ALGEBRA
1. F is a quadratic function whose graph is a parabola opening upward and has a
vertex on the x-axis. The graph of the new function g defined by g(x) = 2 - f(x - 5)
has a range defined by the interval
A. [ -5 , + infinity)
B. [ 2 , + infinity)
C. ( - infinity , 2]
D. ( - infinity , 0]
2. F is a function such that f(x) < 0. The graph of the new function g defined by g(x)
= | f(x) | is a reflection of the graph of F?
A. on the y axis
B. on the x axis
C. on the line y = x
D. on the line y = - x
3. The graphs of the two linear equations a x + b y = c and b x - a y = c, where a, b
and c are all not equal to zero,
A. are parallel
B. intersect at one point
C. intersect at two points
D. perpendicular
4. The graphs of the two equations y = a x 2 + b x + c and y = A x 2 + B x + C, such
that a and A have different signs and that the quantities b 2 - 4 a c and B 2 - 4 A C
are both negative,
A. intersect at two points
B. intersect at one point
C. do not intersect
D. none of the above
5. For x greater than or equal to zero and less than or equal to 2 π, sin x and cos x
are both decreasing on the intervals
A. (0 , π/2)
B. (π/2 , π)
C. (π , 3 π / 2)
D. (3 π / 2 , 2 π)
6. If f(x) is an odd function, then | f(x) | is
A. an odd function
B. an even function
C. neither odd nor even
D. even and odd
7. The period of | sin (3x) | is
A. 2 π
B. 2 π / 3
C. π / 3
D. 3 π
8. The period of 2 sin x cos x is
A. 4 π 2
B. 2 π
C. 4 π
D. π
9. The set of all real numbers under the usual multiplication operation is not a group
since?
A. multiplication is not a binary operation
B. multiplication is not associative
C. identity element does not exist
D. zero has no inverse
10. If (G, .) is a group such that (ab)- 1 = a-1b-1, ∀ a, b ∈ G, then G is a/an
A. commutative semi group
B. abelian group
C. non-abelian group
D. None of these
11. If (G, .) is a group such that a2 = e, ∀a ∈ G, then G is
A. semi group
B. abelian group
C. non-abelian group
D. None of these
12. Let G denoted the set of all n x n non-singular matrices with rational numbers as
entries. Then under multiplication G is a/an
A. subgroup
B. finite abelian group
C. infinite, non-abelian group
D. infinite, abelian
13. Let A be the set of all non-singular matrices over real numbers and let * be the
matrix multiplication operator. Then
A. A is closed under * but < A, * > is not a semi group
B. < A, * > is a semi group but not a monoid
C. < A, * > is a monoid but not a group
D. < A, * > is a group but not an abelian group
14. Which of the following is TRUE?
A. Set of all rational negative numbers forms a group under multiplication
B. Set of all non-singular matrices forms a group under multiplication
C. Set of all matrices forms a group under multiplication
D. Both (b) and (c)
15. The set of all nth roots of unity under multiplication of complex numbers form
a/an?
A. semi group with identity
B. commutative semigroups with identity
C. group
D. abelian group
16. Which of the following statements is FALSE?
A. The set of rational numbers is an abelian group under addition
B. The set of rational integers is an abelian group under addition
C. The set of rational numbers form an abelian group under multiplication
D. None of these
(𝑥 2 +19)(𝑥−4)
17. The function f (x) =
has
𝑥 2 −81
A. two vertical asymptotes; one oblique asymptote.
B. one vertical asymptote; one horizontal asymptote.
C. one vertical asymptote; no horizontal asymptote.
D. no vertical asymptotes; one oblique asymptote.
18. What is not true of the graph of y = f (x) below?
A. This is not a one-to-one function
B. This function has no real roots.
C. The domain of the function has real roots;
D. f (2) is negative.
19. Let (Z, *) be an algebraic structure, where Z is the set of integers and the
operation * is defined by n * m = maximum (n, m). Which of the following
statements is TRUE for (Z, *)?
A. (Z, *) is a monoid
B. (Z, *) is an abelian group
C. (Z, *) is a group
D. None of these
20. Some group (G, 0) is known to be abelian. Then which one of the following is
TRUE for G?
A. g = g-1 for every g ∈ G
B. g = g2 for every g ∈ G
C. (g o h) 2 = g2o h2 for every g,h ∈ G
D. G is of finite order
21. If A = (1, 2, 3, 4). Let ~= {(1, 2), (1, 3), (4, 2)}. Then ~ is
A. not anti-symmetric
B. transitive
C. reflexive
D. symmetric
22. Which of the following statements is false?
A. If R is relexive, then R ∩ R-1 ≠ φ
B. R ∩ R-1 ≠ φ =>R is anti-symmetric.
C. If R, R' are equivalence relations in a set A, then R ∩ R’ also an equivalence
relation in A.
D. If R, R' are relexive relations in A, then R - R' is reflexive.
2
2
23. Without graphing, how many solutions does the linear system y = 7 𝑥 + 8 and y =
6
𝑥 − 9 have?
A. one
B. two
C. no solution
D. infinitely many
16
24. Decide whether the equation is an identity, a conditional equation, or a
contradiction.
A. identity
B. contradiction
C. conditional equation
D. none of these
25. The part of mathematics in which letters and other general symbols are used to
represent numbers and quantities in formulae and equations.
A. algebra
B. geometry
C. trigonometry
D. analytic
26. One where the variable(s) are multiplied by numbers or added to numbers, with
nothing more complicated than that (no exponents, square roots, 1x, or any other
funny business).
A. number equation
B. complicated equation
C. exponential equation
D. linear equation
27. If the first derivative a function is unchanged, then the function is?
A. linear
B. exponential
C. quadratic
D. sinusoidal
28. A ________ is defined as an idea formed from something that is already proved.
A. an axiom
B. a postulate
C. a theorem
D. a corollary
29. A statement that truth of which is admitted without proof is called.
A. an axiom
B. a postulate
C. a theorem
D. a corollary
30. It is the characteristic of a population which is measurable.
A. frequency
B. distribution
C. sample
D. parameter
31. In complex algebra, we use this diagram to represent a complex plane
commonly called.
A. De Moivre ’ s Diagram
B. Funicular Diagram
C. Argand Diagram
D. Venn Diagram
32. A series of numbers which are perfect square numbers (i.e 1, 4, 9, 16, ...) is
called.
A. Fourier series
B. Fermat ‘s number
C. Euler’ s number
D. Fibonacci numbers
33. A sequence of numbers where every term is obtained b adding all the preceding
terms such as (1. 5. 14. 30, …) is called.
A. triangular number
B. pyramidal number
C. tetrahedral number
D. Euler’s number
34. In a proportion of four quantities, the first and the fourth term are referred to as.
A. means
B. consequent
C. extremes
D. discriminants
35. In raw data, the term, which occurs most frequently is know us.
A. mean
B. median
C. mode
D. quartile
36. The number 0,13123123 …. is
A. irrational
B. surd
C. rational
D. transcendental
37. The graphical representation of the cumulative frequency is set of statistical data
is called.
A. ogive
B. histogram
C. frequency
D. mass diagram
38. Convergent series is a sequence of dressing umbers or when the succeeding
term is ______ than the preceding term.
A. ten time more
B. greater
C. equal
D. lesser
39. The characteristics is equal to the exponents of 10, when the number is written
in:
A. exponential
B. scientific notation
C. logarithm
D. irrational
40. Terms that differ only in numeric coefficients are known as:
A. unequal terms
B. unlike terms
C. like terms
D. equal terms
41. ______ is a sequence of terms whose reciprocals form as arithmetic
progression.
A. geometric
B. harmonic
C. algebraic
D. ratio and proportion
42. A sequence of numbers where the succeeding term is greater than the
preceding term is called.
A. dissonant series
B. convergent series
C. isometric series
D. divergent series
43. The logarithm of a number to the base e (2..4524525) is called:
A. Naperian logarithm
B. Characteristic
C. Mantissa
D. Briggsian logarithm
44. The ratio or product of two expressions in direct or inverse relation with each
other is called:
A. ratio and proportion
B. constant of variation
C. means
D. extremes
45. In any square matrix, when the elements of ay two rows are exactly the same
determinant is:
A. zero
B. positive integers
C. negative integers
D. unity
46. A radical expressing an irrational number.
A. surd
B. pure surd
C. mixed surd
D. radical
47. Any permutation ⌅ of {1, 2, . . ., n} is a product of disjoint cycles. The individual
cycles in the decomposition are unique in the sense of being determined by ℴ.
A. permutation
B. Corollary
C. proposition
D. lemma
48. Let ℴ be a permutation of {1, . . ., n}, let (a b) be a transposition, and form the
product ℴ (a b).
A. permutation
B. Corollary
C. proposition
D. lemma
49. This section and the next review row reduction and matrix algebra for rational,
real, and complex matrices. As in Section 3 let F denote Q or R or C. The
members of F are called:
A. variables
B. Scalars
C. row
D. none of these
50. The other x j ’s with j < n will be called:
A. corner variables
B. dependent variables
C. independent variables
D. both B and C
51. A rectangular array of scalars with k rows and n columns is called
A. scalars
B. matrix
C. entry
D. square
52. A square matrix with all entries 0 for i " ≠ j is called:
A. entries
B. diagonal
C. not equal
D. equal
53. The set of all elements a (u, v) for this choice of coset representatives is called a
A. set
B. factor
C. factor set
D. all of the above
𝑓
𝑓(𝑥)
54. The function defined by the formula 𝑔 (𝑥) = 𝑔(𝑥)
A. division
B. quotient
C. both A and B
D. none of these
55. Refers to a change based on a percent of the original amount.
A. Exponential change
B. Exponential decay
C. Percent change
D. rate of change
56. Refers to the original value from the range increases by the same percentage
over equal increments found in the domain.
A. Linear growth
B. Exponential growth
C. Percent change
D. rate of change
57. The process of solving for a variable and substituting in the other can sometimes
be done more efficiently by manipulating the whole equations.
A. Elimination Method
B. Determinant Method
C. Cramer’s Rule
D. Exceptions to Cramer’s Rule
58. There is an even more efficient calculation for two equations in _____ variables
using what is known as the Cramer’s Rule.
A. one
B. two
C. three
D. four
59. Let f(x) = x2 + 2x + 4. Which of the following statements is NOT true?
A. f(x) has a maximum value
B. The graph of f is not a line
C. The graph of f has no x-intercepts.
D. The graph of f has a y-intercept.
60. If f(x) = -2x2 + 8x - 4, which of the following is true?
A. The maximum value of f(x) is - 4.
B. The graph of f opens upward.
C. The graph of f has no x-intercept
D. f is not a one to one function
61. The system of linear equations 2x+2y-3z=1, 4x+4y+z=2, 6x+6y-z=3 has?
A. a unique solution
B. infinite solutions
C. no solution
D. two solutions
62. Following are two statements:
(i) Two finite-dimensional vector spaces over the same field are isomorphic.
(ii) Two finite-dimensional vector spaces over the same field and of the same dimension
are isomorphic.
A. i is true but ii is not true.
B. ii is true, but i is not true.
C. None of them is true
D. All of them are true.
63. Following are three statements:
(i) Any n-dimensional real vector space is isomorphic to R^n.
(ii) Any n-dimensional complex vector space is isomorphic to C^n.
(iii) Any n-dimensional vector space over the field F is isomorphic to F^n.
A. Only i and ii are true.
B. i is true, but ii and iii are not true.
C. None of them is true
D. All of them are true.
64. “Mathematical Expectation of the product of two random variables is equal to the product
of their expectations” is true for
A. any two random variables.
B. if the random variables are independent.
C. if the covariance between the random variables is non zero.
D. if the variance of the random variables is equal.
65. Kinematics is concerned with
A. the physical causes of the motion.
B. the condition under which no motion is apparent
C. the geometry of the motion
D. none of these
66. The sequence {\frac {1}{2}, \frac {2}{3}, \frac {3}{4}, \dots \frac{n}{n+1}} is
A. monotonically increasing
B. increasing and bounded
C. non-increasing and bounded
D. non-increasing, but not bounded
67. An integer greater than 1 that is not a prime is termed
A. Even number
B. Odd number
C. Composite number
D. none of these\
68. The number √2 is
A. Rational
B. Irrational
C. Real number
D. Not a real number
69. 509 is :
A. composite
B. prime
C. even
D. odd
70. If a ≡ b mod n , then
A. a and b leave the same nonnegative remainder when divided by n.
B. a and b leave the different nonnegative remainder when divided by n.
C. and b need not leave the same nonnegative remainder when divided by n.
D. none of these
71. If gcd (a, n) = 1, then the congruence ax ≡ b mod n has
A. Infinitely many solutions modulo n
B. Unique solution modulo n
C. More than one solution modulo n
D. none of these
72. The linear congruence 18x ≡ 6 mod 3 has
A. Infinitely many solutions modulo 3
B. Unique solution modulo 3
C. Exactly 3 solution modulo 3
D. exactly 6
73. σ n = n + 1 if and only if
A. n is an odd number
B. n is an even number
C. n is a prime number
D. all of the above
74. The Euler’s Phi- function is:
A. Multiplicative
B. Not Multiplicative
C. Injective
D. Prime number
75. A number-theoretic function f is said to be multiplicative if f m, n = f m f(n) :
A. For all integers m and n
B. For all positive integers m and n
C. For all relatively prime integers.
D. Whenever gcd m, n = 1
76. A number-theoretic function 𝑓 is said to be multiplicative if 𝑓(𝑚, 𝑛) = 𝑓(𝑚)𝑓(𝑛) :
A. For all integers m and n
B. For all positive integers m and n
C. For all relatively prime integers.
D. Whenever gcd m, n = 1
77. If p is a prime, then
A. p − 1 ! ≡ 1 mod p
B. p − 1 ! ≡ −1 mod p
C. p − 1 ! ≡ 0 mod p
D. none of these
78. For any choice of positive integers, a and b, lcm (a, b) = ab if and only if
A. gcd(a, b) = a
B. gcd(a, b) = ab
C. gcd a, b = a + b
D. gcd a, b = a – b
79. If a and b are given integers, not both zero, then the set T= {ax + by ∶ x, y are
integers} contains:
A. Multiples of d
B. Divisors of d
C. Divisors of a and b
D. none of these
80. Let g c d a, b = d. If c|a and c|b, then
A. c ≤ d
B. c ≥ d
C. c = 1
D. both A and B
81. If the determinant of a matrix A is non zero, then its eigenvalues of A are:
A. 1
B. 0
C. non zero
D. none of these
82. pqr is what type of polynomial?
A. Monomial
B. Binomial
C. Trinomial
D. None of these
83. Evaluate the limit (x – 4)/(x^2 – x – 12) as x approaches 4.
A. undefined
B. 0
C. infinity
D. 1/7
84. What is the limit of cos (1/y) as y approaches infinity?
A. 0
B. -1
C. infinity
D. 1
85. Find the derivative of log a u with respect to x.
A. log u du/dx
B. u du/ln a
C. log a e/u
D. Log a du/dx
86. Logarithms to base e are given the special name of.
A. natural logarithms
B. exponential logarithms
C. logarithms
D. none of these
87. A nonzero algebra A with identity over a field F will be called.
A. center
B. central
C. variable
D. constant
88. In _______ one studies sets of linear equations and their transformation
properties.
A. linear algebra
B. linear algebraic
C. linear equation
D. none of these
89. __________ is one of the divisions in algebra which discovers the truths relating
to algebraic systems independent of specific nature of some operations.
A. Abstract algebra
B. Advanced algebra
C. Algebra
D. Algebraic
90. ______ is defined as the collection of the objects that are determined by some
specific property for a set.
A. Sets
B. Binary Operations
C. Identity Element
D. Inverse Elements
91. When the concept of addition is conceptualized, it gives the binary operations.
The concept of all the binary operations will be meaningless without a set:
A. Sets
B. Binary Operations
C. Identity Element
D. Inverse Elements
92. The idea of Inverse elements comes up with the negative number. For Addition,
we write -an as the inverse of a and for the purpose of multiplication the inverse
form is written as a−1.
A. Sets
B. Binary Operations
C. Identity Element
D. Inverse Elements
93. The numbers 0 and 1 are conceptualized to give the idea of an identity element
for a specific operation.
A. Sets
B. Binary Operations
C. Identity Element
D. Inverse Elements
94. ____________ is one of the branches of algebra that studies the commutative
rings and its ideals.
A. Commutative algebra
B. Linear algebra
C. Algebraic geometry
D. none of these
95. The basic algebra rules are mentioned below except:
A. The Symmetry rule
B. The commutative rules
C. The inverse of adding
D. The linear rules
96. The slope of a vertical line is undefined.
A. True
B. False
C. Both A and B
D. 0
97. Two lines with positive slopes can be perpendicular.
A. True
B. False
C. Both A and B
D. 0
98. The absolute value of a real negative number is negative.
A. True
B. False
C. Both A and B
D. 0
99. 1.5 × 10-5 is the scientific notation of the number 0.0000015.
A. True
B. False
C. Both A and B
D. 0
100.
"x is at most equal to 9" is written mathematically as x < 9.
A. True
B. False
C. Both A and B
D. 0