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 $(A - 2I)(A - 3I) = 0$, where $A = \begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, find the value of $x$.

(ii) Find the value(s) of $k$ so that the line $2x+y+k=0$ may touch the hyperbola $3x^2-y^2=3$.

(iii) Prove that: $tan^{-1}$ $\frac{1}{4}$ $+ tan^{-1}$ $\frac{2}{9}$ $=$ $\frac{1}{2}$ $sin^{-1}$ $\frac{4}{5}$

(iv) Using L’Hospital’s Rule, evaluate:

$\lim \limits_{x \to 0}$ $( \frac{e^x - e^{-x} - 2x}{x - sin \ x})$

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

(vi) Evaluate: $\int \limits_{0}^{1}$ $log$ $(\frac{1}{x}$ $-1)$ $\ dx$

(vii) Two regression lines are represented by $4x+10y=9$ and $6x+3y=4$. Find the line of regression of $y \ on \ x$.

(viii) If $1, \ \omega \ and \ \omega^2$ are the cube roots of unity, evaluate:

$(1-\omega^4+ \omega^8)(1-\omega^8+\omega^{16})$

(ix) Solve the differential equation:

$log$ $(\frac{dy}{dx})$ $= 2x-3y$

(x) If two balls are drawn from a bag containing three red and four blue balls, find the probability that:

(a) They are the same color

(b) They are of different colors

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Question 2:

(a) Using properties of determinants, prove that:

$\left| \begin{array}{ccc} x & y & z \\ x^2 & y^2 & z^2 \\ y+z & z+x & x+y \end{array} \right|= (x-y)(y-z)(z-x)(x+y+z)$    [5]

(b) Find $A^{-1}$, where $A = \begin{bmatrix} 4 & 2 & 3 \\ 1 & 1 & 1 \\ 3 & 1 & -2 \end{bmatrix}$

Hence, solve the system of linear equations:

$4x+2y+3z = 2$

$x+y+z=1$

$3x+y-2z=5$                  [5]

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Question 3:

(a) Solve for $x: sin^{-1} \ x + sin^{-1} \ (1-x) = cos^{-1} \ x$              [5]

(b) Construct a circuit diagram for the following Boolean function:

$(BC+A)(A'B'+C')+A'B'C'$

Using laws of Boolean Algebra, simplify the function and draw the simplified circuit.  [5]

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Question 4:

(a) Verify Lagrange’s Mean Value Theorem for the function $f(x) = \sqrt{x^2 - x}$ in the interval $[1,4]$     [5]

(b) From the following information, find the equation of the Hyperbola and the equation of its Transverse Axis:

Focus: $(-2, 1)$, Directrix: $2x-3y+1 = 0$, $e =$ $\frac{2}{\sqrt{3}}$     [5]

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Question 5:

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

(b) Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}$ cm. (Leave the answer in terms of $\pi$)     [5]

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Question 6:

(a) Evaluate: $\int \limits_{}^{}$ $\frac{cos^{-1} x}{x^2}$ $dx$     [5]

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

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Question 7:

(a) Find the Karl Pearson’s coefficient of correlation between x and y for the following data.     [5]

 $x$ 16 18 21 20 22 26 27 15 $y$ 22 25 24 26 25 30 33 14

(b) The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by students in a school:

 Mathematics Physics Mean 84 81 Standard Deviation 7 4

The correlation co-efficient between the given marks is $0.86$. Estimate the likely marks in Physics in the marks in Mathematics at $92$.     [5]

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Question 8:

(a) $Bag \ A$ contains three red and four white balls. $Bag \ B$ contains two red and three white balls. If one ball is drawn from $Bag \ A$ and two balls are drawn from $Bag \ B$, find the probability that:

(i) One ball is red and two balls are white

(ii) All the three balls are of the same color    [5]

(b) Three persons, Aman, Bipin and Mohan attempt a Mathematics problems independently. The odds in favor of Aman and Mohan solving the problem are $3:2$ and $4:1$ respectively and the odds against Bipin solving the problem are $2:1$. Find:

(i) The probability that all the three will solve the problem

(ii) the probability that the problem will be solved.    [5]

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Question 9:

(a) Find the locus of the complex number $z = x+iy$, satisfying the relation $arg \ (z-1) = \frac{\pi}{4}$ and $|z-2-3i |= 2$. Illustrate the locus on the Argand plane.     [5]

(b) Solve the following differential equation:

$y e^y \ dx = (y^3 + 2 x e^y) \ dy$, given that $x =0, y = 1$    [5]

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Section – B (20 Marks)

Question 10:

(a)  If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and $\theta$ is the angle between them, then show that $|\overrightarrow{a} - \overrightarrow{b} | = 2 \ sin$ $\frac{\theta}{2}$.    [5]

(b) If the value of $\lambda$ for which the four points $A, B, C, D$ with position vectors $-\hat{j} - \hat{k}$; $4\hat{i}+5\hat{j}+ \lambda \hat{k}$; $3\hat{i}+9\hat{j} +4\hat{k}$ and $-4\hat{i}+4\hat{j}+4\hat{k}$ are coplanar.    [5]

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Question 11:

(a) Find the equation of a line passing through the point $(-1, 3, -2)$ and perpendicular to the lines: $\frac{x}{1}$ $=$ $\frac{y}{2}$ $=$ $\frac{z}{-3}$ and $\frac{x+2}{2}$ $=$ $\frac{y-1}{5}$ $=$ $\frac{z+1}{3}$     [5]

(b) Find the equation of planes parallel to the plane $2x-4y+4z=7$ and which are at a distance of five units from the point $(3, -1, 2)$.   [5]

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Question 12:

(a) If the sum and the product of the mean and variance of a Binomial Distribution are $1.8 \ and \ 0.8$ respectively, find the probability distribution and the probability of at least one success.   [5]

(b) For $A, \ B \ and \ C$, the chances of being selected as the manager of  firm are $4:1:2$ respectively. The probabilities for them to introduce a radical chance in the marketing strategy are $0.3, \ 0.8 \ and \ 0.5$ respectively. If a chance takes place; find the probability that it is due to the appointment of B.   [5]

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Section – C (20 Marks)

Question 13:

(a)  If Mr. Nirav deposits $Rs. \ 250$ at the beginning of each month in an account that pays an interest of $6\%$ per annum compounded monthly, how many months will be required for the deposit to at least $Rs. \ 6390$ [5]

(b) A mill owner buys two types of machines $A \ and \ B$ for his mill. $Machine \ A$ occupies $1000$ sqm of area and requires $12$ men to operate it.; while $Machine \ B$ occupies $1200$ sqm of area and requires $8$ men to operate it. The owner has $7600$ sqm of area available and $72$ men to operate the machines. If $Machine \ A$ produces $50$ units and $Machine \ B$ produces $40$ units daily, how many machines of each type should be buy to maximize the daily output? Use linear programming to find the solution.   [5]

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Question 14:

(a) A bill of $Rs. 60000$ was drawn on 1st April 2011 at 4 months and discounted for $Rs. 58560$ at a bank. If the rate of interest was $12\%$ per annum, on what date was the bill discounted?   [5]

(b) A company produces a commodity with $Rs. 24000$ fixed cost. The variable cost is estimated to be $25\%$ of the total revenue recovered on selling the product at a rate of $Rs. 8$ per unit. Find the following:

(i) Cost function

(ii) Revenue function

(iii) Break even point   [5]

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Question 15:

(a) The price index for the following data for the year 2011 taking 2001 as the base year was 127. The simple average of price relative method was used. Find the value of $x$.       [5]

 Items A B C D E F Price (Rs. Per unit) in year 2001 80 70 50 20 18 25 Price (Rs. Per unit) in year 2011 100 87.50 61 22 $x$ 32.50

(b) The profit of a paper bag manufacturing company (in lakhs / millions of Rs.) during each month pf a year are:

 Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sept. Oct. Nov. Dec. Profit 1.2 0.8 1.4 1.6 2.0 2.4 3.6 4.8 3.4 1.8 0.8 1.2

Plot the given data on a graph sheet. Calculate the four monthly moving averages and plot these on the same graph sheet.   [5]

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