Board Paper of Class 12 2019 Mathematics - Solutions

(a) (i) If f : R → R, f(x) = x³ and g : R → R, g(x) = 2x² + 1, and R is the set of real numbers, then find fog (x) and gof (x).

(b) (ii) Solve: Sin (2 tan^–1x) = 1

(c) (iii) Using determinants, find the values of k, if the area of triangle with vertices (−2, 0), (0, 4) and (0, k) is 4 square units.

(d) (iv) Show that (A + Aʹ) is symmetric matrix, if $\mathrm{A}=\left(\begin{array}{cc}2& 4\\ 3& 5\end{array}\right)$.

(e) (v) $f\left(x\right)=\frac{x^2-9}{x-3}$ is not defined at x = 3. What value should be assigned to f(3) for continuity of f(x) at x = 3?

(f) (vi) Prove that the function f(x) = x³ − 6x² + 12x + 5 is increasing on R.

(g) (vii)  Evaluate: $\int \frac{{\sec }^{2}x}{{\mathrm{cosec}}^{2}x}dx$

(h) (viii) Using L’Hospital’s Rule, evaluate: $\underset{x\to 0}{\lim }\frac{8^x-4^x}{4x}$

(i) (ix) Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

(j) (x) If events A and B are independent, such that $\mathrm{P}\left(\mathrm{A}\right)=\frac{3}{5}, \mathrm{P}\left(\mathrm{B}\right)=\frac{2}{3}, \mathrm{find} \mathrm{P}\left(\mathrm{A}\cup \mathrm{B}\right)$. VIEW SOLUTION

If f : A → A and $\mathrm{A}=\mathrm{R}-\left(\frac{8}{5}\right)$, show that the function $f\left(x\right)=\frac{8x+3}{5x-8}$ is one – one onto.
Hence, find f ^–1. VIEW SOLUTION

(a) Solve for x: ${\tan }^{-1}\left(\frac{x-1}{x-2}\right)+{\tan }^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi }{4}$

OR

(b) If ${\sec }^{-1}x={\mathrm{cosec}}^{-1}y, \mathrm{show} \mathrm{that} \frac{1}{x^2}+\frac{1}{y^2}=1$ VIEW SOLUTION

Using properties of determinants prove that:

$\left|\begin{array}{ccc}x& x\left(x^2+1\right)& x+1\\ y& y\left(y^2+1\right)& y+1\\ z& z\left(z^2+1\right)& z+1\end{array}\right|=\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(x+y+z\right)$ VIEW SOLUTION

(a) Show that the function f(x) = |x – 4| , x∈ R is continuous, but not differentiable at x = 4.

OR

(b) Verify the Lagrange’s mean value theorem for the function: VIEW SOLUTION

If $y=e^{{\sin }^{-1}x} \mathrm{and} z=e^{-{\cos }^{-1}x}, \mathrm{prove} \mathrm{that} \frac{dy}{dz}=e^{\frac{\pi }{2}}$ VIEW SOLUTION

A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall? VIEW SOLUTION

(a) Evaluate: $\int \frac{x\left(1+x^2\right)}{1+x^4}dx$

OR

(b) Evaluate: $\underset{-6}{\overset{3}{\int }}\left|x+3\right|dx$ VIEW SOLUTION

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

Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B? VIEW SOLUTION

Solve the following system of linear equations using matrix method:

$$\begin{gathered} \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=9 \\ \frac{2}{x}+\frac{5}{y}+\frac{7}{z}=52 \\ \frac{2}{x}+\frac{1}{y}-\frac{1}{z}=0 \end{gathered}$$ VIEW SOLUTION

(a) The volume of a closed rectangular metal box with a square base is 4096 cm³.

OR

(b) Find the point on the straight line 2x + 3y = 6, which is closest to the origin. VIEW SOLUTION

Evaluate: ${\int }_0^{\pi }\frac{x \tan x}{\sec x+\tan x}dx$ VIEW SOLUTION

(a) Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins.

OR

(b) Determine the binomial distribution where mean is 9 and standard deviation is $\frac{3}{2}$.
Also, find the probability of obtaining at most one success. VIEW SOLUTION