Board Paper of Class 12 2020 Mathematics - Solutions

(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: tan²(sec^–1 2) + cot² (cosec^–1 3) = 11.

(c) (iii) Without expanding at any stage, find the value of the determinant:
 

(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 y² = 4ax.

(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

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

(a) If ${\cos }^{-1}\frac{x}{2}+{\cos }^{-1}\frac{y}{3}=\theta$, then prove that 9x² – 12xy cos θ + 4y² = 36 sin² θ

OR

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

Using properties of determinants, show that
$\left|\begin{array}{ccc}x& p& q\\ p& x& q\\ q& q& x\end{array}\right|=\left(x-p\right)\left(x^2+px-2q^2\right)$ VIEW SOLUTION

Verify Rolle’s theorem for the function, f (x) = −1 + cos x in the interval [0, 2π] VIEW SOLUTION

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

(a) The equation of tangent at (2, 3) on the curve y² = px³ + q is y = 4x − 7. Find the values of  ‘p’ and ‘q’.

OR

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

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

OR

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

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

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

Solve the following system of linear equations using matrices:
x − 2y =10, 2x − y − z = 8, − 2y + z = 7 VIEW SOLUTION

(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

(a) Evaluate: $\int {\tan }^{-1}\sqrt{\frac{1-x}{1+x}}dx$

OR

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

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