1. IIT JEE –Past papers MATHEMATICS- UNSOLVED PAPER - 2000

2. SECTION – I Single Correct Answer Type There are 35 items in this question. For each item four alternative answers are provided. Indicate the choice of the alternative that you think to be the correct answer by writing the corresponding letter from (a), (b), (c), (d), whichever is appropriate, in the answer book, strictly according to the order in which these items appear below.

3. 01 Problem Let 0 only when θ 0 0 for all real θ 0 for all real θ 0 only when θ 0

4. Problem 02 If x + y = k is normal to y2 = 12x, then k is 3 9 -9 -3

5. Problem 03 For a. b. c. d.

6. Problem 04 If , are the roots of the equation x2 + bx + c = 0, where c < 0 < b, then a. 0 < α< β b. α< 0 < β<| α| c. α< β < 0 d. α< 0 <α| <

7. Problem 05 Let f : be any function. Define g : by g (x) = |f(x)| for all x. Then g is Onto if f is onto. One-one if f is one-one. Continuous if f is continuous. Differentiable if f is differentiable.

8. Problem 06 The domain of definition of the function y(x) given by the equation 2x + 2y = 2 is a. 0 x 1 - < x 0 d. - < x < 1

9. Problem 07 If x2 + y2 = 1, then yy" – 2(y’)2 + 1 = 0 yy” + (y’)2 + 1 = 0 yy” + (y’)2 – 1 = 0 yy” + 2(y’)2 + 1 = 0

10. Problem 08 If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) c + d) satisfies the relation 0 M 1 1 M 2 2 M 3 3 M 4

11. Problem 09 If the system of equations x – ky – z = 0, kx – y – z = 0, x + y –z = 0 has a nonzero solution, then the possible value of k are -1, 2 1, 2 0, 1 -1, 1

12. Problem 10 The triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and (-4, 3) respectively, then is equal π/2 π/3 π/4 π/6

13. Problem 11 In a triangle ABC, 2ac a2 + b2 – c2 c2 + a2 – b2 b2 – c2 – a2 c2 – a2 – b2

14. Problem 12 For e e-1 e-5 e5

15. Problem 13 Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4 then a. b. c. d.

16. Problem 14 Let where f is such that . Then g(2) satisfies the inequality a. b. 0 ≤ g(2)<2 c. 3/2<g(2) < 5/2 d. 2 < g (2) < 4

17. Problem 15 In a triangle ABC, let . If r is the inradius and R is thcircumradius of the triangle, then 2(r + R) is equal to a + b b + c c + a a + b + c

18. Problem 16 How many different nine digit numbers can be formed from the number 223355888 by rearranging the digits so that the odd digits occupy even positions? 16 36 60 180

19. Problem 17 If arg (z) < 0, then arg (-z) –arg(z) = a. π b. - π c. - π/2 d. π/2

20. Problem 18 Let PS be the median of the triangle with vertices P(2, 2), Q(6,-1) and R(7, 3). The equation of the line passing through (1, -1) and parallel to PS is 2x – 9y – 7 = 0 2x – 9y – 11 = 0 2x + 9y – 11 = 0 2x + 9y + 7 = 0

21. Problem 19 A pole stands vertically, inside a triangular park ABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ABC the foot of the pole is at the Centroid Circumecentre Incentre Orthocentre

22. Problem 20 If then 0 1 2 3

23. Problem 21 The incentre of the triangle with vertices (1, ), (0, 0) and (2, 0) is a. b. c. d.

24. Problem 22 Consider the following statements S and R: S: both sin x and cos x are decreasing functions in the interval . R : If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b). Which of the following is true? Both S and R are wrong, Both S and R are correct, but R is not the correct explanation for S is correct and R is the correct explanation for S. S is correct and R is wrong.

25. Problem 23 Let f(x) then f decreases in the interval ( - ∞ , -2) (-2, -1) (1, 2) (2, + ∞ )

26. Problem 24 In the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect orthogonally, then k is 2 or – 3/2 2 or – 3/2 2 or 3/2 2 or 3/2

27. Problem 25 If the vectors a, b and c form the sides BC, CA and AB respectively, of a triangle ABC, then a . b + b . c + c . a = 0 a x b = b x c = c x a a . b = b . c = c . a a x b + b x c + c x a = 0

28. Problem 26 If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3π/4 with the positive x-axis, then f’(3) = -1 -3/4 4/3 1

29. 27 Problem Let the vectors a, b, c and d be such that (a x b) x (c x d) = 0. Let P1 and P2 be planes determined by the pairs of vectors a, b and c, d respectively. Then the angle between P1 and P2 is 0 π/3 π/2 π/4

30. Problem 28 Let then at x = 0, f has A local maximum No local maximum A local minimum No extremum

31. Problem 29 If a, b and c are nit coplanar vectors, then the scalar triple product [2a – b, 2b – c, 2c - a] = 0 1 - +

32. Problem 30 If b > a, then the equation (x –a) (x -b) – 1 = 0,m has Both roots in [a, b] Both roots in (-∞, a) Both roots in (b, + ∞) One root in (-∞, a) and other in (b, +∞)

33. Problem 31 If z1, z2, z3 are complex number such that | z1| = |z2| = | z3| =1, then | z1, z2, z3| is Equal to 1 Less than 1 Greater than 3 Equal to 3

34. Problem 32 For the equation 3x2 + px + 3 = 0, p > 0, if one of the roots is square of the other, then p is equal to 1/3 1 3 2/3

35. Problem 33 If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the value of k is 1/8 8 4 1/4

36. Problem 34 For all ex < 1 + x loge (1 + x) < x sin x > x loge x > x

37. Problem 35 The value of the integral 3/2 5/2 3 5

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