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 · Mixed difficulty

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Practice format and chapter quiz

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 110 questions

1 Easy

1 of 110

Application +4 | -1

The surface area of a sphere when its volume is increasing at the same rate as its radius, is

View answer and explanation

Correct option: A

The correct answer is 1 because when the volume and radius increase at the same rate, the surface area of the sphere is equal to 1.

2 Easy

2 of 110

Application +4 | -1

A particle is moving in a straight line has velocity and displacement equation as

v = 4\sqrt {1 + s}

where

v

velocity in m/s and

s

m

. The initial velocity of the particle is

View answer and explanation

Correct option: A

4 m/s

Option A (4 m/s) is correct. It is the initial velocity obtained by substituting s=0 in the given equation.

3 Easy

3 of 110

Application +4 | -1

The diagonal of a square is changing at the rate of 0.5 cm

{s^{ - 1}}

. Then, the rate of change of area , when the area is 400

c{m^2}

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the calculated rate of change of area when the area is 400

c{m^2}

, given the diagonal's rate of change.

4 Easy

4 of 110

Application +4 | -1

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 49 minutes after the leakage began is:

View answer and explanation

Correct option: C

\frac{2}{9}

Option C (

\frac{2}{9}

) is correct because after 49 minutes, the rate at which the radius decreases is calculated to be

\frac{2}{9}

meters per minute.

5 Easy

5 of 110

Application +4 | -1

A solid box is formed by placing a cylinder, having equal height and radius on top of a cube such that, the circular base of cylinder is the inscribed circle for square top of the cube. If the radius of cylinder is changing at the rate

\frac{1}{2 \pi+16}

\mathrm{cm} / \mathrm{sec}

, then the rate of change of volume of the box when radius is

2 \mathrm{~cm}

, is (Assuming that box always remain in the given shape)

View answer and explanation

Correct option: A

Option A (6) is correct because the rate of change of volume is calculated by differentiating the volume equation with respect to time and substituting the given values.

6 Medium

6 of 110

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.

7 Medium

7 of 110

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.

8 Easy

8 of 110

Application +4 | -1

The point on the curve

\sqrt x + \sqrt y = \sqrt a

at which the normal is parallel to the

x

-axis, is

View answer and explanation

Correct option: B

\left( {0,\;a} \right)

Option B (

\left( {0,\;a} \right)

) is correct because it is the point on the curve where the normal is parallel to the x-axis, meaning the derivative dx/dy is zero.

9 Easy

9 of 110

Application +4 | -1

A cylindrical vessel of radius

0.5 \mathrm{~m}

is filled with oil at the rate of

0.25\;\pi {\mathrm{m}^3}

/minute. The rate at which the surface of the oil is rising, is

View answer and explanation

Correct option: A

1 \mathrm{~m} /

minute

The correct answer is A (

1 \mathrm{~m} /

minute) because it represents the rate at which the oil surface is rising, calculated using related rates and the given volume increase.

10 Easy

10 of 110

Application +4 | -1

Side of an equilateral triangle expands at the rate of

2 \mathrm{~cm} / \mathrm{sec}

. The rate of increase of its area when each side is

10 \mathrm{~cm}

View answer and explanation

Correct option: B

10 \sqrt{3} \mathrm{~cm}^2 / \mathrm{sec}

Option B (

10 \sqrt{3} \mathrm{~cm}^2 / \mathrm{sec}

) is correct because it is the rate of increase of the area of the equilateral triangle, calculated using related rates.

11 Medium

11 of 110

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.

12 Easy

12 of 110

Application +4 | -1

The position of a moving car at time

t

is given by

f(t) = a{t^2} + bt + c,t > 0,

where a, b and c are real numbers greater than 1. Then, the average speed of the car over the time interval

[{t_1},{t_2}]

is attained at the point

View answer and explanation

Correct option: C

({t_1} + {t_2})/2

Option C (

({t_1} + {t_2})/2

) is correct because it represents the time at which the instantaneous speed equals the average speed over the given interval, according to the Mean Value Theorem.

13 Easy

13 of 110

Application +4 | -1

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is:

View answer and explanation

Correct option: A

Option B (221) is correct because it is the result of calculating the sum of the determinants of the adjoint matrices, based on the given conditions.

14 Medium

14 of 110

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}}

15 Easy

15 of 110

Application +4 | -1

The height of a cylinder is equal to the radius. If an error of

\alpha \%

is made in the height, then percentage error in its volume is

View answer and explanation

Correct option: C

3\alpha \,\%

Option C (

3\alpha \,\%

) is correct because the percentage error in the volume is three times the percentage error in the height when the height equals the radius.

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16 Easy

16 of 110

Application +4 | -1

A particle is moving along the curve

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

ac = {h^2},

then the particle would be moving with uniform

View answer and explanation

Correct option: C

Acceleration

Option C (Acceleration) is correct because the second derivative of the position function with respect to time is constant, indicating uniform acceleration.

17 Medium

17 of 110

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.

18 Medium

18 of 110

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.

19 Easy

19 of 110

Application +4 | -1

If the surface area of a cube is increasing at a rate of

4.8 \mathrm{~cm}^2 / \mathrm{sec}

. retaining its shape, then the rate of change of its volume (in

\mathrm{cm}^3 / \mathrm{sec}

.) when the length of a side of the cube is

15 \mathrm{~cm}

is:

View answer and explanation

Correct option: A

Option A (18) is correct because differentiating the surface area and volume of the cube with respect to time and using the given rate of change of surface area allows us to find the rate of change of volume.

20 Easy

20 of 110

Application +4 | -1

If the surface area of a cube is increasing at a rate of

3.6\,c{m^2}/\sec ,

retaining its shape; then the rate of change of its volume

(\mathrm{in}\,c{m^3}/\sec ),

when the length of a side of the cube is 10 cm is

View answer and explanation

Correct option: C

Option C (9) is correct because it is derived from finding the rate of change of the volume of the cube using the given rate of change of its surface area.

21 Easy

21 of 110

Application +4 | -1

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

-coordinate is decreasing at the rate of 3unit per second.The rate at which the

x

-coordinate changes at this point is (in unit per second)

View answer and explanation

Correct option: B

3\sqrt 3

Option B (

3\sqrt 3

) is correct because it represents the rate at which the x-coordinate changes at the given point.

22 Easy

22 of 110

Application +4 | -1

A ball weighing 10 gm hits a hard surface vertically with a speed of 4m/s and rebounds with same speed. The ball remains contact with the surface for 0.01s. The average force exerted by the surface on the ball is

View answer and explanation

Correct option: D

8 N

Option D (8 N) is correct because the average force is calculated using the change in momentum divided by the time interval.

23 Easy

23 of 110

Application +4 | -1

If there is

2\%

error in measuring the radius of sphere, then… will be the percentage error in the surface area.

View answer and explanation

Correct option: C

4\%

The correct answer is

4\%

because the percentage error in the surface area is twice the percentage error in the radius.

24 Easy

24 of 110

Application +4 | -1

The minimum acceleration with which a person can slide down a rope whose breaking strength is 60% of his weight

View answer and explanation

Correct option: C

\frac{{2g}}{5}

Option C (

\frac{{2g}}{5}

) is correct because the tension in the rope is less than the person's weight, resulting in a downward acceleration.

25 Easy

25 of 110

Application +4 | -1

In a

\triangle A B C

the sides

b

and

c

are given. If there is an error

\triangle A

in measuring angle

A

, then the error

\triangle

a in side a is given by

View answer and explanation

Correct option: B

\frac{2 S}{a} \Delta A

Option B (

\frac{{2S}}{{a}} \Delta A

) is correct because it represents the error in side a calculated using differentials and the area of the triangle.

26 Medium

26 of 110

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}

27 Medium

27 of 110

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.

28 Medium

28 of 110

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).

29 Easy

29 of 110

Application +4 | -1

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time

t = 0

. The number of bacteria is increased by 20% in

2\,h

. If the population of bacteria is 2000 after

\frac{k}{{{{\log }_e}(6/5)}}h

, then

{\left( {\frac{k}{{{{\log }_e}2}}} \right)^2}

is equal to

View answer and explanation

Correct option: A

Option A (4) is correct because after solving the differential equation for bacterial growth and substituting the given conditions, the value of the expression is found to be 4.

30 Medium

30 of 110

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.

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Key Takeaways

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