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Comprehensive Mathematics for JEE Advanced

Ravi Prakash; Ajay Kumar; Usha Gupta

Chapter 24

Applications of Derivatives - all with Video Answers

Educators

Chapter Questions

06:06

Problem 1

The minimum value of $f(x)=|3-x|+|2+x|+|5-x|$ is
(a) 0
(b) 7
(c) $\underline{8}$
(d) 10

Subash Charan
Subash Charan
Numerade Educator
01:26

Problem 2

If $f(x)=x(x-2)(x-4), 1 \leq x \leq 4$, then a number satisfying the conditions of the mean value theorem is
(a) I
(b) 2
(c) $5 / 2$
(d) $7 / 2$

Aman Gupta
Aman Gupta
Numerade Educator
02:23

Problem 3

The sum of the intercepts of a tangent to $\sqrt{x}+\sqrt{y}$ $=\sqrt{a}, a>0$ upon the coordinate axes is
(a) $2 a$
(b) $a$
(c) $a / 2$
(d) $\sqrt{a}$

Aman Gupta
Aman Gupta
Numerade Educator
01:14

Problem 4

Let $x$ and $y$ be two real numbers such that $x>0$ and $x y=1$. The minimum value of $x+y$ is
(a) 1
(b) $1 / 2$
(c) 2
(d) $1 / 4$

Aman Gupta
Aman Gupta
Numerade Educator
02:05

Problem 5

The maximum value of $\frac{x^{2}-x+1}{x^{2}+x+1}$ for all real val- ues of $x$ is
(a) $1 / 2$
(b) 1
(c) 2
(d) 3

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 6

If $y=2 x+\cot ^{-1} x+\log \left(\sqrt{1+x^{2}}-x\right)$, then $y$
(a) decreases on $(-\infty, \infty)$
(b) decreases on $[0, \infty)$
(c) neither decreases nor increases on $[0, \infty)$
(d) increases on $(-\infty, \infty)$

Aman Gupta
Aman Gupta
Numerade Educator
02:13

Problem 7

Let $g(x)=(\log (1+x))^{-1}-x^{-1}, x>0$ then
(a) $\mid<g(x)<2$
(b) $-1<g(x)<0$
(c) $0<g(x)<1$
(d) $\frac{1}{2}<g(x)<1$

Aman Gupta
Aman Gupta
Numerade Educator
01:20

Problem 8

Given $n$ real numbers $a_{1}, a_{2}, \ldots a_{n}$, the value of $x$ for which sum of the square of all the deviations is least is
(a) $a_{1}+a_{2}+\ldots+a_{n}$
(b) $2\left(a_{1}+a_{2}+\ldots+\sigma_{n}\right)$
(c) $a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}$
(d) $\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$

Aman Gupta
Aman Gupta
Numerade Educator
01:05

Problem 9

The number of solutions of the equation $a^{/ \omega i}+g(x)$ $=0$, where $a>0, g(x) \neq 0$ and $g(x)$ has minimum value $1 / 4$, is
(a) one
(b) two
(c) infinitely many
(d) zero

Aman Gupta
Aman Gupta
Numerade Educator
01:51

Problem 10

Suppose that
$$
\frac{4}{\sin x}+\frac{1}{1-\sin x}=a
$$
has at least one solution on the interval $(0, \pi / 2)$. Then $a$ has minimum value of $x=$
(a) $\sin ^{-1} 2 / 3$
(b) $\sin ^{-1} 1 / 4$
(c) $\cos ^{-1} 4 / 5$
(d) 1

Aman Gupta
Aman Gupta
Numerade Educator
02:15

Problem 11

If the tangent at $(1,1)$ on $y^{2}=x(2-x)^{2}$ meets the curve again at $P$, then $P$ is
(a) $(4,4)$
(b) $(-1,2)$
(c) $(9 / 4,3 / 8)$
(d) $(3 / 4,7 / 4)$

Aman Gupta
Aman Gupta
Numerade Educator
01:19

Problem 12

The function $\frac{\sin (x+\alpha)}{\sin (x+\beta)}$ has no maximum or minimum if (k an integer)
(a) $\beta-\alpha=k \pi$
(b) $\beta-\alpha \neq k \pi$
(c) $\beta-\alpha=2 k \pi$
(d) none of the above

Aman Gupta
Aman Gupta
Numerade Educator
01:06

Problem 13

The two curves $x^{3}-3 x y^{2}+2=0$ and $3 x^{2} y-y^{3}-2=0$
(a) cut at right angles
(b) touch each other
(c) cut at an angle $\pi 3$
(d) cut at an angle $\pi / 4$

Aman Gupta
Aman Gupta
Numerade Educator
02:04

Problem 14

If $x \cos \alpha+y \sin \alpha=p$ touches $x^{2}+a^{2} y^{2}=a^{2}$, then
(a) $p^{2}=a^{2} \sin ^{2} \alpha+\cos ^{2} \alpha$
(b) $p^{2}=a^{2} \cos ^{2} \alpha+\sin ^{2} \alpha$
(c) $1 / p^{2}=\sin ^{2} \alpha+\alpha^{2} \cos ^{2} \alpha$
(d) $1 / p^{2}=\cos ^{2} \alpha+a^{2} \sin ^{2} \alpha$

Aman Gupta
Aman Gupta
Numerade Educator
02:09

Problem 15

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) $(-3 \sqrt{3},-2 \sqrt{3}) \cup(2 \sqrt{3}, 3 \sqrt{3})$

Aman Gupta
Aman Gupta
Numerade Educator
01:29

Problem 16

Three normals are drawn to the parabola $y^{2}=4 x$ from the point $(c, 0)$. These normals are real and distinet when
(a) $c=0$
(b) $c=1$
(c) $c=2$
(d) $c=3$

Aman Gupta
Aman Gupta
Numerade Educator
02:22

Problem 17

The function $f(x)=(\log (x-1))^{2}(x-1)^{2}$ has
(a) local extremum at $x=1$
(b) point of inflection at $x=1$
(c) local extremum at $x=2$
(d) point of inflection at $x=2$

Aman Gupta
Aman Gupta
Numerade Educator
01:39

Problem 18

Given the function $f(x)=x^{2} e^{-2 x}, x>0 .$ Then $f(x)$ has the maximum value equal to
(a) $e^{-t}$
(b) $(2 e)^{-1}$
(c) $e^{-2}$
(d) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:20

Problem 19

Let $f(x)=(x-4)(x-5)(x-6)(x-7)$ then
(a) $f^{\prime}(x)=0$ has four real roots
(b) three roots of $f^{\prime}(x)=0$ lie in $(4,5) \cup(5,6)$ $\cup(6,7)$
(c) the equation $f^{\prime}(x)=$ has only two roots
(d) three roots of $f^{\prime}(x)=0$ lie in $(3,4) \cup(4,5)$ $\cup(5,6)$

Aman Gupta
Aman Gupta
Numerade Educator
03:33

Problem 20

The points on the curve $5 x^{2}-6 x y+5 y^{2}=4$ that are the nearest the origin are
(a) $(1 / 2,-1 / 2),(-1 / 2,1 / 2)$
(b) $(0,2 / \sqrt{5}),(0,-2 / \sqrt{5})$
(c) $(2 \sqrt{5}, 0),(-2 / \sqrt{5}, 0)$
(d) $(2 / \sqrt{3}, 0),\left(\frac{2}{\sqrt{5}}, 1\right)$

Aman Gupta
Aman Gupta
Numerade Educator
01:30

Problem 21

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 $q$. Then $p: q$ is
(a) $9: 4$
(b) $8: 3$
(c) $7: 2$
(d) $5: 3$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:43

Problem 22

The point of intersection of the tangents drawn to the curve $x^{2} y=1-y$ at the points where it is met by the curve $x y=1-y$ is given by
(a) $(0,-1)$
(b) $(1,1)$
(c) $(0,1)$
(d) $(1,-1)$

Aman Gupta
Aman Gupta
Numerade Educator
01:35

Problem 23

The equation of the tangent to the curve $y=$ $(2 x-1) e^{2(1-x)}$ at the point of its maximum is
(a) $y=1$
(b) $x=1$
(c) $x+y=1$
(d) $x-y=-1$

Aman Gupta
Aman Gupta
Numerade Educator
01:43

Problem 24

The distance of the point on $y=x^{4}+3 x^{2}+2 x$ which is nearest to the line $y=2 x-1$ is
(a) $4 / \sqrt{5}$
(b) $3 \sqrt{5}$
(c) $2 \sqrt{5}$
(d) $1 / \sqrt{5}$

Aman Gupta
Aman Gupta
Numerade Educator
01:12

Problem 25

If the function $f(x)=x^{2}+a / x$ has a local minimum at $x=2$, then the value of $a$ is
(a) 8
(b) 16
(c) 18
(d) 12

Aman Gupta
Aman Gupta
Numerade Educator
01:59

Problem 26

The coordinates of the point on the curve $\left(x^{2}+1\right)(y-3)=x$ where a tangent to the curve has the greatest slope are given by
(a) $(\sqrt{3}, 3+\sqrt{3} / 4)$
(b) $(-\sqrt{3}, 3-\sqrt{3} / 4)$
(c) $(0,3)$
(d) $(\sqrt{3}, \sqrt{3} / 4)$

Aman Gupta
Aman Gupta
Numerade Educator
01:08

Problem 27

The critical points of the function $f(x)=$ $(x-2)^{23}(2 x+1)$ are
(a) $-1$ and 2
(b) 1
(c) 1 and $-1 / 2$
(d) $\mid$ and 2

Aman Gupta
Aman Gupta
Numerade Educator
01:37

Problem 28

The function $f(x)=\sin x \cos ^{2} x$ has extremum at
(a) $x=\pi / 2$
(b) $x=\cos ^{-1}(v \sqrt{3})$
(c) $x=\cos ^{-1}(-\sqrt{23})$
(d) $x=\cos ^{-1}(-\sqrt{2 / 3})$

Aman Gupta
Aman Gupta
Numerade Educator
01:43

Problem 29

'The function $f(x)=2 \log (x-2)-x^{2}+4 x+1$
increases in the interval
(a) $(1,2)$
(b) $(2,3)$
(c) $(5 / 2,3)$
(d) $(2,4)$

Aman Gupta
Aman Gupta
Numerade Educator
02:06

Problem 30

The critical points of the function $f^{\prime}(x)$, where $f(x)$ $=\frac{|x-2|}{x^{2}}$ are
(a) 0
(b) 1
(c) 3
(d) $-1$

Aman Gupta
Aman Gupta
Numerade Educator
01:51

Problem 31

$y=\log x$ satisfies for $x>1$, the equality
(a) $x-1>y$
(b) $x^{2}-1>y$
(c) $y>x-1$
(d) $(x-1) / x<y$

Aman Gupta
Aman Gupta
Numerade Educator
03:59

Problem 32

The interval $(s)$ of decrease of the function $f(x)=$ $x^{2} \log 27-6 x \log 27+\left(3 x^{2}-18 x+24\right) \log \left(x^{2}-6 x\right.$
$+8)$ is
(a) $(3-\sqrt{1+13 e}, 2)$
(b) $(4,3+\sqrt{1+1 / 3 e})$
(c) $(3,4+\sqrt{1+1 / 3 e})$
(d) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:37

Problem 33

If the line, $\omega x+b y+c=0$ is a normal to the curve $x y=2$, then
(a) $a<0, b>0$
(b) $a>0, b<0$
(c) $a>0, b>0$
(d) $a<0, b<0$

Aman Gupta
Aman Gupta
Numerade Educator
01:58

Problem 34

The normal to the curve given by $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$ at any
point $\theta$ is such that it
(a) makes a constant angle with x-axis
(b) is at a constant distance from the origin
(c) touches a fixed circle
(d) passes through the origin

Aman Gupta
Aman Gupta
Numerade Educator
02:10

Problem 35

The funetion $f(x)=\frac{\log (\pi+x)}{\log (e+x)}$ is
(a) increasing on $[0, \infty)$
(b) decreasing on $[0, \infty)$
(c) increasing on $\left[0, \pi^{\prime} e\right)$ and decreasing on $[\pi e, \infty)$
(d) decreasing on $[0, \pi e)$ and increasing on $[\pi l e, \infty)$

Aman Gupta
Aman Gupta
Numerade Educator
02:03

Problem 36

$e^{x}>10 x$ is satisfied
(a) for all $x$
(b) for all $x>3$
(c) for $x$ only if $x \geq 20$
(d) for all $x>18$

Aman Gupta
Aman Gupta
Numerade Educator
02:13

Problem 37

An extremum value of the function $f(x)=$ $\left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}(-1<x<1)$ is
(a) $7 \pi^{3} / 8$
(b) $\pi^{3} / 8$
(c) $\pi^{3} / 32$
(d) $\pi^{3} / 16$

Aman Gupta
Aman Gupta
Numerade Educator
01:43

Problem 38

Let $f(x)=\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$ defined on $[-1,1]$ then
(a) the greatest value of $f(x)$ is I
(b) the greatest value of $f(x)$ is $3 \sqrt{8}$
(c) the least value of $f(x)$ is 0
(d) the least value of $f(x)$ is $-1$.

Aman Gupta
Aman Gupta
Numerade Educator
01:16

Problem 39

Let $f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+1$ then
(a) $f$ increases on $[1, \infty)$
(b) $f$ decreases on $[1, \infty)$
(c) $f$ has a minimum at $x=1$
(d) $f$ has neither maximum nor minimum

Aman Gupta
Aman Gupta
Numerade Educator
01:10

Problem 40

The families of curves defined by the equations $y=a x, y^{2}+x^{2}=c^{2}$ are perpendicular for
(a) $a=2, c=4$
(b) $a=-2, c=3$
(c) $a=3, c=2$
(d) $a=3, c=-2$

Aman Gupta
Aman Gupta
Numerade Educator
00:25

Problem 41

(a) $x^{2} e^{-x}$
(b) $\frac{4 x}{x^{2}+4}$
(c) $-x^{2} \sqrt[3]{(x-2)^{2}}$
(d) $\frac{14}{x^{4}-8 x^{2}+2}$
(p) $-1$
(q) 0
(r) $f(2)$
(s) $\rfloor$

Brandon Fox
Brandon Fox
Numerade Educator
03:47

Problem 42

Match the difference of greatest and least value of the functions $f$ on L.H.S
(a) $\sin x \sin 2 x$ on $(-\infty, \infty)$
(b) $2 x^{3}-3 x^{2}-12 x+1$
on $[-2,5 / 2]$
(c) $\cos ^{-1} x^{2}$ on $[-\sqrt{2} / 2, \sqrt{2} / 2]$
(d) $x+\sqrt{x}$ on $[0,4]$

(p) $\pi 6$
(q) $8 / 3 \sqrt{3}$
(r) 6
(s) 27

Ankit Singh
Ankit Singh
Numerade Educator
01:37

Problem 43

For the function on L.H.S
(a) $\cos x-1$
$+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}$
(b) $\cos x-1+\frac{x^{2}}{2 !}$
(c) $x^{4} e^{-x^{2}}$
(d) $\sin 3 x-3 \sin x$

(p) minimum value is $-4$
(q) there is no extremum at $x=0$
(r) the minimum is $f(0)=0$
(s) the functions reaches maximum at $x=\sqrt{2}$

Aman Gupta
Aman Gupta
Numerade Educator
05:07

Problem 44

The functions defined have domain $[-1,1]$ and has values in $[-12,8]$
(a) $2 x^{3}-3 x^{2}-12 x+1$
(b) $\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$
(c) $\frac{1-x+x^{2}}{1+x+x^{2}}$
(d) $x^{3}-3 x^{2}+6 x-2$

(p) is one-one but not onto
(q) is one-one and onto
(r) is ncither onc-one nor onto
(s) is not onc-one but onto

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:32

Problem 45

Statement-1: Let $f(x)=\frac{20}{4 x^{3}-9 x^{2}+6 x}$ then the
range $f=[6,20]$
Statement-2: $\int$ increases on $(1 / 2,1)$ and decreases on $(1, \infty) \cup(-\infty, 1 / 2)$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:12

Problem 46

Statement-1: Let $f(x)=2 \tan ^{-1} \frac{1-x}{1+x}$ on $[0,1]$ then
the range of $f=[0, \pi / 2]$
Statement-2: $f$ decreases from $\pi / 4$ to $0 .$

Rukhmani Jain
Rukhmani Jain
Numerade Educator
01:40

Problem 47

Statement- $1: e^{x}+e^{-x}>2+x^{2}, x \neq 0$
Statement-2: $f(x)=e^{x}+e^{-x}-2-x^{2}$ is an increasing function.

Aman Gupta
Aman Gupta
Numerade Educator
01:12

Problem 48

The profit function represents a
(a) straight line
(b) a parabola with vertex at $(20,700 / 3)$
(c) a parabola with vertex at $\left(\frac{50}{3}, \frac{700}{3}\right)$ and focus $\left(\frac{50}{3}, \frac{2199}{3}\right)$
(d) a parabola with vertex at $\left(\frac{50}{3}, \frac{700}{3}\right)$ and focus $\left(\frac{50}{3}, \frac{2099}{3}\right)$

Aman Gupta
Aman Gupta
Numerade Educator
01:04

Problem 49

Integral value of $x$ for which the profit function is maximum
(a) 24
(b) 22
(c) 17
(d) 16

Aman Gupta
Aman Gupta
Numerade Educator
01:45

Problem 50

If $P$ is the profit function then
(a) $P$ increases on $[8,25]$
(b) $P$ decreases on $[8,25]$
(c) $P$ decreases on $\left[16 \frac{2}{3}, 25\right]$
(d) P increases on $[8,22]$

Aman Gupta
Aman Gupta
Numerade Educator
02:20

Problem 51

The largest term of $a_{n}=n^{2} /\left(n^{3}+200\right)$ is
(a) $29 / 453$
(b) $49 / 543$
(c) $43 / 543$
(d) $41 / 451$

Aman Gupta
Aman Gupta
Numerade Educator
03:11

Problem 52

The largest term of the sequence
$$
a_{a}=n\left(n^{2}+10\right) \text { is }
$$
(a) $3 / 19$
(b) $2 / 13$
(c) 1
(d) $1 / 7$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:16

Problem 53

If $f(x)$ is the function required to find largest term in
Q.51 then
(a) $f$ increases for all $x$
(b) $f$ decreases for all $x$
(c) $f$ has a maximum at $x=\sqrt[3]{400}$
(d) $f$ increases on $[0,9]$.

Aman Gupta
Aman Gupta
Numerade Educator
02:01

Problem 54

If $A$ is the area of the triangle formed by positive $x$ -axis and the normal and the tangent to the circle $x^{2}$ $+y^{2}=4$ at $(1, \sqrt{3})$ then $A \sqrt{3}$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 55

The minimum value of $\sqrt{e^{x^{2}}-1}$ is

Aman Gupta
Aman Gupta
Numerade Educator
01:21

Problem 56

Let $f(x)=\left\{\begin{array}{cl}|x-1|+a & , x \leq 1 \\ 2 x+3 & , x>1\end{array}\right.$
If $f(x)$ has local minimum at $x=1$ and $a \geq 5$ then $a$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
01:18

Problem 57

Let $P$ be a variable point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, with foci $F_{1}$ and $F_{2}$. If $A$ is the area of the triangle $P F_{1} F_{2}$, then the maximum value of $\frac{1}{3} A$ is

Aman Gupta
Aman Gupta
Numerade Educator
01:30

Problem 58

The maximum value of $e \mid x \log x$ / for $0<x \leq 1$ is.

Aman Gupta
Aman Gupta
Numerade Educator
02:18

Problem 59

If $f(x)=\log _{x} 1 / 9-\log _{3} x^{2}(x>1)$ then $\max f(x)$ is
cqual to

Aman Gupta
Aman Gupta
Numerade Educator
02:24

Problem 60

Let $f(x)=\cos ^{2} x+\cos x+3$ then greatest value of $f(x)+4$ least value of $f(x)-9$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
01:22

Problem 61

If $f(x)=e^{2^{2}-4 x+3}$ on $[-5,5]$ then $\frac{1}{12} \log$ (greatest value of $f(x))$ is

Aman Gupta
Aman Gupta
Numerade Educator
02:49

Problem 62

If $(u, v)$ is a point on $4 x^{2}+a^{2} y^{2}=4 a^{2}$, where $4<a^{2}<8$, that is farthest from the point $(0,-2)$ then $u+v$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
03:23

Problem 63

Let $f(x)$ be a polynomial of degree 6, which sutisfies
$\lim _{x \rightarrow 0}\left(1+\frac{f(x)}{x^{3}}\right)^{\sqrt{x}}=e^{2}$ and has local maximum at $x$
$=1$ and local minimum at $x=0$ and 2 then $\frac{5}{112} f(3)$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
02:08

Problem 64

If $y=f(x)$ is represented as $x=g(0)=r^{5}-5 r^{3}-20 r$ $+7$ and $y=h(t)=4 r^{3}-3 r^{2}-18 t+3(-2<t<2)$
then max $f(x)-9$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator
01:23

Problem 65

If $f(x)=x^{2} \log x \operatorname{on}[1, e]$ then $\log$ (greatest of $f(x)$
- Least of $f(x))$ is equal to

Aman Gupta
Aman Gupta
Numerade Educator