MATHEMATICS

(Maximum Marks: 100)

(Time Allowed: Three Hours)

They must NOT start writing during this time)

The Question Paper consists of three sections A, B and C

Candidates are required to attempt all questions from Section A and all question 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 question of four marks each.

Section C: Internal choice has been provided in two question 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 graphs papers are provided.

Section – A (80 Marks)

Question 1:                                                                                                         [10 × 3]

(i) If $A = \begin{bmatrix} 3 & 1 \\ 5 & 7 \end{bmatrix}$, find the values of $x \ and \ y$ such that $A^2 +xI_2=yA$

(ii) Find the eccentricity and the coordinates of foci of the hyperbola $25x^2-9y^2=225$

(iii) Evaluate: $tan \ [ 2\ tan^{-1}$ $\frac{1}{2}$ $- cot^{-1} \ 3]$

(iv) Using L’Hospital’s Rule, evaluate: $\lim \limits_{x \to 0} (1+sin \ x)^{cot \ x}$

(v) Evaluate: $\int \limits_{}^{}$ $e^x$ $\frac{(2+sin \ 2x)}{cos^2 \ x}$ $dx$

(vi) Using the properties of definite integrals, evaluate:

$\int \limits_{0}^{\pi/2}$ $\frac{\sqrt{sin \ x}}{\sqrt{sin \ x}+\sqrt{cos \ x}}$ $dx$

(vii) For the given lines of regression, $3x-2y=5$ and $x-4y=7$, find:

(a) regression coefficient $b_{yx}$ and $b_{xy}$

(b) coefficient of correlation $r(x,y)$

(viii) Express the complex number $\frac{(1+\sqrt{3}i)^2}{\sqrt{3}-i}$ in the form of $a + ib$. Hence, find the modulus and argument of the complex number.

(ix) A bag contains $20$ balls numbered from $1$ to $20$. One ball is drawn at random from the bag. What is the probability that the ball drawn is marked with a number with is multiple of $3$ or $4$?

(x) Solve the differential equation: $(x+1) \ dy - 2xy \ dx = 0$

$\\$

Question 2:

(a) Using properties of determinants, prove that:

$\left| \begin{array}{ccc} a^2+1 & ab & ac \\ ba & b^2+1 & bc \\ ca & cb & c^2+1 \end{array} \right|= a^2 + b^2 + c^2 + 1$    [5]

(b) Solve the following system of linear equations using matrix method:

$x-2y=10, \ \ 2x+y+3z=8, \ \ -2y+z=7$    [5]

$\\$

Question 3:

(a) If $cos^{-1}x +cos^{-1}y+cos^{-1}z = \pi$, prove that $x^2 + y^2 +z^2 + 2xyz = 1$   [5]

(b) $P, Q \ and \ R$ represent switches in ‘on’ position and $P', Q' \ and \ R'$ represent switches in off position. Construct a switching circuit representing the polynomial $PR + Q (Q'+R)(P+QR)$. Using Boolean algebra, simply the polynomial expression and construct the simplified circuit.     [5]

$\\$

Question 4:

(a) Verify Rolle’s Theorem for the function $f(x)=e^x (sin \ x- cox \ x)$ on $[\frac{\pi}{4}, \frac{5 \pi}{4} ]$

(b) Find the equation of the parabola with latus rectum joining points $(4, 6)$ and $(4, -2)$.

$\\$

Question 5:

(a) If $y =$ $\frac{x \ sin^{-1} x}{\sqrt{1-x^2}}$, prove that: $(1-x^2)$ $\frac{dy}{dx}$ $= x +$ $\frac{y}{x}$     [5]

(b) A wire of length $50$ m is cut into two pieces. One piece of the with is bent in the shape of a square and the other int he shape of a circle. What should be the length of each piece so that the combines area of the two is minimum?     [5]

$\\$

Question 6:

(a) Evaluate: $\int \limits_{}^{}$ $\frac{x + sin \ x}{1+cos \ x}$ $dx$     [5]

(b) Sketch the graph of the curves $y^2=x$ and $y^2= 4-3x$ and find the area enclosed between them.      [5]

$\\$

Question 7:

(a) A psychologist selected at random sample of $22$ students. He grouped them in $11$ pairs so that the students in each pair have nearly equal scores in an intelligence test. In each pair, one student was taught by method $A$ and the other by method $B$ and examined after the course. The marks obtained by them after the course are as follows:

 Pairs 1 2 3 4 5 6 7 8 9 10 11 Method A 24 29 19 14 30 19 27 30 20 28 11 Method B 37 35 16 26 23 27 19 20 16 11 21

Calculate Spearman’s Rank correlation.      [5]

(b) The coefficient of correlation between the values denoted by $X$ and $Y$ is $0.5$. The mean of $X$ is $3$ and that of $Y$ is $5$. Their standard deviations are $5$ and $4$ respectively. Find:

(i) the two lines of regression

(ii) the expected value of $Y$, when $X$ is given $14$.

(iii) the expected value of $X$, when $Y$ is given as $9$.       [5]

$\\$

Question 8:

(a) In a college, $70\%$ students pass in Physics, $75\%$ pass in Mathematics and $10\%$ students fail in both. One student is chosen at random. What is the probability that:

(i) He passes in Physics and Mathematics

(ii) He passes in Mathematics given that he passes in Physics

(iii) He passes in Physics given that he passes in Mathematics.       [5]

(b) A bag contains $5$ white and $4$ black balls and another bag contains $7$ white and $9$ black balls. A ball is drawn from the first bag and two balls are drawn from the second bag. What is the probability of drawing $1$ white and $2$ black balls?       [5]

$\\$

Question 9:

(a) Using De Moivre’s theorem, find the least positive integer n such that $(\frac{2i}{1+i})^n$ is a positive integer.       [5]

(b) Solve the following differential equation:

$(3xy+y^2)dx + (x^2+xy)dy=0$       [5]

$\\$

Section – B (20 Marks)

Question 10:

(a) In a triangle $ABC$, using vectors, prove that: $c^2 = a^2+b^2-2ab \ cos \ C$.       [5]

(b) Prove that:

$\overrightarrow{a}.(\overrightarrow{b}+\overrightarrow{c}) \times (\overrightarrow{a}+ 2 \overrightarrow{b} + 3 \overrightarrow{c}) = [ \overrightarrow{a} \overrightarrow{b} \overrightarrow{c} ]$       [5]

$\\$

Question 11:

(a) Find the equation of a line passing through the points $P(-1, 3, 2)$ and $Q(-4, 2, -2)$. Also find the point $R(5, 5, \lambda)$ is collinear with the points $P$ and $Q$, then find the value of $\lambda$.       [5]

(b) Find the equation of the plan passing through the points $(2, -3, 1)$ and $(-1, 1, -7)$ and perpendicular to the plane $x - 2y+5z+1=0$       [5]

$\\$

Question 12:

(a) In a bolt factory, three machines $A, B \ and \ C$ manufacturer $25\%, 35\% \ and \ 40\%$ of the total production respectively. Of their respective outputs, $5\%, 4\% \ and \ 2\%$ are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine $C$.       [5]

(b) On dialing certain telephone numbers, assume that on an average, one telephone number out of five is busy, ten telephone numbers are randomly selected and dialed. Find the probability that at least three of them will be busy.       [5]

$\\$

Section – C (20 Marks)

Question 13:

(a) A person borrows $Rs. 68962$ on the condition that he will repay the money with compound interest at $5\%$  per annum in $4$ equal annual installments, the first one being payable t the end of the first year. Find the value of each installment.       [5]

(b) A company manufactures two types of toys $A$ and $B$. A toy of type $A$ requires $5$ minutes for cutting and $10$ minutes for assembling. A \$ toy of type $B$ requires $8$ minutes of cutting and $8$ minutes of assembling. There are three hours available for cutting and $4$ hours for assembling in a day. The profit is $Rs. 50$ each on a toy of type $A$ and $Rs. 60$ each on a toy of type $B$. How many toys of each type should a company manufacture in a day to maximize the profit? Use linear programming to find the solution.       [5]

$\\$

Question 14:

(a) A firm has the cost function $C =$ $\frac{x^3}{3}$ $- 7x^2+111x+50$ and demand function $x = 100-p$.

(i) Write the total revenue function in terms of $x$

(ii) Formulate the total profit function $P$ in terms of $x$

(iii) Find the profit maximizing level of output.       [5]

(b) A bill of $Rs. \ 5050$ is drawn on 13th April 2013. It was discounted on 4th July 2013 at $5\%$ per annum. If the banker’s gain on the transaction is $Rs. \ 0.50$, find the nominal date of the maturity of the bill.       [5]

$\\$

Question 15:

(a) The price of six different commodities for year 2009 and 2011 are as follows:

 Commodities A B C D E F Price in 2009 (Rs.) 35 80 25 30 80 $x$ Price in 2011 (Rs.) 50 $y$ 45 70 120 105

The index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of $x \ and \ y$ if the total price in 2009 is $Rs. \ 360$.       [5]

(b) The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:

 Year Jan-Mar April-June July-Sept Oct-Dec 2010 70 60 45 72 2011 79 56 46 84 2012 90 64 45 82

Calculate four quarterly moving averages and illustrate them on original figures on one graph using the same axes for both.        [5]

$\\$