Part III : Mathematics

SECTION −I

Straight Objective Type

This section contains 9 multiple choice questions number. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

1. Let α, β be the roots of the equation x² − px + r = 0 and α/2, 2β be the roots of the equation x² − qx + r = 0. Then the value of r is

(A)   

(B)  

(C)   

(D)   

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2. Let f(x) be differentiable on the interval (0, ∞) such that f(1) = 1, and

for each x > 0. Then f(x) is

(A)    

(B)   

(C)   

(D)   

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3. One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

(A)  1/2

(B)  1/3

(C)  2/5

(D)  1/5

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4. The tangent to the curve y = e^x drawn at the point (c, e^c) intersects the line joining the points (c − 1, e^c−1) and (c + 1, e^c+1)

(A)  on the left of x = c

(B)  on the right of x = c

(C)  at no point

(D)  at all points

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5. 

(A)    

(B)    

(C)    

(D)   

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6. A hyperbola, having the transverse axis of length 2 sinθ, is confocal with the ellipse 3x² + 4y² = 12. Then its equation is

(A)  x² cosec²θ − y² sec²θ = 1

(B)  x² sec²θ − y² cosec²θ = 1

(C)  x² sin²θ − y² cos²θ = 1

(D)  x² cos²θ − y² sin²θ = 1

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7. The number of distinct real values of λ, for which the vectors  and  are coplanar, is

(A)  zero

(B)  one

(C)  two

(D)  three

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8. A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks adistance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is

(A)  3e^iπ/4 + 4i

(B)  (3 − 4i) e^iπ/4

(C)  (4 + 3i) e^iπ/4

(D)  (3 + 4i) e^iπ/4

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9. The number of solutions of the pair of equations

2 sin²θ − cos2θ = 0

2 cos²θ − 3 sinθ = 0

in the interval [0, 2π] is

(A)  zero

(B)  one

(C)  two

(D)  four

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SECTION −II

Assertion − Reason Type

This section contains 4 questions number. Each question contains STATEMENT − 1 (Assertion) and STATEMENT -2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

10. Let H₁, H₂, …, H_n be mutually exclusive and exhaustive events with P(H_i) > 0, i = 1, 2, …, n. Let E be any other event with 0 < P(E) < 1.

STATEMENT -1 : P(H_i | E) > P(E | H_i) . P(H_i) for i = 1, 2, …, n

because

 

(A)  Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

(B)  Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)  Statement -1 is True, Statement -2 is False

(D)  Statement -1 is False, Statement -2 is True

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11. Tangents are drawn from the point (17, 7) to the circle x² + y² = 169.

STATEMENT -1 : The tangents are mutually perpendicular.

because

STATEMENT -2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x² + y² = 338

(A)  Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

(B)  Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)  Statement -1 is True, Statement -2 is False

(D)  Statement -1 is False, Statement -2 is True

Show Answer

12. Let the vectors  represents the sides of a regular hexagon.

STATEMENT-1 :  

because

STATEMENT-2 :  

(A)  Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

(B)  Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)  Statement -1 is True, Statement -2 is False

(D)  Statement -1 is False, Statement -2 is True

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13. Let F(x) be an indefinite integral of sin²

STATEMENT-1 : The function F(x) satisfies F(x + π) = F(x) for all real x.

because

STATEMENT-2 : sin²(x + π) = sin²x for all real x.

(A)  Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

(B)  Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)  Statement -1 is True, Statement -2 is False

(D)  Statement -1 is False, Statement -2 is True

Show Answer

SECTION − III

Linked Comprehension Type

This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choice (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for question Nos. 14 to 16

Let V_r denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r − 1). Let

T_r = V_r+1 – V_r – 2 and Q_r = T_r+1 – T_r for r = 1, 2, …

14. The sum V₁ + V₂ + … + V_n is

(A)   

(B)    

(C)    

(D)    

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15. T_r is always

(A)  an odd number

(B)  an even number

(C)  a prime number

(D)  a composite number

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16. Which one of the following is a correct statement?

(A)  Q₁, Q₂, Q₃,… are in A.P. with common difference 5

(B)  Q₁, Q₂, Q_3,… are in A.P. with common difference 6

(C)  Q₁, Q₂, Q₃,…are in A.P. with common difference 11

(D)  Q₁ = Q₂ = Q₃ = …

Show Answer

Paragraph for question Nos. 17 to 19

Consider the circle x² + y² = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

17. The ratio of the areas of the triangles PQS and PQR is

(A)  1 : √2

(B)  1 : 2

(C)  1 : 4

(D)  1 : 8

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18. The radius of the circumcircle of the triangle PRS is

(A)  5

(B)  3√3

(C)  3√2

(D)  2√3

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19. The radius of the incircle of the triangle PQR is

(A)  4

(B)  3

(C)  8/3

(D)  2

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SECTION − IV

Matrix-Match Type

This section contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements (p, q, r, s) in Column II.

20. Consider the following linear equations

ax + by + cz = 0

bx + cy + az = 0

cx + ay + bz = 0

Match the conditions / expressions in Column I with statements in Column II.

(A)  A – p ; B – q ; C – r ; D – p

(B)  A – p ; B – q ; C – r ; D – p

(C)  A – s ; B – s ; C – p ; D – r

(D)  A – r B – q C – p D – s

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21. Match the integrals in Column I with the values in Column II.

(A)  A – s ; B – s ; C – p ; D – r

(B)  A – q ; B – r ; C – q ; D – s

(C)  A – p ; B – q ; C – r ; D – p

(D)  A – s ; B – q ; C – p ; D – s

Show Answer

22. In the following [x] denotes the greatest integer less than or equal to x.

Match the functions in Column I with the properties Column II.

(A)  A – p, q, r ; B – p, s ; C – r, s ; D – p, q

(B)  A – s, p, r ; B – q, r ; C – p, q ; D – s, r

(C)  A – s, r, q ; B – r, q ; C – s, p ; D – q, r

(D)  A – s, q, r ; B – r, p ; C – q, r ; D – s, p

Show Answer

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