# Board Paper of Class 12 2016 Mathematics - Solutions

**Section A**- Answer

**Question 1**(

*compulsory*) and five other questions.

**Section B and Section C**-

*Answer*

**two**

*questions from*

**either**Section B**or**Section C.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.**

Slide rule may be used.Slide rule may be used.

- Question 1
**(a)**(i) Find the matrix*X*for which: $\left[\begin{array}{cc}5& 4\\ 1& 1\end{array}\right]X=\left[\begin{array}{cc}1& -2\\ 1& 3\end{array}\right]$**(b)**Solve for*x*, if: $\mathrm{tan}\left({\mathrm{cos}}^{-1}x\right)=\frac{2}{\sqrt{5}}$**(c)**Prove that the line 2*x*– 3*y*= 9 touches the conics*y*^{2}= – 8*x*. Also, find the point of contact.**(d)**Using L'Hospital's Rule, evaluate: $\underset{x\to 0}{\mathrm{lim}}\left(\frac{1}{{x}^{2}}-\frac{\mathrm{cot}x}{x}\right)$**(e)**Evaluate: $\int {\mathrm{tan}}^{3}xdx$**(f)**Using properties of definite integrals, evaluate: ${\int}_{0}^{\pi /2}\frac{\mathrm{sin}x-\mathrm{cos}x}{1+\mathrm{sin}x\mathrm{cos}x}dx$**(g)**The two lines of regressions are*x*+ 2*y*– 5 = 0 and 2*x*+ 3*y*– 8 = 0 and the variance of*x*is 12. Find the variance of*y*and the coefficient of correlation.**(h)**Express $\frac{2+i}{\left(1+i\right)\left(1-2i\right)}$ in the form of*a*+*ib*. Find its modulus and argument.**(i)**A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8?**(j)**Solve the differential equation: $x.\frac{dy}{dx}+y=3{x}^{2}-2$ VIEW SOLUTION

- Question 2
**(a)**Using properties of determinants, prove that:

$\u25b3=\left|\begin{array}{ccc}b+c& a& a\\ b& a+c& b\\ c& c& a+b\end{array}\right|=4abc$**(b)**Solve the following system of linear equations using matrix method:

3*x*+*y*+*z*= 1, 2*x*+ 2*z*= 0, 5*x*+*y*+ 2*z*= 2 VIEW SOLUTION

- Question 3
**(a)**If ${\mathrm{sin}}^{-1}x+{\mathrm{tan}}^{-1}x=\frac{\pi}{2},\mathrm{prove}\mathrm{that}:2{x}^{2}+1=\sqrt{5}$**(b)**Write the Boolean function corresponding to the switching circuit given below:

A, B and C represent switches in 'on' position and A', B' and C' represent them in off position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit. VIEW SOLUTION

- Question 4
**(a)**Verify the conditions of Rolle's Theorem for the following function:

*f*(*x*) = log(*x*^{2}+ 2) – log 3 on [–1, 1]

Find a point in the interval, where the tangent to the curve is parallel to*x*-axis.**(b)**Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is 10. Also, find its eccentricity. VIEW SOLUTION

- Question 5
**(a)**If log*y*= tan^{–1}*x*, prove that: $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+\left(2x-1\right)\frac{dy}{dx}=0$

**(b)**A rectangle is inscribed in a semicircle of radius*r*with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area. VIEW SOLUTION

- Question 6
**(a)**Evaluate: $\int \frac{\mathrm{sin}x+\mathrm{cos}x}{\sqrt{9+16\mathrm{sin}2x}}dx$**(b)**Find the area of the region bound by the curves*y*= 6*x*–*x*^{2}and*y*=*x*^{2}– 2*x*. VIEW SOLUTION

- Question 7
**(a)**Calculate Karl Pearson's coefficient of correlation between*x*and*y*for the following data and interpret the result:

(1, 6), (2, 5), (3,7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)**(b)**The marks obtained by 10 candidates in English and Mathematics are given below:**Marks in English**20 13 18 21 11 12 17 14 19 15 **Marks in Mathematics**17 12 23 25 14 8 19 21 22 19

Estimate the probable score for Mathematics if the marks obtained in English are 24. VIEW SOLUTION

- Question 8
**(a)**A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.**(b)**An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball. VIEW SOLUTION

- Question 9
**(a)**Find the locus of a complex number,*z*=*x*+*iy*, satisfying the relation $\left|\frac{z-3i}{z+3i}\right|\le \sqrt{2}$. Illustrate the locus of*z*in the Argand plane.**(b)**Solve the following differential equation:*x*^{2}*dy*+ (*xy*+*y*^{2})*dx*= 0, when*x*= l and*y*= 1. VIEW SOLUTION