CBSE Math RevisionCBSE Class 12 Maths Official Sample Paper 2021-22
CBSE's official sample question paper and marking scheme for the 2021-22 session, linked straight to CBSE. Time yourself, mark it against the scheme, then practise the weakest units.
The official sample paper and marking scheme
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Class 12 Mathematics: sample question paper (Term 1) 041 Specimen
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Class 12 Mathematics: sample question paper (Term 2) 041 Specimen
2021-22 had two terms, each with its own sample paper. Term 1 was multiple choice (90 minutes, 40 marks). 2021-22 had two terms, each with its own sample paper. Term 2 was a written paper (2 hours, 40 marks).
Sit it under exam conditions: open the paper, press Start timer, press Finish when time is up, then mark it against the mark scheme.
Try 3 Class 12 questions like these
Written and checked by us, not taken from the CBSE paper. Tap "Worked answer" to see every step.
Continuity and Differentiability · 2 marks
Find the value of \(k\) for which \(f(x) = \begin{cases} \dfrac{\sqrt{x + 7} - 3}{x - 2}, & x \neq 2 \\ k, & x = 2 \end{cases}\) is continuous at \(x = 2\).
Worked answer
\(\displaystyle\lim_{x\to2}\frac{\sqrt{x+7}-3}{x-2}\cdot\frac{\sqrt{x+7}+3}{\sqrt{x+7}+3} = \lim_{x\to2}\frac{1}{\sqrt{x+7}+3}\) M1
\(= \dfrac16\), so \(k = \dfrac16\) A1Integrals · 3 marks
The marginal cost (in ₹ per unit) of producing \(x\) units of a product is \(MC = 30 + 0.6x\), and the fixed cost is ₹\(5000\). Find the total cost function, and the increase in cost when production rises from \(100\) to \(200\) units.
Worked answer
\(C(x) = \displaystyle\int(30 + 0.6x)dx = 30x + 0.3x^2 + K\); \(C(0) = 5000 \Rightarrow C(x) = 5000 + 30x + 0.3x^2\) A1
Increase \(= \displaystyle\int_{100}^{200}(30 + 0.6x)dx = \left[30x + 0.3x^2\right]_{100}^{200}\) M1
\(= (6000 + 12000) - (3000 + 3000) = \)₹\(12\,000\) A1Application of Integrals · 4 marks
The cross-section of a solar cooker reflector is the parabola \(y^2 = 12x\) (units: cm). The reflector is cut off by the straight rim along the line \(x = 6\).
Find the length of the rim (the chord along \(x = 6\)). [1 mark]
The focus of the parabola lies on the line \(x = a\). Find \(a\) and state whether the focus lies inside the region bounded by the parabola and the rim. [1 mark]
Using integration, find the area of the region bounded by the parabola and the rim. [2 marks]
Worked answer
\(y^2 = 72 \Rightarrow y = \pm 6\sqrt2\); length \(= 12\sqrt2\) cm A1
\(4a = 12 \Rightarrow a = 3\); since \(0 \lt 3 \lt 6\), the focus \((3, 0)\) lies inside the region A1
Area \(= 2\displaystyle\int_0^6 \sqrt{12x}\,dx = 4\sqrt3 \cdot \dfrac23 \cdot 6^{3/2}\) M1
\(= \dfrac{8\sqrt3}{3} \cdot 6\sqrt6 = 48\sqrt2\ \text{cm}^2\) A1
More on these topics: Continuity and Differentiability · Integrals · Application of Integrals
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What the paper covers
Class 12 Mathematics (041) is a 3-hour, 80-mark theory paper plus 20 marks of internal assessment. This sample paper is different: 1 h 30 min, 40 marks (see the note above). Unit weightage in the CBSE curriculum:
| Unit | Marks (of 80) |
|---|---|
| Relations and Functions | 8 |
| Algebra | 10 |
| Calculus | 35 |
| Vectors and Three-Dimensional Geometry | 14 |
| Linear Programming | 5 |
| Probability | 8 |
Chapter links will appear once the CBSE course pages are live. We haven't mapped this paper question by question yet.
More practice: our 5 sample papers on the same pattern
Original Class 12 Maths papers written to the CBSE 2026-27 pattern by us (not official CBSE papers): 80 marks, 38 questions, sections A to E, a 3-hour timer, an A4 print view and a step-marking scheme with where toppers lose marks.
Formula booklet
CBSE Mathematics (Class 10 and Class 12) · CBSE · No official formula booklet
CBSE does not publish a formula booklet and none is given in the exam: formulas must be learnt. Use our one-page sheets to revise them.
More Class 12 papers
Questions students ask
Where do I download the Class 12 Maths official sample paper 2021-22?
The buttons above open CBSE's own copy on cbseacademic.nic.in. We link to CBSE rather than host the file, so you always get the current version.
Is there a marking scheme?
Yes: the marking scheme button opens CBSE's own copy (a dashed button opens CBSE's page where it is listed). Mark your answers against it, then enter your mark.
How is the Class 12 Maths paper marked?
Class 12 Mathematics (041) is a 3-hour, 80-mark theory paper plus 20 marks of internal assessment. The unit weightage is in the table on this page.