1. The document contains 30 multiple choice questions from an AIEEE past paper on mathematics. 2. The questions cover topics like trigonometry, geometry, probability, linear algebra, and differential equations. 3. Answer choices ranging from A-D are provided for each question.

1. AIEEE –Past papersMATHEMAATICS- UNSOLVED PAPER - 2010

2. SECTION – ISingle Correct Answer TypeThis Section contains 30 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

3. 01ProblemLet cos(α+ β ) = 4/5 and let sin(α – β) =5/13 where 0 ≤ a, b ≤ π/4 then tan 2a =56/3319/1220/725/16

4. Problem02Let S be a non-empty subset of R. Consider the following statement:P: There is a rational number x S such that x > 0.Which of the following statements is the negation of the statement P ?There is no rational number x S such that x ≤ 0Every rational number x S satisfies x ≤ 0x S and x ≤ 0 x is not rationalThere is a rational number x S such that x ≤ 0

5. Problem03Let Then vector satisfying and is a.b.c.d.

6. Problem04The equation of the tangent to the curve that is parallel to the x-axis, is y= 1y = 2y = 3 y = 0

7. Problem05Solution of the differential equation cos x dy = y(sin x – y) dx, 0 < x < π/2 isy sec x = tan x + c y tan x = sec x + ctan x = (sec x + c)y sec x = (tan x + c)y

8. Problem06The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x =4 +2 4 -14 +14 -2

9. Problem07If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is 2x + 1 = 0x = –12x – 1 = 0 x = 1

10. 08ProblemIf the vectors are mutually orthogonal, then = (1) (2, –3) (2) (–2, 3) (3) (3, –2) (4) (–3, 2)(2, –3) (–2, 3) (3, –2) (–3, 2).

11. Problem09Consider the following relations:R = {(x, y) | x, y are real numbers and x = wy for some rational number w};S = {(m/n,p/q) m, n, p and q are integers such that n, q 0 and qm = pn}. Thenneither R nor S is an equivalence relationS is an equivalence relation but R is not an equivalence relationR and S both are equivalence relationsR is an equivalence relation but S is not an equivalence relation

12. Problem10Let f: R -> R be defined by f(x) = If f has a local minimum at x = –1, then a possible value of k is0 – 1/2–1 1

13. Problem11The number of 3 X 3 non-singular matrices, with four entries as 1 and all other entries as 0, is5 6 at least 7 less than 4

14. Directions: Questions Number 12 to 16 are Assertion – Reason type questions. Each of these questionscontains two statements.Statement-1: (Assertion) and Statement-2: (Reason)Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

15. Problem12Four numbers are chosen at random (without replacement) from the set {1, 2, 3, ....., 20}.Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is 1/85.Statement-2: If the four chosen numbers from an AP, then the set of all possible values of common difference is {±1, ±2, ±3, ±4, ±5}.Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1Statement-1 is true, Statement-2 is falseStatement-1 is false, Statement-2 is trueStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

16. Problem13Statement-1: The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5.Statement-2: The plane x – y + z = 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1Statement-1 is true, Statement-2 is falseStatement-1 is false, Statement-2 is trueStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

17. Problem14Statement-1: S3 = 55 X 29Statement-2: S1 = 90x28 and S2 = 10X 28 .Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1Statement-1 is true, Statement-2 is falseStatement-1 is false, Statement-2 is trueStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

18. Problem15Let A be a 2 X2 matrix with non-zero entries and let A2 = I, where I is 2 ´ 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement-1: Tr(A) = 0Statement-2: |A| = 1Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1Statement-1 is true, Statement-2 is falseStatement-1 is false, Statement-2 is trueStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

19. Problem16Let f: R ->R be a continuous function defined by f(x) =Statement-1: f(c) =1/3, for some c R.Statement-2: 0 < f(x) for all x RStatement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1Statement-1 is true, Statement-2 is falseStatement-1 is false, Statement-2 is trueStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

20. Problem16Let f: R -> R be a continuous function defined by f(x) =Statement-1: f(c) = 1/3 , for some c є R.Statement-2: 0 < f(x) ≤ , for all x є RStatement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1(2) Statement-1 is true, Statement-2 is false(3) Statement-1 is false, Statement-2 is true(4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

21. Problem17For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following isThere is a regular polygon withThere is a regular polygon withThere is a regular polygon withThere is a regular polygon with

22. Problem18If α and β are the roots of the equation x2 – x + 1 = 0, then α 2009 + β 2009 =–1 1 2 –2

23. Problem19The number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals12 ∞0

24. Problem20A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle q with the positive z-axis, then q equals45° 60° 75° 30°

25. 21ProblemThe line L given by passes through the point (13, 32). The line K is parallel to L and has the equation Then the distance between L and K isa.b.c.d.

26. Problem22A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = ...... = a10 = 150 and a10, a11, ...... are in A.P. with common difference –2, then the time taken by him to count all notes is34 minutes 125 minutes 135 minutes 24 minutes

27. Problem23Let f: R ->R be a positive increasing function with then2/33/231

28. Problem24Let p(x) be a function defined on R such that p¢(x) = p¢(1 – x), for all x [0, 1], p(0) = 1 and p(1) = 41. Then p(x) dx equals244142,

29. Problem25Let f: (–1, 1) ->R be a differentiable function with f’(0) = –1 and f’(0) = 1. Let g(x) = [f(2f(x) + 2)]2.Then g’(0) =–4 0 –2 4

30. Problem26There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is36 66 108 3

31. Problem27Consider the system of linear equations:x1 + 2x2 + x3 = 32x1 + 3x2 + x3 = 33x1+ 5x2 + 2x3= 1The system hasexactly 3 solutions a unique solutionno solution infinite number of solutions

32. Problem28An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colour is2/71/212/231/3

33. Problem29For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is 11/2613/25/2

34. Problem30The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if–35 < m < 1515 < m < 65 35 < m < 85 –85 < m < –35

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