CBSE Class 12 Maths Official Sample Paper 2019-20

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. Probability · 2 marks

   The probability distribution of $X$ is $P(X=x)=k{x}^2$ for $x=1,2,3$ (and $0$ otherwise). Find $k$ and $P(X\le 2)$.

   Worked solution

   $k(1+4+9)=1\Rightarrow k={\displaystyle \frac{1}{14}}$ A1
   $P(X\le 2)={\displaystyle \frac{1+4}{14}}={\displaystyle \frac{5}{14}}$ A1

2. Relations and Functions · 2 marks

   On the set $\mathbb{R}-\{0\}$ of non-zero real numbers, define $aRb⟺ab>0$. Show that $R$ is an equivalence relation and write down its equivalence classes.

   Worked solution

   $a^2>0$: reflexive; $ab=ba$: symmetric; $ab>0,bc>0\Rightarrow a{b}^{2}c>0\Rightarrow ac>0$ (as $b^2>0$): transitive A1
   Two classes: the positive reals $(0,\mathrm{\infty })$ and the negative reals $(-\mathrm{\infty },0)$ A1

3. Determinants · 2 marks

   For $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 4\\ 2& 1& 0\end{array}\right]$, find the cofactors $A_{21},A_{22},A_{23}$ of the second row and verify that $a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}=0$.

   Worked solution

   $A_{21}=-\left|\begin{array}{cc}2& 3\\ 1& 0\end{array}\right|=3$, $A_{22}=\left|\begin{array}{cc}1& 3\\ 2& 0\end{array}\right|=-6$, $A_{23}=-\left|\begin{array}{cc}1& 2\\ 2& 1\end{array}\right|=3$ A1
   $1(3)+2(-6)+3(3)=0$ ✓ (elements of one row with cofactors of another row) A1

4. Application of Derivatives · 2 marks

   Using the second derivative test, find the local maximum value of $f(x)={\sin }^{2}x-\cos x$ in $(0,\pi )$.

   Worked solution

   $f^{\prime }(x)=2\sin x\cos x+\sin x=\sin x(2\cos x+1)=0\Rightarrow \cos x=-{\displaystyle \frac{1}{2}},x={\displaystyle \frac{2\pi }{3}}$ (as $\sin x\ne 0$ in $(0,\pi )$); $f^{″}(x)=2\cos 2x+\cos x$, $f^{″}\left(\frac{2\pi }{3}\right)=-1-\frac{1}{2}=-{\displaystyle \frac{3}{2}}<0$ M1
   Local maximum value $={\displaystyle \frac{3}{4}}+{\displaystyle \frac{1}{2}}={\displaystyle \frac{5}{4}}$ at $x={\displaystyle \frac{2\pi }{3}}$ A1

More on these topics: Probability · Relations and Functions · Determinants · Application of Derivatives

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.

• 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
• 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
• 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
• 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

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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