MATHEMATICS. (Three hours) (Candidates are allowed additional 15 minutes for only reading the paper.

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1 [ For more model sample papers visit : MATHEMATICS (Three hours) (Candidates are allowed additional 15 minutes 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. All working, including rough work, should e 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 rackets [ ]. Mathematical tales and graph papers are provided. Slide rule may e used SECTION A Question 1 [10 3] 3 1 (i) If A, find x and y so that A 2 + xi 2 = ya. 7 5 (ii) Evaluate: tan[2tan -1 (1/5) - / 4] (iii) (iv) Find the value(s) of k so that the line 3x 4y + k = 0 is tangent to the hyperola x 2 4y 2 = 5. Evaluate: x 1 Lim x 1 log x x 1 xe x 2 (v) Evaluate: dx ( x 1) (vi) Evaluate 0 2 2cos x sin 2x dx 2(1 sin x) 1

2 (vii) Deepak rolls two dice and gets a sum more than 9. What is the proaility that the numer on the first die is even? (viii) You are given the following two lines of regression. Find the regression of Y on X and X on Y and justify your answer: 3x + 4y = 8; 4x + 2y = 10 (ix) If w is the cue root of unity, then find the value of (1-3w+w 2 ) (1+w-3w 2 ) (x) Solve: (y + xy)dx + y (1-y 2 )dy = 0 Question 2 Using properties of determinants, prove that: 2 c2 a ac a c a c 4a c ca c a 2 2 Using matrix method, solve the following system of linear equations: x 2y 2z 5 = 0; x + 3y + 4 = 0 and 2x + z 4 = 0 Question 3 Verify Rolle s theorem for f(x) = e x (sinx cosx) on the interval where derivative vanishes. 5, 4 4 and find the point in Find the equation of the paraola whose vertex is at the point (4, 1) and focus is (6, -3). Question 4 Prove that: Tan -1 [Sin -1 (1/ 17) + Cos -1 (9/ 85] = ½. A, B and C represent three switches in on position and A, B and C represent the three switches in off position. Construct a switching circuit representing the polynomial: (A + B)( B + C)(A +C ). Using the laws of Boolean Algera, show that the aove polynomial is equivalent to AB + AC + B and construct an equivalent switching circuit. 2

3 Question 5 Using suitale sustitution, find dy 1 x2 1 1 for y dx Tan x Show that the right circular cone of least curved surface area and given volume has an altitude equal to 2 times the radius of the ase. Question 6 Evaluate: tan x tan 2 x dx 1 tan 2 x Sketch the graphs of y = x(4 - x) and find the area ounded y the curve, x axis and the lines x = 0 and x = 5. Question 7 Calculate Spearman s rank correlation for the following data: Advertisement cost ( in thousands) Sales ( in lakhs) Fit a straight line to the following data, treating y as dependent variale: x y Hence, predict the value of y when x = 16. Question 8 Bag I has two red and three lack alls, ag II has four red and one lack all and ag III has three white and two lack alls. A ag is selected at random and a all is drawn at random. What is the proaility of drawing a red all? A man makes attempts to hit a target. The proaility of hitting the target is 3/5. Find the proaility that the man hit the target at least two times in five attempts. 3

4 Question 9 Solve: (Tan -1 y x) dy = (1 + y 2 ) dx. If z = x + iy, = (2 iz) / (2z i) and = 1, find the locus of z and illustrate it in the complex plane. SECTION B Question 10 Find the shortest distance etween the lines whose vector equations are: r ( 1 t) i ( t 2) j (3 2t) k and r ( s 1) i (2s 1) j (2s 1) k Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes x + 2y + 3z 7 = 0 and 2x 3y + 4z = 0. Question 11 Question 12 Show that a a. a a. a.. Using Vector method, prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to the half of it. A manufacturer has three machines operators A, B and C. The first operator A produces 1% defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the jo for 50% of the time, B is on the jo for 30% of the time and C is on the jo for 20% of the time. A defective item is produced. What is the proaility that it was produced y A? If a coin is tossed ten times, find the proaility of getting: (i) Exactly six heads (ii) At least six heads. 4

5 SECTION C Question 13 An aeroplane can carry a maximum of 200 passengers. A profit of 1000 is made on each executive class ticket and a profit of 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least four times as many passengers prefer to travel y economy class than y the executive class. Determine how many tickets of each type must e sold in order to maximise the profit for the airline. What is the maximum profit earned? How much should e a company set aside at the eginning of each year if it has to uy a machine expected to cost 2,00,000 at the end of six years, when the rate of interest is 10% per annum, compounded annually. (Use (1.1) 6 = 1.772) Question 14 Given the total cost function for x units of a commodity as: C(x) = (i) (ii) x3 3 3x2 7x 16, The marginal cost. The average cost. find: (iii) Show that the marginal average cost is given y { x MC C(x)}/ x 2?. A ill of 12,750 was payale 60 days after sight. It was accepted on 4 th March 1990 and discounted on 25 th March What was the discounted value of the ill if the rate of interest was 7/2% per annum? Question 15 Calculate the index numer for 1979 with 1970 as the ase year y using the weighted average of price relative method: Commodity Weight Price (in ) A B C D

6 Calculate the three yearly moving averages of the data given elow and plot them on a graph paper: Year Production (in tonnes) For more model sample papers visit : 6