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Question 1 of 20
1. Question
SECTION – 1 : (Maximum Marks : 32)
1. This section contains EIGHT questions
2. The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive
Question :
The number of distinct solutions of the equation in the interval [0, 2π] is
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Question 2 of 20
2. Question
Let the curve C be the mirror image of the parabola y^{2} = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y= – 5, then the distance between A and B is
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Question 3 of 20
3. Question
The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is
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Question 4 of 20
4. Question
Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is
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Question 5 of 20
5. Question
If the normals of the parabola y^{2} = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)^{2} + (y + 2)^{2} = r^{2}, then the value of r^{2} is
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Question 6 of 20
6. Question
Let f: R → R be a function defined by where [x] is the greatest integer less than or equal to x. If , then the value of (4I – 1) is
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Question 7 of 20
7. Question
A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume of V mm3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of is
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Question 8 of 20
8. Question
Let for all x ∈ R and be a continuous function. For if F(a) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is
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Question 9 of 20
9. Question
SECTION – 2 : (Maximum Marks : 40)
1. This section contains TEN questions
2. 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 X and Y be two arbitrary, 3 × 3, nonzero, skewsymmetric matrices and Z be an arbitrary 3 × 3, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?
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Question 10 of 20
10. Question
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Question 11 of 20
11. Question
In R^{3}, consider the planes P_{1} : y = 0 and P_{2} : x + z = 1. Let P_{3} be a plane, different from P_{1} and P_{2}, which passes through the intersection of P_{1} and P_{2} . If the distance of the point (0, 1, 0) from P_{3} is 1 and the distance of a point (α, β, γ) from P_{3} is 2, then which of the following relation is (are) true ?
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Question 12 of 20
12. Question
In R^{3}, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P_{1} : x + 2y – z + 1 = 0 and P_{2} : 2x – y + z – 1 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following points lie(s) on M?
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Question 13 of 20
13. Question
Let P and Q be distinct points on the parabola y^{2} = 2x such that a circle with PQ as diameter passes through the vertex O of the parabola, If P lies in the first quadrant and the area of the triangle ∆OPQ is 3√2, then which of the following is (are) the coordinates of P ?
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Question 14 of 20
14. Question
Let y(x) be a solution of the differential equation (1 + e^{x})y’ + ye^{x} = 1. If y(0) = 2, then which of the following statements is (are) true?
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Question 15 of 20
15. Question
Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation Py” + Qy’ + 1 = 0, where P, Q are functions of x, y and y’ , then which of the following statements is (are) true?
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Question 16 of 20
16. Question
Let g : R → R be a differentiable function with g(0) = 0, g′(0) = 0 and g′(1) ≠ 0. Let and h(x) = e^{x} for all x ∈ R. Let (foh)(x) denote f(h(x)) and (hof)(x) denote h(f(x)). Then which of the following is (are) true?
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Question 17 of 20
17. Question
Let for all x ∈ R and for all x ∈ R. Let (fog) (x) denote f(g(x)) and (gof) (x) denote g(f(x)). Then which of the following is (are) true?
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Question 18 of 20
18. Question
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Question 19 of 20
19. Question
SECTION – 3 : (Maximum Marks : 16)
1. This section contains TWO questions
2. Each question contains two columns, Column I and Column II
3. Column I has four entries (A),(B), (C) and (D)
4. Column II has five entries (P),(Q), (R), (S) and (T)
5. Match the entries in Column I with the entries in Column II_____________________________________________________________________________________________________
Column – I Column – II
(A) In R^{2}, if the magnitude of the projection vector of the vector (P) 1
value(s) of α is (are)
(B) Let a and b be real numbers such that the function (Q) 2
is differentiable for all x∈R.
value (s) of α is (are)
(C) Let ω ≠ 1 be a complex cube root of unity. If (R) 3
(3 – 2ω + 2ω^{2})^{4n + 3} + (2+ 3ω – 3ω^{2})^{4n + 3} + (−3 + 2ω + 3ω^{2})^{4n + 3} = 0,
then possible value (s) of n is (are)
(D) Let the harmonic mean of two positive real numbers a and b be 4. If q is a (S) 4
positive real number such that a, 5,q, b is an arithmetic progression, then
the value(s) of q – a is (are)
(T) 5
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Question 20 of 20
20. Question
Column – I ColumnII
(A) In a triangle ∆XYZ, let a, b and c be the lengths of the sides opposite to the angles (P) 1
X, Y and Z, respectively. If 2(a^{2} – b^{2}) = c^{2} and then possible
values of n for which cos(nπλ) = 0 is (are)
(B) In a triangle ∆XYZ, let a, b and c be the lengths of the sides opposite to the angles (Q) 2
X, Y and Z, respectively. If 1 + cos 2X – 2 cos 2Y = 2 sin X sin Y, then possible
value (s) of a/b is (are)
(C) In R^{2}, let be the position vectors of (R) 3
X, Y and Z with respect to the origin O, respectively. If the distance of Z
from the bisector of the acute angle of , then
possible value(s) of β is (are)
(D) Suppose that F(α) denotes the area of the region bounded by x = 0, (S) 5
x = 2, y^{2} = 4x and y = αx – 1 + αx – 2 + αx, where α ∈ {0, 1}.
Then the value(s) of
(T) 6
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