NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester

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1 NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester Choose (tick) the appropriate from the following options given below. 1. Find the number of subsets of a set {x : x is a of the week}. a. 127 b. 128 c. 256 d. 64 SECTION A 2. In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects? a. 42 b. 40 c. 41 d The set {x : 1<x<2, x N} is a. Singleton set b. Null set c. Both (a) and (b) 4. When A and B are empty sets, then their Cartesian product is a. Finite b. Infinite c. Empty d. Disjoint 5. If A = {2, 3, 4, 5} and B = {5, 6, 8, 9}, then A B is a. {2, 3, 4} b. {6, 8, 9} c. d. {2, 3, 4, 6, 8, 9} 6. For two sets A and B, which is true statement a. A A = A b. A = A c. A = d. A B B A 7. The number of elements in a set {a : a 2 + 2a + 1 = 0} is a. 2 b. 3 c. 1 d If A = {x : x is an integer, 1 x 40} and B = {x : x is an integer, 15 x 100}, then A B is a. {x : x is an integer, 1 x 15} b. {x : x is an integer, 16 x 40} c. {x : x is an integer, 15 x 40} d. {x : x is an integer, 1 x 100}

2 9. If the sum of n terms of an AP is 2n 2 + 3n, then m th term is a. 4m+2 b. 4m+1 c. 4m 1 d. 4m If the sum of n terms of an AP is 3n 2 + 5n, then which term is 134 a. 20 b. 21 c. 22 d If four numbers are in AP such that their sum is 50, and the greatest number is four times the least, then first number is a. 5 b. 10 c. 15 d In an AP, the first term is 6 and common difference is -2/3, the 6 th term is a. 2/3 b. -8/3 c. 8/3 d. -2/3 13. The first term of an AP is -50 and 50 th term is 48, common difference is a. 3 b. 2 c. 4 d The first term of an AP whose common difference is 1/3 and 25 th term is 11 is a. 2 b. 3 c. 4 d In an AP the first and last term is 10 and -10 and common difference is -5, number of terms in series is a. 4 b. 5 c. 6 d The n th term of a sequence is 2n+4, the common difference is a. 2 b. 3 c. 4 d The value obtained by subtracting 10 th term of an AP t 17 th term is 56, common difference is a. 6 b. 7 c. 8 d In an AP the 9 th term is -72, the 10 th term is 60 less than 4 th term, the first term is a. -8 b. 8 c. 10 d If the p th term of an AP is 1/q and q th term is 1/p, then pq th term is a. 1/pq b. 1

3 c. -1 d. -1/pq 20. The sum of all odd numbers between 1 to 100 is a b c d Sum of three consecutive terms of an AP is 42 and product of first and last is 180, the middle term is a. 10 b. 14 c. 18 d If a, b, c in AP, then b+c, c+a, a+b are in a. AP b. GP c. HP 23. Which term of GP is -384 a. 6 b. 7 c. 8 d If a, b, c are p th, q th and r th terms of a GP, then a q-r b r-p c p-q is a. -1 b. 1 c. 0 d Sum of first 10 terms of the series is a. 121( 3 1) b. 121( 3 + 1) c. 120( 3 1) d. 120( 3 + 1) 26. The sum of series Up to n terms is a. { (10 + 1) n} c. { (10 1) + n} b. { (10 1) n} d. { (10 1) n} 27. Geometric mean of 12 and 48 is a. 24 b. -24 c. Both a and b 28. If a, b, c are in GP then log a, log b, log c in a. AP b. GP c. HP 29. If the 5 th and 9 th term of a HP is 1/10 and 1/15, first term of HP is a. 1/5 b. 1/6 c. 1/8 d. 1/10

4 30. If a, b, c in AP and b, c, d in HO, then a. ab=cd b. ad=bc c. ac=bd 31. In the equation, 2x 2 5x + 7 = 0, the sum of the roots is a. 2 b. 2.5 c. 3 d If roots of the equation ax 2 + bx + c = 0 are α & β, then (α 2 + β 2 ) is a b. 2-2 c d Which function is an odd function? a. f (x) = x 2 + x b. f (x) = x 3 + x c. f (x) = x sin x d. f (x) = x 2 2x 34. If f (x) = 4x and g (x) = x 2, then f(g(2)) is a. -16 b. 14 c. 16 d The range of the function f (x) = x is a. (0, α) b. [0, α) c. (0, α] d. [0, α] 36. The function f(x) = is undefined for the values of x is a. b. c. nπ d. 37. The domain of the function f (x) = a. (-α, 1) (1, α) b. (-α, 1] [1, α) c. (-α, 1] [1, α) d. (-α, 1] [2, α) 38. If f (x) = x - x, then f (x) = 0 for a. All positive values of x b. At x = 0 c. All negative values of x d. Both a and b are true 39. The slope of the line parallel to the line 2x + y 1 = 0 is a. 2 b. -2

5 c. 1 d If the discriminant b 2-4ac = 0, then roots of the equation are a. Real and unequal b. Real and equal c. Complex d. None of these 41.Which statement is false for exponential function a. a x a y = a x + y b. (a x ) y = a xy c. = ax-y d. 42. The value of log 2 8 is a. 1 b. 2 c. 3 d The sum of the series 1+! +! +! +..is (a x ) y = a x + y a. (e e-1 ) b. (e + e-1 ) c. (e e -1 ) d. (e + e -1 ) 44. The sum of the series α is!!! a. 2e 1 b. 2e + 1 c. 2e d. -2e 45. The third term in expansion of increasing powers of x is a. -2x b. 5x 2 c. 2x d. -5x The value of α is... a. log e 2 b. log e 3 c. e 2 d. e log is equal to a. log(1 + x) + log(1 x) b. log(1 + x) log(1 x) c. log(1 + x) - log(1 x) d. log(1 - x) - log(1 + x) 48. If e x2-1 = 1, then value of x is a. 1 b. -1 c. ±1

6 49. The amount of annuity consisting of payment of Rs. 500 at the end of every 3 months for 2 years at the rate of 6% compounded quarterly is a b c d The present value of Rs due at the end of 4 years if the discounting rate is 4% is Rs. a. 855 b. 860 c. 865 d Partial fraction of a. c. + () () - () () is ()() 52. Partial fraction of is a c If, then x is a. 5 b. -5 c. 0 d. 4 b. - () () b. - d If sum of two numbers is 30 and their product is 176, then one of the number is a. 7 b. 8 c. 9 d The roots of the quadratic equation x 2 + (7 + p)x + p = 0 are α and β. If (α β) 2 is a. p p 49 b. p p + 49 c. p 2 10p + 49 d. p 2 10p The quadratic equation x2 + kx + 2 = 0 has roots α and β. If α 2 + β 2 = 5, then value of k is a. 3 b. -3 c. ±3 57. The cubic equation x 3 11x 150 = 0 has roots α, β and γ. Then α + β + γ is a. 0 b. 11 c. 150 d. 150/ If x+ 4 [2 + {x (2 + x)}] =, then, then x is

7 a. 7/2 b. -7/2 c. 7 d For what values of k the equation 9x 2 + kx + 1 = 0 has real and equal roots a. 6 b. -6 c. Both a and b 60. If one root of the quadratic equation ax2 + bx + c = 0 is two times the other, then a. 2b 2 = 9ac b. 2b = 9ac c. 2b = 9ac 2 d. b 2 = 9ac 61. If 3x x + 7 = 9, then x is a. 2, 52 b. 2, 53 c. 2, 54 d. 1, If 4 x -5.2 x + 4 = 0, then one of the root is a. 1 b. 2 c. 3 d If k = 2 is the root of the equation, x 3 (k + 1) x + k = 0, then k is a. 6 b. -6 c. ±6 64. If A and B are disjoint sets, then AxB and BxA are a. Equal b. Disjoint c. Empty 65. If A = {x : x is multiple of 2, x N} and B = {x : x is multiple of 5, x N}, then A B is a. {x : x is multiple of 2, x N} b. {x : x is multiple of 5, x N} c. {x : x is multiple of 10, x N} d. 66. The 15 th term from the end of the series 2, 6, is a. 28 b. 30 c. 32 d The sum of the series 1,,, is a. 1 b. 2 c. 3 d Which function is not a transcendental function a. y = x + sin x b. y = x sin x c. y = x + x 2 + x α d. y = x + x 2 + x 3 +.x n

8 69. The salary of an employee in 2000 was Rs and in 2005 was Rs What will be his salary in 2007? a b c d If in a group of 400 people, 250 can speak English only and 70 can speak Hindi only. How many can speak both English and Hindi only. a. 70 b. 80 c. 90 d If log SECTION B, and log are in AP, where a, b, c are in GP, then a, b, c are the length of sides of a. an isosceles triangle b. an equilateral triangle c. a scalene triangle 72. If the p th, q th and r th terms of an AP are in GP then the common ratio of GP is a. b. c. 73. If a, b, c, d are four numbers such that the first three are in AP while the last three are in HP then a. bc=ad b. ac=cd c. ab = cd 74. If S r denotes the sum of the first r terms of an AP then a. 2r 1 b. 2r + 1 c. 4r + 1 d. 2r If a, b, c are in AP then a +, b +,c + are in a. AP b. GP c. HP 76. If - < < then the minimum value of cos3 + sec 3 is a. 1 b. 2 is equal to

9 c Let S n denote the sum of the cubes of the first n natural numbers and S n denotes the sum of the first n natural numbers. Then a. ()() c. is equal to 78. If a, b, c are in HP then + is equal to a. b. (). b. c. + d. Both (a) and (b) 79. The number of real solutions of x - =2 - is a. 0 b. 1 c The number of real solutions of the equation e x = x is a. 1 b. 2 c The number of real solutions of the equation log 0.5 x = x is a. 1 b. 2 c The equation x + 1 x 1 = 4x + 1 has a. No solution b. One solution c. Two solutions d. More than two solutions 83. The number of solutions of the equation x = cos x is a. One b. Two c. Three d. Zero

10 84. If 3 x+1 = 6log 2 3 then x is a. 3 b. 2 c. log 3 2 d. log If ( 2) x + ( 3) x = (13) x/2 then the number of values of x is a. 2 b. 4 c The number of real solutions of the equations a. Two b. One = 2 + is c. Three 87. The number of real solutions of x 4x x 9 = 4x 14x + 6 is a. One b. Two c. Three 88. If x 2 = x + 2, where x = the greatest integer less than or equal to x, then x, must be such that a. x = 2-1 b. x (2,3) c. x (-1,0) 89. The number of solutions of x - 2x = 4, where x is the greatest integer x, is a. 2 b. 4 c. 1 d. Infinite 90. The set of real values of x satisfying x -1 3 and - 1 x, i a. [2, 4] b. [ - ( -α, 2] 3, + α) c. [-2, 0] [2, 4] 91. The solution set of > 1, x R, is a. (3, +α) b. (-1, 1) 3, +α) c. [-1, 1] 3, +α)

11 92. The number of integral solutions of > is a. 4 b. 5 c The value of the sum (i n + 1 n + 1 ) where I = 1 is a. i b. i-1 c. I d If a + I = α + iβ then b + i 5 is equal to a. β + iα b. α - iβ c. β - iα d. - α iβ 95. If ω is a non real cube rot of unity then the expression 1 ω) 1 ω 2 ) 1- ω 4 ) 1 + ω 8 ) is equal to a. 0 b. 3 c. 1 d If x 2 x + 1 = 0, then the value of (x + )2 is a. 8 b. 10 c If z = 1, then is equal to a. z b.ż c. z + ż 98. = 1 represents a. A circle b. An ellipse c. A straight line a + x a x 99. If a x a x = 0 then x is a x a x a. 0 b. a

12 c. 3 d. 2a 0 p q p q 100. q p 0 q r is equal to r p r q 0 a. p + q +r b. 0 c. p q r d. p + q + r bc ca ab 101. The value of the determinant p q r, where a, b, c are the p th, q th, and r th terms of a HP, is ap + bq + cr a. a + b + b. p + q + r c. 0 a x + b y a x + b y a x + b y 102.The value of b x + a y b y + a y b y + a y is equal to b x + a b x + a b x + a a. x 2 + y 2 b. 0 c. a 1 a 2 a 3 x 2 + b 1 b 2 b 3 y If α, β are non-real numbers satisfying x 3 1 =0 then the value of a. 0 b. λ 3 c. λ λ + 1 α β α λ + β 1 is equal to β 1 λ + α 1 ω 1 + i + ω 104. If 1 = i, and ω is a non-real cube root of unity then the value of i 1 1 i + ω is equal to 1 i ω 1 1 a. 1 b. i c. ω d The system of equations ax + 4y + z = 0, bx + 3y +z = 0, cx + 2y + z = 0 has nontrivial solutions if a, b, c are in a. AP b. GP c. HP 106. The equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + mz = n give infinite number of values of the triplet x, y, z if

13 a. m = 3, n R b. m = 3, n 10 c. m = 3, n = The system of equations 2x +3y = 8, 7x 5y + 3 = 0, 4x 6y + λ = 0 is solvable if λ is a. 6 b. 8 c. -8 d Indefinite Integra if (x) = cot 4 xdx + cot3 x cot x and = then (x) is a. π x b. x - π c. - x 109. x x (1 + log x)dx is equal to a. x x logx + k b. e xx + k c. x x + k 110. is equal to a. tan -1 x 2 + k b. tan-1 x 2 + k c. log(1 + x 4 ) + k 111. / dx is a. log(x7/2 + x + 1) + c b. log + c c x ) + c 112. The primitive of the function x coz x when < x < π is given by a. cos x + x sin x b. cos x + x sin x c. xsinx cosx 113. e -x (1 tan x) see xdx is equal to a. e -x sec + c b. e-x tan x + c c. e -x tan x + c

14 114. is equal to a. log tan + + k b. log tan + k c. log tan + + k 115. In the expansion of (2 x) -5/2, the coefficient of x 4, if it exists, is a.... /.! 4 b....! 4 c The coefficient of x 5 in the expansion of a. -1 b. 2 c. 0 d The coefficient of x n in the expansion of x 2x + 3 is a.! b..! c. a p = a q, x < 1, is 118. The coefficient of x 10 in the expansion of 10 x in ascending powers of x is a. ( )! b.! c. ( )! 119. The constant term in the expansion of is a. log e 3 b. log e 6. log e c. log e6. log e 120. If x <1, the coefficient of x 3 in the expansion of.() is a. c. b..

15 121. If x <1, the coefficient of x 2 in the expansion of () is () a. b. 2 C 1 c If x to α = y then y to α is equal to!! a. x b. x c. x + 1 λ The matrix is a singular matrix if λ is a. b. c If A = λ 3 then A -1 exists i.e., A is invertible if a. λ 4 b. λ 8 c. λ = The rank of the matrix a. λ c. λ = 126. The value of a. 10 c λ 2 4 is 3 if b. λ = 127. The square root of 2x + x 1 is equal to b.. is

16 a. x + 1 x 1 b. x x 1 c. x 1 x If x =, y = then the value of x2 + xy + y = is a. 5 b. 99 c If y = cos -1 cosx) then at x = a. 1 b. -1 is equal to c If x = e y + e y +.. to α then is a. c. b If y = tan -1 then is equal to a. c. b If x y.y x = 16 then at x = e is a. 1 b. 0 c The derivative of tan -1 with respect to tan -1 x is a. b. 1 c.

17 x sin x cos x 134. Let f(x) = 6 1 0, where p is a constant. Then p p p {f (x)} at x = 0 is a. p b. p + p 2 c. p + p 3 d. Independent of p 135. If y = at 2, x = 2at, where a is a constant, then at x = is a. b. 1 c. 2a 136. Let A and B be two sets such that A B = A. Then A B is equal to a. b. B c. A 137. Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic team is a. 43 b. 76 c The relations congruence modulo m its a. Reflexive only b. Transitive only c. Symmetric only d. An equivalence relation 139. Let f : R R such that fx =, x a. Infective b. Surjetive R. Then f is c. Bijective 140. Let R be the relation over the set of integers such that m Rn if and only if m is a multiple of n. Then R is a. Reflexive b. Symmetric c. Transitive d. An Equivalence relation

Objective Mathematics

Objective Mathematics Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given