Application of Derivatives Practice Questions

Class 12 · Mathematics · 1543 free MCQs with instant results and detailed explanations.

1543

Total

511

Easy

689

Medium

343

Hard

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Topics in Application of Derivatives

Maxima and Minima 465

Rate of Change 286

Tangents and Normals 316

Increasing and Decreasing Functions 476

Sample Questions from Application of Derivatives

Here are 10 sample questions. Start a quiz to get randomized questions with scoring.

Easy

If the radius of a circle increases at a rate of 2 cm/s, at what rate is the area of the circle increasing when the radius is 5 cm?

A. 20π cm²/s

B. 10π cm²/s

C. 15π cm²/s

D. 25π cm²/s

Show Answer & Explanation

Correct Answer: A

The area (A) of a circle is given by A = πr². The rate of change of area with respect to time can be found using the chain rule: dA/dt = (dA/dr) * (dr/dt). Here, dA/dr = 2πr and dr/dt = 2 cm/s. At r = 5 cm, dA/dt = 2π(5)(2) = 20π cm²/s.

Easy

If y = x^2 + 3x, what is the rate of change of y with respect to x at x = 2?

A. 10

B. 8

C. 6

D. 4

Show Answer & Explanation

Correct Answer: A

The derivative dy/dx = 2x + 3. At x = 2, dy/dx = 2(2) + 3 = 10.

Easy

The function f(x) = 5x - 3 has a constant rate of change. What is it?

A. 5

B. 3

C. 0

D. 10

Show Answer & Explanation

Correct Answer: A

The rate of change is the coefficient of x in a linear function, which is 5.

Medium

A ball is thrown vertically upwards. Its height is given by h(t) = 20t - 5t^2. What is the instantaneous rate of change of height at t = 3 seconds?

A. 5 m/s

B. 0 m/s

C. 10 m/s

D. -5 m/s

Show Answer & Explanation

Correct Answer: D

To find the instantaneous rate of change, calculate the derivative and evaluate it at t = 3, which results in -5 m/s.

Medium

If the distance travelled by a vehicle is given by d(t) = 5t^2 + 2t, what is the average speed over the interval [1, 3] seconds?

A. 20 m/s

B. 15 m/s

C. 10 m/s

D. 12 m/s

Show Answer & Explanation

Correct Answer: B

Average speed is the change in distance divided by change in time. Evaluate d(3) and d(1) and calculate.

Medium

A company's revenue R is given by R(x) = 100x - 2x^2. Find the rate of change of revenue when x = 25 units.

A. 100

B. 150

C. 75

D. 50

Show Answer & Explanation

Correct Answer: C

The rate of change of revenue is given by R'(x). Evaluating at x = 25 yields 75.

Medium

The temperature of a cup of coffee is given by T(t) = 90 - 10t. What is the rate of change of temperature after 5 minutes?

A. -10 °C/min

B. 0 °C/min

C. -50 °C/min

D. -5 °C/min

Show Answer & Explanation

Correct Answer: A

Differentiate T(t) to find the rate of change. It is constant at -10 °C/min.

Hard

The revenue R (in thousands of rupees) generated by selling x items is given by R(x) = 60x - 0.5x². Find the rate of change of revenue when x = 30.

A. 9000

B. 1800

C. 1500

D. 1200

Show Answer & Explanation

Correct Answer: A

To find the rate of change of revenue, we differentiate R(x). R'(x) = 60 - x. At x = 30, R'(30) = 60 - 30 = 30. Therefore, the rate of change is 30 thousand rupees per item, which equals 9000 rupees.

Hard

The height of a cone is decreasing at a rate of 2 cm/min, while the radius is increasing at a rate of 1 cm/min. At what rate is the volume of the cone changing when the height is 10 cm and radius is 5 cm?

A. 20π cm³/min

B. 40π cm³/min

C. 100π cm³/min

D. 60π cm³/min

Show Answer & Explanation

Correct Answer: B

Volume V = (1/3)πr²h. Using the chain rule, dV/dt = (1/3)π(2rh(dr/dt) + r²(dh/dt)). Substituting values gives the rate of change as 40π cm³/min.

Q10

Hard

The distance s (in meters) covered by a car is described by the function s(t) = 3t² + 5t + 2. Determine the average rate of change of distance from t = 1 to t = 3.

A. 12 m/s

B. 8 m/s

C. 10 m/s

D. 6 m/s

Show Answer & Explanation

Correct Answer: A

The average rate of change is given by (s(3) - s(1))/(3 - 1). Calculating gives average speed as 12 m/s.

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Application of Derivatives — Class 12 Mathematics Practice Questions Online

This page contains 1543 practice MCQs for the chapter Application of Derivatives in Class 12 Mathematics. The questions are organized by difficulty — 511 easy, 689 medium, 343 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit. This chapter covers 4 topics, giving you comprehensive coverage of the entire chapter.