Mathematics

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

1. The number of distinct solutions of the equation in the interval [0, 2π] is

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2. Let the curve C be the mirror image of the parabola y² = 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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3. 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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4. 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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5. If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)² + (y + 2)² = r², then the value of r² is

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6. 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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7. 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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8. 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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SECTION – 2 : (Maximum Marks : 40)

• This section contains TEN questions

•  Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct

9. Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

(A) 

(B) 

(C) 

(D) 

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10. Which of the following values of α satisfy the equation

(A) −4 

(B) 9

(C) −9 

(D) 4

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11. In R³, consider the planes P₁ : y = 0 and P₂ : x + z = 1. Let P₃ be a plane, different from P₁ and P₂, which passes through the intersection of P₁ and P₂ . If the distance of the point (0, 1, 0) from P₃ is 1 and the distance of a point (α, β, γ) from P₃ is 2, then which of the following relation is (are) true ?

(A) 2α + β + 2γ + 2 = 0 

(B) 2α − β + 2γ + 4 = 0 

(C) 2α + β − 2γ − 10 = 0

(D) 2α − β + 2γ − 8 = 0 

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12. In R³, 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 P₁ : x + 2y – z + 1 = 0 and P₂ : 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?

(A) 

(B) 

(C) 

(D) 

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13. Let P and Q be distinct points on the parabola y² = 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 ?

(A) 

(B) 

(C) 

(D) 

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14. 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?

(A) y(−4) = 0

(B) y(−2) = 0 

(C) y(x) has a critical point in the interval (−1, 0) 

(D) y(x) has no critical point in the interval (−1, 0) 

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15. 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?

(A) P = y + x 

(B) P = y – x 

(C) 

(D) 

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16. 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?

(A)  f is differentiable at x = 0 

(B) h is differentiable at x = 0 

(C) foh is differentiable at x = 0 

(D) hof is differentiable at x = 0 

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17. 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?

(A) 

(B) 

(C) 

(D) There is an x ∈ R such that (gof)(x) = 1 

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18. Let ∆PQR be a triangle. Let If then which of the following is (are) true ?

(A) 

(B) 

(C) 

(D) 

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SECTION – 3 : (Maximum Marks : 16)

• This section contains TWO questions

• Each question contains two columns, Column I and Column II

• Column I has four entries (A),(B), (C) and (D)

• Column II has five entries (P),(Q), (R), (S) and (T)

• Match the entries in Column I with the entries in Column II

Column – I 

(A) In R², if the magnitude of the projection vector of the vector   and if  then possible

 value(s) of |α| is (are)

(B)  Let a and b be real numbers such that the function   is differentiable for all x∈R. value (s) of α is (are)

(C) Let ω ≠ 1 be a complex cube root of unity. If (3 – 2ω + 2ω²)^{4n + 3} + (2+ 3ω – 3ω²)^{4n + 3} + (−3 + 2ω + 3ω²)^{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 positive real number such that a, 5,q, b is an arithmetic progression, then the value(s) of |q – a| is (are)

Column – II

(P) 1

(Q) 2

(R) 3

(S) 4

(T) 5

(A) (A) → S, T, Q ; (B) → R ; (C) → R, S ; (D) → Q, R 

(B) (A) → P,R,S ; (B) → P ; (C) → P,Q ; (D) → S, T 

(C) (A) → S, Q, T ; (B) → Q ; (C) → S, T ; (D) → P, Q 

(D) (A) → R, P, S ; (B) → T ; (C) → R, Q ; (D) → P, S 

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20. 

Column – I 

(A)  In a triangle ∆XYZ, let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively. If 2(a² – b²) = c² 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 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 R², let  be the position vectors of  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, x = 2, y² = 4x and y = |αx – 1| + |αx – 2| + αx, where α ∈ {0, 1}. Then the value(s) of

Column – II

(P) 1

(Q) 2

(R) 3

(S) 5

(T) 6

(A)  (A) → P, R, S ; (B) → P ; (C) → P, Q ; (D) → S, T 

(B) (A) → P, Q, T ; (B) → R ; (C) → Q, Q ; (D) → R, P 

(C) (A) → Q, S, T ; (B) → Q ; (C) → R. S ; (D) → P, Q 

(D)  (A) → R, Q, T ; (B) → S ; (C) → R, S ; (D) → P, T, Q 

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