local maxima
(B) Exactly one local maxima and no local minima
(C) Exactly one local maxima and two local minima
(D) Exactly two local maxima and one local minima

Hast Aggarwal

Numerade Educator

Problem 82

Consider $f, g$ and $h$ be three real valued dilierentiable functions defined on $\mathrm{R}$.
Let $g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1)$, $f(x)=x g(x)-12 x+1$ and $f(x)=(h(x))^{2}$ where $h(0)=1$.
Which of the following is are true for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$ ?
(A) $g(x)$ monotonically decreases in $\left(-\infty, 2-\frac{1}{\sqrt{3}}\right)$ and $\left(2+\frac{1}{\sqrt{3}}, \infty\right)$
(B) g(x) monotonically increases in $\left(2-\frac{1}{\sqrt{3}}, 2+\frac{1}{\sqrt{3}}\right)$
(C) There exists exactly one tangent to $\mathrm{y}=\mathrm{g}(\mathrm{x})$ which is parallel to the chord joining the points $(1, g(1))$ and $(3, g(3))$
(D) There exists exactly two distinct Lagrange's Mean Value in $(0,4)$ for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$.

Hast Aggarwal

Numerade Educator

Problem 83

Consider $f, g$ and $h$ be three real valued dilierentiable functions defined on $\mathrm{R}$.
Let $g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1)$, $f(x)=x g(x)-12 x+1$ and $f(x)=(h(x))^{2}$ where $h(0)=1$.
Which one of the following does not hold good for $\mathrm{y}=\mathrm{h}(\mathrm{x})$ ?
(A) Exactly one critical point
(B) No point of inflection
(C) Exactly one real zero in $(0,3)$
(D) Exactly one tangent parallel to $\mathbf{x}$-axis

Hast Aggarwal

Numerade Educator

Problem 84

Let $f(x)=4 x^{2}-4 a x+a^{2}-2 a+2$ and the global minimum value of $f(x)$ for $x \in[0,2]$ is equal to 3 .
The number of values of a for which the global minimum for $\mathrm{x} \in[0,2]$ occurs at the end point of interval $[0,2]$ is
(A) 1
(B) 2
(C) 3
(D) 0

Hast Aggarwal

Numerade Educator

Problem 85

Let $f(x)=4 x^{2}-4 a x+a^{2}-2 a+2$ and the global minimum value of $f(x)$ for $x \in[0,2]$ is equal to 3 .
The number of values of a for which the global minimum for $\mathrm{x} \in[0,2]$ occurs at a value of $\mathrm{x}$ lying in $(0,2)$ is
(A) $\underline{1}$
(B) 2
(C) 3
(D) 0

Hast Aggarwal

Numerade Educator

Problem 86

Let $f(x)=4 x^{2}-4 a x+a^{2}-2 a+2$ and the global minimum value of $f(x)$ for $x \in[0,2]$ is equal to 3 .
The values of a for which $f(x)$ is monotonic for $x \in[0,2]$ are
(A) $a \leq 0$ or a $\geq 4$
(B) $0 \leq a \leq 4$
(C) $a>0$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 87

Consider a polynomial $\mathrm{y}=\mathrm{P}(\mathrm{x})$ of the least degree passing through $\mathrm{A}(-1,1)$ and whose graph has two points of inflexion $\mathrm{B}(1,2)$ and $\mathrm{C}$ with abscissa $0 \mathrm{at}$ which the curve is inclined to the positive axis of abscissas at an angle of $\sec ^{-1} \sqrt{2}$.
If $\int_{-2}^{2} \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{\mathrm{k}}{5}, \mathrm{k} \in \mathrm{N}$ then $\mathrm{k}$ equals
(A) 17
(B) 24
(C) 32
(D) 42

Hast Aggarwal

Numerade Educator

Problem 88

Consider a polynomial $\mathrm{y}=\mathrm{P}(\mathrm{x})$ of the least degree passing through $\mathrm{A}(-1,1)$ and whose graph has two points of inflexion $\mathrm{B}(1,2)$ and $\mathrm{C}$ with abscissa $0 \mathrm{at}$ which the curve is inclined to the positive axis of abscissas at an angle of $\sec ^{-1} \sqrt{2}$.
The value of $\mathrm{P}(-1)$ equals
(A) $-1$
(B) 0
(C) 1
(D) 2

Hast Aggarwal

Numerade Educator

Problem 89

Consider a polynomial $\mathrm{y}=\mathrm{P}(\mathrm{x})$ of the least degree passing through $\mathrm{A}(-1,1)$ and whose graph has two points of inflexion $\mathrm{B}(1,2)$ and $\mathrm{C}$ with abscissa $0 \mathrm{at}$ which the curve is inclined to the positive axis of abscissas at an angle of $\sec ^{-1} \sqrt{2}$.
The area of $\triangle \mathrm{ABC}$ equals
(A) $\frac{1}{2}$
(B) 1
(C) $\frac{1}{8}$
(D) $\frac{1}{12}$

Hast Aggarwal

Numerade Educator

Problem 90

Let $f(x)=x^{3}-3(7-a) x^{2}-3\left(9-a^{2}\right) x+2$
The values of a, if $f(x)$ has a negative point of local minimum, are
(A) $\phi$
(B) $(-3,3)$
(C) $\left(-\infty, \frac{58}{14}\right)$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 91

Let $f(x)=x^{3}-3(7-a) x^{2}-3\left(9-a^{2}\right) x+2$
The values of $\mathrm{a}$, if $\mathrm{f}(\mathrm{x})$ has a positive point of local maxima, are
(A) $\phi$
(B) $(-\infty,-3) \cup(3, \infty)$
(C) $\left(-\infty, \frac{58}{14}\right)$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 92

Let $f(x)=x^{3}-3(7-a) x^{2}-3\left(9-a^{2}\right) x+2$
The values of a if $f(x)$ has points of extrema which are of opposite signs, are
(A) $\bar{\phi}$
(B) $(-3,3)$
(C) $\left(-\infty, \frac{58}{14}\right)$
(D) None of these

Hast Aggarwal

Numerade Educator

Problem 93

Let $f(x)=e^{v+1 / n}-e^{x}$ for real number $p>0$.
The value of $x=s$, for which $f(x)$ is minimum, is
(A) $-\frac{\ln (\mathrm{p}+1)}{\mathrm{p}}$
(B) $-\ln (\mathrm{p}+1)$
(C) $-\ln \mathrm{p}$
(D) $\ln \left(\frac{\mathrm{p}+\mathrm{l}}{\mathrm{p}}\right)$

Hast Aggarwal

Numerade Educator

Problem 94

Let $f(x)=e^{v+1 / n}-e^{x}$ for real number $p>0$.
Let $g(t)=\int_{1}^{1+1} f(x) e^{1-x} d x .$ The value of $t=t_{p}$, for which $\mathrm{g}(\mathrm{t})$ is minimum is
(A) $-\ln \left(\frac{\mathrm{e}^{p}-1}{\mathrm{p}}\right)$
(B) $-\frac{1}{p} \ln \left(\frac{e^{p}-1}{p}\right)$
$(C)-\frac{1}{p} \ln \left(\frac{(p+1)\left(e^{p}-1\right)}{p}\right)$
(D) $-\ln \left((p+1)\left(\mathrm{e}^{p}-1\right)\right)$

Hast Aggarwal

Numerade Educator

Problem 95

Column - I $\quad$ Column - II
(A) If $x^{2}+y^{2}=1$, then minimum value $x+y$ is
(P) $-3$
(B) If the maximum value of $y=a \cos x-\frac{1}{3} \cos 3 x$ occurs
(Q) $-\sqrt{2}$
when $x=\frac{\pi}{6}$, then the value of $^{\prime} a$ ' is
(C) If $f(x)=x-2 \sin x, 0 \leq x \leq 2 \pi$ is increasing in the interval
(R) 3 $[a \pi, b \pi]$, then $a+b$ is
(D) If equation of the tangent to the curve $\mathrm{y}=-\mathrm{e}^{-\mathrm{u}^{2}}$ where it
(S) 2 crosses the y-axis is $\frac{x}{p}+\frac{y}{q}=1$, then $p-q$ is

Hast Aggarwal

Numerade Educator

Problem 96

Column-1 Column - II
A) The greatest value of $f(x)=\frac{x}{4+x+x^{2}}$ on $[0, \infty)$ is
(P) $\frac{18}{\mathrm{e}}$
B) The maximum value of $\frac{\ln x}{x}$ in $[2, \infty)$ is
(Q) $\frac{1}{\mathrm{e}}$
C) Let $x>0, y>0$ and $x y=1$, then minimum value of $\frac{3}{c^{3}} x+27 e y$
(R) e
D) The perimeter of a sector is $4 \mathrm{e}$. The area of the
(S) $\frac{1}{5}$
sector is maximum when its radius is

Hast Aggarwal

Numerade Educator

Problem 97

(A) The intercept of the common tangent to the curves
(P) $-1$ $y^{2}=8 x$ and $x y=-1$ on the axis of $y$ is equal to
(B) Let $f$ be a real function whose derivates upto third order exist and for
(Q) 0 some pair $a, b \in R, a<b \log \frac{f(a)+f^{\prime}(a)+f^{\prime \prime}(a)}{\left. \left.f(b)+f^{\prime}(b)+f^{\prime \prime}\right) b\right)}=a-b$, then there
exists $\mathrm{c}(\mathrm{a}, \in \mathrm{b})$ for which $\frac{\mathrm{f}^{\prime \prime}(\mathrm{c})}{\mathrm{f}(\mathrm{c})}$ is equal to
(C) Let $f(x)=\left(x^{2}-1\right)\left(x^{2}-4\right)$, and $\alpha, \beta, \gamma$ be the roots of the equation
(R) 1 $f^{\prime}(x)=0$ then $[\alpha]+[\beta]+[\gamma]$ is equal to $([t]$ represents the integral part of 1$)$
(D) If three normals can be drawn to the curve $y^{2}=x$
(S) $5 / 4$ from the point $(\mathrm{c}, 0)$ then $\mathrm{c}$ can be equal to $\quad$ (T) 2

Hast Aggarwal

Numerade Educator

Problem 98

Column- I Column-1I
(A) If the maximum and minimum value of the function
(P) 1 $h(y)=y^{3}-6 y^{2}+9 y+1$ on $[0,2]$ are $M$ and $m$ respectively, then $(\mathrm{M}+\mathrm{m})$ is equal to
(B) If the maximum and minimum value of the function $f(x)=\tan ^{-1} x-\frac{1}{2} \ell n x$
(Q) 6 on $\left[\frac{1}{\sqrt{3}}, \sqrt{3}\right]$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then $[\mathrm{M}+\mathrm{m}]$ is equal to (where [.] denotes greatest integer function)
(C) If the maximum and minimum value of the function
(R) $\underline{0}$ $h(y)=\left\{\begin{array}{cc}2 y^{2}+\frac{2}{y^{2}}, & |y| \in[1,2] \\ 1 & |y|<1\end{array}\right.$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then
$\mathrm{M}+\mathrm{m}$ is equal to
(D) For a given $\mathrm{n} \in \mathrm{N}$, number of real solutions of the
(S) $19 / 2$ equation $\min \left(\mathrm{e}^{x}, \mathrm{x}^{2}\right)=\mathrm{n}$ is

Hast Aggarwal

Numerade Educator

Problem 99

Column-I
(A) The value of the sum $\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2}-3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\ldots . .+\frac{29}{14^{2} \cdot 15^{2}}$ is
(P) 1
(B) If the tangents to the graph of $f(x)=x^{3}+a x+b$ at $x=a$ and
(Q) 0
$\mathrm{x}=\mathrm{b}(\mathrm{a} \neq \mathrm{b})$ are parallel then $\mathrm{f}(1)$ is equal to
(C) If the sum of x coordinates of points of extrema and points of inflection
(R) $\frac{224}{225}$
of $f(x)=3 x^{5}-250 x^{3}+735 x$ is $75 k$ then $k$ equals
(D) If a right triangle is drawn in a semi circle of radius $\frac{1}{2}$ with one leg
(S) $\frac{3 \sqrt{3}}{32}$
(not the hypotenuse) along the diameter, the maximum area of the triangle is

Hast Aggarwal

Numerade Educator