8/7/2019 ISC 2011 Mathematics Paper

1/6

MATHEMATICS(Three hours)

(Candidates are allowed additional 15minutes for only reading the paper.They must NOT start writing during this time.)

Section A - Answer Question 1 (compulsory) and five other questions.Section B and Section C - Answer two questions from either Section B or Section C.

A ll 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 [ }.Mathemat ical tables and graph papers are provided.

Slide rule may be used.

SECTION A

Question 1(i) If A = [~ = ~ lind x such that A2 = xA - 2 I.Hence findA -1 (ii) Find the values of k, if the equation 8x2 - 16xy+ ky2 - 22x+ 34y = 12 represents an

ellipse.(iii) Solve for x: sin (2 Tan-I x),= I(iv) Two regression lines are represented by 2x + 3y - 10=0 and 4x + y -5 =O.

Find the line of regression ofy on x.(v) Evaluate: f cosecx dxIOgtan(~)(vi) Evaluate:

y-Tan-1yy-siny

This Paper consists of 6 printed pages.1211-860 NS Copyright reserved.

Turn over

[3 J

[3 J

[3 J[3 J

[3 J

[3 J

8/7/2019 ISC 2011 Mathematics Paper

2/6

(vii) Evaluate: [3]I f x e--dxo ( 1 + x ) 2

(viii) Find the modulus and argument of the complex number 2 + i 2, 4i+(1+i)

(ix) A word consist s of 9 dif ferent alphabets, in which there are 4 consonants and 5 vowels. [3]Three alphabets are chosen at random. What is the probability that more than onevowel will be selected?

[3]

(x) Solve the differential equation: [3]

Question 2(a) Using propert ies of determinants, show that pal + 2 q a + r = 0, given that: , [ 5 1

p, q and r are not in G.P. and

.9 . a + . 9 .p pa+!. =0q

p a+q qo r-r 0

(b) Solve the following system of equat ions us ing matrix method:~+~+!Q=4x y z

1 5 1

~_i+2=1x y z

21211-860 NS

8/7/2019 ISC 2011 Mathematics Paper

3/6

Question 3(a) Prove that:

2 Tan-I _ !_ + COS-I _7_ + 2 Tan-I.!. = 2 :5 s,fi 8 4(b)

. ..P, Q and R represent switches in 'ON position' and pi, QI and RI represent switches in'OFF position' . Construct a switching circuit representing the polynomial:

PcP + Q) Q(Q + RI)Use Boolean Algebra to show that the above circuit is equivalent to a switching circuitin which when P and Q are in 'O N position', the light is on.

Question 4(a) Verify Lagrange's mean value theorem for the function f(x) = sinx - sin2x in the [5]

interval [0, 1 1 : ] .(b) Find the equation of the hyperbola whose foci are (0, 13) and the length of the [5]

conjugate axis i s 20.

Question 5(a) Evaluate:

J X2 -5x-} dxx4 +X2 + 1(b) Draw a rough sketch of the curvesy= (x-1)2 and y = I X - I I . Hence, find the area of the [5 J

region bounded by these curves.

Question 6(a) If the sum of the lengths of the hypotenuse and a side ofa right angled triangle is given, [5]

show that the area of the triangle is maximum when the angle between them is ~.3(b) If y = :i', prove that:

d 2 y _ ! _ _ ( d y ) 2 _l.. =0dxt Y dx X

3121l~860NS Turn over

[5 1

[5]

[5 ]

[5)

' lJ @ e ! >

8/7/2019 ISC 2011 Mathematics Paper

4/6

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'j: when the value of x is 10 and also estimate the value of x when thevalue of j = 5,

(b) Compute Karl Pearson's Coefficient of Correlation between sales and expenditures of a (5)firm for six months.

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

Question 8(a) A purse contains 4 silver and 5 copper coins. A second purse contains 3 silver and 7 [5]

copper coins. If a coin is taken out at random from one of the purses, what is theprobability that it is a copper coin?

(b) Aman and Bhuvan throw a pair of dice alternately. In order to win, they have to get a (5)sum of8. Find thei r respective probabili ties of winning i fAman starts the game.

Question 9(a) Using De Moivre's theorem, find the value of:

(b) Solve the following differential equat ion for a part icular solution:y - x : ; x+ y : , when j.=0 and x= ISECTION B

Question 10(a) Prove that:

[-> -> - > - > -> - > ] [ - > - > - > ]a+ b b+ c c+ a ; 2 abc

(b) If D, E, F are mid-points of the sides of a triangle ABC, prove by vector method that:I IS ]Area of l : : : . DEF ; - (Area of l : : : . ABC).4

41211-860 NS

(5)

IS ]

IS ]

IS ]

"Docs

8/7/2019 ISC 2011 Mathematics Paper

5/6

Question II(a) Find the vector equation of the line passing through the point (-I, 2, I) and parallel to the [5 1

line -; = 2 ; + 3 ] - k + . " ' ( - ; - 2 ) + k ) . Also, find the distance between these lines.(b) Find the equation of the plane passing through the points A (2, I, -3), B (-3, -2, I) and (5 1

C (2, 4, -I).

Question 12(a) A box contains 4 red and 5 black marbles. Find the probability distribution of the red

marbles in a random draw of three marbles. Also find the mean and standard deviationof the distribution.

(b) Bag A contains 2 white, I black and 3 red balls, Bag B contains 3 white, 2 black and 4red balls and Bag C contains 4 white, 3 black and 2 red balls. One Bag is chosen atrandom and 2 balls are drawn at random from that Bag. If the randomly drawn ballshappen to be red and black, what is the probability that both balls come from Bag B?

SECTION CQuestion 13(a) The price of a tape recorder is Rs.I,661. A person purchases it by making a cash (5(

payment of Rs.400 and agrees to pay the balance with due interest in 3 half-yearly equalinstalments. If the dealer charged interest at the rate of 10% per annum compoundedhalf-yearly, f ind the value of the instalment.

(b) A manufacturer manufactures two types of tea-cups, A and B. Three machines are (51needed for manufacturing the tea cups. The time in minutes required for manufacturingeach cup on the machines is given below:

Time in minutesType of Cup

Machine I Machine II Machine IIIA 12 18 6B 6 0 9

Each machine is available for a maximum of six hours per day. If the profit on each cupof type A is Rs.1.50 and that on each cup of type B is Rs.1.00, find the number of cups ofeach type that should be manufactured in a day to get maximum profit.

51211-8.60 NS Turn over

1 5 1

1 5 1

8/7/2019 ISC 2011 Mathematics Paper

6/6

Question 14(a) If the difference between Banker's discount and True discount of a bill for 73 days [5]

at 5% per annum is Rs. I 0, find (i) the amount of the bill (ii) the Banker's discount.(b) Given that the total cost function for x units of a commodity is: [5 ]x 3C{x)=-+3x2 -7x+163

Find the Marginal Cost (MC)i)(ii)(iii)

Find the Average Cost (AC)x ( M C ) - C ( x )Prove that : Marginal Average Cost (MAC) = .x2

Question IS(a) The price quotations of four different commodities for 2001 and 2009 are as given [5 ]

below. Calculate the index number for 2009 with 200 I as the base year by usingweighted average of price relative method.

Commodity Weight Price (in Rs.)2009 2001A 10 9.00 4.00B 49 4.40 5.00C 36 9.00 6.00D 4 3.60 2.00

(b) The profit ofa soft drink firm (in thousands of rupees) during each month of the year is [5]as given below:

Months Profit (in thousands of Rupees)January 3.6February 4.3March 4.3April 3.4May 4.4June 5.4July 3.4August 2.4September 3.4October 1. 8November 0.8December 1.2

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

61 21 1- 86 0 N S