MCQs on Class 12 Maths Chapter Application of Integrals
Check out the Class 12 Maths Chapter 8 Application of Integrals multiple-choice questions. Each MCQ has four options, but only one of them is correct. Students must select the appropriate option and compare their results to the solutions on this page. Also, check important questions for class 12 Maths.
1) The area bounded by the curves y2 = 4x and y = x is equal to
1/3
8/3
35/6
None of these
Answer: (b) 8/3
Explanation:
For the given curves, y2 = 4x and y = x, the intersection points are (0, 0) and (4, 4).
Therefore, the area bounded by the curves,
A =0∫4 [√(4x)-x] dx [since y2 = 4x, y = √(4x) ]
A = 2. 0∫4√x dx – 0∫4 x dx
Integrate the function and apply the limits, we get
A = (4/3)(8) – 8
A = (32-24)/3 = 8/3.
Hence, option (b) is the correct answer.
2) The area of the figure bounded by the curve y = logex, the x-axis and the straight line x = e is
5-e
3+e
1
None of these
Answer: (c) 1
Explanation:
At, x= 1, y = loge (1) = 0
At, x = e, y = loge (e) = 1
Therefore, A = 1∫e logex dx
Using integration by parts,
A = [x loge x – x]1e
Now, apply the limits, we get
A = [e-e-0+1]
A = 1
3) The area of the region bounded by the curve x² = 4y and the straight line x = 4y – 2 is
⅜ sq. units
⅝ sq. units
⅞ sq. units
9/8 sq. units
Answer: (d) 9/8 sq. units
Explanation:
For the curves x2 = y and x = 4y-2, the points of intersection are x = -1 and x = 2.
Hence, the required area, A = -1∫2 {[(x+2)/4]- [x2/4] } dx
Now, integrate the function and apply the limits, we get
A = (¼)[(10/3)-(-7/6)]
A = (¼)(9/2) = 9/8 sq. units
Hence, the correct answer is option (d) 9/8 sq. units
4) The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is
7
14
45/4
None of these
Answer: (c) 45/4
Explanation:
Given curve, y=x3 or x = y1/3.
Hence, the required area, A = 1∫8 y1/3 dy
A = [(y4/3)/(4/3)]18
Now, apply the limits, we get
A = (¾)(16-1)
A = (¾)(15) = 45/4.
Hence, option (c) 45/4 is the correct answer.
5) The area of the region bounded by the curve y² = x, the y-axis and between y = 2 and y = 4 is
52/3 sq. units
54/3 sq. units
56/3 sq. units
None of these
Answer: (c) 56/3
Explanation:
Given: y2 = x
Hence, the required area, A = 2∫4 y2 dy
A = [y3/3]24
A = (43/3) – (23/3)
A = (64/3) – (8/3)
A = 56/3 sq. units.
6) Area of the region bounded by the curve y = cos x between x = 0 and x = π is
1 sq. units
2 sq. units
3 sq. units
4 sq. units
Answer: (b) 2 sq. units
Explanation:
Given: y= cos x and also provided that x= 0 and x = π
Hence, the required area, A =0∫π |cos x| dx
It can also be written as,
A = 20∫π/2 cos x dx
Now, integrate the function, we get
A = 2[sin x]0π/2
Now, apply the limits we get
A = 2 sq. units
Hence, the correct answer is option (b) 2 sq. units.
7) Area of the region bounded by the curve x = 2y + 3, the y-axis and between y = -1 and y = 1 is
6 sq. units
4 sq. units
8 sq. units
3/2 sq. units
Answer: (a) 6 sq. units
Explanation:
Required Area =-1∫1(2y+3)dy
A=[(2y2/2)+3y]-11
Now, apply the limits, we get
A = 1+3-1+3
A = 6 sq. units.
8) The area bounded by the curve y = x3, the x-axis and two ordinates x = 1 and x = 2 is
15/2 sq. units
15/4 sq. units
17/2 sq. units
17/4 sq. units
Answer: (b) 15/4 sq. units
Explanation:
Required Area = 1∫2 x3 dx
A = [x4/4]12
Now, apply the limits, we get
A = [(24/4) – (¼)]
A = (16/4) – (¼)
A = 15/4
Hence, the required area is 15/4 is the correct answer.
9) The area of the region bounded by the circle x² + y² = 1 is
2π sq. units
3π sq. units
4π sq. units
1π sq. units
Answer: (d) 1π sq. units
Explanation: Given:
Given circle equation is x2+y2 =1, whose centre is (0, 0) and radius is 1.
Therefore, y2 = 1-x2
y=√(1-x2)
Hence, the required area, A = 40∫1√(1-x2) dx
Now, integrate the function and apply limits, we get
A = 4(½)(π/2)
A = π sq. units
Hence, option (d) 1π sq. unit is the correct answer.
10) Area bounded by the curve y = sin x and the x-axis between x = 0 and x = 2π is
2 sq. units
3 sq. units
4 sq. units
None of these
Answer: (c) 4 sq. units
Explanation:
Required Area, A = 0∫2π |sin x| dx
A = 0∫π sin x dx + π∫2π(-sin x) dx
Now, substitute the limits, we get
A= 4 sq. units.
Class 12 Maths Chapter 8 Application of Integrals MCQs are provided to help students improve their test scores. Answers and full explanations are provided for these multiple-choice questions. The questions are written in accordance with NCERT guidelines and the CBSE syllabus.
Neeraj Anand, Param Anand
Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
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CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution
The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.
CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24
Max Marks: 80
No.
Units
Marks
I.
Relations and Functions
08
II.
Algebra
10
III.
Calculus
35
IV.
Vectors and Three – Dimensional Geometry
14
V.
Linear Programming
05
VI.
Probability
08
Total Theory
80
Internal Assessment
20
Grand Total
100
Unit-I: Relations and Functions
1. Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Unit-II: Algebra
1. Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Unit-III: Calculus
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
5. Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
dy/dx + py = q, where p and q are functions of x or constants.
dx/dy + px = q, where p and q are functions of y or constants.
Unit-IV: Vectors and Three-Dimensional Geometry
1. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
2. Three – dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.
Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26
Q1
Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?
Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.
Q2
How many units are discussed in the CBSE Class 12 Maths Syllabus?
In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.
Q3
How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?
About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.
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