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Question 1 of 20
1. Question
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Question 2 of 20
2. Question
For every pair of continuous functions f, g:[0, 1] → R such that max {f(x) : x ∈ [0, 1]} = max {g(x) : x ∈ [0, 1}, the correct statement(s) is (are)
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Question 3 of 20
3. Question
Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if
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Question 4 of 20
4. Question
Let be three vectors each of magnitude √2 and the angle between each pair of them is π/3. If is a nonzero vector perpendicular to is a nonzero vector perpendicular to and then
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Question 5 of 20
5. Question
From a point P(λ, λ, λ), perpendiculars PQ and PR are drawn respectively on the lines y = x = z = 1 and y = −x, z = −1. If P is such that ∠QPR is a right angle, then the possible value(s) of λ is (are)
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Question 6 of 20
6. Question
Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N^{2} and M^{2} = N^{4}, then
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Question 7 of 20
7. Question
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Question 8 of 20
8. Question
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Question 9 of 20
9. Question
A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)^{2} + y^{2} = 16 and x^{2} + y^{2} = 1. Then
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Question 10 of 20
10. Question
Let a ∈ R and let f : R → R be given by f(x) = x^{5} – 5x + a. Then
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Question 11 of 20
11. Question
SECTION – 2 : (One Integer Value Correct Type)
This section contains 10 questions. Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
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The slope of the tangent to the curve (y – x^{5})^{2} = x(1 + x^{2})^{2} at the point (1, 3) is
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Question 12 of 20
12. Question
Let f : [0, 4π] → [0, π] be defined by f(x) = cos^{−1} (cos x). The number of points x ∈ [0, 4π] satisfying the equation
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Question 13 of 20
13. Question
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Question 14 of 20
14. Question
Let f : R → R and g : R → R be respectively given by f(x) = x + 1 and g(x) = x^{2} + 1. Define h : R → R by The number of points at which h(x) is not differentiable is
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Question 15 of 20
15. Question
For a point P in the plane, let d_{1}(P) and d_{2}(P) be the distance of the point P from the lines x – y = 0 and x + y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying 2 ≤ d_{1}(P) + d_{2}(P) ≤ 4, is
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Question 16 of 20
16. Question
Let n_{1} < n_{2} < n_{3} < n_{4} < n_{5} be positivie integers such that n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 20. Then the number of such distinct arrangements (n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) is
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Question 17 of 20
17. Question
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Question 18 of 20
18. Question
Let be three noncoplanar unit vectors such that the angle between every pair of them is π/3. If where p, q and r are scalars, then the value of
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Question 19 of 20
19. Question
Let a, b, c be positive integers such that b/a is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of is
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Question 20 of 20
20. Question
Let n ≤ 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Color the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is
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