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 the matrix $A = \begin{bmatrix} 6 & x & 2 \\ 2 & -1 & 2 \\ -10 & 5 & 2 \end{bmatrix}$ is a singular matrix, find the value of $x$.

(ii) Solve: $cos^{-1} \big[ sin (cos^{-1} x) \big] =$ $\frac{\pi}{3}$

(iii) Show that the line $y = x+\sqrt{7}$ touches the hyperbola $9x^2-16y^2=144$

(iv) Evaluate: $\lim \limits_{x \to \frac{\pi}{2}}$ $\Big[ x tan \ x - \frac{\pi}{2} sec \ x \Big]$

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

(vi) Evaluate: $\int \limits_{-3}^{3}$ $|x+2| dx$

(vii) A fair die is thrown once. What is the probability that either an even number or a number greater than three will turn up?

(viii) If the regression equation of $x \ on\ y$ is given by $mx-y+10=0$ and the equation of $y \ on \ x$ is given by $-2x+5y=14$, determine the value of $'m'$ if the coefficient of correlation given between $x \ and \ y$ is $\frac{1}{\sqrt{10}}$

(ix) If $1, \ \omega, \ \omega^2$ are three cubes of unity, then simplify:

$(3 + 5 \omega + 3 \omega^2)^2 (1+2\omega+\omega^2)$

(x) Solve the differential equation: $cosec^3x \ dy - cosec \ y \ dx=0$

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Question 2:

(a) Using the properties of determinants, prove that the determinant:

$\left| \begin{array}{ccc} a & sin \ x & cos \ x \\ -sin \ x & -a & 1 \\ cox \ x & 1 & a \end{array} \right|$  is independent of $x$.     [5]

(b) Using the matrix method, solve the following equations:

$5x+3y+z=16$

$2x+1y+3z=19$

$1x+2y+4z=25$     [5]

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Question 3:

(a) Using Rolle’s theorem, find a point on the curve $y = sin \ x + cos \ x -1 , x \in \Big[ 0, \frac{\pi}{2} \Big]$     [5]

(b) Find the equation of the parabola whose focus is $(-1, -2)$ and the equation of the directrix is given by $4x-3y+2 = 0$. Also find the equation of the axis.     [5]

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Question 4:

(a) Prove that: $sin \ \Big[ 2 tan^{-1} \frac{3}{5} - sin^{-1} \frac{7}{25} \Big] =$ $\frac{304}{425}$     [5]

(b) $x, y \ and \ z$ represent three switches in an ‘ON’ position and $x', y' \ and \ z'$ represent the same three switches in an ‘OFF’ position. Construct a switching circuit representing the polynomial $(x+y)(x'+z)+y(y'+z')$.

Using the laws of Boolean Algebra, show that the above polynomial is equivalent to $xz+y$ and construct an equivalent switching circuit.     [5]

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Question 5:

(a) Using a suitable substitution, find the derivative of $tan^{-1}$ $\sqrt{\frac{a-x}{a+x}}$ with respect to $x$.     [5]

(b) A closed right circular cylinder has volume $\frac{539}{2}$ cubic units. Find the radius and the height of the cylinder so that the total surface area is minimum.      [5]

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Question 6:

(a) Evaluate: $\int \limits_{}^{}$ $\frac{2 \ sin \ 2\theta - cos \ \theta }{6 - cos^2 \theta - 4 \ sin \ \theta}$ $d\theta$     [5]

(b) Draw a rough sketch of the curve $y = x^2-5x+6$ and find the area bounded bu the curve and the $x-axis$.     [5]

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Question 7:

(a) Find the equation of two lines of regression for the following observations: $(3,6), \ (4,5), \ (5,4), \ (6,3), \ (7,2)$. Find the estimate of $y \ for \ x = 2.5$.     [5]

(b) Calculate the Spearman’s coefficient of rank correlation from the following data and interpret the results:     [5]

 x 16 19 22 28 25 31 37 40 43 49 y 25 25 27 31 27 33 35 41 45 41

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Question 8:

(a) Akhil and Vijay appear for an interview for two vacancies. The probability of Akhil’s selection is $\frac{1}{4}$ and Vijay’s selection is $\frac{2}{3}$. Find the probability  that only one of them will selected.     [5]

(b) There are two bags. One bag contains six green and three red balls. The second bag contains five green and four red balls. One ball is transferred from the first bag to the second bag. The one ball is drawn from the second bag. Find the probability that this is a red ball.     [5]

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Question 9:

(a) Solve the differential equation: $(y+log \ x) dx - x \ dy = 0$, given that $y = 0$, when $x = 1$.     [5]

(b) Find the locus of the complex number $z = x+iy$, satisfying the relation $|3z-4i| \leq |3z+2|$. Illustrate the locus in the Argand plane.    [5]

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Section – B (20 Marks)

Question 10:

(a)  Find the value of $\lambda$ for which the four points with position vectors $2\hat{i}+5\hat{j}+\hat{k}, -\hat{j}-4\hat{k}$, $3\hat{i}+\lambda \hat{j}+8\hat{k}$ and $-4\hat{i}+3\hat{j}+4\hat{k}$ are co-planar.    [5]

(b) In any $\triangle ABC$, prove by vector method that $cos \ B =$ $\frac{c^2+a^2-b^2}{2ca}$    [5]

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Question 11:

(a) Find the shortest distance between the lines whose vector equations are: $\overrightarrow{r} = (4\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$ and $\overrightarrow{r}= (2\hat{i}+\hat{j}-\hat{k}) + \mu (3\hat{i}+2\hat{j}-4\hat{k})$    [5]

(b) Find the equation of the plan passing through the line of intersection of the planes $x+2y+3z-4=0$ and $3z-y=0$ and perpendicular to the plane $3x+4y-2z+6=0$    [5]

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Question 12:

(a) A factory has a machine $A, B \ and \ C$ producing $1500, 2500 \ and \ 3000$bulbs per day respectively. $Machine \ A$ produces $1.5\%$ defective bulbs, $Machine \ B$ produces $2\%$ defective bulbs and $Machine \ C$ produces $2.5\%$ defective bulbs. At the end of the day, a bulb is drawn at random and is found defective. What is the probability that the defective bulb has been produced by $Machine \ B$   [5]

(b) Five bad eggs are mixed with $10$ good ones. If three eggs are drawn one by one with replacement, find the probability distribution of the number of good eggs drawn.    [5]

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Section – C (20 Marks)

Question 13:

(a) A company produces two types of items, $P$ and $Q$. Manufacturing of both items requires the metals gold and copper. Each using of item $P$ requires $3$ grams of gold and $1$ gram of copper while that of item $Q$ requires $1$ gram of gold and $2$ grams of copper. The company has $9$ grams of gold and $8$ grams of copper in its stores. If each unit of items $P$ makes a profit of $Rs. 50$ and each unit of item $Q$ makes a profit of $Rs. 60$, determine the number of unites of each item that the company should produce to maximize profit.    [5]

(b) At the beginning of each quarter, a sum of $Rs. 1500$ is deposited into a savings account that pays $12\%$ per annum compounded quarterly. Find the amount in the account at the end of four years.    [5]

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Question 14:

(a) A bill of exchange of $Rs. 722$ was drawn on 3rd April 2009, payable three months after date. It was discounted on 15th April, 2009 at $4.75\%$ per annum. What was the discounted value of the bill?    [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 the:

(i) Total cost and the marginal cost as the functions of $x$

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

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Question 15:

(a) The index number of the following data, for the year 2008, taking 2004 as the base year was found to be $Rs. 116$. The simple aggregate method was used for calculation. Find the numerical values of x and y if the sum of the prices in the year 2008 is $Rs. 203$   [5]

 Commodity Price in Rs in 2004 Price in Rs in 2008 A 20 25 B 10 30 C 30 15 D 25 45 E $x$ 35 F 50 $y$

(b) Consider the following data:

 Dates in the month of April 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Number of units sold 2 5 0 12 13 25 45 13 31 18 11 2 3 1

Calculate three days moving averages and display these and the original figures on the same graph.    [5]

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