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) Find $x$ and $y$ , if $x+y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $x - y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$

(ii) Find the equation of a straight line through origin and passing through the intersection of lines $x - 2y +3 = 0$ and $3x+5y+6 = 0$

(iii) Find the equation of the normal to the ellipse $5x^2+3y^2= 137$ and the point where the ordinate is $2$.

(iv) If $y =$ $\sqrt{\frac{1-cos \ x}{1 + cos \ x}}$, find $\frac{dy}{dx}$

(v) Evaluate $\int \limits_{}^{}$ $\frac{x^2}{x^2-4}$ $dx$

(vi) Find the equation of the tangents to the hyperbola $3x^2-y^2=3$ which are perpendicular to the line $x+3y=2$

(vii) In a single throw of two dice, find the probability of getting a total of at most $9$.

(viii) If the standard deviation of the numbers, $2, 3, 11$ and $x$ is $3$ $\frac{1}{2}$, find the value of $x$.

(ix) Find the value of $x$ and $y$, given that $(x+iy)(2-3i)= 4+i$

(x) Solve the differential equation: $(x+1)$ $\frac{dy}{dx}$ $- y = e^{3x} (x+1)^2$

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Question 2:

(a) Prove that: $\left| \begin{array}{ccc} a & b & ax+by \\ b & c & bx+cy \\ ax+by & bx+cy & 0 \end{array} \right|$ $= (b^2-ac)(ax^2+2bxy+cy^2)$     [5]

(b) If $A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 2 & -3 & -4 \end{bmatrix}$, find $A^{-1}$ and hence solve the following system of equations:

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

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

$3x-3y-4z=11$     [5]

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Question 3:

(a)   (i) Show that the second degree equation $x^2-5xy+4y^2+x+2y-2=0$ represents a pair of straight lines.

(ii) Find the equation of the individual lines and their point of intersection.     [5]

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

(ii) Simplify the expression and construct the switching circuit for the simplified expression.     [5]

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Question 4:

(a) Solve for $x: tan^{-1}(x-1) + tan^{-1}x+tan^{-1}(x+1)=tan^{-1}3x$     [5]

(b) Find $\frac{dy}{dx}$ if $y = tan^{-1}$ $\frac{\sqrt{1+x^2} - 1}{x}$     [5]

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Question 5:

(a) Use Lagrange’s mean value theorem to determine a point $P$ on the curve $y = \sqrt{x-2}$ defined in the interval $\Big[ 2, 3 \Big]$, where the tangent is parallel to the chord joining the end points on the curve.     [5]

(b) An open box with a square base is to be made out of a given quantity of cardboard whose area is $c^2$ square units. Show that the maximum volume of the box is $\frac{c^3}{6\sqrt{3}}$     [5]

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Question 6:

(a) Evaluate $\int \limits_{0}^{9}$ $f(x) \ dx$ where $f(x)$ is defined by

$f(x) = \Bigg\{ \begin{array}{lll} sin \ x; if \ 0 \leq x \leq \frac{\pi}{2} \\ \\ 1; \frac{\pi}{2} \leq x \leq 5 \\ \\ e^{x-5}; 5 \leq x \leq 9 \end{array}$     [5]

(b) Draw a rough sketch of the curve $y^2+1=x, x \leq 2$.Find the area enclosed by the curve and the line $x= 2$.     [5]

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Question 7:

The data for marks in Physics and History obtained by ten students are given below:

Using this data:

 Marks in Physics 15 12 8 8 7 7 7 6 5 3 Marks in History 10 25 17 11 13 17 20 13 9 15

(a) Compute Karl Pearson’s coefficient of correlation between the marks in Physics and History obtained by the 10 students.     [5]

(b)  (i) Find the line of regression in which Physics is taken as the independent variable.

(ii) A candidate had scored 0 marks in Physics but was absent  from the History test. Estimate his probable score for the latter test     [5]

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Question 8:

(a) There are $3 \ Urns \ A, B \ and\ C$. $Urn \ A$ contains $4$ red balls and $3$ black balls. $Urn \ B$ contains $5$ red balls and $4$ black balls. $Urn \ C$ contains $4$ red balls and $4$ black balls. One ball is drawn from each of these urns. What is the probability that the three balls drawn consists of $2$ red balls and $1$ black ball?     [5]

(b) The probability that a teacher will give an unannounced test during any class meeting is $\frac{1}{5}$. If a student is absent twice, find the probability that the student will miss at least one test.     [5]

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Question 9:

(a) If the ratio $\frac{z-i}{z-1}$ is purely imaginary, prove that the point $z$ lies on the circle whose center is the point $\frac{1}{2}$ $(1+i)$ and radius is $\frac{1}{\sqrt{2}}$     [5]

(b) Solve: $(x^2+y^2) \ dx - 2xy \ dy=0$ and $y =0$ and $x = 1$.     [5]

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Section – B (20 Marks)

Question 10:

(a) Find the coordinate of a point where the line joining the points $(1, -2, 3)$ and $(2, -1, 5)$ cuts the plane $x-2y+3z=19$. Hence, find the distance of this point from the point $(5, 4, 1)$     [5]

(b) If $A(-1, 4, -3)$ is one end of the diameter $AB$ of the sphere $x^2+y^2+z^2-2y+2z-15=0$ then find the coordinates of the other end point $B$.     [5]

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Question 11:

(a) A small industrial concern uses three raw materials $A, \ B \ and \ C$ in its manufacturing process. The prices of the materials are as shown below.

 Commodity Price in Rs. In the year 1995 Price in Rs. In the year 2005 A 4 5 B 60 57 C 36 42

Using 1995 as the base year, calculate a simple aggregate price index for 2005.     [5]

(b) Coded monthly sales figures of a particular brand of T.V. for 18 months commencing January 1, 2005 are as follows:

 Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2005 18 16 23 27 28 19 31 29 35 27 28 24 2006 24 28 29 30 29 22

Calculate six monthly moving averages and display these and the original figures on the same graph using the same axes for both.     [5]

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Question 12:

(a) Find $\overrightarrow{a}.\overrightarrow{b}$ is $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|= 5$ and $|\overrightarrow{a} \times \overrightarrow{b} |=8$     [5]

(b)  Given $\overrightarrow{a}= i-2j+k$$\overrightarrow{b}= 2i+j+k$ and $\overrightarrow{c} = i+2j-k$. Find $\overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c})$     [5]

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Section – C (20 Marks)

Question 13:

(a) The banker’s gain on a certain bill due 6 months hence is Rs. 100, the rate of interest being $10\%$ per annum. Find the face value of the bill.     [5]

(b) Mr. Aggarwal buys a house at Rs. 30,00,000 for which he agrees to make make equal payments at the end of each year for 10 years.If money is worth $10\%$ per annum, find the amount of each installment. Take $\Big[ 1^{-10}=0.3855 \Big]$     [5]

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Question 14:

(a) A manufacturer produces two type of steel trunks. He has two machines A and B. The first type of trunk requires 3 hours on Machine A and 3 hours on Machine B. The second type requires 3 hours on Machine A and 2 hours on machine B. Machine A and B can work at most 18 hours and 15 hours respectively. He earns Rs. 30 per trunk on the first type of trunk and Rs. 25 per trunk on the second type. Formulate a linear programming problem to find out how many trunks of each type he should make daily to maximize his profit.     [5]

(b) The average cost function associated with producing and marketing x units of an item is given by $AC=2x-11+$ $\frac{50}{x}$. Find:

(i) The total cost function and marginal cost function

(ii) The range of values of the output $x$ for which $AS$ is increasing.     [5]

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Question 15:

(a) Eight coins are thrown simultaneously.

(i) Show that the probability of getting at least 6 heads is $\frac{37}{256}$

(ii) What is the probability of getting at least 3 heads?     [5]

(b) A class consists of 50 students of which 10 are girls. In the class 2 girls and 5 boys are rank holders in an examination. If a student is selected at random from the class and is found to be a rank holder, what is the probability that the student selected is a girl.     [5]

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