CBSE Class 12 Maths Official Sample Paper 2024-25

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. Continuity and Differentiability · 2 marks

   Find ${\displaystyle \frac{dy}{dx}}$ if $y=\sqrt{1+{\cos }^{2}2x}$.

   Worked solution

   ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{1}{2\sqrt{1+{\cos }^{2}2x}}}\cdot 2\cos 2x\cdot (-\sin 2x)\cdot 2$ M1
   $=-{\displaystyle \frac{\sin 4x}{\sqrt{1+{\cos }^{2}2x}}}$ A1

2. Application of Derivatives · 2 marks

   Find the intervals in which $f(x)=x^4-4x^3+10$ is strictly increasing or strictly decreasing.

   Worked solution

   $f^{\prime }(x)=4x^3-12x^2=4x^2(x-3)$; $4x^2\ge 0$, so the sign of $f^{\prime }$ is that of $x-3$ (zero at $x=0$ without sign change) M1
   Strictly decreasing on $(-\mathrm{\infty },3]$; strictly increasing on $[3,\mathrm{\infty })$ A1

3. Differential Equations · 2 marks

   Verify that $y=Ax+{\displaystyle \frac{B}{x}}$ $(x\ne 0)$, where $A$ and $B$ are constants, is a solution of $x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}-y=0$.

   Worked solution

   $y^{\prime }=A-{\displaystyle \frac{B}{x^2}}$, $y^{″}={\displaystyle \frac{2B}{x^3}}$ M1
   LHS $={\displaystyle \frac{2B}{x}}+Ax-{\displaystyle \frac{B}{x}}-Ax-{\displaystyle \frac{B}{x}}=0$ = RHS A1

4. Three Dimensional Geometry · 2 marks

   Find the vector and Cartesian equations of the line through $(3,-1,2)$ parallel to the vector $2\hat{i}-5\hat{k}$.

   Worked solution

   $\overrightarrow{r}=(3\hat{i}-\hat{j}+2\hat{k})+\lambda (2\hat{i}-5\hat{k})$ A1
   ${\displaystyle \frac{x-3}{2}}={\displaystyle \frac{y+1}{0}}={\displaystyle \frac{z-2}{-5}}$ (i.e. $y=-1$ and ${\displaystyle \frac{x-3}{2}}={\displaystyle \frac{z-2}{-5}}$) A1

More on these topics: Continuity and Differentiability · Application of Derivatives · Differential Equations · Three Dimensional Geometry

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.

• 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 Derivatives: Substituting the particular value before differentiating in a rate problem. Also: Forgetting end points when finding absolute extrema; confusing a local maximum with the absolute maximum. 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
• Three Dimensional Geometry: Reading direction ratios before putting the line in standard form (signs and coefficients of $x,y,z$). Also: Omitting the modulus in the angle formula (the angle between lines is taken acute) or in the distance 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

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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