# Board Paper of Class 12 2020 Mathematics - Solutions

*This Question Paper consists of three sections A, B and C.*

Candidates are required to attempt all questions from

All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.

The intended marks for questions or parts of questions are given in brackets [ ].

Candidates are required to attempt all questions from

**Section A**and all questions**EITHER****from Section****B**__OR__**Section C****Section A:**Internal choice has been provided in three questions of four marks each and two questions of six marks each.**Section B:**Internal choice has been provided in two questions of four marks each.**Section C:**Internal choice has been provided in two questions of four marks each.All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.

The intended marks for questions or parts of questions are given in brackets [ ].

**Mathematical tables and graph papers are provided.**- Question 1
**(a)**(i) Determine whether the binary operation ∗ on R defined by*a*∗*b*= |*a*−*b*| is commutative. Also, find the value of ( ̶ 3)∗2.**(b)**(ii) Prove that: tan^{2}(sec^{–1}2) + cot^{2}(cosec^{–1}3) = 11.**(c)**(iii) Without expanding at any stage, find the value of the determinant:

$\u2206=\left|\begin{array}{ccc}20& a& b+c\\ 20& b& a+c\\ 20& c& a+b\end{array}\right|$**(d)**(iv) If $\left(\begin{array}{cc}2& 3\\ 5& 7\end{array}\right)\left(\begin{array}{cc}1& -3\\ -2& 4\end{array}\right)=\left(\begin{array}{cc}-4& 6\\ -9& x\end{array}\right),\mathrm{find}x.$**(e)**(v) Find $\frac{dy}{dx}\mathrm{if}{x}^{3}+{y}^{3}=3axy$**(f)**(vi) The edge of a variable cube is increasing at the rate of 10 cm/sec. How fast is the volume of the cube increasing when the edge is 5 cm long?**(g)**(vii) Evaluate: $\underset{4}{\overset{5}{\int}}\left|x-5\right|dx$**(h)**(viii) Form a differential equation of the family of the curves*y*^{2 }= 4*ax*.**(i)**(ix) A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white?**(j)**(x) Let A and B be two events such that $P\left(A\right)=\frac{1}{2},P\left(B\right)=p\mathrm{and}P\left(A\cup B\right)=\frac{3}{5}$ find ‘*p*’ if A and B are independent events. VIEW SOLUTION

- Question 2
If the function
*f*: R → R be defined as $f\left(x\right)=\frac{3x+4}{5x-7},\left(x\ne \frac{7}{5}\right)\hspace{0.17em}\mathrm{and}g:\mathrm{R}\to \mathrm{R}\mathrm{be}\mathrm{defined}\mathrm{as}g\left(x\right)=\frac{7x+4}{5x-3},\left(x\ne \frac{3}{5}\right)\hspace{0.17em}$ show that (*gof*)(*x*) = (*f og*)(*x*). VIEW SOLUTION

- Question 3
(a) If ${\mathrm{cos}}^{-1}\frac{x}{2}+{\mathrm{cos}}^{-1}\frac{y}{3}=\theta $, then prove that 9
*x*^{2}– 12*xy*cos*θ*+ 4*y*^{2}= 36 sin^{2}*θ***OR**

(b) Evaluate*: $\mathrm{cos}\left(2{\mathrm{cos}}^{-1}x+{\mathrm{sin}}^{-1}x\right)\mathrm{at}x=\frac{1}{5}$*VIEW SOLUTION

- Question 4
Using properties of determinants, show that

$\left|\begin{array}{ccc}x& p& q\\ p& x& q\\ q& q& x\end{array}\right|=\left(x-p\right)\left({x}^{2}+px-2{q}^{2}\right)$ VIEW SOLUTION

- Question 5

- Question 6
If $y={e}^{m{\mathrm{sin}}^{-1}x}$, prove that $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}={m}^{2}y$ VIEW SOLUTION

- Question 7
(a) The equation of tangent at (2, 3) on the curve
*y*^{2}=*px*^{3}+*q*is*y*= 4*x*− 7. Find the values of ‘*p*’ and ‘*q*’.**OR**

(b) Using L’Hospital’s rule, evaluate : $\underset{x\to 0}{\mathrm{lim}}\frac{x{e}^{x}-\mathrm{log}\left(1+x\right)}{{x}^{2}}$ VIEW SOLUTION

- Question 8
(a) Evaluate : $\int \frac{dx}{\sqrt{5x-4{x}^{2}}}$
**OR**

(b) Evaluate : $\int {\mathrm{sin}}^{3}x{\mathrm{cos}}^{4}xdx$ VIEW SOLUTION

- Question 9
Solve the differential equation $\left(1+{x}^{2}\right)\frac{dy}{dx}=4{x}^{2}-2xy$ VIEW SOLUTION

- Question 10
Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are $\frac{5}{6},\frac{4}{5}\mathrm{and}\frac{3}{4}$ respectively. Find the probability that:

(i) Exactly two persons hit the target.

(ii) At least one person hits the target. VIEW SOLUTION

- Question 11
Solve the following system of linear equations using matrices:

*x*− 2*y*=10, 2*x − y − z*= 8, − 2*y + z*= 7 VIEW SOLUTION

- Question 12
(a) Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height.
**OR**

(b) Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. VIEW SOLUTION

- Question 13
(a) Evaluate: $\int {\mathrm{tan}}^{-1}\sqrt{\frac{1-x}{1+x}}dx$
**OR**

(b) Evaluate: $\int \frac{2x+7}{{x}^{2}-x-2}dx$ VIEW SOLUTION

- Question 14
The probability that a bulb produced in a factory will fuse after 150 days of use is 0·05.

Find the probability that out of 5 such bulbs:

(i) None will fuse after 150 days of use.

(ii) Not more than one will fuse after 150 days of use.

(iii) More than one will fuse after 150 days of use.

(iv) At least one will fuse after 150 days of use. VIEW SOLUTION