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A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 14

Applications of Derivatives - all with Video Answers

Educators

Chapter Questions

02:36

Problem 1

The set of values of $x$ for which log $(1+x)<x$, is
(A) $x<0$
(B) $x>0$
(C) $0<x<1$
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
01:13

Problem 2

Let $f(x)=\cos 2 \pi x+x-[x]$, where $[\cdot]$ denotes the greatest integer function. Then the number of points in $[0,10]$ at which $f(x)$ assumes its local maximum value, is
(A) 10
(B) 9
(C) 0
(D) infinite

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:25

Problem 3

The function $f(x)=\frac{\sin x}{x}$ is decreasing in the interval
(A) $\left(-\frac{\pi}{2}, 0\right)$
(B) $\left(0, \frac{\pi}{2}\right)$
(C) $(0, \pi)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:30

Problem 4

If $a x+\frac{b}{x} \geq c$ for all positive $x$, where $a, b>0$, then
(A) $a b<\frac{c^{2}}{4}$
(B) $a b \geq \frac{c^{2}}{4}$
(C) $a b \geq \frac{c}{4}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:32

Problem 5

$\mathrm{f} 0<\alpha<\beta<\frac{\pi}{2}$ then
(A) $\frac{\tan \beta}{\tan \alpha}<\frac{\alpha}{\beta}$
(B) $\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}$
(C) $\frac{\tan \alpha}{\tan \beta}<\frac{\alpha}{\beta}$
(D) $\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:47

Problem 6

If $a<0$, the function $\left(e^{a x}+e^{-a x}\right)$ is a monotonic decreasing function for all values of $x$, where
(A) $x>0$
(B) $x<0$
(C) $x>1$
(D) $x<1$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:10

Problem 7

The range of values of $a$ for which the function $f(x)=x^{3}+(a+2) x^{2}+3 a x+5$
may be monotonic in $R$, is
(A) $a<1$
(B) $1<a<4$
(C) $a>4$
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
02:19

Problem 8

The values of $k$ for which the function $f(x)=k x^{3}-9 x^{2}+9 x+3$ may be increasing on $R$ are
(A) $k>3$
(B) $k<3$
(C) $k \leq 3$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:45

Problem 9

The least possible value of $k$ for which the function $f(x)=x^{2}+k x+1$ may be increasing on $[1,2]$ is
(A) 2
(B) $-2$
(C) 0
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:45

Problem 10

If $f(x)=2 x^{3}+9 x^{2}+\lambda x+20$ is a decreasing function of $x$ in the largest possible interval $(-2,-1)$ then $\lambda$ is equal to
(A) 12
(B) $-12$
(C) 6
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:14

Problem 11

Let $f^{\prime}(x)>0$ and $g^{\prime}(x)<0$ for all $x \in R$. Then,
(A) $f[g(x)]>f[g(x-1)]$
(B) $f[g(x)]>f^{\prime}[g(x+1)]$
(C) $g[f(x)]>g[f(x-1)]$
(D) $g[f(x)]<g[f(x+1)]$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:15

Problem 12

If the function $f(x)=3 \cos |x|-6 a x+b$ increases for all $x \in R$, then the range of values of $a$ is given by
(A) $a>-\frac{1}{2}$
(B) $a<-\frac{1}{2}$
(C) $a \leq b$
(D) $a \geq b$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:50

Problem 13

The equation $x+e^{x}=0$ has
(A) only one real root
(B) only two real roots
(C) no real root
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:19

Problem 14

The value of $a$ in order that $f(x)=\sin x-\cos x-a x+b$ decreases for all real values is given by
(A) $a \geq \sqrt{2}$
(B) $a<\sqrt{2}$
(C) $a \geq 1$
(D) $a<1$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:05

Problem 15

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
(A) always zero
(B) always negative
(C) always positive
(D) strictly increasing

Patha Sharma
Patha Sharma
Numerade Educator
02:23

Problem 16

If $f^{\prime \prime}(x)<0 \forall x \in(a, b)$, then $f^{\prime}(x)=0$
(A) exactly once in $(a, b)$
(B) atmost once in $(a, b)$
(C) atleast once in $(a, b)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:43

Problem 17

The two tangents to the curve $a x^{2}+2 h x y+b y^{2}=1$, $a>0$ at the points where it crosses $x$-axis, are
(A) parallel
(B) perpendicular
(C) inclined at an angle $\frac{\pi}{4}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:12

Problem 18

The curve $y-e^{x y}+x=0$ has a vertical tangent at the point
(A) $(1,1)$
(B) at no point
(C) $(0,1)$
(D) $(1,0)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
06:52

Problem 19

The set of all values of $a$ for which the function $f(x)=$ $\left(a^{2}-3 a+2\right)\left(\cos ^{2} x / 4-\sin ^{2} x / 4\right)+(a-1) x+\sin 1$ does
not possess critical points is
(A) $[1, \infty)$
(B) $(0,1) \cup(1,4)$
(C) $(-2,4)$
(D) $(1,3) \cup(3,5)$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:35

Problem 20

Let $f(x)=\left\{\begin{array}{cc}-x^{3}+\log _{2} b & 0<x<1 \\ 3 x & x \geq 1\end{array}\right.$. Then set of val-
ues of $b$ for which $f(x)$ has least value at $x=1$ is:
(A) $R$
(B) $(0,16]$
(C) $[16, \infty)$
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
01:23

Problem 21

If at any point on a curve the sub-tangent and sub-normal are equal, then the length of the normal is equal to
(A) $\sqrt{2}$ ordinate
(B) ordinate
(C) $\sqrt{2 \text { ordinate }}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
04:46

Problem 22

Tangent is drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$, where $\theta \in(0, \theta / 2)$. Then, the value
of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{8}$
(D) $\frac{\pi}{4}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:35

Problem 23

The minimum value of $a \tan ^{2} x+b \cot ^{2} x$ equals the maximum value of $a \sin ^{2} \theta+b \cos ^{2} \theta$ where $a>b>0$,
when
(A) $a=b$
(B) $a=2 b$
(C) $a=3 b$
(D) $a=4 b$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:41

Problem 24

A function $f$ is such that $f^{\prime}(a)=f^{\prime \prime}(a)=f^{\prime \prime \prime}(a)=\ldots=$
$f^{(2 n)}(a)=0$ and $f$ has a local maximum value $b$ at $x=$ $a$, if $f(x)$ is
(A) $(x-a)^{2 n+2}$
(B) $b-1-(x+1-a)^{2 n-1}$
(C) $b-(x-a)^{2 n+2}$
(D) $(x-a)^{2 n+2}-b$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:24

Problem 25

If $P=x^{3}-\frac{1}{x^{3}}$ and $Q=x-\frac{1}{x}, x \in(0, x)$ then minimum value of $P / Q^{2}$
(A) is $2 \sqrt{3}$
(B) is $-2 \sqrt{3}$
(C) does not exist
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
07:00

Problem 26

If the area of the triangle included between the axes and any tangent to the curve $x^{n} y=a^{n}$ is constant, then $n$ is equal to
(A) 1
(B) 2
(C) $\frac{3}{2}$
(D) $\frac{1}{2}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:07

Problem 27

If $f(x)$ and $g(x)$ are differentiable functions for $0 \leq x \leq$ 1 such that $f(0)=2, g(0)=0, f(1)=6, g(1)=2$, then in the interval $(0,1)$,
(A) $f^{\prime}(x)=0$ for all $x$
(B) $f^{\prime}(x)=2 g^{\prime}(x)$ for atleast one $x$
(C) $f^{\prime}(x)=2 g^{\prime}(x)$ for atmost one $x$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:28

Problem 28

If $y=\frac{\sin (x+a)}{\sin (x+b)} ; a \neq b$, then $y$ has
(A) maximum at $x=0$
(B) minimum at $x=0$
(C) neither maximum nor minimum
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
01:31

Problem 29

For a differentiable curve $y=f(x)$ having atleast two extremum in the interval $[a, b]$,
(A) two of its maximum values occur successively
(B) two of its minimum values occur successively
(C) maximum and minimum values occuralternatively
(D) None of the above

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:49

Problem 30

The points on the curve $x y^{2}=1$ which are nearest to the origin are
(A) $\left[\left(\frac{1}{2}\right)^{1 / 3}, \pm\left(\frac{1}{2}\right)^{-1 / 6}\right]$
(B) $\left[\left(\frac{1}{2}\right)^{1 / 3}, 2^{-1 / 6}\right]$
(C) $\left(2^{1 / 3}, \pm\left(\frac{1}{2}\right)^{-1 / 6}\right)$
(D) None of these

Anurag Kumar
Anurag Kumar
Numerade Educator
02:27

Problem 31

N characters of information are held on magnetic tape, in batches of $x$ characters each; the batch processing time is $\alpha+\beta x^{2}$ seconds; $\alpha, \beta$ are constants. The optimum value of $x$ for fast processing is
(A) $\frac{\alpha}{\beta}$
(B) $\frac{\beta}{\alpha}$
(C) $\sqrt{\frac{\alpha}{\beta}}$
(D) $\sqrt{\frac{\beta}{\alpha}}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:59

Problem 32

$A B$ is a diameter of a circle and $C$ is any point on the circumference of the circle, then
(A) area of $\triangle A B C$ is maximum when it is an isosceles
(B) area of $\Delta A B C$ is minimum when it is an isosceles(C) the perimeter of $\Delta A B C$ is minimum when it is isosceles
(D) the perimeter of $\Delta A B C$ is maximum when it is isosceles

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:34

Problem 33

Let $f(x)=1+3 x^{2}+3^{2} x^{4}+\ldots+3^{30} \cdot x^{60} .$ Then $f(x)$ has
(A) atleast one maximum
(B) exactly one maximum
(C) atleast one minimum
(D) exactly one minimum

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:38

Problem 34

A function $f$ is such that $f^{\prime}(4)=f^{\prime \prime}(4)=0$ and $f$ has minimum value 10 at $x=4$. Then $f(x)=$
(A) $4+(x-4)^{4}$
(B) $10+(x-4)^{4}$
(C) $(x-4)^{4}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:54

Problem 35

The range of values of $k$ for which the function $f(x)=\left(k^{2}-7 k+12\right) \cos x+2(k-4) x+\log 2$
does not possess critical points, is
(A) $(1,5)$
(B) $(1,5)-\{4\}$
(C) $(1,4)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:05

Problem 36

The minimum value of the function $f(x)=\frac{x^{p}}{p}+\frac{x^{-q}}{q}$, where $\frac{1}{p}+\frac{1}{q}=1, p>1$ is
(A) 1
(B) 0
(C) 2
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:12

Problem 37

If $f(x)=\frac{x^{2}-1}{x^{2}+1}$, for every real number $x$, then the minimum value of $f$
(A) does not exist because $f$ is unboundecd
(B) is not attained even though $f$ is bounded
(C) is equal to 1
(D) is equal to - I

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:53

Problem 38

If a differentiable function $f(x)$ 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 $a>0$
(B) all $b>0$
(C) all $a$ and $b$
(D) all $b$ if $a=0$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:27

Problem 39

On the curve $x^{3}=12 y$, the abscissa changes at a faster rate than the ordinate. Then, $x$ belongs to the interval
(A) $(-4,4)$
(B) $(-3,3)$
(C) $(-2,2)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:52

Problem 40

The maximum value of radius vector where $\frac{c^{4}}{r^{2}}=\frac{a^{2}}{\sin ^{2} t}+\frac{b^{2}}{\cos ^{2} t} ;(a, b>0)$ is
(A) $(a+b)^{2}$
(B) $\frac{c^{4}}{(a+b)^{2}}$
(C) $\frac{c^{2}}{a+b}$
(D) $c^{2}(a+b)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:45

Problem 41

Let $f(x)$ and $g(x)$ be defined and differentiable for $x \geq x_{0}$ and $f\left(x_{0}\right)=g\left(x_{0}\right), f^{\prime}(x)>g^{\prime}(x)$ for $x>x_{0}$, then
(A) $f(x)<g(x), x>x_{0}$
(b) $f(x)=g(x), x>x_{0}$
(C) $f(x)>g(x), x>x_{0}$
(d) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:20

Problem 42

If $\alpha$ and $\beta(\alpha<\beta)$ be two different real roots of the equation $a x^{2}+b x+c=0$, then
(A) $\alpha>-\frac{b}{2 a}$
(B) $\beta<-\frac{b}{2 a}$
(C) $\alpha<-\frac{b}{2 a}<\beta$
(D) $\beta<-\frac{b}{2 a}<\alpha$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:19

Problem 43

If $f^{\prime}(x)=\frac{1}{1+x^{2}}$ for all $x$ and $f(0)=0$, then
(A) $f(2)<0.4$
(B) $f(2)>2$
(C) $0.4<f(2)<2$
(D) $f(2)=2$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 44

The interval in which $\lambda$ should be if $f(x)=\sin ^{3} x+\lambda$ $\sin ^{2} x(-\pi / 2<x<\pi / 2)$ has exactly one maximum and one minimum is
(A) $(-1,1)$
(B) $\left(-\frac{1}{2}, \frac{1}{2}\right)$
(C) $\left(\frac{-3}{2}, \frac{3}{2}\right)$
(D) $\left(\frac{-3}{2}, 0\right) \cup\left(0, \frac{3}{2}\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:41

Problem 45

Twenty metre of wire is available to fence off a flower bed in the form of a sector. If the flower bed has the maximum surface then radius is
(A) 10
(B) $5 / 2$
(C) 5
(D) $15 / 2$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:54

Problem 46

If $f^{\prime \prime}(x)>0, \forall x \in R, f^{\prime}(3)=0$ and $g(x)=f\left(\tan ^{2} x-2\right.$
$\tan x+4), 0<x<\pi / 2$, then $g(x)$ is increasing in
(A) $\left(0, \frac{\pi}{4}\right)$
(B) $\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
(B) $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
(D) $\left(0, \frac{\pi}{2}\right)$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:12

Problem 47

The normal to the curve $x=a(1+\cos \theta), y=a \sin \theta$
at $\theta$ always passes through the fixed point
(A) $(a, a)$
(B) $(a, 0)$
(C) $(0, a)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:39

Problem 48

If the tangent to the curve $2 y^{3}=a x^{2}+x^{3}$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes such that $\alpha^{2}+\beta^{2}=61$, then $a=$
(A) $\pm 30$
(B) $\pm 5$
(C) $\pm 6$
(D) $\pm 61$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:21

Problem 49

If $x \in[0,2]$ and $g(x)=f(x)+f(2-x)$. Also, $f^{\prime \prime}(x)<0$
then $g(x)$
(A) increases in $[0,2]$
(B) decreases in $[0,2]$(C) decreases in $[0,1)$ and increases in $(1,2]$
(D) increases in $[0,1)$ and decreases in $(1,2]$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
04:33

Problem 50

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 the radius of the balloon decreases 40 minutes after the leakage began is
(A) $9 / 7$
(B) $7 / 9$
(C) $2 / 9$
(D) $9 / 2$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:02

Problem 51

Let $a, b \in R$ be such that the fucntion $f$ 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 $\mathbf{2}: a=\frac{1}{2}$ 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- $I$ 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.

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:18

Problem 52

Each side of a square is increasing at the uniform rate of $1 \mathrm{~m} / \mathrm{sec}$. If after some time the area of the square is increasing at the rate of $8 \mathrm{~m}^{2} / \mathrm{sec}$, then the area of square at that time in sq. meters is:
(A) 4
(B) 9
(C) 16
(D) 25

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
06:10

Problem 53

Let $a, b, c \in R, a>0$ and function $f: R \rightarrow R$ be defined by $f(x)=a x^{2}+b x+c$
Statement 1: $b^{2}<4 a c \Rightarrow f(x)>0$, for every value of $x$. Statement 2: $f$ is strictly decreasing in the interval $\left(-\infty, \frac{-b}{2 a}\right)$ and strictly increasing in the interval $\left(\frac{-b}{2 a}, \infty\right) .$
(A) Statement- 1 istrue, Statement- 2 is true, Statement-2 is a correct explanation for Statement-1.
(B) Statement- 1 is true, Statement- 2 is true, Statement-2 is not a correct explanation for Statement-1.
(C) Statement- 1 is true, Statement- 2 is false.
(D) Statement- 1 is false, Statement- 2 is true.

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:51

Problem 54

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

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
05:03

Problem 55

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 $P(-1)<P(1)$, then in the interval $[-1,1]$
(A) $P(-1)$ is the minimum and $P(1)$ is the maximum of $P$
(B) $P(-1)$ is not minimum but $P(1)$ is the maximum of $P$
(C) $P(-1)$ is the minimum and $P(1)$ is not the maximum of $P$
(D) neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:16

Problem 56

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:56

Problem 57

The fraction exceeding its $p$ th power by the greatest number possible, where $p \geq 2$, is
(A)
(B)
(C) $p^{1 / p}$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:34

Problem 58

Let the function $f(x)$ be defined as $f(x)=\left\{\begin{array}{lc}\tan ^{-1} \alpha-3 x^{2}, 0<x<1 \\ -6 x, & x \geq 1\end{array}\right.$
$f(x)$ can have a maximum at $x=1$ if the value of $\alpha$ is
(A) 0
(B) 2
(C) 1
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:45

Problem 59

Let $f(x)=\left\{\begin{array}{ll}|x|, & 0<|x| \leq 2 \\ 1, & x=0\end{array}\right.$. Then, at $x=0, f$ has
(A) a local maximum
(B) nolocal maximum
(C) a local minimum
(D) no extremum

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:08

Problem 60

Let $f(x)=\int_{0}^{x} \frac{\cos t}{t} d t(x>0)$; then for $x=(2 n+1) \frac{\pi}{2}$,
$f(x)$ has
(A) minima when $n=0,2,4, \ldots$
(B) maxima when $n=0,2,4,6, \ldots$
(C) neither max. nor min. when $n=-1,-3,-5, \ldots$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:47

Problem 61

If the equation $a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x=0$ 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) smaller than $\alpha$
(B) greater than $\alpha$(C) equal to $\alpha$
(D) greater than or equal to $\alpha$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
04:14

Problem 62

If $a, b, c$ be non-zero real numbers such that
$$
\begin{aligned}
& \int_{0}^{1}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x \\
=& \int_{0}^{2}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x=0,
\end{aligned}
$$
then the equation $a x^{2}+b x+c=0$ will have
(A) one root between 0 and 1 and other root between 1 and 2
(B) both the roots between 0 and 1
(C) both the roots between 1 and 2
(D) None of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:55

Problem 63

Let $f$ be a function which is continuous and differentiable for all real $x$. If $f(2)=-4$ and $f^{\prime}(x) \geq 6$ for all $x \in[2,4]$, then
(A) $f(4)<8$
(B) $f(4) \geq 8$
(C) $f(4) \geq 12$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:30

Problem 64

If $a x+\frac{b}{x} \geq c$ for all positive $x$, where $a, b>0$, then
(A) $a b<\frac{c^{2}}{4}$
(B) $a b \geq \frac{c^{2}}{4}$
(C) $a b \geq \frac{c}{4}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:20

Problem 65

Let $f$ be $a$ continuous, diferentiable and bijective function. If the tangent to $y=f(x)$ at $x=a$ is also the normal to $y=f(x)$ at $x=b$, then there exists at least one $c \in(a, b)$ such that
(A) $f^{\prime}(c)=0$
(B) $f^{\prime}(c)>0$
(C) $f^{\prime}(c)<0$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:19

Problem 66

The values of $k$ for which the function $f(x)=k x^{3}-9 x^{2}+9 x+3$ may be increasing on $R$ are
(A) $k>3$
(B) $k<3$
(C) $k \leq 3$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:45

Problem 67

The least possible value of $k$ for which the function $f(x)=x^{2}+k x+1$ may be increasing on $[1,2]$ is
(A) 2
(B) $-2$
(C) 0
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
04:21

Problem 68

Let $a+b=4, a<2$ and $g(x)$ be a monotonically increasing function of $x$. Then, $f(a)=\int_{0}^{a} g(x) d x+\int_{0}^{b} g(x) d x$
(A) increases with increase in $(b-a)$
(B) decreases with increase in $(b-a)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:50

Problem 69

The equation $x+e^{x}=0$ has
(A) only one real root
(B) only two real roots
(C) no real root
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:53

Problem 70

The value of $a$ in order that
$f(x)=\sin x-\cos x-a x+b$
decreases for all real values is given by
(A) $a \geq \sqrt{2}$
(B) $a<\sqrt{2}$
(C) $a \geq 1$
(D) $a<1$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:23

Problem 71

If $f^{\prime \prime}(x)<0 \forall x \in(a, b)$, then $f^{\prime}(x)=0$
(A) exactly once in $(a, b)$
(B) at most once in $(a, b)$
(C) at least once in $(a, b)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:35

Problem 72

The minimum value of $a \tan ^{2} x+b \cot ^{2} x$ equals the maximum value of $a \sin ^{2} \theta+b \cos ^{2} \theta$ where $a>b>0$, when
(A) $a=b$
(B) $a=2 b$
(C) $a=3 b$
(D) $a=4 b$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:07

Problem 73

If $f(x)=\frac{x^{2}-1}{x^{2}+1}$, for every real number, then minimum value of $f$
(A) Does not exist
(B) Is note attained even through $f$ is bounded
(C) Is equal to 1
(D) Is equal to $-1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:51

Problem 74

If $y=a \log |x|+b x^{2}+x$ has its extremum values at $x=-1$ and $x=2$, then
(A) $a=2, b=-1$
(B) $a=2, b=-1 / 2$
(C) $a=-2, b=1 / 2$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:04

Problem 75

If $f(x)$ and $g(x)$ are differentiable functions for $0 \leq x \leq 1$ such that $f(0)=2, g(0)=0, f(1)=6, g(1)=2$, then in the interval $(0,1)$,
(A) $f^{\prime}(x)=0$ for all $x$
(B) $f^{\prime}(x)=2 g^{\prime}(x)$ for at least one $x$
(C) $f^{\prime}(x)=2 g^{\prime}(x)$ for at most one $x$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:24

Problem 76

The difference between the greatest and least values of the function $f(x)=\cos x+\frac{1}{2} \cos 2 x-\frac{1}{3} \cos 3 x$ is
(A) $2 / 3$
(B) $8 / 7$
(C) $9 / 4$
(D) $3 / 8$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:38

Problem 77

A function $f$ is such that $f^{\prime}(4)=f^{\prime \prime}(4)=0$ and $f$ has minimum value 10 at $x=4$. Then $f(x)=$
(A) $4+(x-4)^{4}$
(B) $10+(x-4)^{4}$
(C) $(x-4)^{4}$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:54

Problem 78

The range of values of $k$ for which the function $f(x)=\left(k^{2}-7 k+12\right) \cos x+2(k-4) x+\log 2$
does not possess critical points, is
(A) $(1,5)$
(B) $(1,5)-\{4\}$
(C) $(1,4)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:53

Problem 79

If a differentiable function $f(x)$ 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 $a>0$
(B) all $b>0$
(C) all $a$ and $b$
(D) $\mathrm{all} b$ if $a=0$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:45

Problem 80

Let $f(x)$ and $g(x)$ be defined and differentiable for $x \geq x_{0}$ and $f\left(x_{0}\right)=g\left(x_{0}\right), f^{\prime}(x)>g^{\prime}(x)$ for $x>x_{0}$, then
(A) $f(x)<g(x), x>x_{0}$
(b) $f(x)=g(x), x>x_{0}$
(C) $f(x)>g(x), x>x_{0}$
(d) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:20

Problem 81

If $\alpha$ and $\beta(\alpha<\beta)$ be two different real roots of the equation $a x^{2}+b x+c=0$, then
(A) $\alpha>-\frac{b}{2 a}$
(B) $\beta<-\frac{b}{2 a}$
(C) $\alpha<-\frac{b}{2 a}<\beta$
(D) $\beta<-\frac{b}{2 a}<\alpha$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:51

Problem 82

If $p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}$ and $|p(x)| \leq\left|e^{x-1}-1\right|$
for all $x \geq 0$, then $\left|a_{1}+2 a_{2}+3 a_{3}+\ldots+n a_{n}\right|$
(A) $\leq 1$
(B) $\geq 1$
(C) $\geq 0$
(D) $\leq 0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:52

Problem 83

The maximum value of radius vector where $\frac{c^{4}}{r^{2}}=\frac{a^{2}}{\sin ^{2} t}+\frac{b^{2}}{\cos ^{2} t} ;(a, b>0)$ is
(A) $(a+b)^{2}$
(B) $\frac{c^{4}}{(a+b)^{2}}$
(C) $\frac{c^{2}}{a+b}$
(D) $c^{2}(a+b)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:35

Problem 84

Let $f(x)=\left\{\begin{array}{cc}-x^{3}+\log _{2} b & 0<x<1 \\ 3 x & x \geq 1\end{array}\right.$. Then, the set of
values of $b$ for which $f(x)$ has least value at $x=1$ is
(A) $R^{+}$
(B) $(0,16]$
(C) $[16, \infty)$
(D) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
04:35

Problem 85

The second drivative $f^{\prime \prime}(x)$ of the function $f(x)$ exists for all $x$ in $[0,1]$ and satisfies $\left|f^{\prime \prime}(x)\right| \leq 1 .$ If $f(0)=f(1)$, then for all $x$ in $[0,1]$
(A) $\left|f^{\prime}(x)\right|<1$
(B) $\left|f^{\prime}(x)\right|>1$
(C) $\left|f^{\prime}(x)\right|=1$
(D) $f(x)$ is constant

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:58

Problem 86

Let the function $f$ be defined as $f(x)=\left\{\begin{aligned} \frac{P(x)}{x-2}, & x \neq 2 \\ 7, & x=2 \end{aligned}\right.$
where $P(x)$ is a polynomial such that $P^{\prime \prime \prime}(x)$ is identically equal to 0 and $P(3)=9 .$ If $f(x)$ is continuous at $x=2$, then
(A) $P(x)=2 x^{2}-x-6$
(B) $P(x)=2 x^{2}+x-6$
(C) $P(x)=2 x^{2}-x+6$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 87

The equation $x^{5}-3 x-1=0$ has, in the interval $[1,2]$
(A) at least one root
(B) at most one root
(C) no root
(D) a unique root

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:48

Problem 88

If the equation $x-\sin x=k$ has a unique root in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, then the range of values of $k$ are
(A) $\left(1-\frac{\pi}{2}, \frac{\pi}{2}-1\right)$
(B) $\left[1-\frac{\pi}{2}, \frac{\pi}{2}-1\right]$
(C) $\left[0, \frac{\pi}{2}+1\right]$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:11

Problem 89

The largest term in the sequence
$a_{n}=\frac{n}{n^{2}+10}, n \in N$ is
(A) $\frac{4}{26}$
(B) $\frac{3}{19}$
(C) $\frac{7}{18}$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:08

Problem 90

The range of values of $a$ for which all roots of the equation $3 x^{4}+4 x^{3}-12 x^{2}+a=0$ are real and distinct is
(A) $(0,5)$
(B) $(1,4)$
(C) $(-1,5)$
(D) None of these

Patha Sharma
Patha Sharma
Numerade Educator
03:05

Problem 91

If $\phi(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)<0$ in $(-1,1)$, then $\phi(x)$ strictly increases in the interval
(A) $\left(0, \frac{1}{2}\right)$
(B) $\left(\frac{1}{2}, 1\right)$
(C) $(-1,0)$
(D) $(0,1)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:58

Problem 92

$f(x)$ is a cubic function with $f(1)=-6, f(-1)=10$ and has maxima at $x=-1$. If $f^{\prime}(x)$ has minima at $x=1$, then
(A) $f(x)=x^{3}+3 x^{2}-9 x+5$
(B) $f(x)=x^{3}-3 x^{2}-9 x+5$
(C) $f(x)=x^{3}-3 x^{2}+9 x+5$
(D) $f(x)=x^{3}-3 x^{2}-9 x+5$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:07

Problem 93

If the function $f(x)= \begin{cases}-x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3 b+2}, & 0 \leq x<1 \\ 2 x-3, & 1 \leq x \leq 3\end{cases}$
has the least value at $x=1$, then all possible real values of $b$ are
(A) $(-1,1)$
(B) $(-2,-1) \cup[1, \infty)$
(C) $(-2,1)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 94

The function $f(x)=\frac{|x+1|}{x^{2}}$ is strictly decreasing in the interval
(A) $(-\infty,-2) \cup(0,1)$
(B) $(-2,0) \cup(1, \infty)$
(C) $(-2,-1) \cup(0, \infty)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:03

Problem 95

If the equation $a x^{2}+b x+c=0, a, b, c, \in R$ has at least one root in $(0,1)$, then
(A) $2 a+3 b+6 c=0$
(B) $a+3 b+6 c=0$
(C) $2 a+b+6 c=0$
(D) $2 a+3 b+c=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:23

Problem 96

The range of values of $a$ so that the equation $x^{3}-3 x+$ $a=0$ has three real and distinct roots is
(A) $(-\infty,-2) \cup(2, \infty)$
(B) $(-2,0)$
(C) $(-2,0)$
(D) $(-2,2)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:18

Problem 97

The curves $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{a_{1}}+\frac{y^{2}}{b_{1}}=1$ will cut
orthogonally if
(A) $a+b=a_{1}+b_{1}$
(B) $a-b=a_{1}-b_{1}$
(C) $\frac{1}{a}-\frac{1}{b}=\frac{1}{a_{1}}-\frac{1}{b_{1}}$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:54

Problem 98

If $x$ and $y$ are the sides of two squares such that $y=x$ $-x^{2}$, then the rate of change of the area of the second square with respect to the first square is
(A) $2 x^{2}-3 x+1$
(B) $2 x^{2}+3 x+1$
(C) $2 x^{2}-3 x-1$
(D) $2 x^{2}+3 x-1$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:16

Problem 99

The point on the curve $3 x^{2}-4 y^{2}=72$ which is nearest to the line $3 x+2 y+1=0$ is
(A) $(6,-3)$
(B) $(6,3)$
(C) $(-6,3)$
(D) $(-6,-3)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:21

Problem 100

If the function $f(x)=\left(a^{2}-3 a+2\right) \cos \frac{x}{2}+(a-1) x$
possesses critical points, then $a$ belongs to the interval
(A) $(-\infty, 0) \cup(4, \infty)$
(B) $(-\infty, 0] \cup[4, \infty)$
(C) $(-\infty, 0] \cup\{1\} \cup[4, \infty)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 101

If the function $f(x)=\int_{0}^{x} \mid \log _{2}\left(\log _{3}\left(\log _{4}(\cos t\right.\right.$
$+a$ ) ) $\mid d t$, be increasing for all real values of $x$, then
(A) $a \geq 2$
(B) $a \geq 5$
(C) $a<5$
(D) $a<2$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:13

Problem 102

The value of $n$, for which the function $f(x)=\left(x^{2}-4\right)^{n}$ $\left(x^{2}-x+1\right), n \in N$ assumes a local minima at $x=2$, is
(A) an even number
(B) an odd number
(C) an irrational number
(D) cannot be determined

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:42

Problem 103

If the function $f(x)=\left(1-\frac{\sqrt{21-4 b-b^{2}}}{b+1}\right) x^{3}$
$+5 x+\sqrt{16}$ increases for all $x$, then
(A) $b \in(-1,2)$
(B) $b \in(-7,3)-\{-1\}$
(C) $b \in(-7,-1) \cup(2,3)$
(D) None of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:18

Problem 104

The range of parameter $b$, for which the function
$f(x)=\int_{0}^{x}\left(b l^{2}+b+\cos t\right) d t$
is entirely increasing or decreasing for all real values of $x$ is
(A) $[-1,1]$
(B) $(-\infty,-1] \cup[1, \infty)$
(C) $(-\infty,-1) \cup(1, \infty)$
(D) $(-1,1)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:19

Problem 105

Let $f(x)=(x-3)(x-4)(x-4)(x-5)(x-6)$, then
(A) $f^{\prime}(x)$ has four roots
(B) three roots of $f^{\prime}(x)=0$ lie in $(3,4) \cup(4,5) \cup(5,6)$
(C) the equation $f^{\prime}(x)=0$ has only one root
(D) three roots of $f^{\prime}(x)=0$ lie in $(2,3) \cup(3,4) \cup(4,5)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:11

Problem 106

For $a \in[\pi, 2 \pi]$ and $n \in Z$, the critical points of $f(x)=$ $\frac{1}{3} \sin a \tan ^{3} x+(\sin a-1) \tan x+\sqrt{\frac{a-2}{8-a}}$ are
(A) $x=n \pi$
(B) $x=2 n \pi$
(C) $x=(2 n+1) \pi$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 107

Let $f^{\prime \prime}(x)>0 \forall x \in R$ and $g(x)=f(2-x)+f(4+x)$.
Then, $g(x)$ is increasing in
(A) $(-\infty,-1)$
(B) $(-\infty, 0)$
(C) $(-1, \infty)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:52

Problem 108

The curves $x^{2}-4 y^{2}+c=0$ and $y^{2}=4 x$ will cut orthogonally for
(A) $c \in(0,16)$
(B) $c \in(-3,4)$
(C) $c \in(3,4)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:16

Problem 109

Which of the following is not true? The function $f(x)=x^{2}+\frac{\lambda}{x}$ has a
(A) minimum at $x=2$ if $\lambda=16$
(B) maximum at $x=2$ if $\lambda=16$
(C) maximum for no real value of $\lambda$
(D) point of inflexion at $x=1$ if $\lambda=-1$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:41

Problem 110

If the parabola $y=f(x)$, having axis parallel to the $y$-axis, touches the line $y=x$ at $(1,1)$, then
(A) $2 f^{\prime}(0)+f(0)=1$
(B) $2 f(0)+f^{\prime}(0)=1$
(C) $2 f(0)-f^{\prime}(0)=1$
(D) $2 f^{\prime}(0)-f(0)=1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:39

Problem 111

The angle between the tangents at any point $P$ and the line joining $P$ to the origin $O$, where $P$ is a point on the curve $\ln \left(x^{2}+y^{2}\right)=c \tan ^{-1} \frac{y}{x}, c$ is a constant
(A) varies as $\tan ^{-1} x$
(B) varies as $\tan ^{-1} y$
(C) is a constant
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:55

Problem 112

If the equation $a x^{2}+b x+c=0$ has two distinct positive roots, then the equation $a x^{2}+(b+6 a) x+$ $(c+3 b)=0$ has
(A) two positive roots
(B) exactly one positive root
(C) at least one positive root
(D) no positive root

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 113

If $f(x)$ is continuous in $[a, b]$ and differentiable in $(a, b)$ then there exists at least one $c \in(a, b)$ such that $\frac{f(b)-f(a)}{b^{3}-a^{3}}$ equals
(A) $3 c^{2} f^{\prime}(\mathrm{C})$
(B) $\frac{f^{\prime}(c)}{3 c^{2}}$
(C) $f(c) f^{\prime}(C)$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:03

Problem 114

Let $f(x)=\ln x$ and $g(x)=x^{2} .$ If $c \in(4,5)$, then $c \ln \left(\frac{4^{25}}{5^{16}}\right)$ equals
(A) $c \ln 5-8$
(B) $2\left(c^{2} \ln 4-8\right)$
(C) $2\left(c^{2} \ln 5-8\right)$
(D) $c \ln 4-8$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:14

Problem 115

$\mathrm{f} 0<x<\frac{\pi}{2}$, then
(A) $\frac{2}{\pi}>\frac{\sin x}{x}$
(B) $\frac{2}{\pi}<\frac{\sin x}{x}$
(C) $\frac{\sin x}{x}<1$
(D) $\frac{\sin x}{x}>1$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:35

Problem 116

If $0<x<\frac{\pi}{2}$, then
(A) $\cos (\sin x)>\cos x$
(B) $\cos (\sin x)<\cos x$
(C) $\cos (\sin x)>\sin (\cos x)$
(D) $\cos (\sin x)<\sin (\cos x)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:50

Problem 117

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 118

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 $(-\infty,-2)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:29

Problem 119

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 $\left(\frac{1}{2}, \infty\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 120

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 $(-\infty, \infty) \cup(1,2)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:49

Problem 121

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:59

Problem 122

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) $f(x)<g(x) \forall x \in(-\infty, 0)$
(C) $f(x)<g(x) \forall x \in(0, \infty)$
(D) $f(x)>g(x) \forall x \in(-\infty, 0)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:01

Problem 123

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:56

Problem 124

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:39

Problem 125

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

Patha Sharma
Patha Sharma
Numerade Educator
00:58

Problem 126

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

Patha Sharma
Patha Sharma
Numerade Educator
02:12

Problem 127

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:18

Problem 128

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 $\left(0, \frac{1}{2}\right)$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:56

Problem 129

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 $(-\infty, \infty) \cup(1,2)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 130

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:46

Problem 131

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) $f(x)<g(x) \forall x \in(-\infty, 0)$
(C) $f(x)<g(x) \forall x \in(0, \infty)$
(D) $f(x)>g(x) \forall x \in(-\infty, 0)$

Patha Sharma
Patha Sharma
Numerade Educator
01:32

Problem 132

If $f^{\prime}(x)>0$ and $g^{\prime}(x)<0 \forall x \in R$, then
(A) $\int \operatorname{og}(x)>\operatorname{fog}(x+1)$
(B) $\operatorname{fog}(x)>\operatorname{fog}(x-1)$
(C) $\operatorname{gof}(x)>\operatorname{gof}(x+1)$
(D) $g o f(x)>g o f(x-1)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:45

Problem 133

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) $\left(\frac{4 a}{9}, \frac{4 a}{27}\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:27

Problem 134

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) $\sqrt{2} x+y=\frac{89 \sqrt{2}}{27}+1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:34

Problem 135

Let $f(x)=\left\{\begin{array}{l}x+2,-1 \leq x<0 \\ 1, x=0 \\ \frac{x}{2}, 0<x \leq 1\end{array}\right.$
Then, on $[-1,1], f(x)$ has
(A) a minimum
(B) a maximum
(C) neither a maximum nor a minimum
(D) $f^{\prime}(0)$ does not exist

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:47

Problem 136

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

Patha Sharma
Patha Sharma
Numerade Educator
02:06

Problem 137

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 138

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:46

Problem 139

Between any two real roots of the equation $e^{x} \sin$ $x=1$, the equation $e^{x} \cos x=-1$ has
(A) at least one root
(B) exactly one root
(C) at most one root
(D) no rootPassage 3 If two functions $f$ and $g$ defined on $[a, b]$ are
1. continuous on the closed interval $[a, b]$
2. derivable on the open interval $(a, b)$
3. $g^{\prime}(x) \neq 0$ for any $x \in(a, b)$then there exists at least one real number $c \in(a, b)$ such that
$$
\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}
$$
We may write it as
$$
\frac{f(b)-f(a)}{g(b)-g(a)} g^{\prime}(c)=f^{\prime}(c)
$$
Hence, there is an ordinate $x=c$ between $x=a$ and $x=b$ such that the tangents at the points, where $x=c$ cuts the graphs of the functions $f(x)$ and $\frac{f(b)-f(a)}{g(b)-g(a)} g(x)$, are mutually parallel.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:39

Problem 140

The, value of $\mathrm{c}$ for the functions $f(x)=\sqrt{x}$ and $g(x)$ $=\frac{1}{\sqrt{x}}$ in the interval $[a, b]$ is
(A) $\sqrt{a}$
(B) $\sqrt{b}$
(C) $\sqrt{a b}$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:59

Problem 141

$\frac{\sin \alpha-\sin \beta}{\cos \beta-\cos \alpha}=F(\alpha)$, where $0<\alpha<\theta<p<\frac{\pi}{2}$
Then, $F(\theta)=$
(A) $\tan \theta$
(B) $\cot \theta$
(C) $\sin \theta$
(D) $\cos \theta$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:31

Problem 142

The value of $\mathrm{c}$ for the functions $f(x)=e^{x}$ and $g(x)=$ $e^{x}$ in the interval $(a, b)$ is
(A) $\frac{a+b}{2}$
(B) $\frac{a+b}{4}$
(C) $a+b$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:15

Problem 143

$$
\begin{array}{ll}
\text { Column-I } & \text { Column-II } \\
\hline \text { I. Let } f(x)=\left(1+b^{2}\right) x^{2}+2 b x+1 & \text { (A) }(0,1] \\
\text { and } m(b) \text { the minimum value of } \\
f(x) \text { for a given } b . \text { As } b \text { varies, the } \\
\text { range of } m(b) \text { is } \\
\text { II. The set of values of } x \text { for which } & \text { (B) }(0,1) \\
\log (1+x)<x, \text { is }
\end{array}
$$III. If $\frac{a_{0}}{n+1}+\frac{a_{1}}{n}+\frac{a_{2}}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}$
(C) $(0, \infty)$
$=0$, then the equation $a_{0} x^{n}+$ $a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}=0$ has at
least one root in the interval
IV. If $27 a+9 b+3 c+d=0$, then the
(D) $(0,3)$ equation $4 a x^{3}+3 b x^{2}+2 c x+d=$
0 has at least one real root lying in the interval

P Krishnamurthy
P Krishnamurthy
Numerade Educator
05:48

Problem 144

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 toIII. 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 $f(x)=\left|e^{a x}-e^{-a x}\right|$,
(D) $(3, \infty)$
$a>0$ is strictly increasing in the interval

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:25

Problem 145

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 $y=x^{2}-x+1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:11

Problem 146

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 $0<x<\frac{\pi}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:46

Problem 147

Assertion: If $0<x<\frac{\pi}{2}$, then $\frac{2}{\pi}<\frac{\sin x}{x}<1$
Reason: $\tan x<x$ for $0<x<\frac{\pi}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 148

Assertion: If $0<x<\frac{\pi}{2}$, then $\cos (\sin x)>\cos x>\sin$
$(\cos x)$
Reason: $\sin x<x$ for $0<x<\frac{\pi}{2}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:50

Problem 149

Assertion: If $0<\alpha<\beta<, \frac{\pi}{2}$ then $\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}$
Reason: $x \tan x$ is increasing for $0<x<\frac{\pi}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:56

Problem 150

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 $x$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:42

Problem 151

Assertion: If $f^{\prime}(x)=\frac{1}{1+x^{2}}$ for all $x$ and $f(0)=0$, then $0.4<f(2)<2$
Reason: By mean value theorem, there exists a point ce $(0,2)$ such that
$$
f^{\prime}(c)=\frac{f(2)-f(0)}{2-0}
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:10

Problem 152

Assertion: If $f(x)=\tan x, x \in\left[0, \frac{\pi}{7}\right]$, then $\frac{\pi}{7}$
$<f\left(\frac{\pi}{7}\right)<\frac{2 \pi}{7}$
Reason: $\sec ^{2} x$ is strictly increasing in $\left[0, \frac{\pi}{7}\right]$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:04

Problem 153

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 $(a, b)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:01

Problem 154

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:21

Problem 155

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 $0<p \leq 1$ decreases in $(0, \infty)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:19

Problem 156

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:34

Problem 157

Assertion: $\ln (\cos \theta)<\cos (\ln \theta)$,
where $e^{-\theta 2}<\theta<\frac{\pi}{2}$
Reason: $\ln x<x \forall x>0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:35

Problem 158

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 $\frac{\pi}{4}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:13

Problem 159

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

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:52

Problem 160

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) $1 / 3$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:14

Problem 161

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) $\frac{1}{2}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:22

Problem 162

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) $(x+1)^{2}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:12

Problem 163

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) $(a, a)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:52

Problem 164

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

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:51

Problem 165

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) $(-\infty,-4]$
$x^{3}+6 x^{2}+6$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:50

Problem 166

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) $f(6)=5$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:20

Problem 167

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) $\frac{5}{6 \pi} \mathrm{cm} / \mathrm{min}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:14

Problem 168

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

Patha Sharma
Patha Sharma
Numerade Educator
01:57

Problem 169

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) $x=1$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:22

Problem 170

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) $\frac{\pi}{4}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:54

Problem 171

The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of P, then the curve is a
(A) ellipse
(B) parabola
(C) circle
(D) hyperbola

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:22

Problem 172

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) $\log _{e^{e}}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:34

Problem 173

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) $(-2,0)$

Patha Sharma
Patha Sharma
Numerade Educator
01:36

Problem 174

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 $-\sqrt{\frac{p}{3}}$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:51

Problem 175

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

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
03:07

Problem 176

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 $P(-1)<P(1)$, then in the interval $[-1,1]$ $[2009$
(A) $P(-1)$ is the minimum and $P(1)$ is the maximum of $\mathrm{P}$
(B) $P(-1)$ is not minimum but $P(1)$ is the maximum of $\bar{P}$
(C) $P(-1)$ is the minimum and $P(1)$ is not the maximum of $P$
(D) neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$

Patha Sharma
Patha Sharma
Numerade Educator
01:43

Problem 177

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) $\frac{\sqrt{3}}{4}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:06

Problem 178

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) $y=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:21

Problem 179

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

Patha Sharma
Patha Sharma
Numerade Educator
01:37

Problem 180

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) $\frac{9}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:02

Problem 181

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

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:06

Problem 182

If the functions $f$ and $g$ are differentiable functions on $[0,1]$ satisfying $f(0)=2=g(1), g(0)=0$ and $f(1)=6$,
then for some $c \in] 0,1[$
(A) $2 f^{\prime}(C)=g^{\prime}(c)$
(B) $2 f^{\prime}(C)=3 g^{\prime}(c)$
(C) $f^{\prime}(C)=g^{\prime}(c)$
(D) $f^{\prime}(C)=2 g^{\prime}(c)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 183

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) $\alpha=2, \beta=\frac{1}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:29

Problem 184

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.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:02

Problem 185

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) $x=2 r$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:25

Problem 186

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) $\left(\frac{\pi}{6}, 0\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator