KEAM (ENGINEERING) ANSWER KEY 2018

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1 MTHEMTIS KEM KEY 08 PGE: M.O: Kunnumpuram, yurveda ollege Jn., Trivandrum-, (: , Website: KOHI KOLLM RNHES Puthussery uilding, Kaloor Sivajyothi omple, Polayathode (: (: KEM (ENGINEERING) KEY 08 PPER II MTHEMTIS QUESTIONS & S o o (cos75 + i sin75 ). The value of is o o 0.(cos0 + i sin0 ) 5 () ( + i ) () 0 ( + i) 0 ( i) 5 ( i ) ( + i ). If the conjugate of a comple number z is, then z is i () () i i + i D i + i. The value of i () + i + i 8 5 is equal to () + i i i i 4. The modulus of + i i i + i is E () () If z i 4 / = e, then ( z 9 z 94 ) + is equal to () () i i 0

2 MTHEMTIS KEM KEY 08 PGE: 6. If a and b are real numbers and (a + ib) = + i, then (b + ia) is equal to () i + () + i i 0 i E 7. If α βα, = 5α, β = 5β, then the equation having α β and β α as its roots is () 9 = 0 () + 9 = 0 9 9= 0 9+ = 0 E + 9+ = 0 8. The focus of the parabola (), 4 (), 4 D y y 4 + = 0 is, 4 9. If f : R (0, ) is an increasing function and if, f( ) lim f ( ) is equal to 08, 4 f( ) lim f ( ) =, then 08, 4 () () E 0. f() f( ) If f is differentiable at =, then lim is () f '() () f() f '() f () f '() f () + f '() f() + f '(). Eccentricity of the ellipse () () 4 4 y 8 4y = is 8 6. The focus of the parabola ( y + ) = 8( +) is () ( 4, ) () (, 4) (, 4) (4, ) (, 4). Which of the following is the equation of a hyperbola? () 4 + 6y + 7 = 0 () 4 + 4y 6 + 4y 60 = 0 y y = 0 y = 0 D y y + 4 = 0

3 MTHEMTIS KEM KEY 08 PGE: 4. Let f( ) = p + q + r, where p, q, r are constants and p 0. If f(5) = f() and f ( 4) = 0, then the other root of f is () () Let f : f = R R satisfy f ( ) f( y) = f ( y) for all real numbers and y. If f () = 4, then () 0 () 4 6. Sum of last 0 coefficients in the binomial epansion of ( + ) 59 is () 9 () ( + ) ( ) = () 0 6 () D 8. Three players, and play a game. The probability that, and will finish the game are respectively, and. The probability that the game is finished is 4 () 8 () 4 4 D z + z + 9. If z = i and z = + i, then is z z + i () () N sin 0. If f( ) = + cos, then lim f( ) is equal to () () 0

4 MTHEMTIS KEM KEY 08 PGE: 4. The value of sin is () (). The sum of odd integers from to 00 is () () () (0). If (0) (0) sin cos y = + + cot + tan, then y'( ) is equal to (00) () cos () cos cos cos cos 4. The foci of the hyperbola 6 9y y 90 = 0 are 4± 5 45 ± 5 45 (), (), ± 5 45 ± 5 45,, 4± 5 45, 5. 5 If the sum of the coefficients in the epansions of ( a a + ) is zero, then a is equal to () 0 () 6. The mean deviation of the data,9,9,,6,9,4 from the mean is (). () The mean and variance of a binomial distribution are 8 and 4 respectively. What is (X = )? () () The number of diagonals of a polygon with 5 sides is () 90 ()

5 MTHEMTIS KEM KEY 08 PGE: 5 9. 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? () () The eccentricity of the conic + y + y+ = 0 is () 0 (). If the mean of a set of observations,,. 0 is 0, then the mean of + 4, + 8, +..., is () 4 () letter is taken at random from the word GTTISTIS and another letter is taken at random from the word SSISTNT. The probability that they are same letters is () 45 () If sin α and cos α are the roots of the equation a + b+ c = 0, then () a b + ac = 0 () ( a c) = b + c a + b ac = 0 a + b + ac = 0 a + b + c = If the sides of a triangle are 4, 5 and 6 cms. Then the area of triangle is..sq.cms. () () E 5. 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 () 59 () password is set with distinct letters from the word LOGRITHMS. How many such passwords can be formed? () 90 ()

6 MTHEMTIS KEM KEY 08 PGE: 6 7. If 5 97 is divided by 5, the remainder obtained is () () quadratic equation a + b+ c= 0, with distinct coefficients is formed. If a, b, c are chosen from the numbers,, 5, then the probability that the equation has real roots is () () lim () () 9 D is equal to The minimum value of f ( ) = ma{, +, } is () () 0 4. The equations of the asymptotes of the hyperbola y + y 0= 0 are () =, y = () =, y = =, y = = 4, y = =, y = 4 4. If f() = 6 + 6, then f () is equal to () () log (6) The standard deviation of the data 6, 5, 9,,, 8, 0 is 5 () () N cos m 44. lim 0 cos n = m () n () n m 0

7 MTHEMTIS KEM KEY lim 0 = () 0 () D PGE: Let f and g be differentiable functions such that f() = 5, g() = 7, f () =, g () = 6, f (7) = and g (7) = 0. If h() = (fog) (), then h () = () 4 () = o o sin(0 ) cos(0 ) () () D poison variate X satisfies P(X = ) = P( = ). P(X = 6) is equal to 4 () 45 e () 45 e 9 e 4 e 45 e 49. Let a and b be consecutive integers selected from the first 0 natural numbers. The probability that () 9 9 a + b + ab is an odd positive integer is D () n ellipse of eccentricity is inscribed in a circle. point is chosen inside the circle at random. The probability that the point lies outside the ellipse is () () If the vectors 4i + j + mk, 7i + j + 6k and i + 5 j + 4k are coplanar, then m is equal to () 8 ()

8 MTHEMTIS KEM KEY 08 r r r 5. Let a = i + j + k, b = i + j + 5k and c = 7i + 9 j + k. Then the area of the r r r r parallelogram with diagonals a + b and b + c is () 4 6 () 6 6 PGE: r r r r r r rr rr rr If a =, b =, c = 4 and a + b + c = 0, then the value of ab. + bc. + ca. is equal to () () D r r r r 54. IF a b = a = b =, then the angle between a r and b r is equal to () () r r If the vectors a = i j + k, b = i + 4 j + k r and c = λ i + 9 j + µ k are mutually orthogonal, then λ + µ is equal to () 5 () The solutions of = 0 are (), 04 (), 04, 0 04,,0 D 57. If the equations + a + = 0and a = 0 have a real common root b, then the value of b is equal to () 0 () 58. If sin θ cos θ =, then the value of sin θ cos θ is equal to () () Two dice of different colours are thrown at a time. The probability that the sum is either 7 or is () 7 6 ()

9 MTHEMTIS KEM KEY 08 PGE: is equal to 9!!7! 5!5! 7!! 9! 9 0 () () 0! 8! 9! 0 7! 8 9! 6. The order and degree of the differential equation ( y"') ( y") ( y') + y = 0 is () and () and and and 4 and 5 6. d is equal to () 0 () 4 D 6. 0 d + + is equal to () 0 () If 4 f ( d ) = 4 and 4 ( f ( )) d = 7, then f ( d ) is () () 4 5 E e 65. d = ( + ) e () + e + c + + c e () c + e + e + c + e c + e The remainder when 000 is divided by 7 is () () 8 4

10 MTHEMTIS KEM KEY 08 PGE: The coefficient of 5 in the epansion of ( + ) 8 is () 54 () The maimum value of 5 cos θ + cos θ + + is () 5 () The area of the triangle in the comple plane formed by z, iz and z + iz is () z () z z z + iz z + iz 70. Let f : be a differentiable function. If f is even, then f (0) is equal to () () 0 7. The coordinate of the point dividing internally the line joining the points (4, ) and (8, 6) in the ratio 7 : 5 is () (6, 8) () (8, 6) ,, (7, ) 7. The area of the triangle formed by the points (a, b + c), (b, c + a), (c, a + b) is (a) abc () a + b + c ab + bc + ca 0 a(ab + bc + ca) D 7. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then () a + by = 0 () a by = 0 b + ay = 0 b ay = 0 = y D y 74. The equation of the line passing through (a, b) and parallel to the line + = is a b y y y () + = () + = + = 0 a b a b a b y y + + = 0 + = 4 a b a b

11 MTHEMTIS KEM KEY 08 PGE: 75. If the points (a, a), (a, a) and (a, a) enclose a triangle of area 8 square units, then the centroid of the triangle is equal to () (4, 4) () (8, 8) ( 4, 4) ( 4,4 ) (6, 6) 76. The area of a triangle is 5 sq. units. Two of its vertices are (, ) and (, ). The third verte lies on y = +. The coordinates of the third verte can be 7 (), (),,,, If + y + g + fy + = 0 represents a pair of straight lines, then f + g is equal to () 0 () If θ is the angle between the pair of straight lines 5y + 4y + 4 = 0, then tan θ is equal to () 9 () If i + j 5k = i j + k + y i + j k + z i + j k, then () =, y =, z = () =, y =, z = =, y =, z = =, y =, z = =, y =, z = S80. sin 5 o = () () + + (+ ) 8. If a r r and b = i + 6 j + 6k () i + j + k r r are collinear and ab=. 7, then a r is equal to i + j + k i j + k () i + j + k i + j + k

12 MTHEMTIS KEM KEY 08 r 8. If a = r, b = 5 r r and ab=. 0 r r, then a b is equal to PGE: () 0 () 0 5 E If 56 P r + 6 : 54 P r + = 0800 :, then r is equal to () 69 () Distance between two parallel lines y = + 4 and y = is () 5 () ( ) + ( ) ( ) = () 8 () The coefficient of in the epansion of ( + 7 ) ( ) 6 is () 7 () D 87. The equation of the circle with centre (, ) which passes through (4, 5) is () + y 4 + 4y 77 = 0 () + y 4 4y 5 = 0 + y + + y 59 = 0 + y y = 0 + y + 4 y 6 = The point in the y-plane which is equidistant from (, 0, ), (0,, ) and (0, 0, ) is () (,, ) () (,, 0) (,, 0) (,, 0) (,, ) D 89. Let f : be such that f() = and f ( + y) = f() f(y) for all natural numbers and y. n If f ( a+ k) = 6 ( n ), then a is equal to k = () () If n r = 6, n r = 84 and n r + = 6, then n = () () D

13 MTHEMTIS KEM KEY 08 PGE: 9. Let f : (-, ) (-, ) be continuous, f() = f( ) for all (, ) and f (0) =, then the value of 4 f 4 is () () lim θ + = () () If f is differentiable at = and lim h 0 f ( + h) = 5, then f '() = h () 0 () 4 5 E 94. The maimum value of the function is attained at () 0 () 4 5 D 95. If f( )cos d = { f ( )} + c, then f is () c () + c c + + c c d = /4 + sin /4 () ( ) () ( + ) ( ) ( + ), E sin sin cos d = () () 4 0

14 MTHEMTIS KEM KEY 08 PGE: lim sin tdt 0 0 = () () D 99. The area bounded by y = sin, = and = is () () The differential equation of the family of curves y = e ( cos+ sin ). where and are arbitrary constants is () y'' y' + y = 0 () y'' + y' y = 0 y'' + y' y = 0 y'' y' y = 0 y y y '' + ' + = 0 0. The real part of ( i ) is () 0 () 0 N + e 0. lim 0 () = () 0 (sin+ cos )( sin ) 0. d = sin () sin + cos sin cos + c () + c sin sin sin sin cos + c sin cos sin + cos + c sin sin + cos + c

15 MTHEMTIS KEM KEY 08 PGE: plane is at a distance of 5 units from the origin; and perpendicular to the vector $ i + $ j + k$ r. The equation of the plane is r () r.$ $ $ ( i + j k ) = 5 () r.$ $ $ ( i + j k ) = 5 r r r.$ $ $ ( i + j + k ) = 5 r. $ $ ( $ i + j + k ) = 5 r r.$ i $ j + k$ = 5 ( ) 05. sin sin is equal to cos+ cos + () sin tan D () tan( + ) cot cot( + ) 06. If sin4,then d = cos 4t + t dt = () () n n n n 07. The arithmetic mean of o,,,..., n is () n n + () n n n n + n n n n The variance of first 0 natural numbers is () 99 () 79 D If S is a set with 0 elements and = {( y, ): y, S, y}, then the number of elements in is () 00 ()

16 MTHEMTIS KEM KEY 08 PGE: 6 0. coin is tossed and a die is rolled. The probability that the coin shows head and the die shows is () () If =, then the sum of all the diagonal entries of is () () 4 4 E. Let 7 f( ) =. If = 9 is a root of f( ) = 0, then the other root are 7 6 () and 7 () and 6 7 and 6 and 6 and 7. If [ ] 5 = 0 5, then can be () () 4 4 0, D 0 4. If = and = 0 =, then = () () 0 5. If 4 a b c d e 6 6 = , then 5a + 4b + c + d + e is equal to () ()

17 MTHEMTIS KEM KEY 08 PGE: 7 6. a b + c b c + a = c a + b () () 0 ( a)( b)( c) a + b + c ( a + b + c) 7. If 8. If f( ) = ( ) ( )( ) ( ), then f (50) = () 0 () 4 cos cos ( ) = + sin cos + sin cos, then () sin sin () / 0 ( d ) = 0 9. The equation of the plane passing through the points (,,), (, 4, ) and (,,) is () 5 + y + z = 0 () 5 + 6y + z= 5 + 6y + z= + y + z = + 6y + 5z = 7 0. In an arithmetic progression, if the k th term is 5k +, then the sum of first 00 terms is () 50(507) () 5(506) 50(506) 5(507) 5(506)

KEAM (ENGINEERING) ANSWER KEY 2017

KEAM (ENGINEERING) ANSWER KEY 2017 MTHMTICS KM KY 07 PG: KM (NGINRING) KY 07 PPR II MTHMTICS QUSTIONS & S. p q r p q r + is equal to () q p () q + p (C) q () p () 0 5 0. Let = 0 5 5 () 0 and () 0 = 0. If + 5 C = 0, then C is 0 5 5 5 5 0