1. VIT PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2008

2. SECTION I Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them iscorrect. Indicate you choice of the correct answer for each part in your answer-book bywriting the letter (a), (b), (c) or (d) whichever is appropriate 3. 01 Problem 1 If a, b, c be three unit vectors such that a x b x c b, b and c being non-2 parallel. If 1 is the angle between is a and b and 2 the angle between a and c , then a. 1 , 263 b. 1 , 23 6 c. 1 , 22 3 d. 1 , 23 2 4. 02 Problem 2 The rr . c h 0, c h , represents a. circle b. ellipse c.cone d. sphere 5. 03 Problem The simplified expression of sin tan 1 x , for any real number x is given by1 a.1 x2 b.1 x21 c.1 x21 d. 1 x2 6. 04 Problem If z 255 , find the value ofz .z 1 a. 3 b. 4 c. 5 d. 6 7. 05 Problem Argument of the complex number 1 3i is:2 i a. 45 b. 135 c. 225 d. 240 8. 06 Problem In a triangle ABC, the sides band c are the roots of the equation4x2 61x 8200 and A tan 1, then a2 is equal to:3 a. 1098 b. 1096 c.1097 d. 1095 9. 07 Problem The shortest distance between the straight lines through the points A1 6, 2, 2 and A2 4, 0, 1 , in the directions of (1,-2, 2) and (3, - 2,- 2) is a. (a) 6 b. (b) 8 c. (c) 12 d. (d) 9 10. 08 Problem 2 The centre and radius of the sphere x y2 z2 3x 4z 1 0 are: a. 321, 0, 2 ;223 b. , 0, 2 ; 212321 c. , 0, 2 ;22321, 2, 0 ; d. 22 11. 09 Problem Let A and B are two fixed points in a plane, then locus of another point C on the same plane such that CA+ CB = constant, (> AB) is a. circle b. ellipse c. parabola d. nyperbola 12. 10 Problem The directrix of the parabola y2 4x 3 0 is a. 4x 034 b. x 033 c. x 041 d. x 04 13. 11 Problem If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y)+ g(xy) - 2 for all real x and y and g(2) = 5, thenlim g x is: x 3 a. 9 b. 10 c. 25 d. 20 14. 12 Problem The value of f(0) so that ex 2x may be x continuous at x = 0 isx a. log 12 b. 0 c. 4 d. - 1 + log 2 15. 13 Problem Let [ ] denotes the greatest integer function and f x tan2 x Then, a. lim f x does not existx 0 b. f(x) is continuous at x = 0 c. f (x) is not differentiable at x = 0 d. f(x) = 1 16. 14 Problem A spherical balloon is expanding. If the radius is increasing at the rate of 2 cm/min, the rate at which the volume increases (in cubic centimeters per minute) when the radius, is 5 cm, is a. 10 b. 100 c. 200 d. 50 17. 15 Problem2 The length of the parabola y 12x cut off by the latus rectum is a. 6 2 + log 12 b. 3 2log 1 2 c. 62 - log 1 23 2 - log 12 d. 18. 16 Problem If I x5 dx , then I is equal to1 x3 5 3 a. 2 1x 3 221x 3 2 c9 3 b. log x1x3 c c. log x1 x3c 3 123 223 2 d. 1x1x c9 3 19. 17 Problem2 Area enclosed by the curve 4 x 2 y2 8 is: a. sq unit b. 2 sq unit c. 3sq unit d. 4 sq unit 20. 18 Problema a x The value of dx is:0 x a. a2a b.4 a c.2 a d.4 21. 19 Problem Lety be the number of people in a village at time t. Assume that the rate of change of the population is proportional to the number of people in the village at any time, and further assume that the population never increases in time. Then, the population of the village at any fixed time t is given by a. yektc, for some constants, c 0 and k 0 b. yce ct , for some constants c0 and k 0 c. y ectk, for some constants c 0 and k 0yke ct , for some constants c0 and k 0 d. 22. 20 Problem The differential equation of all straight lines touching the circle x 2 y2 a2 is:22 dy2dy a.y a 1 dx dx22 dy 2 dy y a 1 b.dx dxdydy c.yxa2 1dxdxdy dy d.y a2 1dx dx 23. 21 Problem The differential equation dyadmits y 3 0 dx a. infinite number of solutions b. no solutions c. a unique solution d. many solutions 24. 22 Problem Solution of the differential equation x dy y dx x2 y2 dx 0 is: a. yx2y2cx2 b. y+x2y2 cx2 c.y+x2y2cy2 d. xx2y2 cy2 25. 23 Problem Let p, q, rand s be statements and suppose that p q r p If ~s r , then a. s~q b. ~q~s c. ~s~q d. q~s 26. 24 Problem In how many number of ways can 10 students be divided into three teams, one containing four students and the other three? a. 400 b. 700 c. 1050 d. 2100 27. 25 Problem If R be a relation defined as aRb iff a b 0 ,then the relation is a. reflexive b. symmetric c. transitive d. symmetric and transitive 28. 26 Problem Let S be a finite set containing n elements. Then the total number of commutative binary operation on S is n n 1 a.2n nn 1 b.2n c. n2nn2 d. 2 29. 27 Problem A manufacturer of cotter pins knows that 5% of his product is defective. He sells pins in boxes of 100and guarantees that not more than one pin will be defective in a box. In order to find the probability that a box will fail to meet theguaranteed quality, the probability distribution one has to employ is a. binomial b. poisson c. normal d. exponential 30. 28 Problem The probability that a certain kind ot component will survive a given shock test3 is 4 .The probability that exactly 2 of the next 4 components tested survive is9 a.41 25 b. 1281 c. 5 27 d. 128 31. 29 Problem Mean and standard deviation from the following observations of marks of 5 students of a tutorial group (marks out of 25) 8 12 13 15 22 are a. 14, 4.604 b. 15, 4.604 c. 14, 5.604 d. None of these 32. 30 Problem A random variable X follows binomial distribution with meanand variance. Then a. 0 b. 0 c.0 d.0 33. 31 Problem 31.The system of equations x + y + z=0 2x + 3y + z = 0 And x + 2y = 0 Has a. a unique solution; x = 0,y =0, z = 0 b. infinite solutions c.no solution d. finite number of non-zero solutions 34. 32 Problem40 a , thenb 0 a. a = 1 = 2b b. a = b c. ab2 d. ab = 1 35. 33 Problem If Ddiag d1 , d2 , dn , where di 0 , for i = 1, 2, ... , n, then is equal to a. b. D c. adj (D) d. diag (d 1 , d 1 ,........d1) 1 2n 36. 34 Problem a b-y c-z If x,y, z are different from zero and a-x b c-z =0,,then the value ofa b ca-x b-yc expression is:x y z a. 0 b. -1 c. 1 d. 2 37. 35 Problem Probability of getting positive integral roots of the equation x2 n 0 for the integer n 1 n 40 , is: 1 a.5 1 b. 10 3 c. 201 d. 20 38. 36 Problem The number of real roots of the equation x 4 x4 20 22 is: a. 4 b. 2 c. 0 d. 1 39. 37 Problem Let , be the roots of the equationx2 ax b 0 and A n n nthen A n+1 -aA n +bA n-1 is equal to: a. - a b. b c.0 d. a - b 40. 38 Problem If the sides of a right angle triangle form an AP, the sin of the acute angles are a. 3 4 ,5 51 b.3,3 5 15 1 , c. 223 1 3 1, d.2 2 41. 39 Problem The plane through the point (- 1, - 1, - 1) and containing the line of intersection of r i 3j k 0 and r j2k 0 the planes is: i r. 2j + 3k 0 a. ij r. 4 + k 0 b. c.ij r. 5 5k 0 d.r. i j+3k 0 42. 40 Problem ai j and bk2i 4j3k are one of the sides and medians respectively, of a triangle through the same vertex, then area of the triangle is1 a. 832 b.831 c. 852 d. 86 43. 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