CBSE Class 12 Maths Board Paper 2025

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. Determinants · 2 marks

   Find the values of $x$ for which $\left|\begin{array}{ccc}x& -1& 2\\ 3& x& 0\\ 1& 2& 1\end{array}\right|=18$.

   Worked solution

   Expanding along $R_1$: $x(x-0)+1(3-0)+2(6-x)=x^2-2x+15$ M1
   $x^2-2x+15=18\Rightarrow x^2-2x-3=0\Rightarrow x=3$ or $x=-1$ A1

2. Continuity and Differentiability · 2 marks

   Examine whether $f(x)=\{\begin{array}{ll}{x}^2+1,& x\le 1\\ 3x-1,& x>1\end{array}$ is differentiable at $x=1$.

   Worked solution

   $f(1)=2$ and ${\displaystyle \underset{x\to 1^{+}}{lim}(3x-1)=2}$, so $f$ is continuous at $1$.
   LHD $={\displaystyle \underset{h\to 0^{-}}{lim}\frac{(1+h{)}^2+1-2}{h}=\underset{h\to 0^{-}}{lim}(2+h)=2}$; RHD $={\displaystyle \underset{h\to 0^{+}}{lim}\frac{3(1+h)-1-2}{h}=3}$ M1
   LHD $\ne$ RHD, so $f$ is not differentiable at $x=1$. A1

3. Application of Integrals · 2 marks

   The area of the region in the first quadrant bounded by the parabola $y^2=9x$, the $x$-axis and the line $x=c$ $(c>0)$ is $16$ sq units. Find $c$.

   Worked solution

   ${\displaystyle {\int }_0^{c}3\sqrt{x}dx=2c^{3/2}}$ M1
   $2c^{3/2}=16\Rightarrow c^{3/2}=8\Rightarrow c=4$ A1

4. Vector Algebra · 2 marks

   For $\overrightarrow{a}=3\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}-\hat{k}$, find the real values of $\lambda$ for which $|\overrightarrow{a}+\lambda \overrightarrow{b}|=2\sqrt{3}$.

   Worked solution

   $\overrightarrow{a}+\lambda \overrightarrow{b}=(3+\lambda )\hat{i}+(\lambda -1)\hat{j}+(1-\lambda )\hat{k}$; $|\overrightarrow{a}+\lambda \overrightarrow{b}{|}^2=(3+\lambda {)}^2+2(\lambda -1{)}^2=3{\lambda }^2+2\lambda +11=12$ M1
   $3{\lambda }^2+2\lambda -1=0\Rightarrow \lambda =\frac{1}{3}$ or $\lambda =-1$ A1

More on these topics: Determinants · Continuity and Differentiability · Application of Integrals · Vector Algebra

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.

• 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
• Continuity and Differentiability: Declaring a function differentiable because the two formulas "join up" — continuity is not enough. Also: Using $\log (u+v)=\log u+\log v$ (false). Take logs only of products, quotients and powers. More on one page
• Application of Integrals: Reporting a negative or zero “area” because part of the curve is below the axis. Also: Forgetting to double when using symmetry — or doubling a region that is not symmetric. More on one page
• Vector Algebra: Middle term of the cross-product determinant: it is $-\hat{j}(a_1{b}_3-a_3{b}_1)$. Also: Using $\frac{m\overrightarrow{a}+n\overrightarrow{b}}{m+n}$ (weights swapped) in the section formula. 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

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