(Three Hours)

(Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time)

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 graphs papers are provided. Slide rule may be used.

Section – A (80 Marks)

Question 1.                                                                                                   $[10 \times 3]$

(i)  If the matrix $\begin{bmatrix} 6 & -x^2 \\ 2x-15 & 10 \end{bmatrix}$ is symmetric, find the value of $x$.

(ii) If $y - 2x - k = 0$ touches the conic $3x^2-5y^2 = 15$, find the value of $k$.

(iii) Prove that $\frac{1}{2}$ $cos^{-1} \ ($ $\frac{1-x}{1+x}$ $)= tan^{-1} \sqrt{x}$

(iv) Using L’Hospital’s rule, evaluate: $\lim \limits_{x \to \frac{\pi}{2}} \ (x \ tan \ x - \frac{\pi}{4} sec \ x)$

(v) Evaluate: $\int \limits_{}^{}$ $\frac{1}{x^2}$ $\ sin^2 \ ($ $\frac{1}{x}$ $) \ dx$

(vi) Evaluate: $\int \limits_{0}^{\frac{\pi}{4}} log \ (1 + tan \ \theta) \ d \theta$

(vii) By using the data $\bar{x} = 25, \bar{y} = 30, b_{yx}=1.6$ and $b_{xy}=0.4$, find:

(a) The regression equation $y$ on $x$

(b) What is the most likely value of $y$ when $x = 60$?

(c) What is the coefficient of correlation between $x$ and $y$?

(viii) A problem is given to three students whose chances of solving it are $\frac{1}{4}$$\frac{1}{5}$ and $\frac{1}{3}$ respectively. Find the probability that the problem is solved.

(ix) If $a + ib =$ $\frac{x+iy}{x-iy}$  prove that $a^2 + b^2 = 1$ and $\frac{b}{a}$ $=$ $\frac{2xy}{x^2 -y^2}$

(x) Solve: $\frac{dy}{dx}$ $= 1 - xy + y - x$

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Question 2:

(a) Using properties of determinants, prove that:

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

(b) Given that: $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$. Find $AB$.

Using this result, solve the following system of equation:

$x-y = 3, 2x+3y+4z=17$ and $y +2z=7$    [5]

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Question 3:

(a) Solve the equation for $x$:

$sin^{-1} \ x + sin^{-1} \ (1-x) = cos^{-1} \ x, x \neq 0$    [5]

(b) If $A, B$ and $C$ are elements of Boolean Algebra, simplify the expression $(A' + B')(A + C') + B'(B + C)$. Draw the simplified circuit.    [5]

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Question 4:

(a) Verify Lagrange’s mean value theorem for the function: $f(x) = x(1-log \ x)$ and find the value of $c$ in the interval $[1, 2]$    [5]

(b) Find the coordinates of center, foci and equation of directrix of hyperbola: $x^2 - 3y^2 -4^x = 8$    [5]

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Question 5:

(a) If $y = cos \ (sin \ x)$, show that:

$\frac{d^2y}{dx^2}$ $+ tan \ x$ $\frac{dy}{dx}$ $+ y \ cos^2 \ x = 0$    [5]

(b) Show that the surface are of a closed cuboid with square base and a given volume is minimum when it is a cube.    [5]

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Question 6:

(a) Evaluate:

$\int \limits_{}^{}$ $\frac{sin \ 2x}{(1 + sin \ x)(2 + sin \ x)}$ $dx$    [5]

(b) Draw a rough sketch of the curve $y^2 = 4x$ and find the area of the region enclosed by the curve and the line $y = x$.     [5]

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Question 7:

(a) Calculate the Spearman’s rank correlation coefficient for the following data and interpret the results:     [5]

 X 35 54 80 95 73 73 35 91 83 81 Y 40 60 75 90 70 75 38 95 75 70

(b) Find the line of best fit for the following data, treating $x$ as a dependent variable (Regression equation $x$ on $y$):

 X 14 12 13 14 16 10 13 12 Y 14 23 17 24 18 25 23 24

Hence, estimate the value of $x$ when $y= 16$.     [5]

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Question 8:

(a) In a class of $60$ students, $30$ opted for Mathematics, $32$ opted for Biology and $24$ opted for both mathematics and Biology. If one of these students is selected at random, find the probability that:

(i) The student opted for Mathematics or Biology

(ii) The student has opted neither for Mathematics nor Biology

(iii) The student has opted Mathematics but not Biology     [5]

(b) Bag $A$ contains $1$ white, $2$ blue and $3$ red balls. Bag $B$ contains $3$ white, $3$ blue and $2$ red balls. Bag $C$ contains $2$ white, $3$ blue and $4$ red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red.     [5]

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Question 9:

(a) Prove that locus of $z$ is circle and find its center and radius if $\frac{z-i}{z-1}$ is purely imaginary.     [5]

(b) Solve: $(x^2 - yx^2)dy + (y^2+xy^2)dx = 0$     [5]

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Section – B (20 Marks)

Question 10:

(a) If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three mutually perpendicular vectors of equal magnitude, prove that $(\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c})$ is equally inclined with vectors $\overrightarrow{a}, \overrightarrow{b}$ and  $\overrightarrow{c}$     [5]

(b) Find the value of $\lambda$ for which the four points with position vectors $6 \hat{i} - 7 \hat{j}, 16 \hat{i} - 19 \hat{j} - 4 \hat{k}$, $\lambda \hat{j} - 6 \hat{k} \ and \ 2 \hat{i} - 5 \hat{j}+ 10 \hat{k}$ are co-planar.     [5]

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Question 11:

(a) Show that the lines $\frac{x-4}{1}$ $=$ $\frac{y+3}{-4}$ $=$ $\frac{z+1}{7}$ and $\frac{x-1}{2}$ $=$ $\frac{y+1}{-3}$ $=$ $\frac{z+10}{8}$ intersect. Find the coordinate of their point of intersection.     [5]

(b) Find the equation of the plane passing through the point $(1, -2, 1)$ and perpendicular to the line joining the points $A (3, 2, 1)$ and $B (1, 4, 2)$.     [5]

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Question 12:

(a) A fair die is rolled. If face $1$ turns up, a ball is drawn from $Bag \ A$. If face $2 \ or \ 3$ turns up, a ball is drawn from $Bag \ B$. If face $4 \ or \ 5 \ or \ 6$turns up, a ball is drawn from $Bag \ C$. $Bag \ A$ contains $3 \ red$ and $2 \ white$ balls, $Bag \ B$ contains $3 \ red$ and $4$ white balls and $Bag \ C$ contains $4 \ red$ and $5 \ white$ balls. The die is rolled, a Bag is picked and a ball is drawn. If the drawn ball is red, what is the probability that is it drawn from $Bag \ B$.     [5]

(b) An urn contains $25$ balls of which $10$ balls are red and the remaining are green. A ball is drawn at random from the urn, the color is noted and the ball is replaced. If $6$ balls are drawn in this way, find the probability that:

(i) All the balls are red

(ii) Not more than $2$ balls are green

(iii) Number of red balls and green balls are equal     [5]

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Section – C (20 Marks)

Question 13:

(a) A machine costs $Rs. \ 60000$ and its effective life is estimated to be $25$ years. A sinking fund is to be created for replacing the machine at the end of its life when its scrap value is estimated as $Rs. \ 5000$. The price of the new machine is estimated to be $100\%$ more than the price of the present one. Find the amount that should be set aside at the end of each year, out of the profits, for the sinking fund it is accumulates at an interest of $6\%$ per annum compounded annually.     [5]

(b) A farmer has a supply of chemical fertilizer of $Type \ A$ which contains $10\%$ nitrogen and $6\%$ phosphoric acid and of $Type \ B$ which contains $5\%$ nitrogen and $10\%$ phosphoric acid. After soil test, it is found that at least $7 \ kg$ of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of $Type \ A$ costs $Rs. 5.00$ per kg and the $Type \ B$ costs $Rs. \ 8.00$ per kg. Using linear programming, find how many kilograms of each fertilizer should be brought to meet the requirement and and for the cost to be minimum. Find the feasible region in the graph.     [5]

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Question 14:

(a) The demand for a certain product is represented by the equation $p = 500 + 25x -$ $\frac{x^2}{3}$ in Rupees where $x$ is the number of units and $p$ is the price per unit. Find:

(i) Marginal revenue function

(ii)  The marginal revenue when $10$ units are sold.     [5]

(b) The bill of $Rs. \ 60000$ payable $10$ months after date was discounted for $Rs. \ 57300$ on 30th June 2007. If the rate of interest was $11$ $\frac{1}{4}$ $\%$ per annum, on what date was the bill drawn?     [5]

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Question 15:

(a) The price relative and weights of a set of commodities are given below:

 Commodity A B C D Price Relatives 125 120 127 119 Weights $x$ $2x$ $y$ $y+3$

If the sum of the weights is $40$ and weighted average of price relative index number is $122$, find the numerical value of $x$ and $y$.     [5]

(b) Construct three yearly moving averages from the following data and show on a graph against the original data:     [5]

 Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Annual Sales in Lakhs 18 22 20 26 30 22 24 28 32 35