CBSE Class 12 Maths calculus: 35 marks, five chapters, worked examples

Class 12 Maths unit marks, 2026-27

Source: CBSE Mathematics (041) syllabus 2026-27. The question paper design states there is no chapter-wise weightage and all chapters must be covered.

What the five calculus chapters include in 2026-27

Worked example 1: maxima and minima

Find the local maximum and minimum values of f(x) = x^3 - 6x^2 + 9x + 1.

1. Differentiate: f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3).
2. Critical points where f'(x) = 0: x = 1 and x = 3.
3. Second derivative: f''(x) = 6x - 12. At x = 1, f''(1) = -6 < 0, so a local maximum. At x = 3, f''(3) = 6 > 0, so a local minimum.
4. Values: f(1) = 1 - 6 + 9 + 1 = 5 (local maximum); f(3) = 27 - 54 + 27 + 1 = 1 (local minimum).

Worked example 2: integration by parts

Evaluate the integral of x e^x from 0 to 1.

1. Choose u = x (it simplifies when differentiated) and dv = e^x dx, so du = dx and v = e^x.
2. By parts: integral of x e^x dx = x e^x - integral of e^x dx = x e^x - e^x + C = e^x(x - 1) + C.
3. Apply the limits: [e^x(x - 1)] from 0 to 1 = e^1(0) - e^0(-1) = 0 + 1 = 1.
4. The answer is exactly 1. A numerical check of the area agrees to many decimal places.

Worked example 3: a linear differential equation

Solve dy/dx + y = e^(-x), given y = 2 when x = 0.

1. This is linear with P = 1 and Q = e^(-x). Integrating factor: e^(integral of 1 dx) = e^x.
2. Multiply through: e^x dy/dx + e^x y = 1, and the left side is d/dx (y e^x).
3. Integrate: y e^x = x + C, so y = (x + C) e^(-x).
4. Use y(0) = 2: 2 = (0 + C) × 1, so C = 2. Particular solution: y = (x + 2) e^(-x).
5. Check: dy/dx = e^(-x) - (x + 2)e^(-x), and adding y gives e^(-x). Correct.

Worked example 4: implicit differentiation

Find dy/dx for x^2 + y^2 = 25 at the point (3, 4).

1. Differentiate both sides with respect to x: 2x + 2y (dy/dx) = 0.
2. Rearrange: dy/dx = -x/y.
3. At (3, 4): dy/dx = -3/4. This is the gradient of the tangent to the circle at that point, which is perpendicular to the radius of gradient 4/3, as expected.

Common mistakes that cost marks

• Leaving out the constant of integration in indefinite integrals, or carrying it into a definite integral.
• Choosing u and dv the wrong way round in integration by parts, which makes the integral harder.
• Stating a maximum without justifying it with the first or second derivative test.
• Area questions: not sketching the region, and so integrating the wrong function or between the wrong limits.
• Differential equations: forgetting to divide through so the coefficient of dy/dx is 1 before finding the integrating factor.
• Chain rule slips with inverse trig and log functions, such as differentiating log(sin x) as 1/sin x instead of cot x.

Exam technique and how a tutor helps

The 2026-27 sample paper has 38 questions: 18 MCQs and 2 assertion-reason questions of 1 mark, 5 questions of 2 marks, 6 of 3 marks, 4 long answers of 5 marks and 3 case studies of 4 marks. Calculus appears at every level, so speed on standard derivatives and integrals in Section A buys time for the long answers. In long answers, write the method line by line: CBSE marking schemes give step marks. The syllabus typology puts 44 marks on remembering and understanding, 20 on applying and 16 on analysing, evaluating and creating.

In one-to-one lessons a tutor watches the working on the shared whiteboard and finds the step that fails, whether it is choosing a substitution, setting up an optimisation problem from words, or drawing the region for an area question. The tutor then sets short targeted questions on that step before moving to full board-style questions. Calculus also underpins entrance exams for engineering, so lessons can stretch beyond the board level where a student needs it.

Self-check

• Differentiate y = sin^(-1)(2x). (Answer: 2/sqrt(1 - 4x^2))
• Find the integral of 1/(x^2 + 9) dx. (Answer: (1/3) tan^(-1)(x/3) + C)
• Find the area under y = x^2 from x = 0 to x = 3. (Answer: 9 square units)
• Find the order and degree of (d2y/dx2)^2 + dy/dx = 0. (Answer: order 2, degree 2)
• Is f(x) = e^x increasing for all x? Justify. (Answer: yes, f'(x) = e^x > 0)