# Board Paper of Class 12 2018 Mathematics - Solutions

*The 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)**The binary operation ∗ : R × R → R is defined as*a*∗*b*= 2*a*+*b*. Find (2 ∗ 3) ∗ 4.**(b)**If $A=\left(\begin{array}{cc}5& a\\ b& 0\end{array}\right)$and A is symmetric matrix, show that 𝑎 =*b***(c)**Solve: 3 tan^{–1}*x*+ cot^{–1}*x*= π**(d)**Without expanding at any stage, find the value of:

$\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$**(e)**Find the value of constant ‘*k*’ so that the function*f*(*x*) defined as:

$f\left(x\right)=\left\{\begin{array}{ll}\frac{{x}^{2}-2x-3}{x+1},& x\ne -1\\ k,& x=-1\end{array}\right.$

is continuous at*x*= −1.**(f)**Find the approximate change in the volume ‘𝑉’ of a cube of side*x*metres caused by decreasing the side by 1%.**(g)**Evaluate : $\int \frac{{x}^{3}+5{x}^{2}+4x+1}{{x}^{2}}dx.$**(h)**Find the differential equation of the family of concentric circles*x*^{2}+*y*^{2 }=*a*^{2}**(i)**If A and B are events such that $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$ and $P\left(A\cap B\right)=\frac{1}{4},$ then find:

(a) P(*A*⁄*B*)

(b) P(*B*⁄*A*)**(j)**In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the probability of neither of them winning the race. VIEW SOLUTION

- Question 2
If the function $f\left(x\right)=\sqrt{2x-3}$ is invertible then find its inverse. Hence prove that (𝑓𝑜𝑓
^{–1}) (*x*) =*x*. VIEW SOLUTION

- Question 3

- Question 4
Use properties of determinants to solve for
*x*:

$\left|\begin{array}{ccc}x+a& b& c\\ c& x+b& a\\ a& b& x+c\end{array}\right|=0\mathrm{and}x\ne 0.$ VIEW SOLUTION

- Question 5
Show that the function $f\left(x\right)=\left\{\begin{array}{ll}{x}^{2},& x\le 1\\ \frac{1}{x},& x>1\end{array}\right.$is continuous at
*x*= 1 but not differentiable.**OR**

Verify Rolle’s theorem for the following function:*f*(*x*) =*e*^{–x }sin*x*on [0, π] VIEW SOLUTION

- Question 6
If $x=\mathrm{tan}\left(\frac{1}{a}\mathrm{log}y\right),$ prove that $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+\left(2x-a\right)\frac{dy}{dx}=0$ VIEW SOLUTION

- Question 7
Evaluate: $\int {\mathrm{tan}}^{-1}\sqrt{x}dx$ VIEW SOLUTION

- Question 8
Find the points on the curve
*y*= 4*x*^{3}– 3*x*+ 5 at which the equation of the tangent is parallel to the*x*-axis.**OR**

Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi}{4}$ at the uniform rate of 2 cm^{2}/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water. VIEW SOLUTION

- Question 9
Solve: $\mathrm{sin}x\frac{dy}{dx}-y=\mathrm{sin}x\xb7\mathrm{tan}\frac{x}{2}$
**OR**

The population of a town grows at the rate of 10% per*y*ear. Using differential equation, find how long will it take for the population to grow 4 times.

VIEW SOLUTION

- Question 10
Using matrices, solve the following system of equations:

2*x*– 3*y*+ 5*z*= 11

3*x*+ 2*y*– 4*z*= −5

*x*+*y*– 2*z*= – 3**OR**

Using elementary transformation, find the inverse of the matrix:

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$ VIEW SOLUTION

- Question 11
speaks truth in 60% of the cases, while*A*in 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? VIEW SOLUTION*B*

- Question 12
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. VIEW SOLUTION

- Question 13
Evaluate: $\int \frac{x-1}{\sqrt{{x}^{2}-x}}dx$
**OR**

Evaluate:${\int}_{0}^{\pi /2}\frac{{\mathrm{cos}}^{2}x}{1+\mathrm{sin}x\mathrm{cos}x}dx$ VIEW SOLUTION

- Question 14
From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable
*X*denote the number of defective items in the sample. If the sample is drawn without replacement, find:

(a) The probability distribution of*X*

(b) Mean of*X*

(c) Variance of*X*VIEW SOLUTION