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 value of $k$ if $M = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ and $M^2-kM-I_2=0$

(ii) Find the equation of an ellipse whose latus rectum is $8$ and eccentricity is $\frac{1}{3}$

(iii) Solve: $cos^{-1} \ (sin \ cos^{-1} \ x) =$ $\frac{\pi}{6}$

(iv) Using L’Hospital’s rule, evaluate: $\lim \limits_{x \to 0}$ $\frac{x-sin \ x}{x^2 sin \ x}$

(v) Evaluate: $\int \limits_{}^{}\frac{2y^2}{y^2+4}$ $\ dy$

(vi) Evaluate: $\int \limits_{0}^{3} f(x) \ dx$, where $f(x) = \Bigg \{ \begin{matrix} cos \ 2x, 0 \leq x \leq \frac{\pi}{2} \\ \\ 3, \ \ \ \ \ \ \frac{\pi}{2} \leq x \leq 3 \end{matrix}$

(vii)  The two lines of regressions are $4x+2y-3=0$ and $3x+6y+5 = 0$. Find the correlation co-efficient between $x \ and \ y$.

(viii) A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a space or an ace or both?

(ix) If $1, \omega \ and \ \omega^2$ are the cube roots of unity, prove that $\frac{a + b\omega + c\omega^2}{c+a\omega + b\omega^2}$ $= \omega^2$

(x) Solve the differential equation: $sin^{-1}$ $(\frac{dy}{dx})$ $= x + y$

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Question 2:

(a) Using properties of determinants, prove that:

$\left| \begin{array}{ccc} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{array} \right|= (1+a^2+b^2)^3$    [5]

(b) Given two matrices $A \ and \ B$

$A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 4 & 1 \\ 1 & -3 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 11 & -5 & -14 \\ -1 & -1 & 2 \\ -7 & 1 & 6 \end{bmatrix}$. Find $AB$.

Using this result, solve the following system of equation:

$x-2y+3z=6, \ x+4y+z=12, \ x-3y+2z=1$    [5]

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Question 3:

(a) Solve the equation for $x$:

$sin^{-1}$ $\frac{5}{x}$ $+ sin^{-1}$ $\frac{12}{x}$ $=$ $\frac{\pi}{2}$ $, x \neq 0.$     [5]

(b) $A, B \ and\ C$ represent switches in ‘on’ position and $A', B' \ and\ C'$ represent them in ‘off’ position. Construct a switching circuit representing the polynomial $ABC + AB'C + A'B'C$. Using Boolean Algebra, prove that the given polynomial can be simplified to $C(A + B')$. Construct an equivalent switching circuit.     [5]

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Question 4:

(a) Verify Lagrange’s Mean Value Theorem for the following function:

$f(x) = 2 \ sin \ x + sin \ 2x \ on \ [0, \pi]$     [5]

(b) Find the equation of the hyperbola whose foci are $(0, \pm \sqrt{10})$ and passing through the point $(2, 3)$   [5]

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Question 5:

(a) If $y = e^{m \ cos^{-1} \ x}$, prove that $(1-x^2)$ $\frac{d^2y}{dx^2}$ $- x$ $\frac{dy}{dx}$ $= m^2y$     [5]

(b) Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius $10 \ cm$ is a square of side $10\sqrt{2}$     [5]

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Question 6:

(a) Evaluate: $\int \frac{sec \ x}{1+ cosec \ x}$ $dx$     [5]

(b) Find the smaller are enclosed by the circle $x^2+y^2$ and the line $x+y = 2$.     [5]

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Question 7:

(a) Given that the observations are: $(9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8)$. Find the two lines of regression and estimate the value of $y$ when $x = 13.5$.     [5]

(b) In a contest the competitors are awarded marks out of $20$ by two judges. The scores of the $10$ competitors are given below. Calculate Spearman’s rank correlation.      [5]

 Competitors A B C D E F G H I J Judge A 2 11 11 18 6 5 8 16 13 15 Judge B 6 11 16 9 14 20 4 3 13 17

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Question 8:

(a) An urn contains $2$ white and $2$ black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same color. The process is repeated. Find the probability that the third ball drawn is black.     [5]

(b) Three person $A, B \ and \ C$ shoot to hit a target. If $A$ hits the target $4$ times out of $5$ trials, $B$ hits it $3$ times in $4$ trials and $C$ hits is $2$ times in $3$ trials, find the probability that:

(i) Exactly two person hit the target.

(ii) At least two person hit the target.

(iii) Non hit the target.     [5]

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Question 9:

(a) If $z = x + iy, \omega =$ $\frac{2-iz}{2z-i}$ and $|\omega|=1$, find the locus of $z$ and illustrate it in Argand Plane.     [5]

(b) Solve the differential equation:

$e^{x/y}(1 -$ $\frac{x}{y}$ $) + (1 + e^{x/y})$ $\frac{dx}{dy}$ $= 0 \ when \ x=0, y=1$     [5]

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Section – B (20 Marks)

Question 10:

(a) Using vectors, prove that an angle in a semicircle is a right angle.     [5]

(b) Find the volume of a parallelopiped whose edges are represented by the vectors: $\overrightarrow{a} = 2 \hat{i} - 3 \hat{j}- 4\hat{k}, \overrightarrow{b} = \hat{i} + 2 \hat{j} - \hat{k}, and \overrightarrow{c} = 3 \hat{i}+\hat{j} + 2\hat{k}$    [5]

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Question 11:

(a) Find the equation of the plane passing through the intersection of the planes $x+y+z+1 = 0$ and $2x-3y+5z-2=0$ and the point $(-1, 2, 1)$.     [5]

(b) Find the shortest distance between the lines $\overrightarrow{r}=\hat{i} + 2 \hat{j} + 3 \hat{k} + \lambda (2\hat{i} + 3 \hat{j} + 4 \hat{k})$  and $\overrightarrow{r}= 2\hat{i} + 4 \hat{j} + 5 \hat{k} + \mu (4\hat{i} + 6 \hat{j} + 8 \hat{k})$    [5]

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Question 12:

(a) $Box \ I$ contains $2$ white and $3$ black balls. $Box \ II$ contains $4$ white and $1$ black balls and $Box \ III$ contains $3$ white and $4$ black balls. A dice having $3$ red, $2$ yellow and $1$ green face, is thrown to select a box. If red face turns up we pick $Box \ I$, if yellow face turns up we pick $Box \ II$ otherwise we pick $Box \ III$. The we draw a ball from a selected box. If the ball drawn is white, what is the probability that the dice had turned up with a Red face.     [5]

(b) 5 dices are thrown simultaneously. If the occurrence of an odd number in a single die is considered a success, find the probability of maximum three successes.     [5]

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Section – C (20 Marks)

Question 13:

(a) Mr. Nirav borrowed $Rs. 50000$ from the bank for $5$ years. The rate of interest is $9\%$ per annum compounded monthly. Find the payment he makes if he pays back at the beginning of each month.     [5]

(b) A dietitian wishes to mix two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$, $12$ units of vitamin $B$ and $8$ units of vitamin $C$. The vitamin contents of one kg of food is given below:

 Food Vitamin A Vitamin B Vitamin C X 1 unit 2 units 3 units Y 2 units 2 units 1 unit

One kg of food $X$ costs $Rs. \ 24$ and one kg of food $Y$ costs $Rs. \ 36$. Using linear programming, find the least cost of the total mixture which will contain the required vitamins.     [5]

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Question 14:

(a) A bill of $Rs. \ 7650$ was drawn on 8th March 2013, at $7$ months. It was discounted on 18th May, 2013 and the holder of the bill received $Rs. \ 7497$. What is the rate of interest charged by the bank?     [5]

(b) The average cost function, $AC$ for a commodity is given by $AC = x + 5 +$ $\frac{36}{x}$ in terms of output $x$. Find:

(i) The total cost, $C$ and marginal cost, $MC$ as a function of $x$

(ii) The outputs for which $AC$ increases.     [5]

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Question 15:

(a) Calculate the index number for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data:        [5]

 Commodity Price in Rs. Weight 2010 2014 A 2 4 8 B 5 6 10 C 4 5 14 D 2 2 19

(b) The quarterly profits of a small scale industry (in thousands of Rupees) is as follows:

Calculate the quarterly moving averages. Display these and the original figures graphically on the same graph sheet.     [5]

 Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 2012 39 47 20 56 2013 68 59 66 72 2014 88 60 60 67

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