PART-III Mathematics

SECTION – I

Straight Objective Type

This section contains 9 multiple choice questions numbered 1 to 9. Each question has 4 choices (A), (B), (C) and (D), out of which only one is correct.

1. Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

(A) 

(B) 

(C) 

(D) 

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2. If |z| = 1 and z ≠ ± 1, then all the values of  lie on

(A)   a line not passing through the origin

(B)   |z| = √2

(C)   the x-axis

(D)   the y-axis

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3. Let E^c denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and P(E∩F∩G) = 0. Then P(E^C∩F^C | G) equals

(A)   P(E^C) + P(F^C)

(B)   P(E^C) − P(F^C)

(C)   P(E^C) − P(F)

(D)   P(E) − P(F^C)

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4. 

(A) 

(B) 

(C) 

(D) 

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5. The differential equation  determines a family of circles with

(A)   variable radii and a fixed centre at (0, 1)

(B)   variable radii and a fixed centre at (0, −1)

(C)   fixed radius 1 and variable centres along the x-axis

(D)   fixed radius 1 and variable centres along the y-axis

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6. Let  be unit vectors such that  Which one of the following is correct ?

(A) 

(B) 

(C) 

(D) 

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7. Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

(A)   3

(B)   2

(C)   3/2

(D)   1

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8. Let  for n ≥ 2 and  Then ∫x^{n – 2} g(x) dx equals

(A) 

(B) 

(C) 

(D) 

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9. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

(A)   360

(B)   192

(C)   96

(D)   48

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SECTION −II

Assertion − Reason Type

This section contains 4 questions numbered 54 to 57. 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. Consider the planes 3x − 6y − 2z = 15 and 2x + y − 2z = 5.

STATEMENT -1 : The parametric equations of the line of intersection of the given planes are x = 3 + 14t, y = 1 + 2t, z = 15t

because

STATEMENT -2 : The vectors  is parallel to the line of intersection of the given planes.

(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. STATEMENT -1 : The curve  is symmetric with respect to the line x = 1.

because

STATEMENT -2 : A parabola is symmetric about its axis.

(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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12. Let f(x) = 2 + cosx for all real x.

STATEMENT -1 : For each real t, there exists a point c in [t, t + π] such that f′(c) = 0.

because

STATEMENT -2 : f(t) = f(t + 2π) for each real t.

(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. Lines L₁ : y − x = 0 and L₂ : 2x + y = 0 intersect the line L₃ : y + 2 = 0 at P and Q, respectively. The bisector of the acute angle between L₁ and L₂ intersects L₃ at R.

STATEMENT -1 : The ratio PR : RQ equals 2√2 : √5

because

STATEMENT -2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

(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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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 A₁, G₁, H₁ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let A_n−1 and H_n−1 has arithmetic, geometric and harmonic means as A_n, G_n, H_n respectively.

14. Which one of the following statements is correct?

(A)   G₁ > G₂ > G₃ >…

(B)   G₁ < G₂ < G₃ < …

(C)   G₁ = G₂ = G₃ = …

(D)   G₁ < G₃ < G₅ < … and G₂ > G₄ > G₆ > …

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15. Which of the following statements is correct?

(A)   A₁ > A₂ > A₃ > …

(B)   A₁ < A₂ < A₃ < …

(C)   A₁ > A₃ > A₅ > … and A₂ < A₄ < A₆ < …

(D)   A₁ < A₃ < A₅ < … and A₂ > A₄ > A₆ > …

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16. Which of the following statements is correct?

(A)   H₁ > H₂ > H₃ > …

(B)   H₁ < H₂ < H₃ < …

(C)   H₁ > H₃ > H₅ > … and H₂ < H₄ < H₆ < …

(D)   H₁ < H₃ < H₅ < … and H₂ > H₄ > H₆ > …

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Paragraph for Question Nos. 17 to 19

If a continuous f defined on the real line R, assumes positive and negative values in R then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative then the equation f(x) = 0 has a root in R.

Consider f(x) = ke^x − x for all real x where k is a real constant.

17. The line y = x meets y = ke^x for k ≤ 0 at

(A)   no point

(B)   one point

(C)   two points

(D)   more than two points

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18. The positive value of k for which ke^x − x = 0 has only one root is

(A)   1/e

(B)   1

(C)   e

(D)   log_e 2

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19. For k > 0, the set of all values of k for which k e^x − x = 0 has two distinct roots is

(A) 

(B) 

(C) 

(D) 

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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. 

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

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

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

(C)   A – s, B – p ; C – s, r ; D – r

(D)   A – r, B – p ; C – q, s ; D – s

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21. Let (x, y) be such that

sin⁻¹(ax) + cos⁻¹(y) + cos⁻¹(bxy) = π/2

Match the statements in Column I with the statements in Column II

(A)   A – p ; B – q ; C – p ; D – s

(B)   A – r, B – p ; C – q, s ; D – s

(C)   A – s, B – p ; C – r ; D – r

(D)   A – q, B – r ; C – s ; D – p

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22. Match the statements in Column I with the properties Column II

(A)   A – r, B – p ; C – q, s ; D – s

(B)   A – q, B – r ; C – s ; D – p

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

(D)   A – q, B – r ; C – s ; D – p

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