# Board Paper of Class 12 2019 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) If*f*:*R*→*R*,*f*(*x*) =*x*^{3}and*g*:*R*→*R*,*g*(*x*) = 2*x*^{2}+ 1, and*R*is the set of real numbers, then find*fog*(*x*) and*gof*(*x*).**(b)**(ii) Solve: Sin (2 tan^{–1}*x*) = 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*) =*x*^{3}− 6*x*^{2}+ 12*x*+ 5 is increasing on R.**(g)**(vii) Evaluate: $\int \frac{{\mathrm{sec}}^{2}x}{{\mathrm{cosec}}^{2}x}dx$**(h)**(viii) Using L’Hospital’s Rule, evaluate: $\underset{x\to 0}{\mathrm{lim}}\frac{{8}^{x}-{4}^{x}}{4x}$**(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

- Question 2
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

- Question 3
(a) Solve for
*x*: ${\mathrm{tan}}^{-1}\left(\frac{x-1}{x-2}\right)+{\mathrm{tan}}^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$**OR**

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

- Question 4
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

- Question 5
(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:$f\left(x\right)=x+\frac{1}{x}$ in the interval [1, 3]VIEW SOLUTION

- Question 6
If $y={e}^{{\mathrm{sin}}^{-1}x}\mathrm{and}z={e}^{-{\mathrm{cos}}^{-1}x},\mathrm{prove}\mathrm{that}\frac{dy}{dz}={e}^{\frac{\pi}{2}}$ VIEW SOLUTION

- Question 7
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

- Question 8
(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

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

- Question 10
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

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

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=9\phantom{\rule{0ex}{0ex}}\frac{2}{x}+\frac{5}{y}+\frac{7}{z}=52\phantom{\rule{0ex}{0ex}}\frac{2}{x}+\frac{1}{y}-\frac{1}{z}=0$ VIEW SOLUTION

- Question 12
(a) The volume of a closed rectangular metal box with a square base is 4096 cm
^{3}.The cost of polishing the outer surface of the box is ₹ 4 per cm^{2}. Find the dimensions of the box for the minimum cost of polishing it.**OR**

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

- Question 13
Evaluate: ${\int}_{0}^{\pi}\frac{x\mathrm{tan}x}{\mathrm{sec}x+\mathrm{tan}x}dx$ VIEW SOLUTION

- Question 14
(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.A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.
**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