A spherical iron ball of radius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100 $\pi \;{\rm{c}}{{\rm{m}}^3}/{\rm{min}}$.The rate at which the thickness of decreases when the thickness of ice is 5 cm, is

$\frac{1}{{376}}{\rm{cm}}/{\rm{min}}$

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JEE • MATHEMATICS • RATE MEASURE

Rate measure practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 37 questions

1 Medium

1 of 37

Application +4 | -1

If the area of an expanding circular region increases at a constant rate with respect to time, then the rate of marease of the perimeter with respect to the time

View answer and explanation

Correct option: A

Varies inversely as radius

Option B (

4 \pi

) is correct because it is the rate at which V is increasing when r=10 and dr/dt=0.01.

2 Medium

2 of 37

Application +4 | -1

A spherical balloon is expanding. If the radius is increasing at the rate of 2 cm/min, the rate at which the volume increase (in cubic centimeters per minute)when the radius is 5 cm, is

View answer and explanation

Correct option: C

200\pi

Option C (

200\pi

) is correct because it represents the rate of volume increase when the radius is 5 cm, calculated using the derivative of the volume formula.

3 Medium

3 of 37

Application +4 | -1

From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of

48 \mathrm{~m} / \mathrm{s}

. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration

\mathrm{g}=

32 \mathrm{~m} \mathrm{~s}^2

, is:

View answer and explanation

Correct option: D

100

Option D (100) is correct because the greatest height is the sum of the tower's height and the height the stone reaches above the tower, calculated using kinematics.

4 Medium

4 of 37

Application +4 | -1

If a metallic circular plate of radius

50 \mathrm{~cm}

is heated so that its radius increases at the rate of

1 \mathrm{~mm}

per hour, then the rate at which, the area of the plate increases (in

\mathrm{cm}^2 /

hour) is

View answer and explanation

Correct option: B

10 \pi

Option B (

10 \pi

) is correct because the rate at which the area of the plate increases is calculated to be

10 \pi \text{{ cm}}^2/\text{{hour}}

5 Medium

5 of 37

Application +4 | -1

The cost of running a bus from

\mathrm{A}

\mathrm{B}

, is

₹\left(a v+\frac{b}{v}\right)

, where

v \mathrm{~km} / \mathrm{h}

is the average speed of the bus. When the bus travels at

30 \mathrm{~km} / \mathrm{h}

, the cost comes out to be

₹ 75

while at

40 \mathrm{~km} / \mathrm{h}

, it is

₹ 65

. Then the most economical speed(in

\mathrm{km} / \mathrm{h}

) of the bus is :

View answer and explanation

Correct option: C

Option C (60) is correct because it represents the speed at which the cost function is minimized, found by taking the derivative and setting it to zero.

6 Medium

6 of 37

Application +4 | -1

There is an error of

\pm 0.04

cm in the measurement of the diameter of a sphere. when the radius is 10 cm, the percentage error in the volume of the sphere is

View answer and explanation

Correct option: D

\pm 0.6

Option D (

\pm 0.6

) is correct because it represents the percentage error in the volume of the sphere, calculated using the formula for volume and error propagation.

7 Medium

7 of 37

Application +4 | -1

The rate of change of the surface area of the sphere of radius

r

when the radius is increasing at the rate of

2\,cm/s

is proportional to

View answer and explanation

Correct option: C

{r^2}

Option D (

r

) is incorrect. The rate of change of the surface area is proportional to the square of the radius, not the radius itself. The correct answer should be

{r^2}

. However, since the provided answer is

{r^2}

and the explanation states that the answer is

{r^2}

, there seems to be a typo in the options. The correct answer should be proportional to

{r^2}

8 Medium

8 of 37

Application +4 | -1

A spherical iron ball of radius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100

\pi \;\mathrm{c}{\mathrm{m}^3}/\mathrm{min}

.The rate at which the thickness of decreases when the thickness of ice is 5 cm, is

View answer and explanation

Correct option: A

1cm/min

Option A (1cm/min) is correct because it represents the rate at which the thickness of the ice decreases, calculated using related rates.

9 Medium

9 of 37

Analyzing +4 | -1

A particle is moving in a straight line such that the distance described ‘

s

’ and the time taken ‘

t

’ are given by

t = a{s^2} + bs + c,a > 0.

v

is the velocity of the particle at any time

t

, then acceleration is

View answer and explanation

Correct option: C

- 2a{v^3}

Option C (

-2a{v^3}

) is correct because it is derived by differentiating the given equation with respect to time and solving for the acceleration (dv/dt).

10 Medium

10 of 37

Application +4 | -1

A particle moves in a straight line so that

s = \sqrt t

,then its acceleration is proportional to

View answer and explanation

Correct option: A

{\left( {\mathrm{velocity}} \right)^3}

Option C (

\frac{2}{{\sqrt 5 }}

) is correct because it is the perpendicular distance from the origin to the normal of the curve at x=0.

11 Medium

11 of 37

Application +4 | -1

The rate of change of area of a square

S

is equal to its side length. The rate of change of side of

S

is same as that of a cube

C

. The rate of change of volume of

C

when its side length is 2 units, is

View answer and explanation

Correct option: C

Option C (6) is correct because it is the rate of change of the volume of the cube when its side length is 2 units, calculated using related rates.

12 Medium

12 of 37

Application +4 | -1

A balloon is pumped at the rate of

\mathrm{cm}^3 /

minute. The rate of increase of its surface area when the radius is b

\mathrm{cm}

, is -

View answer and explanation

Correct option: C

(2 \mathrm{a} / \mathrm{b}) \mathrm{cm}^2 / \mathrm{min}

Option C (

(2a / b) \text{{ cm}}^2 / \text{{min}}

) is correct because differentiating the volume and surface area formulas with respect to time and then substituting r=b leads to this rate of increase.

13 Medium

13 of 37

Application +4 | -1

A particle is moving in a straight line. At time

t

, the distance between the particle from its starting point is given by

x = t - 6{t^2} + {t^3}

.Its acceleration will be zero at

View answer and explanation

Correct option: B

t = 2

unit time

Option B (

t = 2

unit time) is correct because it's the time at which the second derivative of the position function (acceleration) is zero.

14 Medium

14 of 37

Application +4 | -1

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

View answer and explanation

Correct option: D

\frac{1}{2 \sqrt{\pi}}

unit

Option A (10.08 ft) is correct because it is the distance traveled by the car before it stops, calculated using the given equation.

15 Medium

15 of 37

Application +4 | -1

The sides of an equilateral triangle are increasing at the rate of 2cm/s.The rate at . Which the area increases, when the side is 10cm is

View answer and explanation

Correct option: C

10\sqrt 3 \mathrm{c}{\mathrm{m}^2}/\mathrm{s}

Option C (

10\sqrt 3 \mathrm{c}{\mathrm{m}^2}/\mathrm{s}

) is correct because it represents the rate of change of the area of the equilateral triangle when the side is 10cm, calculated using related rates.

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16 Medium

16 of 37

Application +4 | -1

A spherical balloon is being inflated at the rate of 30

c{c^{\;\;}}/\mathrm{min}

. The rate of increase of the surface area of the balloon when its diameter is 14cm, is

View answer and explanation

Correct option: B

10 sq cm/min

Option B (10 sq cm/min) is correct because it is the rate of increase of the surface area calculated using related rates, given the rate of inflation and the diameter.

17 Medium

17 of 37

Application +4 | -1

A spherical raindrop evaporates at a rate proportional to its surface area at any instant

t

. The differential equation giving the rate of change of the radius of the raindrop is

View answer and explanation

Correct option: A

\frac{d r}{d t}+k=0\,\,\,(k > 0)

Option A (

\frac{d r}{d t}+k=0

k > 0

) is correct because the rate of change of the radius is proportional to the surface area, and the radius is decreasing.

18 Medium

18 of 37

Application +4 | -1

Moving along the x-axis there are two points with

x = 10 + 6t,x = 3 + {t^2}.

The speed with which they are reaching from each other at the time of encounter is in cm and t is in seconds)

View answer and explanation

Correct option: C

8 cm/s

Option C (8 cm/s) is correct because it represents the relative speed between the two points at the time of their encounter, calculated by finding the difference between their individual velocities.

19 Medium

19 of 37

Application +4 | -1

A point moves along the curve

12y = {x^3}

in such a way that the rate of increase of its ordinate is more than the rate of increase of abscissa. The abscissa of the point lies in the interval

View answer and explanation

Correct option: B

\left( { - \infty ,\; - 2} \right) \cup \left( {2,\;\infty } \right)

Option B (

\left( { - \infty ,\; - 2} \right) \cup \left( {2,\;\infty } \right)

) is correct because it represents the interval where the rate of increase of the ordinate is greater than the rate of increase of the abscissa.

20 Medium

20 of 37

Application +4 | -1

A particle moves along a straight line with the law of motion given by

{s^2} = a{t^2} + 2bt + c.

then the acceleration varies are

View answer and explanation

Correct option: A

\frac{1}{{{s^3}}}

Option A (

\frac{1}{{{s^3}}}

) is correct because the acceleration is inversely proportional to the cube of the displacement, derived from differentiating the given equation of motion.

21 Medium

21 of 37

Application +4 | -1

If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of the side is

View answer and explanation

Correct option: B

2 units

Option D (

f(6) \ge 8

) is correct because it is the lower bound for f(6) based on the Mean Value Theorem and the given conditions.

22 Medium

22 of 37

Application +4 | -1

The edge of a cube is equal to the radius of a sphere. If the edge and the radius increase at the same rate, then the ratio of the increases in surface areas of the cube and sphere is

View answer and explanation

Correct option: B

3\;:2\pi

Option B (

3\;:2\pi

) is correct because it is the ratio of the rates of change of the surface areas of the cube and sphere, given that their edge and radius increase at the same rate.

23 Medium

23 of 37

Application +4 | -1

If the radius of a circle be increasing at a uniform rate of 2cm/s. The area of increasing of area of circle ,at the instant when the radius is 20 cm, is

View answer and explanation

Correct option: C

80\pi c{m^2}/s\;

Option C (

80\pi c{m^2}/s\;

) is correct because it represents the rate of increase of the area of the circle when the radius is 20 cm, calculated using related rates.

24 Medium

24 of 37

Application +4 | -1

The distance

s

meters covered by a body in t seconds,is given by

s = 3{t^2} - 8t + 5.

The body will stop after

View answer and explanation

Correct option: C

\frac{4}{3}s

Option C (

\frac{4}{3}s

) is correct because it represents the time at which the velocity of the body becomes zero, indicating that the body has stopped.

25 Medium

25 of 37

Application +4 | -1

If a particle moves such that the displacement is proportional to the square of the velocity acquired , then its acceleration is

View answer and explanation

Correct option: D

A constant

Option D (A constant) is correct because if displacement is proportional to the square of the velocity, then the acceleration is constant.

26 Medium

26 of 37

Application +4 | -1

The pressure

P

and volume

V

of a gas are connected by the relation

P{V^{1/4}} = \mathrm{constant}.

The percentage increase in the pressure corresponding to a deminition of

1/2\;\%

in the volume is

View answer and explanation

Correct option: C

\frac{1}{8}\%

Option C (

\frac{1}{8}\%

) is correct because it represents the percentage increase in pressure corresponding to a diminution of 1/2 % in volume, calculated using the given relation.

27 Medium

27 of 37

Application +4 | -1

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of

\mathrm{50 c}{\mathrm{m}^\mathrm{3}}/\min

. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is

View answer and explanation

Correct option: B

\frac{1}{{18\pi }}

Option B (

\frac{1}{{18\pi }}

) is correct because it represents the rate at which the thickness of the ice decreases, calculated using related rates and the volume of a sphere.

28 Medium

28 of 37

Application +4 | -1

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

r

, is

View answer and explanation

Correct option: D

2{r^2}

Option D (

2{r^2}

) is correct because it represents the maximum area of a rectangle inscribed in a circle, derived through calculus.

29 Medium

29 of 37

Application +4 | -1

A missile is fired from the ground level rises

x

metres vertically upwards in

t

seconds where

x = 100t - \frac{{25}}{2}{t^2}.

The maximum height reached is

View answer and explanation

Correct option: A

200 m

Option A (200 m) is correct because it is the maximum height reached by the missile, calculated by finding when the velocity is zero and substituting that time back into the height equation.

30 Medium

30 of 37

Application +4 | -1

The distance travelled by a particle upto time

x

is given by

f\left( x \right) = {x^3} - 2x + 1.

The time

c

at which the velocity of the particle is equal to its average velocity between times

x = 1

sec and

x = 2

sec, is

View answer and explanation

Correct option: C

\sqrt 3 \;\mathrm{sec}

Option C (

\sqrt 3 \;\mathrm{sec}

) is correct because it represents the time at which the instantaneous velocity equals the average velocity over the given interval, as determined by the Mean Value Theorem.

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Key Takeaways

✓ This set contains 37 single-correct-answer MCQs on RATE MEASURE.
✓ Try each question before opening its answer. Read the explanation where one is available.
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