KEAM (ENGINEERING) ANSWER KEY 2017

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1 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 () Let = () 0 and () 0 = 0. If + 5 C = 0, then C is (C) 0 5 () 5 () If U =, then U is () U T () U (C) I () 0 () U. 0 0 If = 0 0, then is 0 0 () T () (C) () I () y y 0 0 If = + z + z, then the values of, y and z are respectively () 0, 0, (),, 0 (C), 0, 0 () 0, 0, 0 (),,

2 MTHMTICS KM KY 07 PG: is equal to () 7 () 6 (C) 6 5 () 6 5 () If 5 is singular, then the value of a is a () a = 6 () a = 5 (C) a = 5 () a = 6 () a = 0 8. If 0 5 y =, then (, y, z) is equal to 0 0 z () (, 6, 6) () (, 6, ) (C) (,, 6) () (6,, ) () (, 6, ) 9. 5 If = 0, then () + I = 0 () + I = 0 (C) 5 + I = 0 () + I = 0 () + + I = y + y If = p q p + q 0 0, then (, y, p, q) equals () 0,, 0, 0 () 0,, 0, 0 (C), 0, 0, 0 () 0,, 0, (), 0,, 0. The value of + is () () (C) () () 5

3 MTHMTICS KM KY 07 PG:. The value of 8 / 6 / 9 / is () () (C) () () 5. Let = be a root of y = + q = 0. Then y is equal to () ( ) ( 6) () ( ) ( + 6) (C) ( ) ( 6) () ( ) ( 6) () ( ) ( + ). If and are the roots of 6 = 0, then () 50 9 () 0 9 (C) is equal to () 0 9 () Let and be the roots of the equation + p = 0. If + = 0, then the value of p is equal to () or () or (C) or () or () 0 C 6. If the product of roots of the equation m (m ) = 0 is ; then the value of m is () () (C) () () 7. If f( ), then is equal to () () (C) () () 8. If and y are the roots of the equation + b + = 0, then the value of + is + b y + b () b () b (C) b () b () 9. The equations 5 + a + = 0 and 6 + a + = 0 have a common root. Then a is equal to () () (C) () () 0

4 MTHMTICS KM KY 07 PG: 0. The roots of a + + = 0, where a 0, are in the ratio :. Then a is equal to () () (C) () () 0. If z + z + = 0 where z is a comple number, then the value of. Let. If z + + z + + z + z z z equals () () 5 (C) 6 () 7 () 8 C = w w, where w is a comple number such that w =. Then w w equals () w + w () w (C) (w w ) () w () w + i 9i 9i = + iy, then 0 9 i () =, y = () = 0, y = (C) =, y = 0 () = 0, y = 0 () =, y = 0. If z = cos i sin, then z z + is equal to () 0 () (C) () () 5. + cos + isin cos isin + 7 is equal to () 0 () (C) () C ()

5 MTHMTICS KM KY 07 PG: 5 6. If k = 0 k k and det () = 56, then k equals 0 0 k k () () 5 (C) 6 () 7 () If = n + ni is equal to () I () n (C) I + n () I n () n I C 8. If z = 5 and w = z z + 5 5, then Re(w) is equal to () 0 () 5 (C) 5 () () 9. If =, then 07 is equal to () 05 () 06 (C) 0 () 07 () If a = e iθ, then + a is equal to a θ θ () cot () tan θ (C) i cot θ () i tan () tan θ C. Three numbers, y and z are in arithmetic progression. If + y + z = and yz = 8, then + y + z is equal to () 9 () 0 (C) () 0 (). The 0 th term of the arithmetic progression 0, 7, is () 97 () 87 (C) 77 () 67 () 57 C. The arithmetic mean of two numbers and y is and geometric mean is. Then + y is equal to () 0 () (C) () ()

6 MTHMTICS KM KY 07 PG: 6. The solution of = 8 is () () 6 (C) 7 6 () 5 6 () 5. The sith term in the sequence is,,,... is () 7 () 9 C (C) 8 () 7 () 7 6. Three numbers are in arithmetic progression. Their sum is and the product of the first number and the third number is 5. Then the product of these three numbers is () 5 () 90 (C) 80 () 70 () If a +, a +, a are in arithmetic progression, then the value of a is () () (C) () () 5 8. Two numbers and y have arithmetic mean 9 and geometric mean. Then and y are the roots of () 8 6 = 0 () = 0 (C) = 0 () = 0 () = 0 9. Three unbiased coins are tossed. The probability of getting at least tails is () () (C) () () C 0. single letter is selected from the word TRICKS. The probability that it s either T or R is () 6 () (C) () (). From red balls, white balls and black balls, four balls are selected. The probability of getting red balls is () 7 () 8 C (C) 9 () 0 ()

7 MTHMTICS KM KY 07 PG: 7. In a class, 60% of the students know lesson I, 0% know lesson II and 0% know lesson I and lesson II. student is selected at random. The probability that the student does not know lesson I and lesson II is () 0 () 5 (C) 5 () 5 () 5. Two distinct numbers and y are chosen from,,,, 5. The probability that the arithmetic mean of and y is an integer is () 0 () 5 (C) 5 () 5 () 5. The number of matrices with entries or + is () () 5 (C) 6 () 7 () 9 5. Let S be the set of all symmetric matrices whose entries are either zero or one. matri X is chosen from S. The probability that the determinant of X is not zero is () () (C) () () 9 6. The number of words that can be formed by using all the letters of the word PROLM only once is () 5! () 6! (C) 7! () 8! () 9! C 7. The number of diagonals in a heagon is () 8 () 9 (C) 0 () () 8. The sum of odd integers from to 00 is () 00 () 000 (C) 00 () 00 () Two balls are selected from two black and two red balls. The probability that the two balls will have no black ball is () 7 () 5 (C) () () 6

8 MTHMTICS KM KY 07 PG: If z = i 9 + i 9, then z is equal to () 0 + 0i () + 0 i (C) 0 + i () + i () + i 5. The mean for the data 6, 7, 0,,,, 8, is () 9 () 8 (C) 7 () 6 () 5 5. The set of all real numbers satisfying the inequality < is () (, ) () [, ) (C) [, ) () (, ) () (, ) 5. If > 0, then () (, ) () (, ) (C) (, ) () (, ) () (, ) 5. The mode of the data 8,, 9,8,,9, 7,8,7,,, 8 is () () 9 (C) 8 () () 7 C 55. If the mean of si numbers is, then the sum of these numbers is () 6 () 6 (C) 6 () 6 () If f 0 (t)dt = + e ( > 0), then f () is equal to () + e () + e (C) + e () e () / + d = / / / / / () + + c () (C) + + c () / / / + + c () C 58. In a flight 50 people speak Hindi, 0 speak nglish and 0 speak both nglish and Hindi. The number of people who speak at least one of the two languages is () 0 () 50 (C) 0 () 80 () 60

9 MTHMTICS KM KY 07 PG: If f ( ) =, then the value of f (f () is equal to () () 0 (C) () () 60. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even? () () (C) () () 6 6. lim + 0 is equal to () () (C) 0 () oes not eist () 6. d is equal to e + e + () + c () + c e + e + (C) + c + e () + c e () + c e θ 6. tan + + tan θ + is equal to () sec θ () sec θ (C) sec θ () sin θ () cos θ 6. 0 d + + is equal to () () (C) () 0 () No nswer 65. sin d is equal to 0 sin + cos () 0 () (C) () ()

10 MTHMTICS KM KY 07 PG: If (, y) is equidistant from (a + b, b a) and (a b, a + b), then () + y = 0 () b ay = 0 (C) a by = 0 () b + ay = 0 () a + by = If the points (, 0), (0, ) and (, 8) are collinear, then the value of is equal to () 5 () 6 (C) 6 () 7 () The minimum value of the function ma {, } is equal to () 0 () (C) () () 69. Let f ( +y) = f() f(y) for all and y. If f(0) =, f() = and f '(0) =, then f '() is equal to () () (C) () () 55 C 70. If f (9) = f ' lim f( ) (9) = 0, then is equal to 9 () 0 () f(0) (C) f '() () f(9) () * 7. The value of cos + + cos is () sin () sin (C) cos () cos () cos 7. rea of the triangle with vertices (, ), (, 5) and (6, ) is () 5 () 9 (C) () 5 () 5 7. The equation of the line passing through (, 5) and perpendicular to the line through the points (, 0) and (, ) is () 5+ y+ 0 = 0 () 5 y+ 0 = 0 (C) 5 y 0 = 0 () 5+ y+ 0 = 0 () 5y 0 = 0 7. The coefficient of 5 in the epansion of ( + ) 5 ( + ) is () 0 () 60 (C) 0 () 0 () 5

11 MTHMTICS KM KY 07 PG: 75. The coefficient of in the epansion of ( ) 5 is equal to () 0 () 0 (C) 0 () () The equation 5 + y + y = 8 represents () an ellipse () a parabola (C) a hyperbola () a circle () a straight line 77. The center of the ellipse + y 8+ y 8 =0 is () (0, ) () (, ) (C) (,) () (, ) () (, ) 78. The area bounded by the curves y = + and y = 0 is () + () (C) () 5 () 6 C d y d y dy 79. The order of the differential equation + + d d d = 0 is () () (C) () 5 () If f( ) = +, then f '() is equal to () 0 () (C) () () 8. The area of the circle + y 0y + k = 0 is 5. The value of k is equal to () () (C) 0 () () d is equal to () () (C) 07 () () The solution of dy + y tan = sec, y(0) = 0 is d () y sec = tan () y tan = sec (C) tan = y tan () sec = tan y () y cot = sec

12 MTHMTICS KM KY 07 PG: 8. If the vectors i + j+ 6 k,i+ λ j+ 6 k,i j + k are coplanar, then the value of λ is () 0 () (C) 0 () 0 () 85. The distance between (,, 0) and + y+ z+ 5 = 0 is () 0 () 0 (C) 0 9 () 5 () 86. The equation of the hyperbola vertices ( 0, ±5) and foci ( 0, ±0) is y y () = () = (C) y y () = () = y = The value of () is equal to + (6) + 6(6) + (6) + 96 () 7 9 (C) 6 7 () 9 () The equation of the plane that passes through the points (, 0, ), (,, ) (5, 0, ) is () + y z + 7 = 0 () + y z + 7 = 0 (C) y + z + 7 = 0 () y z 7 + = 0 () + y + z + 7 = The verte of the parabola y y + = 0 is () (, ) () (, ) (C) (, ) () (, ) () (, ) 90. If a, b, c are vectors such that a + b + c = 0 and a = 7, b = 5, c =, then the angle between c and b () () 6 is (C) () () 0

13 MTHMTICS KM KY 07 PG: 9. Let f() = 9a + a +, where a > 0. The minimum of f is attained at a point q and the maimum is attained at a point p. If p = q, then a is equal to () () (C) () () 9. For all real number and y, it is known that the real valued function f satisfies f() 00 = f(y) = f( + y). If f() = 7, then f ( ) is equal to r= () () (C) () () The eccentricity of the ellipse ( ) + y + = 6 () () (C) is () () 9. ma{, } () () d is equal to (C) () () If 0,, y 0, and sin + cos y =, then the value of + 0is equal to () () (C) () () Let a, a + r and a + r be positive real numbers such that their product is 6. Then the maimum value of a + r is equal to () () (C) () () 97. The sum 0 () 8! 9 () 0! S = is equal to 9!!7! 7!! 9! (C) 7 0! () 6 0! () 5 8!

14 MTHMTICS KM KY 07 PG: 98. If f() = 0 6, then f' ( ) is equal to () + 6 () 6 (C) () 6 () d is equal to ( ) + () () tan c tan c + () + () C tan c tan + c tan + c + (C) ( ) 00. Let f n () be then n th derivative of f(). The least value of n so that f n = f n + where f() = + e is () () 5 (C) () () 6 0. sin 765º is equal to () () 0 (C) () () 0. The distance of the point (, 5) from the line y 6 = 0 is () 7 () 5 (C) 7 5 () 5 () 0. The difference between the maimum and minimum value of the function f( ) = ( t + t+ dt ) on [, ] is 0 () 9 () 9 (C) 59 () C () If a and b are the non zero distinct roots of + a + b = 0, then the minimum value of + a + b is () () 9 9 (C) () () C

15 MTHMTICS KM KY 07 PG: If the straight line y = + c touches the ellipse + y = then c is equal to () 0 () ± 65 (C) ± 6 () ± () ± 06. The equation λ y =, y = λ and y = are consistent for () λ= () λ=, (C) λ=, () λ=, () λ= 07. The set {(, y) : + y = } in the y plane represents () a square () a circle (C) an ellipse () a rectangle which is not a square () a rhombus which is not a square 08. The value of costan is () 5 () 5 (C) () 5 () Let (6, ), (, ) and C (, 8) be three points such that = C. The value of are (), 5 (), 5 (C), 5 (), 5 (), 5 0. In an eperiment with 5 observations on, the following results were available = 80 = 70 One observation that was 0, was found to be wrong and was replaced by the correct value 0. Then the corrected variance is () 9. () 8. (C) 88.6 () 77. () 78. The angle between the pair of lines is () cos () cos 9 8 y z+ = = and + y z = = () cos () cos 9 8 (C) cos 9 8

16 MTHMTICS KM KY 07 PG: 6. Let a be a unit vector. If a. a + =, then the magnitude of is () 8 () 9 (C) 0 () (). The area of the triangular region whose sides are y = +, y = + and = is () 5 () 6 (C) 7 () 8 () 9. If nc r 6 ncr = 8 and ncr+ = 6, then the value of r is () 9 () (C) () 5 () 6 5. Let f( + y) = f() f(y) an f() = + sin () g(), where g is differentiable. Then f' is equal to ( ) () f( ) () g( 0 ) (C) f( ) g0 ( ) () g( ) () f( ) g0 ( ) 6. The roots of the equation = 0 (), (), (C), (), (), 7. If the 7the and 8 th term of the binomial epansion (a b) n are equal, then is equal to () n n + n + () n (C) 6 n n 8. Standard deviation of first n odd natural numbers is () n () C ( n+ )( n+ ) (C) n are n () n () n a+ b a b () n n () n 9. Let S = {,,,.., 0}. The number of subsets of S containing only odd numbers is () 5 () (C) 6 () 7 () 5 0. The area of the parallelogram with vertices (0, 0), (7,, (5, 9) and (, ) is () 50 () 5 (C) 5 () 5 () 5

02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by =

02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by = 0. π/ sin d 0 sin + cos (A) 0 (B) π (C) 3 π / (D) π / (E) π /4 0. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) + y = 0 (B) b ay = 0 (C) a by = 0 (D) b + ay = 0 (E) a + by = 0 03.