1. The document provides an unsolved mathematics test with 35 multiple choice questions covering topics like trigonometry, vectors, complex numbers, functions, and integrals. 2. For each question, 4 possible answers are provided and test takers must select the correct answer. 3. The questions cover a wide range of mathematical concepts to test the test taker's understanding of different areas of mathematics.

1. IIT JEE –Past papersMATHEMATICS- UNSOLVED PAPER - 2000

2. SECTION – ISingle Correct Answer TypeThere 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. 01ProblemLet 0 only when θ 0 0 for all real θ 0 for all real θ 0 only when θ 0

4. Problem02If x + y = k is normal to y2 = 12x, then k is 39-9-3

5. Problem03For a.b. c.d.

6. Problem04If , are the roots of the equation x2 + bx + c = 0, where c < 0 < b, then a. 0 < α< βb. α< 0 < β<| α|c. α< β < 0d. α< 0 <α| <

7. Problem05Let 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. Problem06The domain of definition of the function y(x) given by the equation 2x + 2y = 2 isa.0 x 1- < x 0d. - < x < 1

9. Problem07If x2 + y2 = 1, then yy" – 2(y’)2 + 1 = 0yy” + (y’)2 + 1 = 0yy” + (y’)2 – 1 = 0yy” + 2(y’)2 + 1 = 0

10. Problem08If 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 11 M 22 M 33 M 4

11. Problem09If 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, 21, 20, 1-1, 1

12. Problem10The 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/2π/3π/4π/6

13. Problem11In a triangle ABC, 2aca2 + b2 – c2c2 + a2 – b2b2 – c2 – a2c2 – a2 – b2

14. Problem12For ee-1e-5e5

15. Problem13Consider 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. Problem14Let 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. Problem15In a triangle ABC, let . If r is the inradius and R is thcircumradius of the triangle, then 2(r + R) is equal to a + bb + cc + aa + b + c

18. Problem16How many different nine digit numbers can be formed from the number 223355888 by rearranging the digits so that the odd digits occupy even positions? 163660180

19. Problem17If arg (z) < 0, then arg (-z) –arg(z) = a. πb. - πc. - π/2 d. π/2

20. Problem18Let 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 = 02x – 9y – 11 = 02x + 9y – 11 = 02x + 9y + 7 = 0

21. Problem19A 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 CentroidCircumecentreIncentreOrthocentre

22. Problem20If then 0123

23. Problem21The incentre of the triangle with vertices (1, ), (0, 0) and (2, 0) is a.b.c.d.

24. Problem22Consider 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. Problem23Let f(x) then f decreases in the interval( - ∞ , -2)(-2, -1)(1, 2)(2, + ∞ )

26. Problem24In the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect orthogonally, then k is 2 or – 3/22 or – 3/22 or 3/22 or 3/2

27. Problem25If 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 = 0a x b = b x c = c x aa . b = b . c = c . aa x b + b x c + c x a = 0

28. Problem26If 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) = -1-3/44/31

29. 27ProblemLet 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 0π/3π/2π/4

30. Problem28Let then at x = 0, f hasA local maximum No local maximum A local minimum No extremum

31. Problem29If a, b and c are nit coplanar vectors, then the scalar triple product [2a – b, 2b – c, 2c - a] = 01-+

32. Problem30If b > a, then the equation (x –a) (x -b) – 1 = 0,m hasBoth roots in [a, b]Both roots in (-∞, a)Both roots in (b, + ∞)One root in (-∞, a) and other in (b, +∞)

33. Problem31If z1, z2, z3 are complex number such that | z1| = |z2| = | z3| =1, then | z1, z2, z3| is Equal to 1Less than 1Greater than 3Equal to 3

34. Problem32For the equation 3x2 + px + 3 = 0, p > 0, if one of the roots is square of the other, then p is equal to 1/3132/3

35. Problem33If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the value of k is 1/8841/4

36. Problem34For all ex < 1 + xloge (1 + x) < xsin x > xloge x > x

37. Problem35The value of the integral 3/25/235

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