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JEE • MATHEMATICS • RATE MEASURE

Rate measure practice · Easy

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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 72 questions

1 Easy

1 of 72

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 72

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 72

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 72

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 72

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 Easy

6 of 72

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.

7 Easy

7 of 72

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.

8 Easy

8 of 72

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.

9 Easy

9 of 72

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.

10 Easy

10 of 72

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.

11 Easy

11 of 72

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.

12 Easy

12 of 72

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.

13 Easy

13 of 72

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.

14 Easy

14 of 72

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.

15 Easy

15 of 72

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.

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16 Easy

16 of 72

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.

17 Easy

17 of 72

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.

18 Easy

18 of 72

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.

19 Easy

19 of 72

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.

20 Easy

20 of 72

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.

21 Easy

21 of 72

Application +4 | -1

The rate of change of surface area of a sphere of raditis

r

when the radius is increasing at the rate of

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

is proportional to

View answer and explanation

Correct option: D

r

The correct answer is D (

r

) because the rate of change of the surface area of a sphere is directly proportional to its radius when the radius is increasing at a constant rate.

22 Easy

22 of 72

Application +4 | -1

Rays from the sun are focused by a lens of diameter 5m on to a block of ice

\left( {l = 80\,kcal/10g} \right)

and 10 g of ice is melted in 20 minutes. Calculate the heat received from the sun on

1\,c{m^2}

per minute

View answer and explanation

Correct option: D

2 cal

Option D (2 cal) is correct because it is the heat received per unit area per minute, calculated from the latent heat of fusion and the area of the lens.

23 Easy

23 of 72

Application +4 | -1

A particle moves along the curve

y = {x^2} + 2x

. Then, The point on the curve such that

x\,and\,y

coordinates of the particle change with the same rate is

View answer and explanation

Correct option: C

\left( { - \frac{1}{2}, - \frac{3}{4}} \right)

Option C

\left( { - \frac{{1}}{{2}}, - \frac{{3}}{{4}}} \right)

is correct because it satisfies the condition where the rates of change of x and y coordinates are equal.

24 Easy

24 of 72

Application +4 | -1

A vehicle of 200 kg moving with a velocity of 6 m/sec . To stop if in

\frac{1}{{15}}\sec

the required force in opposite direction

View answer and explanation

Correct option: D

8000 N

Option D (8000 N) is correct because the required force is calculated using the change in momentum divided by the time interval.

25 Easy

25 of 72

Application +4 | -1

The distance covered by a particle in

t

second is given by

x = 3 + 8t - 4{t^2}.\;

After 1s its velocity will be

View answer and explanation

Correct option: A

0 unit

Option A (0 unit) is correct because the velocity after 1 second is calculated by differentiating the position function and substituting t=1.

26 Easy

26 of 72

Application +4 | -1

the speed v of particle moving along a straight line is given by

a + {y^2} = {x^2}

(where

x

is its distance from the origin).The acceleration of the particle is

View answer and explanation

Correct option: C

\frac{x}{b}

Option B ((

- \infty ,\;0

)) is correct because the function is increasing when its derivative is positive, which occurs for x < 0.

27 Easy

27 of 72

Application +4 | -1

A cone whose height is always equal to its diameter is increasing in volume at the rate of

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

. At what rate is the radius increasing when its circular base area is

1 \mathrm{~m}^2

View answer and explanation

Correct option: D

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

The correct answer is D (

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

) because it is the rate of change of the radius calculated using related rates and the given volume increase.

28 Easy

28 of 72

Application +4 | -1

A sphere of

10 \mathrm{~cm}

radius has a uniform thickness of ice around it. If ice is melting at the rate of

50 \mathrm{~cm}^3 / \mathrm{min}

when thickness is

5 \mathrm{~cm}

, then rate of change of thickness is

View answer and explanation

Correct option: B

\frac{1}{18 \pi}

The correct answer is

\frac{1}{18 \pi}

because this is the calculated rate of change of the ice thickness based on the given melting rate and thickness.

29 Easy

29 of 72

Application +4 | -1

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

50\,c{m^3}/\min .

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

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the rate at which the thickness of the ice decreases when the thickness is 5 cm, calculated using related rates.

30 Easy

30 of 72

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

{(\mathrm{velocity})^3}

Option A (

{(\mathrm{velocity})^3}

) is correct because the acceleration is proportional to the cube of the velocity, derived from the given relation

s = \sqrt{t}

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Key Takeaways

✓ This set contains 72 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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