CBSE Class 12 Maths Official Sample Paper 2023-24

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. Application of Derivatives · 2 marks

   Show that $f(x)={\tan }^{-1}x-x$ is strictly decreasing on $\mathbb{R}$.

   Worked solution

   $f^{\prime }(x)={\displaystyle \frac{1}{1+x^2}}-1=-{\displaystyle \frac{x^2}{1+x^2}}$ M1
   $f^{\prime }(x)<0$ for all $x\ne 0$ and $f^{\prime }(0)=0$ at a single point only, so $f$ is strictly decreasing on $\mathbb{R}$ R1

2. Integrals · 2 marks

   Find ${\displaystyle \int x{\sec }^{2}2xdx}$.

   Worked solution

   $=x\cdot {\displaystyle \frac{\tan 2x}{2}}-{\displaystyle \int \frac{\tan 2x}{2}dx}$ M1
   $={\displaystyle \frac{x\tan 2x}{2}}-{\displaystyle \frac{1}{4}}\log |\sec 2x|+C$ A1

3. Vector Algebra · 2 marks

   The points $A(1,2,-1)$, $B(3,-2,5)$ and $C(k,-4,8)$ are collinear. Using vectors, find $k$ and the ratio $AB:BC$.

   Worked solution

   $\overrightarrow{AB}=2\hat{i}-4\hat{j}+6\hat{k}$, $\overrightarrow{AC}=(k-1)\hat{i}-6\hat{j}+9\hat{k}$; collinear $\Rightarrow \overrightarrow{AC}=t\overrightarrow{AB}$: $-6=-4t$, $9=6t$ give $t=\frac{3}{2}$ M1
   $k-1=3\Rightarrow k=4$; $\overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}=\frac{1}{2}\overrightarrow{AB}$, so $AB:BC=2:1$ A1

4. Linear Programming · 2 marks

   A potter makes mugs and bowls. A mug needs $3$ minutes on the wheel and $2$ minutes of glazing; a bowl needs $2$ minutes on the wheel and $4$ minutes of glazing. At most $120$ minutes of wheel time and $160$ minutes of glazing time are available each day. The profit is ₹$40$ per mug and ₹$55$ per bowl. Formulate the problem of maximising the daily profit as an LPP (do not solve).

   Worked solution

   Let $x$ mugs and $y$ bowls be made; maximise $Z=40x+55y$ B1
   subject to $3x+2y\le 120,2x+4y\le 160,x\ge 0,y\ge 0$ B1

More on these topics: Application of Derivatives · Integrals · Vector Algebra · Linear Programming

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.

• 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
• 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
• 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
• Linear Programming: Missing a corner (e.g. where a constraint meets an axis) or including an intersection point that violates another constraint. Also: Declaring a maximum/minimum on an unbounded region without the half-plane check. 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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