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 $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$, find $x$ such that $A^2 = xA - 2 I$. Hence find $A^{-1}$

(ii) Find the value of $k$, if the equation $8x^2-16xy+ky^2-22x+34y=12$ represents an ellipse.

(iii) Solve for $x: sin(2 \ tan^{-1} x) = 1$

(iv) Two regression lines are represented by $2x+3y-10 = 0$ and $4x+y-5=0$

(v) Evaluate: $\int \limits_{}{} \frac{cosec \ x}{log \ tan(\frac{x}{2})}$ $dx$

(vi) Evaluate: $\lim \limits_{y \to 0}$ $\frac{y - tan^{-1} y}{y - sin \ y}$

(vii) Evaluate: $\int \limits_{0}^{1} \frac{xe^x}{(1+x)^2}$ $dx$

(viii) Find the modulus and argument of the complex number $\frac{2+i}{4i+(1+i)^2}$

(ix) A word consists of $9$ different alphabets, in which $4$ are consonants and $5$ are vowels. Three alphabets are chosen at random. What is the probability that more than one vowel will be selected?

(x) Solve the differential equation: $\frac{dy}{dx}$ $= e^{x+y}+x^2 e^y$

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Question 2:

(a) Using the properties of determinants, show that $p \alpha^2 + 2 q \alpha + r = 0$, given that: $p, q, \ and \ r$ are not in G.P. and

$\left| \begin{array}{ccc} 1 & \frac{p}{q} & \alpha + \frac{q}{p} \\ 1 & \frac{r}{q} & \alpha + \frac{r}{q} \\ p \alpha + q & q \alpha + r & 0 \end{array} \right|$ $= 0$     [5]

(b) Solve the following system of equations using the matrix method:     [5]

$\frac{2}{x}$ $+$ $\frac{3}{y}$ $+ \frac{10}{z}$ $= 4$

$\frac{4}{x}$ $-$ $\frac{6}{y}$ $+$ $\frac{5}{z}$ $= 1$

$\frac{6}{x}$ $+$ $\frac{9}{y}$ $-$ $\frac{20}{z}$ $= 2$

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Question 3:

(a) Prove that: $2 \ tan^{-1}$ $\frac{1}{5}$ $+ cos^{-1}$ $\frac{7}{5\sqrt{2}}$ $+ 2 \ tan^{-1}$ $\frac{1}{8}$ $=$ $\frac{\pi}{4}$     [5]

(b) $P, Q \ and \ R$ represent switches in ‘ON’ position and $P', Q' \ and \ R'$ represent switches in ‘OFF’ position. Construct a switching circuit representing the polynomial. $P(P+Q) Q(Q+R')$

Using Boolean algebra to show that the above circuit is equivalent to a switching circuit in which when $P \ and \ Q$ are in ‘ON’ position, the light is on.     [5]

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Question 4:

(a) Verify Lagrange’s mean value theorem for the function $f(x) = sin \ x - sin \ 2x$ in the interval $\big[ 0, \pi \big]$     [5]

(b) Find the equation of the hyperbola whose foci are $(0, \pm 13)$ and the length of the conjugate axis is $20$.     [5]

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Question 5:

(a) Evaluate: $\int \limits_{}{} \frac{x^2-5x-1}{x^4+x^2+1}$ $dx$     [5]

(b) Draw a rough sketch of the curves $y = (x-1)^2$ and $y = |x-1 |$. Hence, find the are of the region bounded by these curves.     [5]

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Question 6:

(a) If the sum of the length of the hypotenuse and a side of a right angles triangle is given, show that the area of the triangle is maximum when the angle between them is $\frac{\pi}{3}$.     [5]

(b) If $y = x^x$, prove that: $\frac{d^2y}{dx}$ $-$ $\frac{1}{y} (\frac{dy}{dx})^2$ $-$ $\frac{y}{x}$ $= 0$     [5]

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Question 7:

(a) The following observations are given:

$(1,4), \ (2,8),\ (3,2), \ (4, 12), \ (5, 10), \ (6, 14), \ (7, 16),\ (8, 6), \ (9, 18)$

Estimate the value of $y$ when the value of $x$ is $10$ and also estimate the value of $x$ when the value of $y = 5$.     [5]

(b) Compute Karl Pearson’s coefficient of correlation between sales and expenditure of a firm for six months.     [5]

 Sales (in lakhs of Rs) 18 20 27 20 21 29 Expenditures (in lakhs of Rs) 23 27 28 28 29 30

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Question 8:

(a) A purse contains $4$ silver and $5$ copper coins. A second purse contains $3$ silver and $7$ copper coins. If a coin is taken out at random from one of the purses, what is the probability that it is a copper coin?     [5]

(b) Aman and Bhuvan throw a pair of dice alternately. In order to win, they have get a sum of $8$. Find their respective probabilities of winning if Aman starts the game.     [5]

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Question 9:

(a) Using De Moivre’s theorem, find the value of: $(1+i\sqrt{3})^6+(1-i\sqrt{3})^6$     [5]

(b) Solve the following differential equation for a particular solution:

$y - x$ $\frac{dy}{dx}$ $= x+ y$ $\frac{dy}{dx}$ , when $y = 0 \ and \ x = 1$     [5]

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Section – B (20 Marks)

Question 10:

(a) Prove that: $\begin{bmatrix} \overrightarrow{a}+ \overrightarrow{b} & \overrightarrow{b} + \overrightarrow{c} & \overrightarrow{c} + \overrightarrow{a} \end{bmatrix} = 2 \begin{bmatrix} \overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c} \end{bmatrix}$     [5]

(b) If $D, E, \ and \ F$ are mid points of a $\triangle ABC$, prove by vector method that: $Area \ of \ \triangle DEF =$ $\frac{1}{4}$ $(Area \ of \ \triangle ABC)$     [5]

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Question 11:

(a) Find the vector equation of the line passing through the point $(-1, 2, 1)$ and parallel to the line  $\overrightarrow{r} = 2 \hat{i} + 3 \hat{j} - \hat{k} + \lambda (\hat{i}-2\hat{j}+ \hat{k})$. Also, find the distance between the lines.     [5]

(b) Find the equation of the plane passing through the points $A(2, 1, -3), \ B(-3, -2, 1) \ and \ C(2, 4, -1)$.     [5]

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Question 12:

(a) A box contains $4$ red and $5$ black marbles. Find the probability distribution of red marbles in a  random draw of three marbles. Also, find the mean and standard deviation of the distribution.     [5]

(b) $Bag \ A$ contains $2$ white, $1$ black and $3$ red balls. $Bag \ B$ contains $3$ white, $2$ black and $4$ red balls and $Bag \ C$ contains $4$ white, $3$ black and $2$ red balls. One Bag is chosen at random and two balls are drawn at random from that Bag. If the randomly drawn balls happen to be red and black, what is the probability that both balls come from $Bag \ B$.     [5]

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Section – C (20 Marks)

Question 13:

(a) The price of a tape recorder is $Rs. \ 1661$. A person purchases it by making a cash payment of $Rs. \ 400$ and agrees to pay the balance with due interest in 3 half-yearly equal installments. If the dealer charged interest at the rate of $10\%$ per annum compounded half-yearly, find the value of the installments.     [5]

(b) A manufacturer manufactures two types of tea cups, A and B. Three machines are needed for manufacturing the tea cups. The time in minutes required for manufacturing each cup on the machines is given below:     [5]

 Type of Cup Times in minutes Machine I Machine II Machine III A 12 18 6 B 6 0 9

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Question 14:

(a) If the difference between the Banker’s discount and True discount of a bill for $73$ days at  $5\%$ per annum is $Rs. \ 10$, find (i) the amount of the bill (ii) the Banker’s discount.     [5]

(b) Given that the total cost function for $x$ units of a commodity is:

$C(x) =$ $\frac{x^3}{3}$ $+ 3x^2-7x+16$

(i) Find the Marginal Cost (MC)

(ii) Find the Average Cost (AC)

(iii) Prove that:

$Marginal \ Average \ Cost (MAC) =$ $\frac{x(MC)-C(x)}{x^2}$     [5]

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Question 15:

(a) The price quotations of four different commodities for 2001 and 2009 are given as below. Calculate the index form 2009 with 2001 as the base year by using weighted average of price relative method.     [5]

 Commodity Weight Price (in Rs) 2009 2001 A 10 9.00 4.00 B 49 4.40 5.00 C 36 9.00 6.00 D 4 3.6 2.00

(b) The profit of a soft drink firm (in thousand of Rupees) during each month of the year is as given below:

Calculate the four monthly moving averages and plot these and the original data on a graph sheet.

 Months Profits (in thousands of Rupees) January 3.6 Feburary 4.3 March 4.3 April 3.4 May 4.4 June 5.4 July 3.4 August 2.4 September 3.4 October 1.8 November 0.8 December 1.2

Calculate four months moving averages and plot these and the original data on a graph sheet.     [5]

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