CBSE Class 12 Maths Official Sample Paper 2026-27

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. Relations and Functions · 2 marks

   Five seats in a row are numbered $1$ to $5$. On $A=\{1,2,3,4,5\}$ define $aRb$ if seats $a$ and $b$ are the same seat or next to each other, i.e. $|a-b|\le 1$. Determine whether $R$ is reflexive, symmetric or transitive.

   Worked solution

   $|a-a|=0\le 1$: reflexive; $|a-b|=|b-a|$: symmetric A1
   $(1,2),(2,3)\in R$ but $|1-3|=2$, so $(1,3)\notin R$: not transitive A1

2. Inverse Trigonometric Functions · 2 marks

   Find the value of $3{\text{cosec}}^{-1}(-2)+2{\cot }^{-1}(-{\displaystyle \frac{1}{\sqrt{3}}})$.

   Worked solution

   ${\text{cosec}}^{-1}(-2)=-{\displaystyle \frac{\pi }{6}}$ (range $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]-\{0\}$); ${\cot }^{-1}(-{\displaystyle \frac{1}{\sqrt{3}}})={\displaystyle \frac{2\pi }{3}}$ (range $(0,\pi )$) M1
   $3(-{\displaystyle \frac{\pi }{6}})+2\cdot {\displaystyle \frac{2\pi }{3}}=-{\displaystyle \frac{\pi }{2}}+{\displaystyle \frac{4\pi }{3}}={\displaystyle \frac{5\pi }{6}}$ A1

3. Continuity and Differentiability · 2 marks

   Find the value of $k$ for which $f(x)=\{\begin{array}{ll}{\displaystyle \frac{\sqrt{x+7}-3}{x-2}},& x\ne 2\\ k,& x=2\end{array}$ is continuous at $x=2$.

   Worked solution

   ${\displaystyle \underset{x\to 2}{lim}\frac{\sqrt{x+7}-3}{x-2}\cdot \frac{\sqrt{x+7}+3}{\sqrt{x+7}+3}=\underset{x\to 2}{lim}\frac{1}{\sqrt{x+7}+3}}$ M1
   $={\displaystyle \frac{1}{6}}$, so $k={\displaystyle \frac{1}{6}}$ A1

4. Integrals · 2 marks

   Find ${\displaystyle \int \frac{x^3-x+3}{x^2+1}dx}$.

   Worked solution

   ${\displaystyle \frac{x^3-x+3}{x^2+1}}=x+{\displaystyle \frac{3-2x}{x^2+1}}$ M1
   $={\displaystyle \frac{x^2}{2}}-\log (x^2+1)+3{\tan }^{-1}x+C$ A1

More on these topics: Relations and Functions · Inverse Trigonometric Functions · Continuity and Differentiability · Integrals

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.

• Relations and Functions: Checking reflexivity for only the elements that appear in $R$ instead of all of $A$. Also: Showing onto by solving for $x$ but not checking that $x$ lies in the domain (e.g. $x\in \mathbb{N}$, $x\ne 2$). More on one page
• Inverse Trigonometric Functions: Writing ${\sin }^{-1}(\sin {\displaystyle \frac{2\pi }{3}})={\displaystyle \frac{2\pi }{3}}$ — the answer must lie in the principal range. Also: Giving ${\cos }^{-1}(-x)$ a negative value: ${\cos }^{-1}$ is never negative. 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
• 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

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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