CBSE Class 12 Maths Official Sample Paper 2025-26

4 Class 12 questions like these, with worked solutions

Written and checked by us, not taken from the CBSE paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Matrices · 2 marks

   Find matrices $A$ and $B$ if $A+B^T=\left[\begin{array}{cc}3& 5\\ 1& 2\end{array}\right]$ and $A^T-B=\left[\begin{array}{cc}1& -1\\ 3& 0\end{array}\right]$.

   Worked solution

   Transposing the second equation: $A-B^T=\left[\begin{array}{cc}1& 3\\ -1& 0\end{array}\right]$; adding to the first, $2A=\left[\begin{array}{cc}4& 8\\ 0& 2\end{array}\right]$ M1
   $A=\left[\begin{array}{cc}2& 4\\ 0& 1\end{array}\right]$, $B^T=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, so $B=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ A1

2. Determinants · 2 marks

   Evaluate $\left|\begin{array}{ccc}2& -1& 3\\ 0& 4& 1\\ 5& 2& -2\end{array}\right|$ by expanding along the first column.

   Worked solution

   $=2\left|\begin{array}{cc}4& 1\\ 2& -2\end{array}\right|-0+5\left|\begin{array}{cc}-1& 3\\ 4& 1\end{array}\right|$ M1
   $=2(-10)+5(-13)=-85$ A1

3. Integrals · 2 marks

   Find ${\displaystyle \int \frac{\sin 2x}{1+{\cos }^{2}x}dx}$.

   Worked solution

   Put $t=1+{\cos }^{2}x$, $dt=-2\cos x\sin xdx=-\sin 2xdx$ M1
   ${\displaystyle -\int \frac{dt}{t}=-\log (1+{\cos }^{2}x)+C}$ A1

4. Differential Equations · 2 marks

   Find the order and the degree (if defined) of the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}={[1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2]}^{1/3}$.

   Worked solution

   Cubing: ${\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^3=1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2$, a polynomial in the derivatives M1
   Order $2$, degree $3$ A1

More on these topics: Matrices · Determinants · Integrals · Differential Equations

Marks often lost in these chapters

Check your answers to the questions above, and this paper's questions on the same chapters, for these slips.

• Matrices: Writing $(AB{)}^T=A^T{B}^T$ — the order must reverse. Also: Expanding $(A+B{)}^2$ as $A^2+2AB+B^2$ or cancelling: $AB=AC$ does not give $B=C$ unless $A$ is invertible. More on one page
• Determinants: Sign pattern of cofactors $\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}$ forgotten, especially for $A_{12},A_{21},A_{23},A_{32}$. Also: Writing the matrix of cofactors instead of its transpose as $\mathrm{adj}A$. More on one page
• Integrals: Forgetting $+C$ in indefinite integrals, or keeping old limits after a substitution. Also: $\int \frac{dx}{2x+3}=\frac{1}{2}\log |2x+3|$, not $\log |2x+3|$. More on one page
• Differential Equations: Stating the degree without first clearing a radical or fractional power. Also: Forgetting to write the equation in standard form (coefficient of ${\displaystyle \frac{dy}{dx}}$ equal to $1$) before finding $P$. More on one page

What the paper covers

The unit weightage, with each unit's chapters, is on the Class 12 Maths papers page. We haven't mapped this paper question by question yet.

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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