CBSE Class 12 Maths Official Sample Paper 2020-21

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. Linear Programming · 2 marks

   The corner points of the unbounded feasible region of the constraints $x+y\ge 3,x+3y\ge 5,x,y\ge 0$ are $(0,3)$, $(2,1)$ and $(5,0)$. Find the minimum value of $Z=4x+5y$, justifying that it is the minimum.

   Worked solution

   $Z=15,13,20$; smallest corner value $13$ at $(2,1)$ M1
   The open half-plane $4x+5y<13$ has no point in common with the region (it only touches it at $(2,1)$), so the minimum is $13$ A1

2. Probability · 2 marks

   Two fair dice are thrown. $A$: the first die shows an odd number; $B$: the sum of the numbers is $7$. Are $A$ and $B$ independent? Justify.

   Worked solution

   $P(A)={\displaystyle \frac{1}{2}}$, $P(B)={\displaystyle \frac{6}{36}}={\displaystyle \frac{1}{6}}$; $A\cap B=\{(1,6),(3,4),(5,2)\}$, $P(A\cap B)={\displaystyle \frac{3}{36}}={\displaystyle \frac{1}{12}}$ M1
   $P(A)P(B)={\displaystyle \frac{1}{12}}=P(A\cap B)$: independent A1

3. Matrices · 2 marks

   Show that $B=\left[\begin{array}{cc}3& -1\\ -5& 2\end{array}\right]$ is the inverse of $A=\left[\begin{array}{cc}2& 1\\ 5& 3\end{array}\right]$. Hence find the column matrix $X$ such that $AX=\left[\begin{array}{c}1\\ 2\end{array}\right]$.

   Worked solution

   $AB=\left[\begin{array}{cc}6-5& -2+2\\ 15-15& -5+6\end{array}\right]=I$ and $BA=\left[\begin{array}{cc}6-5& 3-3\\ -10+10& -5+6\end{array}\right]=I$, so $B=A^{-1}$ M1
   $X=A^{-1}\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}3-2\\ -5+4\end{array}\right]=\left[\begin{array}{c}1\\ -1\end{array}\right]$ A1

4. Continuity and Differentiability · 2 marks

   If $e^{x-y}=x+y$, show that ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{x+y-1}{x+y+1}}$.

   Worked solution

   Differentiating: $e^{x-y}(1-{\displaystyle \frac{dy}{dx}})=1+{\displaystyle \frac{dy}{dx}}$ M1
   Replace $e^{x-y}$ by $x+y$: $(x+y)-1=(x+y+1){\displaystyle \frac{dy}{dx}}\Rightarrow {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{x+y-1}{x+y+1}}$ A1

More on these topics: Linear Programming · Probability · Matrices · Continuity and Differentiability

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.

• 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
• Probability: Dividing by the wrong event: in $P(A\mid B)$ the given event $B$ goes in the denominator. Also: Assuming independence without justification, or confusing it with mutually exclusive. More on one page
• 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
• 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

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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