Application of Derivatives · Mathematics · MHT CET

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MCQ (Single Correct Answer)

1

A manufacturer produces $x$ items per week at a total cost of ₹ $\left(x^2+78 x+2500\right)$. The price per unit is given by $8 x=600-\mathrm{p}$ where ' p ' is the price of each unit. Then the maximum profit obtained is

MHT CET 2025 20th April Evening Shift
2

If $2 \mathrm{f}(x)+3 \mathrm{f}\left(\frac{1}{x}\right)=x^2+1, x \neq 0$ and $y=5 x^2 \mathrm{f}(x)$, then $y$ is strictly increasing in

MHT CET 2025 20th April Evening Shift
3

If the curve $y=a x^2-6 x+b$ passes through $(0,4)$ and has its tangent parallel to the X-axis at $x=\frac{3}{2}$, then the values of $a$ and $b$ respectively are

MHT CET 2025 20th April Evening Shift
4

20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are

MHT CET 2025 20th April Morning Shift
5

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

MHT CET 2025 20th April Morning Shift
6

If the curves $y^2=6 x$ and $9 x^2+b y^2=16$ intersect each other at right angles, then the value of $b$ is

MHT CET 2025 20th April Morning Shift
7

The position of a point in time $t$ is given by $x=\mathrm{a}+\mathrm{bt}-\mathrm{ct}^2, y=\mathrm{at}+\mathrm{bt}^2$. It's resultant acceleration at time $t$ in seconds is given by

MHT CET 2025 19th April Evening Shift
8

The equation of tangent to the curve $y=\cos (x+y)$ where $-2 \pi \leq x \leq 2 \pi$ and which is parallel to the line $x+2 y=0$, is

MHT CET 2025 19th April Evening Shift
9

If two curves $x^2-4 y^2=2$ and $8 x^2=40-\mathrm{m} y^2$ are orthogonal to each other then $\mathrm{m}=$

MHT CET 2025 19th April Evening Shift
10
A population $p(t)$ of 1000 bacteria introduced into a nutrient medium grows according to the relation $\mathrm{p}(\mathrm{t})=1000+\frac{1000 \mathrm{t}}{100+\mathrm{t}^2}$. The maximum size of this bacterial population is
MHT CET 2025 19th April Morning Shift
11
By dropping a stone in a quiet lake, a wave in the form of circle is generated. The radius of the circular wave increases at the rate of $2.1 \mathrm{~cm} / \mathrm{sec}$. Then the rate of increase of the enclosed circular region, when the radius of the circular wave is 10 cm , is (Given $\pi=\frac{22}{7}$)
MHT CET 2025 19th April Morning Shift
12
The angle between the curves $x y=6$ and $x^2 y=12$ is
MHT CET 2025 19th April Morning Shift
13
In the mean value theorem, $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$, if $\mathrm{a}=0, \mathrm{~b}=\frac{1}{2}$ and $\mathrm{f}(x)=x(x-1)(x-2)$, then the value of $c$ is
MHT CET 2025 19th April Morning Shift
14

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is _________ $\mathrm{cm}^3 / \mathrm{cm}^2$.

MHT CET 2024 16th May Evening Shift
15

Water is being poured at the rate of $36 \mathrm{~m}^3 / \mathrm{min}$ into a cylindrical vessel, whose circular base is of radius 3 meters. Then the water level in the cylinder is rising at the rate of

MHT CET 2024 16th May Evening Shift
16

The equation of the normal to the curve $y=x \log x$ parallel to $2 x-2 y+3=0$ is

MHT CET 2024 16th May Evening Shift
17

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$, then

MHT CET 2024 16th May Evening Shift
18

If $\mathrm{f}(1)=1, \mathrm{f}^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

MHT CET 2024 16th May Morning Shift
19

If $\theta$ denotes the acute angle between the curves $y=10-x^2$ and $y=2+x^2$, at a point of the intersection, then $|\tan \theta|$ is equal to

MHT CET 2024 16th May Morning Shift
20

If $y=a \log x+b x^2+x$ has its extremum values at $x=-1$ and $x=2$, then

MHT CET 2024 16th May Morning Shift
21

The set of all points, for which $f(x)=x^2 e^{-x}$ strictly increases, is

MHT CET 2024 16th May Morning Shift
22

The abscissa of the point on the curve $y=\mathrm{a}\left(\mathrm{e}^{\frac{x}{a}}+\mathrm{e}^{-\frac{x}{a}}\right)$ where the tangent is parallel to the X -axis is

MHT CET 2024 16th May Morning Shift
23

The Number of values of C that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\sin 2 \pi x, x \in[-1,1]$ is

MHT CET 2024 15th May Evening Shift
24

An open tank with a square bottom, to contain 4000 cubic cm . of liquid, is to be constructed. The dimensions of the tank, so that the surface area of the tank is minimum, are

MHT CET 2024 15th May Evening Shift
25

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

MHT CET 2024 15th May Evening Shift
26

The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} / \mathrm{sec}$ and the acceleration is $10 \mathrm{~m} / \mathrm{sec}^2$, then

MHT CET 2024 15th May Evening Shift
27

If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

MHT CET 2024 15th May Morning Shift
28

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq.m) of the flowerbed is

MHT CET 2024 15th May Morning Shift
29

A ladder 5 m in length is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of $2 \mathrm{~m} / \mathrm{sec}$. How fast is the height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

MHT CET 2024 15th May Morning Shift
30

If $y=\mathrm{a} \log x+\mathrm{b} x^2+x$ has its extreme values at $x=-1$ and $x=2$, then the value of $\left(\frac{a}{b}+\frac{b}{a}\right)$ is

MHT CET 2024 11th May Evening Shift
31

The curve $y=a x^3+b x^2+c x+5$ touches the X - axis at $(-2,0)$ and cuts the Y -axis at a point Q where its gradient is 3 , then values of $a, b, c$ respectively, are

MHT CET 2024 11th May Evening Shift
32

The maximum value of the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^3-15 x^2+36 x-48$ on the set $A=\left\{x / x^2+20 \leq 9 x\right\}$ is

MHT CET 2024 11th May Evening Shift
33

If the normal to the curve $y=\mathrm{f}(x)$ at the point $(3,4)$ makes an angle of $\left(\frac{3 \pi}{4}\right)$ with the positive $X$-axis, then the value of $f^{\prime}(3)$ is

MHT CET 2024 11th May Evening Shift
34

If $\mathrm{f}(x)=x^3-6 x^2+9 x+3$ is monotonically decreasing function, then $x$ lies in

MHT CET 2024 11th May Morning Shift
35

If equation of normal to the curve $x=\sqrt{t}$, $y=\mathrm{t}-\frac{1}{\sqrt{\mathrm{t}}}$ at $\mathrm{t}=4$ is

MHT CET 2024 11th May Morning Shift
36

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is

MHT CET 2024 11th May Morning Shift
37

The distance ' $s$ ' in meters covered by a body in $t$ seconds is given by $s=3 t^2-8 t+5$. The body will stop after

MHT CET 2024 11th May Morning Shift
38

The volume of a ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. The rate of increase of the radius, when the volume is $288 \pi \mathrm{cc}$, is

MHT CET 2024 10th May Evening Shift
39

Let $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$. Values of C will be __________ if L.M.V.T. (Lagrange's Mean Value Theorem) can be applied.

MHT CET 2024 10th May Evening Shift
40

If $y=4 x-5$ is a tangent to the curve $y^2=p x^3+q$ at $(2,3)$, then the values of $p$ and $q$ are respectively

MHT CET 2024 10th May Evening Shift
41

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter $=10 \mathrm{~cm}$)

MHT CET 2024 10th May Morning Shift
42

A point moves along the arc of parabola $y=2 x^2$. Its abscissa increases uniformly at the rate of 2 units $/ \mathrm{sec}$. At the instant, the point is passing through ( 1,2 ), its distance from origin is increasing at the rate of

MHT CET 2024 10th May Morning Shift
43

The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is

MHT CET 2024 10th May Morning Shift
44

The function $\mathrm{f}(x)=2 x^3-6 x+5$ is an increasing function, if

MHT CET 2024 9th May Evening Shift
45

A square plate is contracting at the uniform rate $3 \mathrm{~cm}^2 / \mathrm{sec}$, then the rate at which the perimeter is decreasing, when the side of the square is 15 cm , is

MHT CET 2024 9th May Evening Shift
46

A poster is to be printed on a rectangular sheet of paper of area $18 \mathrm{~m}^2$. The margins at the top and bottom of 75 cm each and at the sides 50 cm each are to be left. Then the dimensions i.e. height and breadth of the sheet so that the space available for printing is maximum, are _______ respectively.

MHT CET 2024 9th May Evening Shift
47

The equation of the tangent to the curve $x=\operatorname{acos}^3 \theta, y=\operatorname{asin}^3 \theta$ at $\theta=\frac{\pi}{4}$ is

MHT CET 2024 9th May Evening Shift
48

Let C be a curve given by $y(x)=1+\sqrt{4 x-3}$, $x>\frac{3}{4}$. If P is a point on C , such that the tangent at P has slope $\frac{2}{3}$, then a point through which the normal at P passes, is

MHT CET 2024 9th May Morning Shift
49

If $\mathrm{f}(x)=\frac{\log x}{x}(x>0)$, then it is increasing in

MHT CET 2024 9th May Morning Shift
50

The maximum value of $\frac{\log x}{x}$ is

MHT CET 2024 9th May Morning Shift
51

If the curves $y^2=6 x, 9 x^2+\mathrm{b} y^2=16$ intersect each other at right angles, then the value of $b$ is

MHT CET 2024 9th May Morning Shift
52

If Mean value theorem holds for the function $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$ then the values of $c$ as per the theorem are

MHT CET 2024 9th May Morning Shift
53

A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{sec}$., then the foot of the ladder is sliding at the rate of _________ $\mathrm{m} / \mathrm{sec}$., when it is 4 m away from the wall.

MHT CET 2024 4th May Evening Shift
54

The equation of the tangent to the curve $y=1-\mathrm{e}^{\frac{x}{3}}$ at the point of intersection with Y -axis is

MHT CET 2024 4th May Evening Shift
55

If $8 \mathrm{f}(x)+6 \mathrm{f}\left(\frac{1}{x}\right)=x+5$ and $y=x^2 \mathrm{f}(x)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=-1$ is

MHT CET 2024 4th May Morning Shift
56

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is

MHT CET 2024 4th May Morning Shift
57

The sum of intercepts on coordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

MHT CET 2024 4th May Morning Shift
58

A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side $x$ units and a circle of radius of r units. If the sum of the areas of square and the circle so formed is minimum, then

MHT CET 2024 4th May Morning Shift
59

If sum of two numbers is 3 , then the maximum value of the product of first number and square of the second number is

MHT CET 2024 4th May Morning Shift
60

The curve $y=a x^3+b x^2+c x+5$ touches the $x$-axis at $(-2,0)$ and cuts the $y$-axis at a point Q where its gradient is 3 , then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is

MHT CET 2024 3rd May Evening Shift
61

If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$, then the value of $a+b$ is

MHT CET 2024 3rd May Evening Shift
62

If $\mathrm{f}(x)=x^3+b x^2+c x+d$ and $0< b^2< c$, then in $(-\infty, \infty)$

MHT CET 2024 3rd May Evening Shift
63

The minimum value of the function $\mathrm{f}(x)=2 x^3-15 x^2+36 x-48$ on the set $\mathrm{A}=\left\{x \mid x^2+20 \leqslant 9 x\right\}$ is

MHT CET 2024 3rd May Evening Shift
64

The value of c for which Rolle's theorem for the function $\mathrm{f}(x)=x^3-3 x^2+2 x$ in the interval $[0,2]$ are

MHT CET 2024 3rd May Evening Shift
65

A triangular park is enclosed on two sides by a fence and on the third side a straight river bank. The two sides having fence are of same length $x$. The maximum area (in sq. units) enclosed by the park is

MHT CET 2024 3rd May Morning Shift
66

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circular wave is 12 cm . how fast is the enclosed area increasing?

MHT CET 2024 3rd May Morning Shift
67

A bullet is shot horizontally and its distance S cm at time t second is given by $\mathrm{S}=1200 \mathrm{t}-15 \mathrm{t}^2$, then the distance covered by the bullet when it comes to the rest, is

MHT CET 2024 3rd May Morning Shift
68

The equation of the normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

MHT CET 2024 3rd May Morning Shift
69

The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2 y=3$ are

MHT CET 2024 2nd May Evening Shift
70

The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is

MHT CET 2024 2nd May Evening Shift
71

The maximum value of the function

$$f(x)=3 x^3-18 x^2+27 x-40$$

on the set $\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$ is

MHT CET 2024 2nd May Evening Shift
72

The equation of normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

MHT CET 2024 2nd May Evening Shift
73

The length of the longest interval, in which the function $3 \sin x-4 \sin ^3 x$ is increasing, is

MHT CET 2024 2nd May Morning Shift
74

If Rolle's theorem holds for the function $\mathrm{f}(x)=x^3+\mathrm{bx}{ }^2+\mathrm{ax}+5$ on $[1,3]$ with $\mathrm{c}=2+\frac{1}{\sqrt{3}}$, then the values of $a$ and $b$ respectively are

MHT CET 2024 2nd May Morning Shift
75

The normal to the curve, $y(x-2)(x-3)=x+6$ at the point, where the curve intersects the Y-axis, passes through the point

MHT CET 2024 2nd May Morning Shift
76

The function $$\mathrm{f}(x)=x^3-6 x^2+9 x+2$$ has maximum value when $$x$$ is

MHT CET 2023 14th May Evening Shift
77

If $$y=4 x-5$$ is a tangent to the curve $$y^2=\mathrm{p} x^3+\mathrm{q}$$ at $$(2,3)$$, then $$\mathrm{p}-\mathrm{q}$$ is

MHT CET 2023 14th May Evening Shift
78

The diagonal of a square is changing at the rate of $$0.5 \mathrm{~cm} / \mathrm{sec}$$. Then the rate of change of area when the area is $$400 \mathrm{~cm}^2$$ is equal to

MHT CET 2023 14th May Evening Shift
79

Let $$x_0$$ be the point of local minima of $$\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$ where $$\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$$, then value of $$\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$$ at $$x=x_0$$ is

MHT CET 2023 14th May Evening Shift
80

Let the curve be represented by $$x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)$$. Then normal at any point '$$t$$' of the curve is at a distance of ______ units from the origin.

MHT CET 2023 14th May Morning Shift
81

Let $$\mathrm{B} \equiv(0,3)$$ and $$\mathrm{C} \equiv(4,0)$$. The point $$\mathrm{A}$$ is moving on the line $$y=2 x$$ at the rate of 2 units/second. The area of $$\triangle \mathrm{ABC}$$ is increasing at the rate of

MHT CET 2023 14th May Morning Shift
82

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}$$ is

MHT CET 2023 14th May Morning Shift
83

Slope of the tangent to the curve $$y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$$, where $$0 \leq x \leq 2 \pi$$ is minimum at $$x=$$

MHT CET 2023 13th May Evening Shift
84

If slope of the tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then the value of $$3 a+b$$ is

MHT CET 2023 13th May Evening Shift
85

$$A(1,-3), B(4,3)$$ are two points on the curve $$y=x-\frac{4}{x}$$. The points on the curve, the tangents at which are parallel to the chord $$A B$$, are

MHT CET 2023 13th May Evening Shift
86

Water is running in a hemispherical bowl of radius $$180 \mathrm{~cm}$$ at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is $$120 \mathrm{~cm}$$ ? (1 decimeter $$=10 \mathrm{~m}$$)

MHT CET 2023 13th May Evening Shift
87

If Rolle's theorem holds for the function $$f(x)=x^3+b x^2+a x+5$$ on $$[1,3]$$ with $$c=2+\frac{1}{\sqrt{3}}$$, then the values of $$a$$ and $$b$$ respectively are

MHT CET 2023 13th May Evening Shift
88

If $$\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$$ and $$0<\mathrm{b}^2<\mathrm{c}$$, then in $$(-\infty, \infty)$$

MHT CET 2023 13th May Morning Shift
89

The slope of the normal to the curve $$x=\sqrt{t}$$ and $$y=t-\frac{1}{\sqrt{t}}$$ at $$t=4$$ is

MHT CET 2023 13th May Morning Shift
90

Values of $$c$$ as per Rolle's theorem for $$f(x)=\sin x+\cos x+6$$ on $$[0,2 \pi]$$ are

MHT CET 2023 13th May Morning Shift
91

A poster is to be printed on a rectangular sheet of paper of area $$18 \mathrm{~m}^2$$. The margins at the top and bottom of $$75 \mathrm{~cm}$$ each and at the sides $$50 \mathrm{~cm}$$ each are to be left. Then the dimensions i.e. height and breadth of the sheet, so that the space available for printing is maximum, are ________ respectively.

MHT CET 2023 12th May Evening Shift
92

The equation of the normal to the curve $$3 x^2-y^2=8$$, which is parallel to the line $$x+3 y=10$$, is

MHT CET 2023 12th May Evening Shift
93

If the curves $$y^2=6 x$$ and $$9 x^2+b y^2=16$$ intersect each other at right angle, then value of '$$b$$' is

MHT CET 2023 12th May Evening Shift
94

A ladder, 5 meters long, rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle between the ladder and the floor is decreasing at the rate of __________ radians/second when it's lower end is $$4 \mathrm{~m}$$ away from the wall.

MHT CET 2023 12th May Evening Shift
95

A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹ 100 per square meter for the base and ₹ 50 per square meter for the sides, then the cost of least expensive tank is

MHT CET 2023 12th May Evening Shift
96

The angle between the tangents to the curves $$y=2 x^2$$ and $$x=2 y^2$$ at $$(1,1)$$ is

MHT CET 2023 12th May Morning Shift
97

The function $$\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$$ is increasing in

MHT CET 2023 12th May Morning Shift
98

A ladder 5 meters long rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle between the ladder and the floor is decreasing at the rate of ________ rad./s when it's lower end is $$4 \mathrm{~m}$$ away from the wall.

MHT CET 2023 12th May Morning Shift
99

If the function $$f$$ is given by $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$\mathrm{a} \in \mathbb{R}$$, is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equation $$\frac{\mathrm{f}(x)-14}{(x-1)^2}=0(x \neq 1)$$ is

MHT CET 2023 11th May Evening Shift
100

If $$a$$ and $$b$$ are positive number such that $$a>b$$, then the minimum value of $$a \sec \theta-b \tan \theta\left(0 < \theta < \frac{\pi}{2}\right)$$ is

MHT CET 2023 11th May Evening Shift
101

$$A$$ rod $$A B, 13$$ feet long moves with its ends $$A$$ and $$B$$ on two perpendicular lines $$O X$$ and $$O Y$$ respectively. When $$A$$ is 5 feet from $$O$$, it is moving away at the rate of $$3 \mathrm{feet} / \mathrm{sec}$$. At this instant, $$\mathrm{B}$$ is moving at the rate

MHT CET 2023 11th May Evening Shift
102

The equation of the tangent to the curve $$y=\sqrt{9-2 x^2}$$, at the point where the ordinate and abscissa are equal, is

MHT CET 2023 11th May Evening Shift
103

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production $$\mathrm{P}$$ w.r.t. additional number of worker $$x$$ is given by $$\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}$$. If the firm employees 9 more workers, then the new level of production of items is

MHT CET 2023 11th May Evening Shift
104

Value of $$c$$ satisfying the conditions and conclusions of Rolle's theorem for the function $$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$ is

MHT CET 2023 11th May Morning Shift
105

If $$\mathrm{f}(x)=x \mathrm{e}^{x(1-x)}, x \in \mathrm{R}$$, then $$\mathrm{f}(x)$$ is

MHT CET 2023 11th May Morning Shift
106

The value of $$\mathrm{c}$$ for the function $$\mathrm{f}(x)=\log x$$ on [$$1$$, e] if LMVT can be applied, is

MHT CET 2023 10th May Evening Shift
107

The displacement '$$\mathrm{S}$$' of a moving particle at a time $$t$$ is given by $$S=5+\frac{48}{t}+t^3$$. Then its acceleration when the velocity is zero, is

MHT CET 2023 10th May Evening Shift
108

If the surface area of a spherical balloon of radius $$6 \mathrm{~cm}$$ is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate of increase in its volume in $$\mathrm{cm}^3 / \mathrm{sec}$$ is

MHT CET 2023 10th May Evening Shift
109

The value of $$\alpha$$, so that the volume of parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{\mathrm{i}}+\hat{\mathrm{k}}$$ becomes minimum, is

MHT CET 2023 10th May Evening Shift
110

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are

MHT CET 2023 10th May Evening Shift
111

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for minimum area of the sheet metal used in its construction, are

MHT CET 2023 10th May Morning Shift
112

A square plate is contracting at the uniform rate $$4 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate at which the perimeter is decreasing, when side of the square is $$20 \mathrm{~cm}$$, is

MHT CET 2023 10th May Morning Shift
113

A ladder of length $$17 \mathrm{~m}$$ rests with one end against a vertical wall and the other on the level ground. If the lower end slips away at the rate of $$1 \mathrm{~m} / \mathrm{sec}$$., then when it is $$8 \mathrm{~m}$$ away from the wall, its upper end is coming down at the rate of

MHT CET 2023 10th May Morning Shift
114

If the line $$a x+b y+c=0$$ is a normal to the curve $$x y=1$$, then

MHT CET 2023 10th May Morning Shift
115

A kite is $$120 \mathrm{~m}$$ high and $$130 \mathrm{~m}$$ of string is out. If the kite is moving away horizontally at the rate of $$39 \mathrm{~m} / \mathrm{sec}$$, then the rate at which the string is being out, is

MHT CET 2023 10th May Morning Shift
116

If slope of a tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then a - b is

MHT CET 2023 9th May Evening Shift
117

The equation $$x^3+x-1=0$$ has

MHT CET 2023 9th May Evening Shift
118

The range of values of $$x$$ for which $$f(x)=x^3+6 x^2-36 x+7$$ is increasing in

MHT CET 2023 9th May Evening Shift
119

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$$ is

MHT CET 2023 9th May Evening Shift
120

A water tank has a shape of inverted right circular cone whose semi-vertical angle is $$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of water is rising, at the instant when depth of water in the tank is $$10 \mathrm{~m}$$ is

MHT CET 2023 9th May Evening Shift
121

Let $$\mathrm{f}(0)=-3$$ and $$\mathrm{f}^{\prime}(x) \leq 5$$ for all real values of $$x$$. The $$\mathrm{f}(2)$$ can have possible maximum value as

MHT CET 2023 9th May Evening Shift
122

The value of $$c$$ of Lagrange's mean value theorem for $$f(x)=\sqrt{25-x^2}$$ on $$[1,5]$$ is

MHT CET 2023 9th May Morning Shift
123

The value of $$\alpha$$, so that the volume of the parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{i}+\hat{k}$$ becomes maximum, is

MHT CET 2023 9th May Morning Shift
124

The maximum value of xy when x + 2y = 8 is

MHT CET 2023 9th May Morning Shift
125

An object is moving in the clockwise direction around the unit circle $$x^2+y^2=1$$. As it passes through the point $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$, its $$y$$-co-ordinate is decreasing at the rate of 3 units per sec. The rate at which the $$x$$-co-ordinate changes at this point is

MHT CET 2023 9th May Morning Shift
126

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is $$3 \mathrm{~mm}$$ and 1 hour later it reduces to $$2 \mathrm{~mm}$$, then the expression for the radius $$R$$ of the raindrop at any time $$t$$ is

MHT CET 2023 9th May Morning Shift
127

If the function $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$a \in I R$$ is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equation $$\frac{f(x)-14}{(x-1)^2}=0(x \neq 1)$$ is

MHT CET 2022 11th August Evening Shift
128

A firm is manufacturing 2000 items. It is estimated that the rate of change of production $$P$$ with respect to additional number of workers $$x$$ is given by $$\frac{\mathrm{d} P}{\mathrm{~d} x}=100-12 \sqrt{x}$$. If the firm employs 25 more workers, then the new level of production of items is

MHT CET 2022 11th August Evening Shift
129

If the normal to the curve $$y=f(x)$$ at the point $$(3,4)$$ makes an angle $$\left(\frac{3 \pi}{4}\right)^c$$ with positive $$X$$-axis, then $$f^{\prime}(3)$$ is equal to

MHT CET 2022 11th August Evening Shift
130

If $$y=\cos \left(\sin x^2\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ at $$x=\sqrt{\frac{\pi}{2}}$$ is

MHT CET 2022 11th August Evening Shift
131

A spherical iron ball of $$10 \mathrm{~cm}$$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $$50 \mathrm{~cm}^3 / \mathrm{min}$$. If the thickness of ice is $$5 \mathrm{~cm}$$, then the rate at which the thickness of ice decreases is

MHT CET 2022 11th August Evening Shift
132

The distance 's' in meters covered by a particle in t seconds is given by $$s=2+27 t-t^3$$. The particle will stop after _________ distance.

MHT CET 2021 24th September Evening Shift
133

The curve $$y=a x^3+b x^2+c x+5$$ touches $$X$$-axis at $$P(-2,0)$$ and cuts $$Y$$-axis at a point $$Q$$, where its gradient is 3, then

MHT CET 2021 24th September Evening Shift
134

The minimum value of the function f(x) = x log x is

MHT CET 2021 24th September Evening Shift
135

The maximum area of the rectangle that can be inscribed in a circle of radius $$r$$ is

MHT CET 2021 24th September Morning Shift
136

$$f(x)=\log |\sin x|$$, where $$x \in(0, \pi)$$ is strictly increasing on

MHT CET 2021 24th September Morning Shift
137

The velocity of a particle at time $$t$$ is given by the relation $$v=6 t-\frac{t^2}{6}$$. Its displacement S is zero at $$\mathrm{t}=0$$, then the distance travelled in $$3 \mathrm{~sec}$$ is

MHT CET 2021 24th September Morning Shift
138

The radius of a circular plate is increasing at the rate of $$0.01 \mathrm{~cm} / \mathrm{sec}$$, when the radius is $$12 \mathrm{~cm}$$. Then the rate at which the area increases is

MHT CET 2021 23rd September Evening Shift
139

If $$x=a(\theta+\sin \theta)$$ and $$y=a(1-\cos \theta)$$ then $$\left(\frac{d^2 y}{d x^2}\right)_{at~ \theta=\pi / 2}=$$

MHT CET 2021 23rd September Evening Shift
140

The equation of tangent to the curve $$y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)$$ at $$x=\frac{\pi}{4}$$, is

MHT CET 2021 23rd September Evening Shift
141

Function $$f(x)=e^{-1 / x}$$ is strictly increasing for all $$x$$ where

MHT CET 2021 23th September Morning Shift
142

If $$x=-2$$ and $$x=4$$ are the extreme points of $$y=x^3-\alpha x^2-\beta x+5$$, then

MHT CET 2021 23th September Morning Shift
143

10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively

MHT CET 2021 23th September Morning Shift
144

The function $$f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}$$ is increasing, if

MHT CET 2021 22th September Evening Shift
145

If $$f(x)=x^2+a x+b$$ has minima at $$x=3$$ whose value is 5 , then the values of $$a$$ and $$b$$ are respectively.

MHT CET 2021 22th September Evening Shift
146

The slant height of a right circular cone is $$3 \mathrm{~cm}$$. The height of the cone for maximum volume is

MHT CET 2021 22th September Evening Shift
147

The point on the curve $$y^2=2(x-3)$$ at which the normal is parallel to the line $$y-2 x+1=0$$ is

MHT CET 2021 22th September Morning Shift
148

A sperical snow ball is forming so that its volume is increasing at the rate of $$8 \mathrm{~cm}^3 / \mathrm{sec}$$. Find the rate of increase of radius when radius is $$2 \mathrm{~cm}$$.

MHT CET 2021 22th September Morning Shift
149

The abscissa of the points, where the tangent to the curve $$y=x^3-3 x^2-9 x+5$$ is parallel to $$X$$ axis are

MHT CET 2021 21th September Evening Shift
150

For all real $$x$$, the minimum value of the function $$f(x)=\frac{1-x+x^2}{1+x+x^2}$$ is

MHT CET 2021 21th September Evening Shift
151

The function $$f(x)=\log (1+x)-\frac{2 x}{2+x}$$ is increasing on

MHT CET 2021 21th September Evening Shift
152

A body at an unknown temperature is placed in a room which is held at a constant temperature of $$30^{\circ} \mathrm{F}$$. If after 10 minutes the temperature of the body is $$0^{\circ} \mathrm{F}$$ and after 20 minutes the temperature of the body is $$15^{\circ} \mathrm{F}$$, then the expression for the temperature of the body at any time $$\mathrm{t}$$ is

MHT CET 2021 21th September Morning Shift
153

A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^2$$/sec.

MHT CET 2021 21th September Morning Shift
154

The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

MHT CET 2021 21th September Morning Shift
155

The curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect each other orthogonally, then $$a^2=$$

MHT CET 2021 21th September Morning Shift
156

The surface area of a spherical balloon is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the balloon is , when the radius of the balloon is $$6 \mathrm{~cm}$$.

MHT CET 2021 20th September Evening Shift
157

If $$f(x)=2x^3-15x^2-144x-7$$, then $$f(x)$$ is strictly decreasing in

MHT CET 2021 20th September Evening Shift
158

The equation of tangent to the circle $$x^2+y^2=64$$ at the point $$\mathrm{P\left(\frac{2\pi}{3}\right)}$$ is

MHT CET 2021 20th September Evening Shift
159

Water is being poured at the rate of 36 m$$^3$$/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cylinder is rising at the rate of

MHT CET 2021 20th September Evening Shift
160

A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these parts is

MHT CET 2021 20th September Morning Shift
161

A particle is moving on a straight line. The distance $$\mathrm{S}$$ travelled in time $$\mathrm{t}$$ is given by $$\mathrm{S=a t^2+b t+6}$$. If the particle comes to rest after 4 seconds at a distance of $$16 \mathrm{~m}$$. from the starting point, then the acceleration of the particle is.

MHT CET 2021 20th September Morning Shift
162

The equation of the tangent to the curve $$y = 4x{e^x}$$ at $$\left( { - 1,{{ - 4} \over e}} \right)$$ is

MHT CET 2021 20th September Morning Shift
163

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is

MHT CET 2020 19th October Evening Shift
164

The equation of the normal to the curve $2 x^2+y^2=12$ at the point $(2,2)$ is

MHT CET 2020 19th October Evening Shift
165

The area of the square increases at the rate of $0.5 \mathrm{~cm}^2 / \mathrm{sec}$. The rate at which its perimeter is increasing when the side of the square is 10 cm long, is

MHT CET 2020 19th October Evening Shift
166

The equation of normal to the curve $$y=\sin \left(\frac{\pi x}{4}\right)$$ at the point $$(2,5)$$ is

MHT CET 2020 16th October Evening Shift
167

$$\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=$$

MHT CET 2020 16th October Evening Shift
168

For every value of $$x$$, the function $$f(x)=\frac{1}{a^x}, a>$$ 0 is,

MHT CET 2020 16th October Evening Shift
169

If $$f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$, then value of '$$c$$' by applying LMVT is

MHT CET 2020 16th October Morning Shift
170

The equation of tangent at $$P(-4,-4)$$ on the curve $$x^2=-4 y$$ is

MHT CET 2020 16th October Morning Shift
171

- The edge of a cube is decreasing at the rate of $0.04 \mathrm{~cm} / \mathrm{sec}$. If the edge of the cube is 10 cms , then rate of decrease of surface area of the cube is...

MHT CET 2019 3rd May Morning Shift
172

If $r$ is the radius of spherical balloon at time $t$ and the surface area of balloon changes at a constant rate $K$, then......

MHT CET 2019 3rd May Morning Shift
173

The slope of normal to the curve $x=\sqrt{t}$ and $y=t-\frac{1}{\sqrt{t}}$ at $t=4$ is ..........

MHT CET 2019 3rd May Morning Shift
174

If $f(x)=x+\frac{1}{x}, x \neq 0$, then local maximum and minimum values of function $f$ are respectively.......

MHT CET 2019 3rd May Morning Shift
175

The function $f(x)=x^3-3 x$ is ............

MHT CET 2019 2nd May Evening Shift
176

Using differentiation, approximate value of $f(x)=x^2-2 x+1$ at $x=2.99$ is ............

MHT CET 2019 2nd May Evening Shift
177

A particle moves so that $x=2+27 t-t^3$. The direction of motion reverses after moving a distance of ....... units.

MHT CET 2019 2nd May Evening Shift
178

A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of $5 \mathrm{~cm} / \mathrm{sec}$. Then area increased after 2 seconds is ............

MHT CET 2019 2nd May Morning Shift
179

The equation of normal to the curve $y=\log _\theta x$ at the point $P(1,0)$ is ............

MHT CET 2019 2nd May Morning Shift
180

If $f(x)=3 x^3-9 x^2-27 x+15$, then the maximum value of $f(x)$ is ...........

MHT CET 2019 2nd May Morning Shift
EXAM MAP