CBSE Class 12 Maths Official Sample Paper 2022-23

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.

1. Inverse Trigonometric Functions · 3 marks

   Simplify \(\cot^{-1}\left(\dfrac{1 + \cos 2x}{\sin 2x}\right)\) (a) for \(0 \lt x \lt \dfrac{\pi}{2}\), and (b) for \(\pi \lt x \lt \dfrac{3\pi}{2}\).

   Worked answer

   \(\dfrac{1 + \cos 2x}{\sin 2x} = \dfrac{2\cos^2 x}{2\sin x\cos x} = \cot x\) M1
   (a) \(x\) lies in \((0, \pi)\), the principal range of \(\cot^{-1}\): value \(= x\) A1
   (b) \(\cot x = \cot(x - \pi)\) and \(x - \pi \in \left(0, \dfrac{\pi}{2}\right)\): value \(= x - \pi\) A1

2. Matrices · 5 marks

   A non-zero matrix \(A = \begin{bmatrix}a & b\\ c & d\end{bmatrix}\) with real entries satisfies \(A^2 = O\). (a) Show that \(a + d = 0\) and \(bc = -a^2\). (b) Write down one such matrix with \(a = 3\) and integer entries. (c) Prove that \(I + A\) is invertible, with \((I + A)^{-1} = I - A\).

   Worked answer

   (a) \(A^2 = \begin{bmatrix}a^2 + bc & b(a + d)\\ c(a + d) & d^2 + bc\end{bmatrix} = O\) M1
   If \(a + d \ne 0\) then \(b = c = 0\), so \(a^2 = d^2 = 0\), giving \(a = d = 0\), which contradicts \(a + d \ne 0\). Hence \(a + d = 0\), and then \(a^2 + bc = 0\), i.e. \(bc = -a^2\) R1
   (b) \(d = -3\), \(bc = -9\): e.g. \(\begin{bmatrix}3 & 9\\ -1 & -3\end{bmatrix}\) (any valid choice) A1
   (c) \((I + A)(I - A) = I - A + A - A^2 = I - A^2 = I\) M1
   and similarly \((I - A)(I + A) = I\), so \(I + A\) is invertible with inverse \(I - A\) A1

3. Application of Derivatives · 2 marks

   Find the intervals in which \(f(x) = x\left(\log_e x\right)^2\), \(x \gt 0\), is strictly increasing or strictly decreasing.

   Worked answer

   \(f'(x) = (\log x)^2 + 2\log x = \log x\,(\log x + 2) = 0 \Rightarrow x = 1\) or \(x = e^{-2}\) M1
   \(f' \gt 0\) on \((0, e^{-2})\) and \((1, \infty)\), \(f' \lt 0\) on \((e^{-2}, 1)\): strictly increasing on \(\left(0, e^{-2}\right]\) and \([1, \infty)\); strictly decreasing on \(\left[e^{-2}, 1\right]\) A1

More on these topics: Inverse Trigonometric Functions · Matrices · Application of Derivatives

What the paper covers

Class 12 Mathematics (041) is a 3-hour, 80-mark theory paper plus 20 marks of internal assessment. Unit weightage in the CBSE curriculum:

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.

• Sample paper 1 (Board standard)
• Sample paper 2 (Board standard)
• Sample paper 3 (Above board)
• Sample paper 4 (Above board)
• Sample paper 5 (95+ challenge)
• All Class 12 sample papers

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.

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