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:                                                                                            $[10 \times 3]$

(i) Find the matrix $X$ for which:

$\begin{bmatrix} 5 & 4 \\ 1 & 1 \end{bmatrix} \ X = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix}$

(ii) Solve for $x$, if : $tan \ (cos^{-1} \ x) =$ $\frac{2}{\sqrt{5}}$

(iii) Prove that the line $2x-3x=9$ touches the conic $y^2 = -8x$. Also, find the point contact.

(iv) Using L’Hospitals rule, evaluate: $\lim \limits_{x \to 0}$ $(\frac{1}{x^2} - \frac{cot \ x}{x})$

(v) Evaluate: $\int \limits_{}{} tan^3 \ x \ dx$

(vi) Using properties of definite integrals, evaluate: $\int \limits_{0}^{\frac{\pi}{2}}$ $\frac{sin \ x - cos \ x}{1 + sin \ x \ cos \ x}$ $dx$

(vii) The two lines of regression are $x+2y-5=0$ and $2x+3y-8=0$ and the variance of $x \ is \ 12$. Find the variance of y and the coefficient of correlation.

(viii) Express $\frac{2+i}{(1+i)(1-2i)}$ in the form of $a+ib$. Find its modulus and argument.

(ix) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of $8$?

(x) Solve the differential equation: $x$ $\frac{dy}{dx}$ $+ y = 3x^2-2$

$\\$

Question 2:

(a) Using properties of determinants, prove that:

$A = \left| \begin{array}{ccc} b+c & a & a \\ b & a+c & b \\ c & c & a+b \end{array} \right|= 4abc$    [5]

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

$3x+y+z = 1, \ \ 2x+2z=0, \ \ 5x+y+2z=2$    [5]

$\\$

Question 3:

(a) If $sin^{-1} \ x + tan^{-1} \ x =$ $\frac{\pi}{2}$, prove that: $2x^2 + 1 = \sqrt{5}$     [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 circuit.     [5]

$\\$

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 $x-axis$.     [5]

(b) Find the equation of a standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose lengths of latus rectum is $10$. Also find its eccentricity.     [5]

$\\$

Question 5:

(a) If $log \ y = tan^{-1} \ x$, prove that:

$(1+x^2)$ $\frac{d^2y}{dx^2}$ $+ (2x-1)$ $\frac{dy}{dx}$ $= 0$     [5]

(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 are.      [5]

$\\$

Question 6:

(a) Evaluate: $\int \limits_{}{}$ $\frac{sin \ x + cos \ x}{\sqrt{9 + 16 \ sin \ 2x}}$ $\ dx$     [5]

(b) Find the area of the region bounded by the curve $y = 6x-x^2$ and $y = x^2-2x$     [5]

$\\$

Question 7:

(a) Calculate Carl Pearson’s coefficient of correlation between $x \ and \ y$ for the following data and interpret the result:

$(1,6), (2, 5), (3, 7), (4,9), (5,8), (6,10), (7, 11), (8,13), (9,12)$     [5]

(b) The marks obtained by 10 students 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 of the marks obtained in English are 24.     [5]

$\\$

Question 8:

(a) A committee of $4$ person has to be chosen from $8$ boys and $6$ girls, consisting at least of one girl. Find the probability that the committee consists of more girls than boys.      [5]

(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.    [5]

$\\$

Question 9:

(a) Find the locus of a complex number, $z=x+iy$, satisfying $|\frac{z-3i}{z+3i} |$ $\leq \sqrt{2}$.

Illustrate the locus of $z$ in the Argand plane.     [5]

(b) Solve the following differential equation:

$x^2 \ dy + (xy+y^2) \ dx = 0$, when $x = 1 \ and \ y = 1$   [5]

$\\$

Section – B (20 Marks)

Question 10:

(a) For any three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$, show that $\overrightarrow{a} - \overrightarrow{b}, \overrightarrow{b}- \overrightarrow{c}, \overrightarrow{c} - \overrightarrow{a}$ are co-planar.   [5]

(b) For a unit vector perpendicular to each of the vectors $\overrightarrow{a}+\overrightarrow{b} and \overrightarrow{a} - \overrightarrow{b}$ where $\overrightarrow{a} = 3 \hat{i} + 2 \hat{j} + 2\hat{k}$ and $\overrightarrow{b} = \hat{i} + 2\hat{j} -2\hat{k}$   [5]

$\\$

Question 11:

(a) Find the image of the point $(2, -1, 5)$ in the line $\frac{x-11}{10} = \frac{y+2}{-4} = \frac{z+8}{-11}$. Also find the length of the perpendicular from point $(2, -1, 5)$ to the line.     [5]

(b) Find the Cartesian equation of the plane, passing through the line of intersection of the planes: $\overrightarrow{r}.(2\hat{i}+3\hat{j}-4\hat{k})+5=0$ and $\overrightarrow{r}.(\hat{i}-5\hat{j}+7\hat{k})+2=0$ and intersecting $y-axis$ at $(0,3)$.     [5]

$\\$

Question 12:

(a) In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. One a given day, one of the three persons $A, B \ and \ C$ carries out this task. $A$ has $45\%$ chance, $B$ has $35\%$ change and $C$ has $20\%$ chance of doing the task. The probability that $A, B \ and \ C$ will take more than the allotted time is $\frac{1}{6}, \frac{1}{10}$  and  $\frac{1}{20}$ respectively.  If it is found that the time taken is more than the allotted time, what is the probability that A has done the task?    [5]

(b) The difference between mean and variance of a binomial distribution is $1$ and the difference of their square is $11$. Find the distribution.    [5]

$\\$

Section – C (20 Marks)

Question 13:

(a) A man borrows $Rs. 20000$ at $12\%$ per annum, compounded semi-annually and agrees to pay it in $10$ equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years.    [5]

(b) A company manufacturers two types of products $A$ and $B$. Each unit of $A$ requires $3$ grams of nickel and $1$ gram of chromium, while each unit of $B$ requires $1$ gram of nickel and $2$ grams of chromium. The firm can produce $9$ grams of nickel and $8$ grams of chromium. The profit is $Rs. \ 40$ on each product of type $A$ and $Rs. \ 50$ on each unit of type $B$. How  many units of each type should the company manufacture so as to earn maximum profit. Use linear programming to find the solution.    [5]

$\\$

Question 14:

(a) The demand function is $x =$ $\frac{24-2p}{3}$ where $x$ is the number of units demanded and $p$ is the price per unit. Find:

(i) The revenue function $R$ in terms of  $p$.

(ii) The price and the number of units demanded for which the revenue is maximum.    [5]

(b) A bill of $Rs. \ 1800$ drawn on 10th September, 2010 at 6 months was discounted for $Rs. \ 1782$ at a bank. If the rate of interest was $5\%$ per annum, on what date was the bill discounted.    [5]

$\\$

Question 15:

(a) The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be $116$. If the sum of the price in year 2000 is $Rs. 300$, find the value of $x \ and \ y$ in the data given below.    [5]

 Commodity A B C D E F Price in Base year 2000 (Rs.) 50 $x$ 30 70 116 20 Price in the year 2010 (Rs.) 60 24 $y$ 80 120 28

(b) From the details given below, calculate the five yearly moving averages of the number of students who have studies in a school. Also, plot these and original data on the same graph paper.    [5]

 Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Number of Students 332 317 357 392 402 405 410 427 405 438

$\\$