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1. Aglasem Address:- PAGE: 1 MATHEMATICS KEAM ANSWER KEY 2018 PAPER II – MATHEMATICS 804, Park Centra, Block A, Sector 30 Gurgaon, HaryanaPin code 122003 Contact Number 01244717818 M.O: Kunnumpuram, Ayurveda College Jn., Trivandrum-1, ( : 0471-2573040, 2473040 E-mail: info@zephyrentrance.in, Website: www.zephyrentrance.in BRANCHES Website: https://admission.aglasem.com/keam-2019/ KOCHI Puthussery Building, Kaloor ( : 0484-2531040 KOLLAM Sivajyothi Complex, Polayathode ( : 0474-2743040 KEAM (ENGINEERING) ANSWER KEY 2018 QUESTIONS & ANSWERS The value of 2(cos75 + o o o sin75 ) sin30 ) i (B) 10(1 i 1. is + o 0.2(cos30 (A) 5(1 (C) 10(1 + + − ) ) ) i i i 2 2 2 (D) 5(1 (E) 1(1 − + ) ) i i 2 2 ANSWER B 1 2. If the conjugate of a complex number z is , then z is i − − − 1 − + 1 1 1 1 (E) 1 (A) (B) (C) (D) i i− i+ 1 1 1 1 i i ANSWER D 3         25       1 i + 18 i 3. The value of is equal to + − (A) 1 (C) 1 i i + (B) 2 2i 2 2 − − (D) 2 ANSWER (E) 2 2i 2i E + − − + The modulus of 1 1 1 i i i i − 4. is 1 (A) 2 ANSWER (B) 2 A (C) 4 (D) 8 (E) 10 5. , then ( (B) –1 ) 3 + 4 /3 π = 192 194 i z z If (A) –2 ANSWER is equal to z e (C) –i (D) –2i (E) 0 B

2. PAGE: 2 MATHEMATICS 6. KEAM ANSWER KEY 2018 If a and b are real numbers and (a + ib)11 = 1 + 3i, then (b + ia)11 is equal to (A) 3 i + (B) 1 3i + (D) 0 (E) i − − ANSWER E − (C) 1 3i 3 − , then the equation having α β and β (C) x α ≠ β α = α − β = β 2 2 7. If α as its roots is 19 3 0 x + = , 5 3, 5 3 − − 3 0 − = 19 0 − = + − 3 0 − = 3 0 + = + 2 2 2 (A) (D) ANSWER 3 3 19 19 (B) (E) 3 3 19 19 x x x x x x x x 2 2 E 8. − − + =    2 The focus of the parabola 3,2 4   ANSWER 4 3 0 3 2,4 is y    y x − −                       3, 2 4 D 3,2 4 3 − (A) (B) (C) (D) (E) 2, 4 (3 ) f x f R→ ∞ is an increasing function and if , = : (0, ) 9. If lim 1 , then ( ) f x → 2018 x (2 ) f x limx is equal to ( ) f x → 2018 (A) 2 (B) 3 3 2 (C) 2 (D) 3 (E) 1 ANSWER E − − '(1) 2 (1) x f − ( ) f x x f 10. If f is differentiable at x = 1, then lim is → 1 x 1 − − + + (A) ANSWER Eccentricity of the ellipse '(1) (B) (1) C '(1) (C) 2 (1) (D) 2 (1) '(1) (E) (1) '(1) f f f f f f f f + − + 3 2 8 0 − = 2 2 11. 4 8 4 is x y x y 3 3 3 3 (A) (B) (C) (D) (E) 16 2 4 8 ANSWER A The focus of the parabola ( y + 1)2 = –8(x +2) is (A) (–4, –1) (B) (–1, –4) ANSWER A 12. (C) (1, 4) (D) (4, 1) (E) (–1, 4) 13. Which of the following is the equation of a hyperbola? (A) 4 16 17 0 x x y − + + = (C) 2 4 2 27 x y x y + + + − (E) 4 6 2 0 x x y + + − = ANSWER D + − 3 + y − = 2 2 2 (B) (D) 4 x 4 16 x 4 60 = 0 x y x 2 y = − + − − 2 2 2 2 0 43 0 y 2

3. PAGE: 3 MATHEMATICS 14. + , where p, q, r are constants and , then the other root of f is (B) –7 A KEAM ANSWER KEY 2018 ( ) f x f − p≠ = − = + 2 0 . If (5) 3 (2) f f Let and ( 4) (A) 3 ANSWER px = qx r 0 (C) –2 (D) 2 (E) 6 15. → = = R R satisfy Let f      : ( ) ( ) f x f y ( ) for all real numbers x and y. If (2) 4, then f f xy f 1 2 = (B) 1 (C) 1 (A) 0 4 2 (D) 1 (E) 2 ANSWER B 16. Sum of last 30 coefficients in the binomial expansion of (1 + x)59 is (A) 229 (B) 259 ANSWER C (C) 258 (D) 259 – 229 (E) 260 ( (A) 20 6 (D) 40 6 ANSWER Three players A, B and C play a game. The probability that A, B and C will finish the game are respectively 1 1 , 4. The probability that the game is finished is (A) 1 8 ANSWER D ) ( ) 4 4 + − − = 17. 3 2 3 2 (B) 30 6 (E) 10 6 (C) 5 10 D 18. 2 3 and 1 (C) 1 (D) 3 (E) 1 (B) 1 4 4 2 + − + + 1 i z z z 19. If z1 = 2 – i and z2 = 1 + i, then is 1 z 2 1 2 (A) 2 ANSWER (B) 2 (C) 3 (D) 3 (E) 1 NA − sin cos x x = 20. If ( ) f x , then lim ( ) f x is equal to →∞ x + 2 x x (C) 1 (E) ∞ (A) 1 (B) 2 2 (D) 0 ANSWER A

4. PAGE: 4 MATHEMATICS KEAM ANSWER KEY 2018 The value of sin 31 3π is (B) 1 21. − − 1 2 3 3 (E) 1 (A) (C) (D) 2 2 2 2 ANSWER A The sum of odd integers from 1 to 2001 is (A) (1121) (D) (1021) ANSWER C 22. 2 2 2 (B) (E) (1101) (1011) (C) (1001) 2 2 2 2 sin 1 cot + cos 1 tan + x x = + 23. If , then '( ) y x is equal to y x x (A) 2 cos2x (D) cos 2x ANSWER (B) 2 cos3x (E) 3 cos x (C) –cos 2x C 24. The foci of the hyperbola 16x2 – 9y2 – 64 x + 18 y – 90 = 0 are 24 5 145,1 12   21 5 145 1, 2   ANSWER A                           ± ± ± 21 5 145,1 12 24 5 145 2 1, (A) (B) (C)           ± ± 21 5 145, 1 2 − (D) (E) 25. − + 2 2 51 If the sum of the coefficients in the expansions of to (A) 0 (B) 1 ANSWER B is zero, then a is equal ( 2 1) a x ax (C) –1 (D) –2 (E) 2 26. The mean deviation of the data 2,9,9,3,6,9,4 from the mean is (A) 2.23 (B) 3.23 ANSWER C (C) 2.57 (D) 3.57 (E) 1.03 27. The mean and variance of a binomial distribution are 8 and 4 respectively. What is (X = 1) ? 1 2 2 ANSWER B 1 1 2 1 2 1 2 (A) (B) (C) (D) (E) 8 12 6 4 5 28. The number of diagonals of a polygon with 15 sides is (A) 90 (B) 45 ANSWER A (C) 60 (D) 70 (E) 10

5. PAGE: 5 MATHEMATICS 29. KEAM ANSWER KEY 2018 In a class, 40% of students study maths and science and 60% of students study maths. What is the probability of a student studying science given the student is already studying maths? (A) 1 3 ANSWER C (B) 1 (C) 2 (D) 1 (E) 1 6 3 5 4 30. + − + 2 0 + = is 2 2 The eccentricity of the conic 2 2 3 (C) 1 x y x y (B) 1 (A) 0 2 (D) 2 (E) 1 2 ANSWER B 31. If the mean of a set of observations x1, x2, ……….x10 is 20, then the mean of 4, 8, 12..., 40 x x x x + + + + is (A) 34 (B) 32 ANSWER C 1 2 3 10 (C) 42 (D) 38 (E) 40 32. A letter is taken at random from the word “GTATISTICS” and another letter is taken at random from the word “ASSISTANT”. The probability that they are same letters is (A) 1 45 ANSWER C (B) 13 (C) 19 (D) 5 (E) 9 90 90 18 10 33. If sin α and cos α are the roots of the equation (A) 2 0 a b ac − + = (D) 2 0 a b ac + + = ANSWER A + bx c + = , then (C) a 2 0 ax + 0 a c − + = + − = 2 2 2 2 2 2 2 (B) (E) ( a ) 2 0 b c b ac + = 2 2 b c 34. If the sides of a triangle are 4, 5 and 6 cms. Then the area of triangle is ……..sq.cms. π (B) 7 4 ANSWER E π (C) 4 (D) 4 (D) 15 (A) 4 15 7 7 15 4 35. In a group of 6 boys and 4 girls, a team consisting of four children is formed such that the team has atleast one boy. The number of ways of forming a team like this is (A) 159 (B) 209 (C) 200 ANSWER B (D) 240 (E) 212 36. A password is set with 3 distinct letters from the word LOGARITHMS. How many such passwords can be formed? (A) 90 (B) 720 (C) 80 ANSWER B (D) 72 (E) 120

6. PAGE: 6 MATHEMATICS 37. KEAM ANSWER KEY 2018 If 597 is divided by 52, the remainder obtained is (A) 3 (B) 5 ANSWER B (C) 4 (D) 0 (E) 1 38. + bx c + = 2 A quadratic equation chosen from the numbers 2, 3, 5, then the probability that the equation has real roots is (A) 1 3 ANSWER A 0, with distinct coefficients is formed. If a, b, c are ax (B) 2 (C) 1 (D) 1 (E) 2 5 4 5 3 + − + 3 2 3 2 7 − 9 x x + x 39. lim is equal to →∞ x 3 4 9 2 x x − (A) 2 (B) 1 9 (D) 3 (E) 9 9 2 (C) 4 2 2 ANSWER D 40. The minimum value of ( ) (A) 1 2 ANSWER = + − max{ ,1 ,2 } is f x x x x (B) 3 2 (C) 1 (D) 0 (E) 2 B 41. + 3 − − 10 0 = are (C) The equations of the asymptotes of the hyperbola (A) 2, 3 x y =− = − (D) 4, 3 x y = = ANSWER B 3 2 xy = − 4 = x y = = = = (B) (E) 2, 3, 2, 3 x x y y x y 42. If f(x) = x6 + 6x, then f’(x) is equal to (A) 12 x (B) x + 4 ANSWER C (C) 6x5 + 6x log (6) (D) 6x5 + x6x – 1 (E) x6 43. The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is 52 7 7 ANSWER NA (B) 52 (D) 53 53 7 (A) (C) (E) 6 7 − − 01 cos cos mx nx = 44. lim → x 1 2 2 m n n m (C) ∞ (D) –∞ (A) (B) (E) 0 2 2 ANSWER A

7. PAGE: 7 MATHEMATICS x KEAM ANSWER KEY 2018 + − 1 2 x 1 = 45. limx → 0 − (C) 1 1 (A) 0 (B) –1 2 (D) 1 (E) 2 ANSWER D 46. Let f and g be differentiable functions such that f(3) = 5, g(3) = 7, f’(3) = 13, g’(3) = 6, f’(7) = 2 and g’(7) = 0. If h(x) = (fog) (x), then h’(3) = (A) 14 (B) 12 (C) 16 ANSWER B (D) 0 (E) 10 3 1 − = 47. o o sin(20 ) cos(20 ) (B) 1 (A) 1 (C) 2 (D) 4 (E) 0 2 ANSWER D 48. A poison variate X satisfies P(X = 1) = P(x = 2). P(X = 6) is equal to 4 45e− ANSWER A 1 45e− 1 9e− 1 4e− 1 45e− 2 1 2 2 2 (A) (B) (C) (D) (E) 49. Let a and b be 2 consecutive integers selected from the first 20 natural numbers. The probability that a b a b + + is an odd positive integer is (A) 9 19 ANSWER D 2 2 2 2 (B) 10 (C) 13 19 19 (D) 1 (E) 0 An ellipse of eccentricity 2 2 50. is inscribed in a circle. A point is chosen inside the circle at 3 random. The probability that the point lies outside the ellipse is (A) 1 3 ANSWER B (B) 2 (C) 1 (D) 2 (E) 1 3 9 9 27 ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ + + + + + + If the vectors 4 equal to (A) 38 ANSWER 11 , 7 2 6 5 4 i j mk i j k i j k 51. and are coplanar, then m is (B) 0 (C) 10 (D) –10 (E) 25 C

8. PAGE: 8 MATHEMATICS KEAM ANSWER KEY 2018 ∧ ∧ + and c b + ∧ = r r r ∧ ∧ ∧ ∧ ∧ ∧ + + = + = + + 52. Let a parallelogram with diagonals a (B) 1 , 3 5 r and b 7 is 9 11 . Then the area of the i j k b i j r k i j k r r + c 6 2 (E) 1 21 (A) 4 6 (C) (D) 6 2 6 ANSWER A 53. r r r r r r r r r r r r = = = + + = + + If | | to (A) 13 ANSWER and , then the value of . is equal 3, | | 1, | | 4 0 . . a b c a b c a b b c c a (B) 26 D (C) –29 (D) –13 (E) –26 54. r r r r r r − = = = (B) 3 IF | | | | a | | b 1 , then the angle between a π and b is equal to a π b π (E) π (A) 3 ANSWER (C) 2 (D) 0 4 A r r r ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ = − + + is equal to (B) –9 = + + = λ + + µ 55. If the vectors orthogonal, then λ (A) 5 ANSWER 2 , 2 4 and 9 are mutually a i j k b i j k c i j k µ (C) –1 (D) 0 (E) –5 B 2 1 + − = 56. The solutions of (A) 1, 1024 (D) –1024, 1 ANSWER are (B) –1, 1024 (E) –1,1031 3 4 0 x x 5 5 (C) 1, 1031 D 57. + + = − − = 2 2 If the equations value of b is equal to (A) 0 ANSWER and have a real common root b, then the 1 0 0 x a x x x a (B) 1 (C) –1 (D) 2 (E) 3 C 58. If sin θ – cos θ = 1, then the value of sin3θ – cos3θ is equal to (A) 1 (B) –1 ANSWER A (C) 0 (D) 2 (E) –2 59. Two dice of different colours are thrown at a time. The probability that the sum is either 7 or 11 is (A) 7 36 ANSWER B (B) 2 (C) 2 (D) 5 (E) 6 9 3 9 7

9. PAGE: 9 MATHEMATICS 1 1 KEAM ANSWER KEY 2018 1 9! 10 2 8! 1 3!7! 92 10! ANSWER 1 9! + + + + 60. is equal to 5!5! 7!3! 11 10 7! 82 9! 2 9! 2 (A) (B) (C) (D) (E) C 61. The order and degree of the differential equation 2 3 4 ( "') ( ") ( ') y y y + − + (A) 3 and 2 (D) 1 and 4 ANSWER A = 5 is 0 y (B) 1 and 2 (E) 3 and 5 (C) 2 and 3 2 − ∫ 62. 2| | x dx is equal to (E) 1 (A) 0 (B) 1 (C) 2 (D) 4 2 ANSWER D dx 0 ∫ 63. is equal to + + 2 2 2 x x − 1 π − π π − π (A) 0 (B) 4 (C) (D) 2 (E) 4 2 ANSWER B 4 4 2 ∫ 2(3 ∫ (B) 2 − ∫ = − = 64. If ( ) f x dx 4 and ( )) f x dx 7 , then ( ) f x dx is − 1 1 (A) 1 ANSWER (C) 3 (D) 4 (E) 5 E x xe + ∫ = 65. dx 2 (1 ) x 2 xe + e + x x e + e + x+ + + (A) 1 (B) 1 (C) c c c x x 1 e e − − 2 x x e + + (D) 1 ANSWER (E) c c + 1 x x A 66. The remainder when 22000 is divided by 17 is (A) 1 (B) 2 ANSWER A (C) 8 (D) 12 (E) 4

10. PAGE: 10 MATHEMATICS 67. KEAM ANSWER KEY 2018 The coefficient of x5 in the expansion of (x + 3)8 is (A) 1542 (B) 1512 ANSWER B (C) 2512 (D) 12 (E) 4 π       The maximum value of 5 cos θ + 3 cos θ + + 68. 3 is 3 (A) 5 ANSWER (B) 11 (C) 10 (D) –1 (E) 2 C 69. The area of the triangle in the complex plane formed by z, iz and z + iz is 1| | 2 1| 2 + + 2 2 2 | (E) | | z z iz z iz (A) |z| (B) | | z (C) (D) ANSWER C 70. → ¡ ¡ be a differentiable function. If f is even, then f’(0) is equal to Let : f (E) 1 (A) 1 (B) 2 (C) 0 (D) –1 2 ANSWER C 71. The coordinate of the point dividing internally the line joining the points (4, –2) and (8, 6) in the ratio 7 : 5 is (C) 19 8             8 19 , 3 (A) (16, 18) (B) (18, 16) , (D) (E) (7, 3) 3 3 3 ANSWER C 72. The area of the triangle formed by the points (a, b + c), (b, c + a), (c, a + b) is (a) abc (B) a2 + b2 + c2 (D) 0 (E) a(ab + bc + ca) ANSWER D (C) ab + bc + ca 73. If (x, y) is equidistant from (a + b, b – a) and (a – b, a + b), then (A) ax + by = 0 (D) bx – ay = 0 ANSWER D (B) ax – by = 0 (E) x = y (C) bx + ay = 0 x a y b + = is 1 74. The equation of the line passing through (a, b) and parallel to the line x a x a y b y b x a x a y b y b x a y b + = + = + = (A) 3 (B) 2 (C) 0 + + = + = (D) 2 0 (E) 4 ANSWER B

11. PAGE: 11 MATHEMATICS 75. KEAM ANSWER KEY 2018 If the points (2a, a), (a, 2a) and (a, a) enclose a triangle of area 18 square units, then the centroid of the triangle is equal to (A) (4, 4) (B) (8, 8) (C) (–4, –4) (D) ( ) 4 2, 4 2 (E) (6, 6) ANSWER B 76. The area of a triangle is 5 sq. units. Two of its vertices are (2, 1) and (3, –2). The third vertex lies on y = x + 3. The coordinates of the third vertex can be 3 3 , 2 2   4 2   ANSWER C − − − −                           3 3 7 13 , 2 1 1 , 4 3 3 , 2 2 (A) (B) , (C) (D) (E) 2 2 77. If x2 + y2 + 2gx + 2fy + 1 = 0 represents a pair of straight lines, then f2 + g2 is equal to (A) 0 (B) 1 (C) 2 ANSWER B (D) 4 (E) 3 78. If θ is the angle between the pair of straight lines x2 – 5xy + 4y2 + 3x – 4 = 0, then tan2θ is equal to (A) 9 16 ANSWER C (B) 16 (C) 9 (D) 21 (E) 25 25 25 25 9                , then ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ + − = − + + + − + − + − 79. If 3 2 5 2 3 2 2 3 i j k x i j k y i   j k z i j k (A) x = 1, y = 2, z = 3 (D) x = 1, y = 3, z = 2 ANSWER (B) x = 2, y = 3, z = 1 (E) x = 2, y = 2, z = 3 (C) x = 3, y = 1, z = 2 C S80. sin 15o = − + − + − + 3 2 2 1 3 2 2 1 (C) 1 3 (D) 1 3 (1 3) (A) (B) (E) 2 2 2 2 2 ANSWER A r r r r r ∧ ∧ ∧ = +    + a b = 3 6 6 b i j k 81. If a and    ∧ + are collinear and . 27 , then a is equal to ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ + + + + + + (A) 3 i 2 2 (C) 2 2 2 j k i j k i j k (B) ∧ ∧ ∧ ∧ ∧ + − + (D) ANSWER 3 3 (E) 3 2 i j k i j k B

12. PAGE: 12 MATHEMATICS 82. KEAM ANSWER KEY 2018 30 a b = , then | (C) 30 33 13 r r r r r r a = b = × If | | (A) 30 (B) 30 , | | and . is equal to (D) 65 5 | a b (E) 65 233 193 493 133 25 E 23 13 ANSWER 83. If 56Pr + 6 : 54Pr + 3 = 30800 : 1, then r is equal to (A) 69 (B) 41 ANSWER B (C) 51 (D) 61 (E) 49 84. Distance between two parallel lines y = 2x + 4 and y = 2x – 1 is (E) 3 (D) 1 (A) 5 (B) 5 5 (C) 5 5 5 ANSWER C 85. (7C0 + 7C1) + (7C2 + 7C3) + ... + (7C6 + 7C7) = (A) 28 – 2 (B) 27 – 1 ANSWER C (C) 27 (D) 28 – 1 (E) 27 – 2 86. The coefficient of x in the expansion of (1 – 3x + 7x2) (1 – x)16 is (A) 17 (B) 19 ANSWER D (C) –17 (D) –19 (E) 20 87. The equation of the circle with centre (2, 2) which passes through (4, 5) is (A) x2 + y2 – 4x + 4y – 77 = 0 (C) x2 + y2 + 2x + 2y – 59 = 0 (E) x2 + y2 + 4x – 2y – 26 = 0 ANSWER B (B) x2 + y2 – 4x – 4y – 5 = 0 (D) x2 + y2 – 2x – 2y – 23 = 0 88. The point in the xy-plane which is equidistant from (2, 0, 3), (0, 3, 2) and (0, 0, 1) is (A) (1, 2, 3) (B) (–3, 2, 0) (D) (3, 2, 0) (E) (3, 2, 1) ANSWER D (C) (3, –2, 0) 89. → ( f a ¥ ¥ be such that f(1) = 2 and f (x + y) = f(x) f(y) for all natural numbers x and y. + = 16 (2n – 1), then a is equal to (B) 4 (C) 5 A Let f : ∑ (A) 3 ANSWER n If ) k = 1 k (D) 6 (E) 7 90. If nCr – 1 = 36, nCr = 84 and nC r + 1 = 126, then n = (A) 3 (B) 4 ANSWER D (C) 8 (D) 9 (E) 10

13. PAGE: 13 MATHEMATICS KEAM ANSWER KEY 2018 1 2 Let f : (-1, 1) → (-1, 1) be continuous, f(x) = f(x2) for all x ∈ (–1, 1) and = 91. (0) , then the f f      1 4 value of 4 is (A) 1 ANSWER (B) 2 (C) 3 (D) 4 (E) 5 B 92. + − − = 2 2 lim (A) –1 ANSWER 1 1 x x → θ x (B) 1 (C) 0 (D) 2 (E) 4 C 1 h + = 93. If f is differentiable at x = 1 and lim (1 ) 5 , then ' f (1) = f h → 0 h (A) 0 ANSWER (B) 1 (C) 3 (D) 4 (E) 5 E 94. The maximum value of the function 2x3 – 15x2 + 36x + 4 is attained at (A) 0 (B) 3 ANSWER D (C) 4 (D) 2 (E) 5 π       1 2 ∫ = + 2 95. If ( )cos f x { ( )} f x , then is xdx c f 2 π+ (D) 2π + c (A) c (B) 2 (C) c + 1 (E) c + 2 c ANSWER C x 3 /4 π ∫ (A) ( = 96. dx ) + /41 π sin x π (B) ( ) ( ) − π + π − ) (C) 2 1 2 1 2 2 1 π ( π + (D) (E) 2 2 1 + 2 1 ANSWER A, E sin x 2 x π ∫ = 2 97. xdx + sin cos 2 2 0 π (B) π (D) 2π (A) 2 (C) 4 (E) 0 ANSWER C

14. PAGE: 14 MATHEMATICS KEAM ANSWER KEY 2018       =    2 x ∫ sin tdt 98. lim 0 2 x → 0 x (A) 2 (B) 2 (C) 1 (E) 1 3 9 3 (D) 0 6 ANSWER D x = = π 2 99. The area bounded by y = sin , is x x and x 2 π π π π (E) 2π (A) 2 ANSWER (B) 4 (C) 8 (D) 16 B 100. = + x The differential equation of the family of curves are arbitrary constants is (A) '' 2 ' 2 0 y y y − + = (D) '' 2 ' 0 y y y + − = ANSWER A ( cos A sin ). where A and B y e x B x + − − − = = + + = 2 (B) '' (E) '' 2 ' y 2 ' y 2 2 0 0 (C) '' ' 0 y y y y y y y The real part of ( (A) 2–10 ANSWER ) 13 i − 101. is 3 (B) 212 (C) 2–12 (D) –2–12 (E) 210 NA + − x 1 x e x = 102. lim → 0 x 2 − (A) 1 1 2 (B) (C) 1 (D) –1 (E) 0 2 ANSWER B + − (sin cos ) (2 sin2 ) sin 2 x + + x x xdx ∫ = 103. 2 − (A) sin cos x (B) sin cos x sin x x x x x x + x+ (C) c c c + sin2 sin2 − sin cos sin x (E) sin cos cos + x x x x x+ + (D) c c − sin cos sin x ANSWER B

15. PAGE: 15 MATHEMATICS 104. KEAM ANSWER KEY 2018 A plane is at a distance of 5 units from the origin; and perpendicular to the vector $ $ 2 2 i j k + + . The equation of the plane is $ $ ( (C) $ $ ( $ $ ( ANSWER C $ r r ) ) ) ( ( ) i $ i $ $ $ j − + − = + = (A) (B) . 2 2 15 . 2 r$ 15 r r j k r k ) i $ $ j $ k + + = + + = . 2 2 15 (D) . 2 15 r r j k r i i $ − + = (E) . 2 2 15 r j k − + sin cos sin cos       A A B B + 105. is equal to A B −             A B + (A) sin (B) 2tan( ) (C) cot A B 2 − 2 A B + (D) tan (E) 2cot( ) A B 2 ANSWER D 2 sin4 , thend x t = + = 106. If cos 4 x A t B 2 dt (A) x ANSWER (B) –16x B (C) 15x (D) 16x (E) –15x 107. n n n n The arithmetic mean of , , , ..., C is Co C C 1 2 n 1 − − + 1 1 n (B) 2n n 2n n 2n n 2 2 (A) (C) (D) (E) n+ + 1 1 n n ANSWER A 108. The variance of first 20 natural numbers is (A) 399 2 ANSWER D If S is a set with 10 elements and A = {( , ): , in A is (A) 100 (B) 90 ANSWER B (B) 379 (C) 133 (D) 133 (E) 169 12 2 4 2 x y S x ∈ ≠ , }, x y y 109. then the number of elements (C) 80 (D) 150 (E) 45

16. PAGE: 16 MATHEMATICS 110. KEAM ANSWER KEY 2018 A coin is tossed and a die is rolled. The probability that the coin shows head and the die shows 3 is (A) 1 6 ANSWER B (B) 1 (C) 1 (D) 11 (E) 1 12 9 12 11           0 1 3 1 2 1 2 3 1 , then the sum of all the diagonal entries of A–1 is 111. If A = (A) 2 ANSWER (B) 3 (C) –3 (D) –4 (E) 4 E 3 x 7 2 x x = x = − is a root of ( ) f x = ( ) f x 2 7 9 0 112. Let . If , then the other root are 6 (A) 2 and 7 (D) 6 and 2 ANSWER (B) 3 and 6 (E) 6 and 7 (C) 7 and 3 A      =       (B) 2 1 3 5 2 1 2 1 2 x 113. If [1 1] 2    0 , then x can be x 15 3 (A) –2 ANSWER (C) 14 (D) –14 (E) 0 A, D           2 0 x 1 − 0 x and A–1 = = = 114. If , then x = A A 1 2 x (B) 1 (A) 2 2 (C) 1 (D) 3 (E) 0 ANSWER B 2 x x x x x x = + + + dx e + + + + + 2 4 3 2 115. If 6 6 , then 5 4 3 2 is equal to ax bx cx a b c d e (A) 11 ANSWER (B) –11 (C) 12 (D) –12 (E) 13 B

17. PAGE: 17 MATHEMATICS KEAM ANSWER KEY 2018 b c a + + + 1 1 1 (A) 1 (D) a + b + c ANSWER a b c c a b = 116. − − − (B) 0 (E) 2( ) (C) (1 )(1 )(1 ) a b c + + a b c B 1 1 − 1 x = = 117. If ( ) f x 2 1 x , then (50) x x f − − − − 3 ( x x 1) ( 1)( 2) ( (C) 4 1) x x x (A) 0 ANSWER (B) 2 A (D) 1 (E) 3 − 1 sin cos cos sin 1 cos sin x x x x π /2 ∫ ∆ = + + − ∆ = ( ) x 1 1 cos x x x 118. If , then ( ) x dx 0 sin 1 x − 1 (B) 1 (A) 2 (C) 1 (D) –1 (E) 0 2 ANSWER A 119. The equation of the plane passing through the points (1,2,3), (–1, 4, 2) and (3, 1,1) is (A) 5 12 23 0 x y z + + − = (C) 5 6 2 23 x y z + + = (E) 2 6 5 7 x y z + + = ANSWER B In an arithmetic progression, if the kth term is 5k + 1, then the sum of first 100 terms is (A) 50(507) (B) 51(506) (D) 51(507) (E) 52(506) ANSWER A + + z = (B) 5 (D) 6 2 = 23 x y + z 13 + x y 120. (C) 50(506)