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) Solve for $x \ and \ y$ if  $\begin{bmatrix} x^2 \\ y^2 \end{bmatrix} + 2 \begin{bmatrix} 2x \\ 3y \end{bmatrix} = 3 \begin{bmatrix} 7 \\ -3 \end{bmatrix}$

(ii) Prove that $sec^2 \ (tan^{-1} 2) + cosec^2 \ (cot^{-1} 3) = 15$

(iii) Find the equation of the hyperbola whose Transverse and Conjugate axes are the $x \ and \ y$ axis respectively, given that the length of the conjugate axis is $5$ and distance between the foci is $13$.

(iv) From the equation of two regression lines, $4x+3y+7=0$ and $3x+4y+8=0$, find:

a) Mean of $x \ and \ y$

b) Regression coefficient

c) Coefficient of correlation

(v) Evaluate: $\int \limits_{}^{}$ $e^x (tan \ x+log \ sec \ x) \ dx$

(vi) Evaluate: $\lim \limits_{x \to \pi/2} (cos \ x. log \ tan \ x)$

(vii) Find the locus of the complex number, $Z = x+iy$ given $|\frac{x+iy-2i}{x+iy+2i} |$ $= \sqrt{2}$

(viii) Evaluate: $\int \limits_{1}^{2}$ $\frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}$ $dx$

(ix) Three persons $A, \ B \ and \ C$ shoot to hit a target. If in trials, $A$ hits the target 4 times out of 5 shots, $B$ hits 3 times in 4 shots and $C$ hits 2 times in 3 trials. Find the probability that:

(a) Exactly two persons hit the target

(b) At least two persons hit the target

(x) Solve the differential equation: $(xy^2+x) dx + (x^2y+y) dy = 0$

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Question 2:

(a) Using properties of determinants, prove that:

$\left| \begin{array}{ccc} a & a+b & a+b+c \\ 2a & 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c \end{array} \right|= a^3$    [5]

(b) Find the product of matrix $A \ and \ B$ where:

$A = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix}$$B = \begin{bmatrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$ Hence, solve the following equations by matrix method:

$x+y + 2z = 1$

$3x+2y+z= 7$

$2x+y+3z = 2$                  [5]

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Question 3:

(a) Prove that: $cos^{-1}$ $\frac{63}{65}$ $+2 \ tan^{-1}$ $\frac{1}{5}$ $= sin^{-1}$ $\frac{3}{5}$      [5]

(b) (i) Write the Boolean expression corresponding to the circuit below:

(ii) Simplify the expression using laws of Boolean algebra and construct simplified circuit.         [5]

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Question 4:

(a) Verify Rolle’s theorem for the function:

$f(x) = log \$ $\{ \frac{x^2+ab}{(a+b)x} \}$ in the interval $[a, b]$ where $0 \notin [a, b]$         [5]

(b) Find the equation of the ellipse with its center at $(4, -1)$, focus at $(1, -1)$ and given that it passes through $(8,0)$.         [5]

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Question 5:

(a) If, $e^y(x+1)=1$, then show that $\frac{d^2y}{dx^2} = (\frac{dy}{dx})^2$         [5]

(b) A printed page is to have a total area of $80$ sq. cm with a margin of $1$ cm at the top and on each side and a margin of $1.5$ cm at the bottom. What should be the dimensions of the page so that the printed area will be maximum.         [5]

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Question 6:

(a) Evaluate: $\int \limits_{}^{}$ $\frac{dx}{x\{ 6(log \ x^2)+7 \ log \ x + 2 \}}$        [5]

(b) Find the area of the region bounded by the curves $x = 4y - y^2$ and the $y-axis$.        [5]

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Question 7:

(a) 10 candidates received percentage marks in two subjects as follows:

 Candidate A B C D E F G H I J Mathematics Marks 80 88 76 74 68 65 40 43 40 80 Statistics Marks 72 84 90 66 54 50 54 38 30 43

Calculate Spearman’s rank correlation coefficient and interpret your results.        [5]

(b) The following results were obtained with respect to two variables x and y:

$\Sigma x = 30, \ \Sigma y = 42 \ \Sigma xy = 199, \ \Sigma x^2 = 184, \ \Sigma y^2 = 318, \ n = 6$

Find the following:

(i) The regression coefficient

(ii) Correlation coefficient between $x \ and \ y$

(iii) Regression equation of $y \ on \ x$

(iv) The likely value of $y \ when \ x = 10$        [5]

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Question 8:

(a) A bag contains $8$ red and $5$ white balls. Two successive draws of $3$ balls are make at random form the bag without replacements. Find the probability that the firs draw yields $3$ white balls and the second draw yields $3$ red balls.        [5]

(b) A box contains $30$ bolts and $40$ nuts. Half of the bolts and half of the nuts are rusted. If two items are drawn at random from the box, what is the probability that either both are rusted or both are bolts.        [5]

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Question 9:

(a) Using De Moivre’s theorem, prove that:

$(\frac{1+ cos \ \theta + i \ sin \ \theta}{1+ cos \ \theta - i \ sin \ \theta})$ $= cos \ n\theta + i \ sin \ n\theta$, where $i = \sqrt{-1}$        [5]

(b) Solve the differential equation:

$\frac{dy}{dx}$ $- 3y \ cot\ x= sin \ 2x$, given $y = 2$, when $x =$ $\frac{\pi}{2}$        [5]

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Question 10:

(a)  For any of the vectors $\overrightarrow{a}, \overrightarrow{b}, \ and \ \overrightarrow{c}$, prove:

$\begin{bmatrix} \overrightarrow{a} - \overrightarrow{b} & \overrightarrow{b} - \overrightarrow{c} & \overrightarrow{c} - \overrightarrow{a} \end{bmatrix} = 0$        [5]

(b) In any triangle $ABC$, prove by vector method:

$\frac{a}{sin \ A} = \frac{b}{sin \ B} =\frac{c}{sin \ C}$        [5]

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Question 11:

(a) Find the shortest distance between the lines:

$\frac{x-8}{3}$ $=$ $\frac{y+9}{-16}$ $=$ $\frac{z-10}{7}$ and $\frac{x-15}{3}$ $=$ $\frac{y-29}{8}$ $=$ $\frac{5-z}{5}$         [5]

(b) Find the equation of the plane passing through the line of intersection of the planes $x+2y+3z-5=0$ and $3x-2y-z+1=0$ and cutting off equal intercepts on the $x \ and \ z$ axes.        [5]

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Question 12:

(a) In a class of $75$ students, $15$ are above average, $45$ are average and the rest below average achievers. The probability that an above average achieving student fails is $0.005$, that an average achieving students fails is $0.05$ and the probability of a below average achieving student failing is $0.15$. If a students is know to have passes, what is the probability that he is a below average achiever.        [5]

(b) The probability that a bulb produced by a factory will fuse in $100$ days of use is $0.05$. Find the probability that out of $5$ such bulbs, after $100$ days of use:

(i) None fuse.

(ii) Not more than one fuses.

(iii) More than one fuses.

(iv) At least one fuses.        [5]

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Question 13:

(a) Two tailors $P$ and $Q$ earn $Rs. 150$ and $Rs. 200$ per day respectively. $P$ can stitch $6$ shirts and $4$ trousers a day, while $Q$ can stitch $10$ shirts and $4$ trousers per day. How many days should each work to produce at least $60$ shirts and $32$ trousers at minimum labor cost?       [5]

(b) A machine costs $Rs. 97000$ and its effective life is estimated to be $12$ years. If scrap realizes $Rs. 2000$ only, what amount should be retained out of the profits at the end of each year to accumulate at compound interest of $5\%$ per annum in order to buy a new machine after $12$ years? (user $1.05^{12}=1.769$)       [5]

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Question 14:

(a) A bill of $Rs. 1000$ drawn on 7th May 2011 for six months was discounted on 29th August 2011 for cash payment of $Rs. 988$. Find the rate of interest charged by the bank.       [5]

(b) If total cost function is given by $C = a + bx + cx^2$, where $x$ is the quantity of output, show that:

$\frac{d}{dx}$ $(AC) =$ $\frac{1}{x}$ $(MC-AC)$, where $MC$ is the marginal cost and $AC$ is the average cost.       [5]

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Question 15:

(a) Find the consumer price index number for the year 2010 using year 2000 as the base year by using method of weighted aggregates.       [5]

 Commodity A B C D E Year 2000 price (in Rs. Per unit) 16 40 0.50 5.12 2.00 Year 2010 price (in Rs. Per unit) 20 60 0.50 6.25 1.50 Weights 40 25 5.00 20.00 10.00

(b) Calculate the 5 yearly moving averages of the number of students in a college from the following data and plot them on the graph paper.        [5]

 Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 Number 332 317 357 392 402 405 510 427 405 438

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