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Question 1 of 20
1. Question
SECTION – 1 : (Only One Option Correct Type)
This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct.
Question:
The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has
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Question 2 of 20
2. Question
Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
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Question 3 of 20
3. Question
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
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Question 4 of 20
4. Question
In a triangle the sum of two sides is x and the product of the same two sides is y. If x^{2} – c^{2} = y, where c is the third side of the triangle, then the ratio of the inradius to the circumradius of the triangle is
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Question 5 of 20
5. Question
The common tangents to the circle x^{2} + y^{2} = 2 and the parabola y^{2} = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area of the quadrilateral PQRS is
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Question 6 of 20
6. Question
The function y = f(x) is the solution of the differential equation in (−1, 1) satisfying f(0) = 0. Then is
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Question 7 of 20
7. Question
Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let for x ∈ [0, 2]. If F′(x) = for all x ∈ (0, 2), then F(2) equals
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Question 8 of 20
8. Question
Coefficient of x^{11} in the expansion of (1 + x^{2})^{4} (1 + x^{3})^{7} (1 + x^{4})^{12} is
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Question 9 of 20
9. Question
For x (0, π ), the equation sin x + 2 sin 2x – sin 3x = 3 has
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Question 10 of 20
10. Question
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Question 11 of 20
11. Question
SECTION – 2 : Comprehension Type (Only One Option Correct)
This section contains 3 paragraphs, each describing theory, experiments, data etc. Six questions relate to the three paragraphs with two questions on each paragraph. Each question has only one correct answer among the four given options (A),(B),(C) and (D).
________________________________________________________________________________________________
Paragraph For Questions 51 and 52
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let xi be the number on the card drawn from the ith box, i = 1, 2, 3.
______________________________________________________________________________________________
The probability that x_{1} + x_{2} + x_{3} is odd, is
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Question 12 of 20
12. Question
The probability that x_{1}, x_{2}, x_{3} are in an arithmetic progression, is
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Question 13 of 20
13. Question
Paragraph For Questions 53 and 54
Let a, r, s, t be nonzero real numbers. Let P(at^{2}, 2at), Q, R (ar^{2}, 2ar) and (as^{2}, 2as) be distinct points on the parabola y^{2} = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0)
____________________________________________________________________________________________
The value of r is
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Question 14 of 20
14. Question
If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is
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Question 15 of 20
15. Question
Paragraph For Questions 55 and 56
Given that for each a ∈ (0, 1)
exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).
_______________________________________________________________________________________________
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Question 16 of 20
16. Question
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Question 17 of 20
17. Question
SECTION – 3 : Matching List Type (Only One Option Correct)
This section contains four questions, each having two matching lists. Choices for the correct combination of elements from ListI and ListII are given as options (A),(B),(C) and (D), out of which ONE is correct.
_______________________________________________________________________________________________
List I ListII
P. The number of polynomials f(x) with nonnegative 1. 8
integer coefficients of degree ≤ 2, satisfying f(0) = 0
Q. The number of points in the interval 2. 2
at which f(x) = sin(x^{2}) + cos(x^{2})
attains its maximum value, is
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Question 18 of 20
18. Question
ListI ListII
P. Let y(x) = cos(3cos^{−1}x), x ∈ [−1, 1], 1. 1
equals
Q. Let A_{1}, A_{2}, ……, A_{n} (n > 2) be the vertices of a 2. 2
regular polygon of n sides with its its centre at
the origin. Let be the position vector of the
point A_{k}, k = 1, 2, ….. , n. If
R. If the normal from the point P(h, 1) on the ellipse 3. 8
is perpendicular to the line x + y = 8,
then the value of h is
S. Number of positive solutions satisfying the equation 4. 9
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Question 19 of 20
19. Question
Let f_{1} : R → R, f_{2} : [0, ∞) → R, f_{3} : R → R and f_{4} : R → [0, ∞) be defined by
List – I ListII
P. f_{4} is 1. onto but not oneone
Q. f_{3} is 2. neither continuous nor oneone
R. f_{2} o f_{1} is 3. differentiable but not oneone
S. f_{2} is 4. continuous and oneone
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Question 20 of 20
20. Question
ListI ListII
P. For each z_{k} there exists a z_{j} such 1. True
that z_{k} . z_{j} = 1
Q. There exists a k ∈ {1, 2,……,9} such 2. False
that z_{1} . z = z_{k} has no solution z in the
set of complex numbers.
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