CBSE Class 12 Maths 2024-25: Chapter 8 Application of Integrals Competency-Based Questions with Answers; Download Free PDF

As the CBSE Class 12 board exams get closer, students need to understand the new exam pattern. Starting in the 2024-25 school year, CBSE will include 50% more competency-based questions. These questions will be both multiple choice and written, focusing on how to use what students have learned in real-life situations.

This article explores Chapter 8 Application of Integrals highlights key competency-based questions and provides answers to help students succeed.

Understanding Competency-Based Questions in Chapter 8 Application of Integrals

Competency-based questions are designed to see how well students can apply their knowledge in everyday life. They can come in different forms, such as case studies, true-false questions, gap-filling tasks, and long or short answer questions.

These questions are different from regular memorization. They encourage students to think critically and solve problems, helping them understand the concepts in Chapter 8 Application of Integrals better.

CBSE Class 12 Maths Chapter 8 Application of Integrals Important Competency-Based Questions

Q.1 Shown below are partial graphs of two distinct functions, f ( y ) and g ( y ). The region between them is shaded.

Answer. (3)

Q.2 Shown below is the graph of the function x ² + y ² = 16, 0 ≤ x ≤ 4.

Answer. (1)

Q.3 If the temperature of an electric oven is increasing at the rate of R ( t ) °C per minute, then what does the following expression represent?

1. Rate of increase of temperature from the 0-minute mark to the 15-minute mark
2. Average increase in the temperature from the 0-minute mark to the 15 minute mark
3. Average rate of increase of temperature from the 0-minute mark to the 15-minute mark
4. Difference in the rate of increase of temperature at the 15-minute mark and the 0-minute mark

Answer. (4)

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CBSE Class 12 Maths Chapter 1: Relations and Functions Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 2 : Inverse Trigonometric Functions Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 3 : Matrices Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 4: Determinants Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 5: Continuity & Differentiability Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 6: Application of Derivatives Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 7: Integrals Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 8: Application of Integrals Important Competency-Based Questions	Click Here

Q.4 Shown below is the graph of (-x = y² - 3).

Who is correct?
1. only Ravi
3. both Ravi and Kanika
2. only Kanika
4. neither Ravi nor Kanika

Answer. (3)

Q.5 The shaded region shown below is bounded by the curve y = ex, the y-axis and the line y = e²,

Answer. (2)

Q.6 The area under the curve y = x² between the line x = 0 and x = k is 9 square units. Which of the following could be the value of k?
1. 3
2. 4.5
3. 9
4. 27

Answer. (1)

Q: 7 Water from an overhead tank is flowing down through different pipes. The volume of water (in litres) transferred at time t (in hours) is given by the function:

w(t) = e²-e^t

Shown below is the graph of w(t).

Approximately, what is the total volume of water transferred from the overhead tank in 2 hours?

(Note: Take e² = 7.389.)

1. 6.4 litres
2. 8.4 litres
3. 64 litres
4. 84 litres

Answer. (4)

Q.8 Shown below is the graph of the function y = x³- 1.

Write an expression to find the area of the shaded region by integrating with respect to the x-axis.

Answer. Writes the expression to find the area of the shaded region by integrating with respect to the x-axis as:

(Award 0.5 marks if only the correct expression with correct limits is written without the modulus symbol.)

Q.9 Shown below are two curves.

Answer. Writes the expression for the area of the shaded region as:

Concludes that the area of the shaded region is 21.33 sq units.

(Award full marks if the area in the first quadrant is found and then multiplied by 4.)

Q.10 A taxi company projected its revenue for 6 years using the graph shown below. The shaded region represents the total revenue that can be generated over 6 years.

Find the total revenue that will be generated over the last two years based on the projection. Show your steps.
(Note: Take e² = 7.39.)

Answer. Writes the expression for the total revenue that will be generated over the last two years based on the projection as:

Simplifies the above expression as [ e² - 0.02 x 6) - ( e° - 0.02 x 4) = 7.39 -0.12 - 1 + 0.08 = 6.35.

Finds the total revenue that will be generated over the last two years based on the projection as 6.35 x 1000000 = Rs 6350000.

Q.11 In economics, a learning curve is defined as a curve that shows the relation between the number of units produced and the time taken to produce them over a given period. Shown below is the learning curve of Hunar Cosmetics Private Limited.

Find the total hours required to produce the units from the 3600^th to the 10000^th unit. Show your steps.

Answer. Writes the expression for the total hours required to produce the units from 3600^th to 10000^th unit as:

Solves the above integral to find the hours required to produce the units from the 3600^th to the 10000^th unit as 101760 hours.

Q.12 In economics, the equilibrium price is the point of intersection of the supply curve and demand curve. Consumer surplus is defined as the area covered between the demand curve and the equilibrium price line (see the shaded region in the image below).

Determine

i) the equilibrium price for rice.
ii) the consumer surplus.

Show your steps.

(Note: Use In(2) = 0.69, In(2.25) = 0.81, In(4.5) = 1.50)

Answer. i) Writes the equation to calculate equilibrium price as:

Solves the equation in step 1 and finds r as 2.5 tonnes and equilibrium price as 40 rupees.

ii) Frames the consumer surplus as the area between the demand curve, the y axis and y = 40 as:

Q.13 The area under the velocity-time graph gives the displacement of a moving object. Shown below is the velocity-time graph of a bird's flight.

Find the displacement of the bird in the first 40 seconds. Show your work.

Answer. Writes the integral to find the displacement of the bird in the first 40 seconds as:

Q.14 In economics, the number of years for which any company will be profitable can be found at the point of intersection of the marginal revenue (R') curve and the marginal cost (C') curve. The maximum profit a company can earn during this time is defined as the area between the R' curve, the C' curve and the y-axis as shown in the figure.

where t is the number of years and R' and C' are in crores of rupees. Find:

i) the number of years for which Indian Collection Limited will be profitable.
ii) the maximum profit that Indian Collection Limited can earn during that period. Show your steps.

Answer. i) Writes the equation to find the number of years for which Indian Collection Limited will be profitable as:

Solves the above equation for t to find the total number of years for which Indian Collection Limited will be profitable as 8 years.

ii) Writes the expression for the maximum profit that Indian Collection Limited can earn during that period as:

Solves the above integral to find the maximum profit that Indian Collection Limited can earn during that period as 16 crore rupees.

Q.15 A function f ( x ) is graphed and defined below for x E [0, 6].

Answer. Writes the total area to be a sum of two integrals as:

👉 CBSE Class 12 Study Materials

CBSE Class 12 Syllabus 2024-25	CBSE Class 12 Previous Year Papers
NCERT Books For Class 12 Books	NCERT Class 12 Solutions
CBSE Class 12 Full Study Material	CBSE Class 12 Sample Paper 2023-24