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Question 1 of 30
1. Question
Let cos (α + β) = 4/5 and let sin (α β) = 5/13, where 0 ≤α, β≤π/4, then tan 2α =
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Question 2 of 30
2. Question
Let S be a nonempty 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?
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Question 3 of 30
3. Question
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Question 4 of 30
4. Question
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Question 5 of 30
5. Question
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Question 6 of 30
6. Question
The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x =3π/2 is
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Question 7 of 30
7. Question
If two tangents drawn from a point P to the parabola y² = 4x are at right angles, then the locus of P is
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Question 8 of 30
8. Question
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Question 9 of 30
9. Question
Consider the following relations:
R = {(x, y)  x, y are real numbers and x = wy for some rational number w};
Then
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Question 10 of 30
10. Question
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Question 11 of 30
11. Question
The number of 3×3 nonsingular matrices, with four entries as 1 and all other entries as 0, is
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Question 12 of 30
12. Question
Directions: Questions Number 12 to 16 are Assertion – Reason type questions. Each of these questions contains two statements.
Statement1: (Assertion) and Statement2: (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.
_________________________________________________________________________________
Four numbers are chosen at random (without replacement) from the set {1, 2, 3, ….., 20}.
Statement1: The probability that the chosen numbers when arranged in some order will form an AP is 1/ 85.
Statement2: If the four chosen numbers from an AP, then the set of all possible values of common difference is {±1, ±2, ±3, ±4, ±5}.
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Question 13 of 30
13. Question
Statement1: The point A (3, 1, 6) is the mirror image of the point B (1, 3, 4) in the plane x – y + z = 5.
Statement2: The plane x – y + z = 5 bisects the line segment joining A (3, 1, 6) and B (1, 3, 4).
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Question 14 of 30
14. Question
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Question 15 of 30
15. Question
Let A be a 2 × 2 matrix with nonzero 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.
Statement1: Tr(A) = 0
Statement2: A = 1
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Question 16 of 30
16. Question
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Question 17 of 30
17. Question
For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is
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Question 18 of 30
18. Question
If αand β are the roots of the equation x² – x + 1 = 0, then α²⁰⁰⁹ + β²⁰⁰⁹ =
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Question 19 of 30
19. Question
The number of complex numbers z such that z – 1 = z + 1 = z – i equals
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Question 20 of 30
20. Question
A line AB in threedimensional space makes angles 45° and 120° with the positive xaxis and the positive yaxis respectively. If AB makes an acute angle θ with the positive zaxis, then θ equals
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Question 21 of 30
21. Question
The 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 is
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Question 22 of 30
22. Question
A person is to count 4500 currency notes. Let a_{n} denote the number of notes he counts in the n^{th} 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 is
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Question 23 of 30
23. Question
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Question 24 of 30
24. Question
Let 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.
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Question 25 of 30
25. Question
Let 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) =
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Question 26 of 30
26. Question
There 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 is
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Question 27 of 30
27. Question
Consider the system of linear equations:
x₁ + 2x₂ + x₃ = 3
2x₁ + 3x₂ + x₃ = 3
3x₁ + 5x₂ + 2x₃ = 1
The system has
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Question 28 of 30
28. Question
An 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 is
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Question 29 of 30
29. Question
For 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
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Question 30 of 30
30. Question
The circle x² + y² = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if
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