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Question 1 of 20
1. Question
SECTION – 1 : (Maximum Marks : 32)
This section contains EIGHT questions
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive
Question :
The coefficient of x^{9} in the expansion of (1 + x) (1 + x^{2}) (1 + x^{3})…..(1 + x^{100}) is
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Question 2 of 20
2. Question
Suppose that the foci of ellipse are (f_{1}, 0) and (f_{2}, 0) where f_{1} > 0 and f_{2} < 0. Let P_{1} and P_{2} be two parabolas with a common vertex at (0, 0) and with foci at (f_{1}, 0) and (2f_{2}, 0), respectively. Let T_{1} be a tangent to P_{1} which passes through (2f_{2}, 0) and T_{2} be a tangent to P_{2} which passes through (f_{1}, 0). If m_{1} is the slope of T_{1} and m_{2} is the slope of T_{2}, then the value of is
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Question 3 of 20
3. Question
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Question 4 of 20
4. Question
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Question 5 of 20
5. Question
Let f: R → R be a continuous odd function, which vanishes exactly at one point and f(1) = 1/2. Suppose that for all x ∈ [−1, 2] and for all x ∈ [−1, 2]. If then the value of f(1/2) is
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Question 6 of 20
6. Question
Suppose that are three noncoplanar vectors in R^{3}. Let the components of a vector along be 4, 3 and 5, respectively. If the components of this vector along are x, y and z, respectively, then the value of 2x + y + z is
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BONUS MARK Question

Question 7 of 20
7. Question
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Question 8 of 20
8. Question
Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is
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Question 9 of 20
9. Question
SECTION – 2 : (Maximum Marks : 32)
This section contains EIGHT questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct
__________________________________________________________________________________________________________
Question :
Let f, g : [–1, 2] → R be continuous function which are twice differentiable on the interval (–1, 2). Let the values of f and g at the points –1, 0 and 2 be as given in the following table :
In each of the intervals (–1, 0) and (0, 2) the function (f – 3g)” never vanishes. Then the correct statement(s) is (are)
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Question 10 of 20
10. Question
Let f(x) = 7tan^{8}x + 7tan^{6}x – 3tan^{4}x – 3tan^{2}x all Then the correct expression(s) is (are)
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Question 11 of 20
11. Question
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Question 12 of 20
12. Question
Let S be the set of all nonzero real numbers α such that the quadratic equation αX^{2} – X + α = 0 has two distinct real roots x_{1} and x_{2} satisfying the inequality x_{1} – x_{2} < 1. Which of the following intervals is(are) a subset() of S?
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Question 13 of 20
13. Question
If where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
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Question 14 of 20
14. Question
Let E_{1} and E_{2} be two ellipses whose centers are at the origin. The major axes of E_{1} and E_{2} lie along the xaxis and the yaxis, respectively. Let S be the circle x^{2}+ (y – 1)^{2} = 2. The straight line x + y = 3 touches the curves S, E_{1} and E_{2} at P, Q and R, respectively. Suppose that If e_{1} and e_{2} are the eccentricities of E_{1} and E_{2}, respectively, then the correct expression(s) is(are)
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Question 15 of 20
15. Question
Consider the hyperbola H : x^{2} – y^{2} = 1 and a circle S with center N(x_{2}, 0). Suppose that H and S touch each other at a point P(x_{1}, y_{1}) with x_{1} > 1 and y_{1} > 0. The common tangent to H and S at P intersects the xaxis at point M. If (l, m) is the centroid of the triangle ∆PMN, then the correct expression(s) is(are)
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Question 16 of 20
16. Question
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Question 17 of 20
17. Question
SECTION – 3 : (Maximum Marks : 16)
This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions.
Each equation has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct
____________________________________________________________________________________________________________
PARAGRAPH – 1
57 and 58
Let n_{1} and n_{2} be the number of red and black balls, respectively, in box I. Let n_{3} and n_{4} be the number of red and black balls, respectively, in box II.
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Question :
One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3, then the correct option(s) with the possible values of n_{1}, n_{2}, n_{3} and n_{4} is(are)
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Question 18 of 20
18. Question
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 1/3, then the correct option(s) with the possible values of n_{1} and n_{2} is(are)
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Question 19 of 20
19. Question
PARAGRAPH2
19 and 20
Let F : R → R be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ∈ (1/2, 3). Let f(x) = xF(x) for all x ∈ R.
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Question :
The correct statement(s) is(are)
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Question 20 of 20
20. Question
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