# Application of Derivatives · Mathematics · MHT CET

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MHT CET 2023 14th May Evening Shift
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
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
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 $$400 \mathr... MHT CET 2023 14th May Evening Shift Let$$x_0$$be the point of local minima of$$\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$where$$...
MHT CET 2023 14th May Morning Shift
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 ____...
MHT CET 2023 14th May Morning Shift
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....
MHT CET 2023 14th May Morning Shift
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 13th May Evening Shift
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 $$0 \leq ... MHT CET 2023 13th May Evening Shift If slope of the tangent to the curve$$x y+a x+b y=0$$at the point$$(1,1)$$on it is 2, then the value of$$3 a+b$$is MHT CET 2023 13th May Evening Shift$$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$$,... MHT CET 2023 13th May Evening Shift Water is running in a hemispherical bowl of radius$$180 \mathrm{~cm}$$at the rate of 108 cubic decimeters per minute. How fast the water level is ri... MHT CET 2023 13th May Evening Shift If Rolle's theorem holds for the function$$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$$re... MHT CET 2023 13th May Morning Shift If$$\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$$and$$0
MHT CET 2023 13th May Morning Shift
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
Values of $$c$$ as per Rolle's theorem for $$f(x)=\sin x+\cos x+6$$ on $$[0,2 \pi]$$ are
MHT CET 2023 12th May Evening Shift
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 ...
MHT CET 2023 12th May Evening Shift
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
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
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 be...
MHT CET 2023 12th May Evening Shift
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....
MHT CET 2023 12th May Morning Shift
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
The function $$\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$$ is increasing in
MHT CET 2023 12th May Morning Shift
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 betw...
MHT CET 2023 11th May Evening Shift
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 i...
MHT CET 2023 11th May Evening Shift
If $$a$$ and $$b$$ are positive number such that $$a>b$$, then the minimum value of $$a \sec \theta-b \tan \theta\left(0 ... MHT CET 2023 11th May Evening Shift$$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$$is 5 feet ... MHT CET 2023 11th May Evening Shift The equation of the tangent to the curve$$y=\sqrt{9-2 x^2}$$, at the point where the ordinate and abscissa are equal, is MHT CET 2023 11th May Evening Shift At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production$$\mathrm{P}$$w.r.t. additional number of worker... MHT CET 2023 11th May Morning Shift Value of$$c$$satisfying the conditions and conclusions of Rolle's theorem for the function$$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$is MHT CET 2023 11th May Morning Shift If$$\mathrm{f}(x)=x \mathrm{e}^{x(1-x)}, x \in \mathrm{R}$$, then$$\mathrm{f}(x)$$is MHT CET 2023 10th May Evening Shift The value of$$\mathrm{c}$$for the function$$\mathrm{f}(x)=\log x$$on [$$1$$, e] if LMVT can be applied, is MHT CET 2023 10th May Evening Shift The displacement '$$\mathrm{S}$$' of a moving particle at a time$$t$$is given by$$S=5+\frac{48}{t}+t^3$$. Then its acceleration when the velocity i... MHT CET 2023 10th May Evening Shift If the surface area of a spherical balloon of radius$$6 \mathrm{~cm}$$is increasing at the rate$$2 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate o... MHT CET 2023 10th May Evening Shift The value of$$\alpha$$, so that the volume of parallelopiped formed by$$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$and$$\alpha \hat{\...
MHT CET 2023 10th May Evening Shift
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 1...
MHT CET 2023 10th May Morning Shift
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...
MHT CET 2023 10th May Morning Shift
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 o...
MHT CET 2023 10th May Morning Shift
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...
MHT CET 2023 10th May Morning Shift
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
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 $$39 \mathrm{~m} /... MHT CET 2023 9th May Evening Shift If slope of a tangent to the curve$$x y+a x+b y=0$$at the point$$(1,1)$$on it is 2, then a - b is MHT CET 2023 9th May Evening Shift The equation$$x^3+x-1=0$$has MHT CET 2023 9th May Evening Shift The range of values of$$x$$for which$$f(x)=x^3+6 x^2-36 x+7$$is increasing in MHT CET 2023 9th May Evening Shift The maximum value of the function$$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\}$$is MHT CET 2023 9th May Evening Shift A water tank has a shape of inverted right circular cone whose semi-vertical angle is$$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it ... MHT CET 2023 9th May Evening Shift Let$$\mathrm{f}(0)=-3$$and$$\mathrm{f}^{\prime}(x) \leq 5$$for all real values of$$x$$. The$$\mathrm{f}(2)$$can have possible maximum value as... MHT CET 2023 9th May Morning Shift The value of$$c$$of Lagrange's mean value theorem for$$f(x)=\sqrt{25-x^2}$$on$$[1,5]$$is MHT CET 2023 9th May Morning Shift The value of$$\alpha$$, so that the volume of the parallelopiped formed by$$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$and$$\alpha \h...
MHT CET 2023 9th May Morning Shift
The maximum value of xy when x + 2y = 8 is
MHT CET 2023 9th May Morning Shift
An object is moving in the clockwise direction around the unit circle $$x^2+y^2=1$$. As it passes through the point $$\left(\frac{1}{2}, \frac{\sqrt{3... MHT CET 2023 9th May Morning Shift A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is$$3 \mathrm{~mm}$$and 1 hour later it reduces... MHT CET 2022 11th August Evening Shift If the function$$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)$$, then a root of the equ... MHT CET 2022 11th August Evening Shift A firm is manufacturing 2000 items. It is estimated that the rate of change of production$$P$$with respect to additional number of workers$$x$$is ... MHT CET 2022 11th August Evening Shift If the normal to the curve$$y=f(x)$$at the point$$(3,4)$$makes an angle$$\left(\frac{3 \pi}{4}\right)^c$$with positive$$X$$-axis, then$$f^{\pr...
MHT CET 2022 11th August Evening Shift
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
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 $$50 \mathrm{~cm}^3 /... MHT CET 2021 24th September Evening Shift The distance 's' in meters covered by a particle in t seconds is given by$$s=2+27 t-t^3$$. The particle will stop after _________ distance. MHT CET 2021 24th September Evening Shift The curve$$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$$, where its gradient is 3, then MHT CET 2021 24th September Evening Shift The minimum value of the function f(x) = x log x is MHT CET 2021 24th September Morning Shift The maximum area of the rectangle that can be inscribed in a circle of radius$$r$$is MHT CET 2021 24th September Morning Shift$$f(x)=\log |\sin x|$$, where$$x \in(0, \pi)$$is strictly increasing on MHT CET 2021 24th September Morning Shift The velocity of a particle at time$$t$$is given by the relation$$v=6 t-\frac{t^2}{6}$$. Its displacement S is zero at$$\mathrm{t}=0$$, then the di... MHT CET 2021 23rd September Evening Shift The radius of a circular plate is increasing at the rate of$$0.01 \mathrm{~cm} / \mathrm{sec}$$, when the radius is$$12 \mathrm{~cm}$$. Then the rat... MHT CET 2021 23rd September Evening Shift If$$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}=$$MHT CET 2021 23rd September Evening Shift The equation of tangent to the curve$$y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)$$at$$x=\frac{\pi}{4}$$, is MHT CET 2021 23th September Morning Shift Function$$f(x)=e^{-1 / x}$$is strictly increasing for all$$x$$where MHT CET 2021 23th September Morning Shift If$$x=-2$$and$$x=4$$are the extreme points of$$y=x^3-\alpha x^2-\beta x+5$$, then MHT CET 2021 23th September Morning Shift 10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively MHT CET 2021 22th September Evening Shift The function$$f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}$$is increasing, if MHT CET 2021 22th September Evening Shift If$$f(x)=x^2+a x+b$$has minima at$$x=3$$whose value is 5 , then the values of$$a$$and$$b$$are respectively. MHT CET 2021 22th September Evening Shift The slant height of a right circular cone is$$3 \mathrm{~cm}$$. The height of the cone for maximum volume is MHT CET 2021 22th September Morning Shift The point on the curve$$y^2=2(x-3)$$at which the normal is parallel to the line$$y-2 x+1=0$$is MHT CET 2021 22th September Morning Shift A sperical snow ball is forming so that its volume is increasing at the rate of$$8 \mathrm{~cm}^3 / \mathrm{sec}$$. Find the rate of increase of radi... MHT CET 2021 21th September Evening Shift The abscissa of the points, where the tangent to the curve$$y=x^3-3 x^2-9 x+5$$is parallel to$$X$$axis are MHT CET 2021 21th September Evening Shift For all real$$x$$, the minimum value of the function$$f(x)=\frac{1-x+x^2}{1+x+x^2}$$is MHT CET 2021 21th September Evening Shift The function$$f(x)=\log (1+x)-\frac{2 x}{2+x}$$is increasing on MHT CET 2021 21th September Morning Shift A body at an unknown temperature is placed in a room which is held at a constant temperature of$$30^{\circ} \mathrm{F}$$. If after 10 minutes the tem... MHT CET 2021 21th September Morning Shift A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the... MHT CET 2021 21th September Morning Shift The function$$f(x)=\cot ^{-1} x+x$$is increasing in the interval. MHT CET 2021 21th September Morning Shift The curves$$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$and$$y^3=16 x$$intersect each other orthogonally, then$$a^2=$$MHT CET 2021 20th September Evening Shift The surface area of a spherical balloon is increasing at the rate$$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the ball... MHT CET 2021 20th September Evening Shift If$$f(x)=2x^3-15x^2-144x-7$$, then$$f(x)$$is strictly decreasing in MHT CET 2021 20th September Evening Shift The equation of tangent to the circle$$x^2+y^2=64$$at the point$$\mathrm{P\left(\frac{2\pi}{3}\right)}$$is MHT CET 2021 20th September Evening Shift Water is being poured at the rate of 36 m$$^3$$/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cyli... MHT CET 2021 20th September Morning Shift A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these par... MHT CET 2021 20th September Morning Shift A particle is moving on a straight line. The distance$$\mathrm{S}$$travelled in time$$\mathrm{t}$$is given by$$\mathrm{S=a t^2+b t+6}$$. If the p... MHT CET 2021 20th September Morning Shift The equation of the tangent to the curve$$y = 4x{e^x}$$at$$\left( { - 1,{{ - 4} \over e}} \right)$$is MHT CET 2020 16th October Evening Shift The equation of normal to the curve$$y=\sin \left(\frac{\pi x}{4}\right)$$at the point$$(2,5)$$is MHT CET 2020 16th October Evening Shift$$\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=$$MHT CET 2020 16th October Evening Shift For every value of$$x$$, the function$$f(x)=\frac{1}{a^x}, a>$$0 is, MHT CET 2020 16th October Morning Shift If$$f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$, then value of '$$c$$' by applying LMVT is MHT CET 2020 16th October Morning Shift The equation of tangent at$$P(-4,-4)$$on the curve$$x^2=-4 y is
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