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Question.txt
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Which of the following is the set of odd positive integers less than 4?
The cardinality of the set {1, 2, 2, 3, 3, 3} is
Which of the following is a subset of {a, b, c}?
Let A = {a, b, c, d, . . . , x, y, z} be the set of letters of the English alphabet. Consider the following subsets of A: X = {* : * ∈ A is a letter in the word 'mathematics'}, Y = {* : * ∈ A is a letter in the word 'statistics'}. Find |X|
Let A = {a, b, c, d, . . . , x, y, z} be the set of letters of the English alphabet. Consider the following subsets of A: X = {* : * ∈ A is a letter in the word 'mathematics'}, Y = {* : * ∈ A is a letter in the word 'statistics'}. Find |X∪Y|
Let A = {a, b, c, d, . . . , x, y, z} be the set of letters of the English alphabet. Consider the following subsets of A: X = {* : * ∈ A is a letter in the word 'mathematics'}, Y = {* : * ∈ A is a letter in the word 'statistics'}. Find |X∩Y|
Let A = {a, b, c, d, . . . , x, y, z} be the set of letters of the English alphabet. Consider the following subsets of A: X = {* : * ∈ A is a letter in the word 'mathematics'}, Y = {* : * ∈ A is a letter in the word 'statistics'}. Find |X−Y|
Which of the following denotes the set {x ∈ ℝ : 3 ≤ x ≤ 5}?
Which of the following denotes the set {x ∈ ℝ : x ≤ 3 and x ≥ 5}?
Which of the following denotes the set {x ∈ ℝ : 3 < x < 5}?
Which of the following denotes the set {x ∈ ℝ : 3 < x ≤ 5}?
Which of the following denotes the set {x ∈ ℝ : 3 < x}?
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of A ∪ B ∪ C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of A ∩ B ∩ C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of (A ∪ B) ∩ C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of (A ∩ B) ∪ C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of (A ∪ B) − C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of (A − B) ∪ C is
Suppose A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}. The cardinality of (A∆B)∆C is
Let A = {a, b} and B = {a, c}. Which of the following is an element of A × B?
Let A = {a, b} and B = {a, c}. Which of the following is a subset of A × B?
Let A = {a, b} and B = {a, c}. The cardinality of (A × B) ∪ (B × A) is
Let A = {a, b} and B = {a, c}. The cardinality of (A × B) ∩ (B × A) is
Let A = {a, b} and B = {a, c}. The cardinality of (A × B) − (B × A) is
The statement A ∩ B ⊆ A ∪ C is
The statement A ∩ (B ∪ C) ⊆ A ∪ C is
A − (B ∪ C) =
A − (B ∩ C) =
Choose the correct option to fill in the blank that results in a true statement. A ∪ (B ∪ C) ___ (A ∪ B) ∪ C
Choose the correct option to fill in the blank that results in a true statement. A ∪ (B ∩ C) ___ (A ∪ B) ∩ C
Choose the correct option to fill in the blank that results in a true statement. A ∩ (B ∪ C) ___ (A ∩ B) ∪ C
Choose the correct option to fill in the blank that results in a true statement. A − (B − C) ___ (A ∩ B) ∩ C
(A ∪ B) − B =
((A ∩ B) − C) ∩ ((A ∩ C) − B) =
Let f : {a, b, c, d} → {a, b, c, d} be the function given by f(a) = b, f(b) = d, f(c) = a, f(d) = b. The image of f is
Let f : {a, b, c, d} → {a, b, c, d} be the function given by f(a) = b, f(b) = d, f(c) = a, f(d) = b. f^(-1)({a, c}) =
Let f : {a, b, c, d} → {a, b, c, d} be the function given by f(a) = b, f(b) = d, f(c) = a, f(d) = b. (f ◦ f ◦ f ◦ f)(a) =
Let f, g, h : ℝ → ℝ be the functions given by f(x) = sin(x), g(x) = πx, h(x) = x^(100). Then(f ◦ g ◦ h)(√2) =
Let f : A → B and g : B → C be two functions. If the composite g ◦ f : A → C is surjective then
Let f : A → B and g : B → C be two functions. If the composite g ◦ f : A → C is injective then
Given f : ℝ → ℝ such that the solutions to f(x) = 0 are 1 and −2. Then solutions tof(x/2)= 0 are
Functions f, g : ℝ → ℝ satisfy f(x) = ln(x^2), f(g(x)) = ln((x^2) + 1), and g(x) < 0 for all x ∈ ℝ. Then g(x) =
Let f, g : ℝ → ℝ be the functions defined by f(x) = 2x + 1 and g(x) = x^2. Then the set of real numbers where f◦g and g◦f have the same image is
If f : ℝ → ℝ is defined by f(x) = (x^5) + 7 and g : ℝ → ℝ is its inverse, then g(x) =
How many functions from {a, b, c} to {1, 2, 3, 4} are there?
How many injective functions from {a, b, c} to {1, 2, 3, 4, 5, 6} are there?
How many surjective functions from {a, b, c} to {1, 2, 3, 4, 5, 6} are there?
How many subsets of {1, 2, 3, 4, 5, 6, 7, 8} are there?
Let A = {1, 2, 3, 4, 5, 6}. What is the cardinality of {(x, y, z) ∈ (A^3): x, y, z are all even}?
A bookshelf contains 4 Algebra books, 5 Calculus books and 3 Statistics books. In how many ways can 3 books be chosen so that there is one of each type?
A bookshelf contains 4 Algebra books, 5 Calculus books and 3 Statistics books. In how many ways can 4 books be chosen so that there are two Algebra books and one each of Calculus and Statistics?
A bookshelf contains 4 Algebra books, 5 Calculus books and 3 Statistics books. In how many ways can 4 books be chosen so that there is at least one of each type?
How many anagrams of MATHS are there?
How many anagrams of SHEFFIELD are there?
How many anagrams of ABRACADABRA are there?
How many three letter words can be formed using only P, Q, R, S, T, X?
How many three letter words can be formed using only P, Q, R, S, T, X with no letter repeated?
How many six letter words can be formed using A, B, C, D, E, F with no letter repeated and the vowels in alphabetical order?
An exam paper has 13 questions out of which 10 are to be attempted including at least 4 from the first 5 questions. What is the number of choices available?
For which n ≥ 2 does (n + 1)C3 = 2* nC2 hold?
Let T_(n) be the number of triangles that can be formed using n given points on a circle as vertices. If T_(n+1) − T_(n) = 21 then n =
If x ≠ 1 then 1 + x + (x^2) + (x^3) =
If x ≠ −1 then 1 − x + (x^2) − (x^3) =
If x ≠ 1 then (x^3) + (x^4) + (x^5) =
If r ≠ ±1 then (r^4) + (r^6) + (r^8) + (r^10) =
(3^2) + (3^3) + · · · + (3^20) =
100 + (100 + 4) + (100 + 8) + . . .(100 + 4 · 99) =
For each positive integer n, let P(n) be the statement “1 + 3 + · · · + (2n − 1) = (n^2)”. What is P(1)?
For each positive integer n, let P(n) be the statement “1 + 3 + · · · + (2n − 1) = (n^2)”. Assume that 1 + 3 + · · · + (2k − 1) = (k^2) is true i.e. P(k) is true. Which of the following options continues a valid inductive argument?
An arithmetic progression has a, b, c as three consecutive terms with b ≠ 0. Then (a + c) / b =
Suppose a, b, c, d are four consecutive terms of an arithmetic progression with (b^2) ≠ (c^2). What is the value of [(d^2) − (a^2)]/[(c^2) − (b^2)]?
If 1, x, y, 2 are in geometric progression then xy =
sin(x + π/2) =
cos(x + π/2) =
sin(x - π/2) =
cos(x - π/2) =
sin(x + π) =
sin(x - π) =
cos(x + π) =
cos(x - π) =
sin(π/2 - x) =
sin(π - x) =
cos(π/2 - x) =
cos(π - x) =
sin(x + 3π/2) =
cos(x + 3π/2) =
sin(x - 3π/2) =
cos(x - 3π/2) =
tan(x + π/2) =
cot(x + π/2) =
tan(x - π/2) =
cot(x - π/2) =
tan(x + π) =
tan(x - π) =
cot(x + π) =
cot(x - π) =
tan(π/2 - x) =
tan(π - x) =
cot(π/2 - x) =
cot(π - x) =
tan(x + 3π/2) =
cot(x + 3π/2) =
tan(x - 3π/2) =
cot(x - 3π/2) =
Choose the correct value without using a calculator. sin(7π/6)
Choose the correct value without using a calculator. sin(5π/6)
Choose the correct value without using a calculator. sin(7π/3)
Choose the correct value without using a calculator. sin(11π/6)
Choose the correct value without using a calculator. sin(11π/3)
Choose the correct value without using a calculator. cos(7π/6)
Choose the correct value without using a calculator. cos(5π/6)
Choose the correct value without using a calculator. cos(7π/3)
Choose the correct value without using a calculator. cos(11π/6)
Choose the correct value without using a calculator. cos(11π/3)
Choose the correct value without using a calculator. tan(7π/6)
Choose the correct value without using a calculator. tan(5π/6)
Choose the correct value without using a calculator. tan(7π/3)
Choose the correct value without using a calculator. tan(11π/6)
Choose the correct value without using a calculator. tan(11π/3)
Choose the correct value without using a calculator. sin(-7π/6)
Choose the correct value without using a calculator. sin(-5π/6)
Choose the correct value without using a calculator. sin(-7π/3)
Choose the correct value without using a calculator. sin(-11π/6)
Choose the correct value without using a calculator. sin(-11π/3)
Choose the correct value without using a calculator. cos(-7π/6)
Choose the correct value without using a calculator. cos(-5π/6)
Choose the correct value without using a calculator. cos(-7π/3)
Choose the correct value without using a calculator. cos(-11π/6)
Choose the correct value without using a calculator. cos(-11π/3)
Choose the correct value without using a calculator. tan(-7π/6)
Choose the correct value without using a calculator. tan(-5π/6)
Choose the correct value without using a calculator. tan(-7π/3)
Choose the correct value without using a calculator. tan(-11π/6)
Choose the correct value without using a calculator. tan(-11π/3)
If sin θ = 3/5 and π/2 < θ < π then cos θ =
If sin θ = 3/5 and π/2 < θ < π then cos(π + θ) =
If sin θ = 3/5 and π/2 < θ < π then cos(π/2 + θ) =
If sin θ = 3/5 and π/2 < θ < π then sin(2θ) =
sin x(cot x + tan x) =
(sec x + 1)(sec x − 1) =
csc x − cos^2 x csc x =
(1/[1-sin^2(x)]) - 1 =
([cos^2(x) - tan^2(x)]/[sin^2(x)]) + sec^2(x) =
(1 − tan θ)(1 − cot θ) + sec θ csc θ =
sin^2(α)cot^2(α) + cos^2(α)tan^2(α) =
([sin(2θ)]/[1 + cos(2θ)]) =
([cos(x + y)]/[cos(x)sin(y)]) =
csc θ − tan(θ/2) =
sec θ − tan θ tan(θ/2) =
([1 + tan(x)]/[1 - tan(x)]) =
[cos(x/2) - sin(x/2)]^2 =
[cos(x/2) + sin(x/2)]^2 =
([sin(A + B) + sin(A - B)]/[cos(A + B) + cos(A - B)]) =
If tan^−1(m) = x and tan^−1(n) = y then tan(x + y) =
If tan α = ([√3]/[4 − √3]) and tan β = ([√3]/[4 + √3]) then tan(α − β) =
The exact value of sin(π/12) is
The exact value of sin(7π/12) is
Suppose sin x = 3/5 and sin y = 12/13, and that x is in the first quadrant while y is in the second quadrant. sin(x + y)
Suppose sin x = 3/5 and sin y = 12/13, and that x is in the first quadrant while y is in the second quadrant. cos(x + y)
Suppose sin x = 3/5 and sin y = 12/13, and that x is in the first quadrant while y is in the second quadrant. tan(x - y)
Suppose sin α = 1/√10 and cos β = 3/5. If 0 < α, β < π/2 then tan(α + β) =
Suppose sin α = 1/√10 and cos β = 3/5. If 0 < α < π/2 and 3π/2 < β < 2π then tan(α + β) =
Suppose sin α = 1/√10 and cos β = 3/5. If π/2 < α < π and 0 < β < π/2 then tan(α + β) =
Suppose sin α = 1/√10 and cos β = 3/5. If π/2 < α < π and 3π/2 < β < 2π then tan(α + β) =