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 $M(\theta) = \begin{bmatrix} cos \ \theta & sin \ \theta \\ -sin \ \theta & cos \ \theta \end{bmatrix}$, show that $M(x) \ M(y) = M(x+y)$

(ii) The lines $x-2y+6=0$ and $2x-y-10=0$ intersect at a point $A$. Find the equation of the line making an angle $45^o$ with the positive direction of $x-axis$ and passing through the point $A$.

(iii) Find the equations of the tangents of the parabola $y^2+12x=0$ from the point $(3, 8)$.

(iv) Find the derivative of $sin \ x^2$ with respect to $x^3$.

(v) Evaluate the following integral: $\int \limits_{}^{}$ $\frac{e^{2x}}{2+e^x}$ $dx$

(vi) Evaluate the following limit: $\lim \limits_{x \to \frac{\pi}{4}}$ $\frac{1 - tan \ x}{cos \ 2x}$

(vii) Two horses are considered in a race. The probability of selection of first horse is $\frac{1}{4}$ and that of the second is $\frac{1}{3}$. What is the probability that:

(a) both of them will be selected

(b) only one of them will be selected

(c) none of them will be selected

(viii) The mean weight of $70$ students in a class of $60$ kg. The mean weight of girls in the class is $53$ kg and that of the boys is $70.5$ kg. Find he number of girls in the class.

(ix) If $(-2+\sqrt{-3})(-3+2\sqrt{-3}) = a+ib$, find the real numbers a and b. With these values of a and b, also find he modulus of $a+ib$.

(x) Solve the following differential equation: $(x \ cos \ y) \ dy = e^x (x \ log \ x+1) \ dx$

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Question 2:

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

$\left| \begin{array}{ccc} 1+ sin \ x^2 & cos^2 \ x & 4 sin \ 2x \\ sin^2 \ x & 1+cos^2 \ x & 4 sin\ 2x \\ sin^2 \ x & cos^2x & 1+4 sin\ 2x \end{array} \right| = 2 + 4 sin\ 2x$     [5]

(b)  Solve the following linear equations using the matrix method:

$x+y+z=9$

$2x+5y+7z=52$

$2x+y-z=0$     [5]

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Question 3:

(a) Prove that the following equation represent a pair of straight lines. Find their point of intersection and the angle between them:

$2x^2+7xy+3y^2+2(4x+7y+4)=0$     [5]

(b) $P, Q \ and \ R$ represent switches in ‘on’ positions and $P', Q, \ and \ R'$represent switches in ‘off’ positions. Construct a switching circuit representing the polynomial $PR + Q (Q'+R)(P+QR)$

Use Boolean algebra to prove that the above circuit can be simplified to an expression in which, when $P \ and \ R$ are ‘on’ or $Q \ and \ R$ are ‘on’, the light is on.     [5]

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Question 4:

(a) If $sin^{-1}x + sin^{-1}y + sin^{-1}z = \pi$ prove that

$x^2-y^2-z^2+2yz\sqrt{1-x^2}=0$     [5]

(b) Using a suitable substitution find the derivative of $tan^{-1}$ $\frac{4\sqrt{x}}{1-4x}$ with respect to $x$.     [5]

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Question 5:

(a) It is given that Rolle’s theorem holds good for function:

$f(x)= x^3+ax^2+bx, x \in \big[ 1, 2 \big]$ at the point $x =$ $\frac{4}{3}$. Find the value of $a \ and \ b$.     [5]

(b) A wire of length $20$ meter is available to fence off a flower bed in the form of a section of a circle. What must be the radius of the circle, if we wish to have a flower bed with the greatest possible area?     [5]

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Question 6:

(a)

(i) Evaluate $\int \limits_{0}^{\frac{\pi}{2}}$ $log \ (tan \ x) dx$

(ii) Evaluate $\int \limits_{0}^{5}$ $(x+\frac{1}{2}) dx$ as a limit of sum.     [5]

(b) Draw a rough sketch of the curve $x^2+y=9$ and find the area enclosed by the curve, the x-axis and the lines $x+1=0$ and $x-2=0$.     [5]

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Question 7:

(a) An examination of 8 applicants for a clerical post was taken by a firm. The marks obtained by the applicants in the reasoning and aptitude tests are given below:

 Applicant A B C D E F G H Reasoning Test 20 28 15 60 40 80 20 12 Aptitude Test 30 50 40 20 10 60 30 30

Calculate the Spearman’s coefficient of rank correlation from the data given above.     [5]

(b) If the two regression lines of a bivariate distribution are $4x-5y+33=0$ and $20x-9y-107=0$,

(i) calculate $\overline{x}$ and $\overline{y}$, the arithmetic means of $x \ and \ y$ respectively

(ii) estimate the value of $x$ when $y = 7$

(iii) find the variance of $y$ when $\sigma_x$ $= 3$     [5]

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Question 8:

(a) $Bag \ A$ contains $5$ white and $4$ black balls, and $Bag \ B$ contains $7$ white and $6$ black balls. One ball is drawn from the $Bag \ A$ and without noticing its color, is put in the $Bag \ B$. If a ball is drawn from $Bag \ B$, find the probability that it is black in color.     [5]

(b) An article manufactured by a company consists of two parts $A \ and \ B$. In the process of manufacturer of part $A$, $9$ out of $104$ parts may be defective. Similarly, $5$ out of $100$ are likely to be defective in the manufacturer of part $B$. Calculate the probability that the article manufactured will not be defective.     [5]

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Question 9:

(a) If $z =$ $\frac{13-5i}{4-9i}$, prove by using De Moivre’s theorem that $z^6 = -8i$     [5]

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

$dy = (5x-4y) \ dx$; when $y=0$ and $x=0$     [5]

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Section – B (20 Marks)

Question 10:

(a) Find the equation of the plane which contains the line $\frac{x-1}{2}$ $=$ $\frac{y+1}{-1}$ $=$ $\frac{z-3}{4}$ and its perpendicular to the plane $x+2y+z=12$     [5]

(b) Find the equation of the sphere which passes through the circle $x^2+y^2+z^2-6z-4=0$, $x+2y+2z=0$ and whose center lies on the plane $2x-y+z=1$     [5]

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Question 11:

(a) Find the area of a parallelogram whose diagonals are determined by the vectors:

$\overrightarrow{a} = 3i+j-2k$ and $\overrightarrow{b} = i - 3j+4k$     [5]

(b)   (i) Prove by vector method that the diameter of a circle will subtend a right angle at a point on its circumference.

(ii) If $\overrightarrow{a} , \overrightarrow{b}$  and $\overrightarrow{c}$ represent a position vector of the points with coordinates $(2, -10, 2), (3, 2, 1)$ and $(2, 1, 3)$ respectively, find the value of $\overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c} )$     [5]

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Question 12:

(a) The mean and the variance of the binomial distribution are $4$ and $2$ respectively. Find the probability of at least 6 successes.     [5]

(b) An insurance company insured $4000$ doctors, $8000$ teachers and $12000$ engineers. The probability of a doctor, a teacher and an engineer dying before the age of 58 years are $0.01, 0.03$ and $0.05$ respectively. If one of the insured person dies before the age of 58 years, find the probability that he is a doctor.     [5]

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Section – C (20 Marks)

Question 13:

(a) A bill of $Rs. 84150$ is drawn on 22nd April 2002 at 11 months and is discounted on 11th January 2003. Find the banker’s gain if the rate of interest is $10\%$.     [5]

(b) An iPod is purchased on installment basis, such that $Rs. 8000$ is to be paid on the signing of the contract and four yearly installments of $Rs. 3000$ each, payable at the end of the first, second, third and fourth years. If the compound interest is charged at $5\%$ per annum, what would be the cash price of the iPod. Take $(1.05)^{-4}=0.8227$     [5]

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Question 14:

(a) A new cereal, formed of a mixture of barn and rice, contains 88 grams of protein and at least 36 mill grams of iron. Knowing that barn contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of proteins and 30 milligrams of iron per kilogram, find the minimum cost of producing a kilogram of this new cereal if barn costs $Rs. 28$ per kilogram and rice costs $Rs. 25$ per kilogram.     [5]

(b) The cost of manufacturing of certain items consist of $Rs. 1600$ as overheads, $Rs. 30$ per item as the cost of the material and the labor cost $Rs.$ $\frac{x^2}{100}$ for $x$ items produced. How many items are to be produced to have a minimum average cost?     [5]

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Question 15:

(a) Calculate the index number for the year 2006 with 1996 as the base year by the weighted average of price relatives method from the following data.     [5]

 Commodity A B C D E Weight 40 25 5 20 10 Price (Rs. Per unit) year 1996 32.00 80.00 1.00 10.24 4.00 Price (Rs. Per unit) year 2006 40.00 120.00 1.00 15.36 3.00

(b) A propeller costs $Rs. 180000$ and its effective life is estimated to be 10 years. A sinking fund is created for replacing the propeller by a new model at the end of its life time, when its scrap realizes a sum of $Rs. 34000$ only.  The price of the new model is estimated to be $30\%$ more than the price of the present one. What amount should be put in sinking fund at the end of each year, if it accumulates at $4\%$ per annum compound interest? Take $(1.04)^10 = 1.480$     [5]

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