Let A = {1, 2, 3} and B = {p, q}. The number of relations from A to B is
- 9
- 8
- 64 Correct answer.
- 32
Explanation
A relation from A to B is any subset of A × B. Since |A × B| = 3 × 2 = 6, the number of subsets is 2^6 = 64.
Assam ASSEB Higher Secondary Mathematics practice questions
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Which of the following relations on A = {1, 2, 3} is reflexive?
- {(1,1), (2,2), (1,3), (3,1)}
- {(1,2), (2,3), (3,1)}
- {(1,1), (1,2), (2,1), (3,3)}
- {(1,1), (2,2), (3,3), (1,2)} Correct answer.
Explanation
A relation on A is reflexive when (a, a) belongs to R for every a in A. Only the relation containing (1,1), (2,2) and (3,3) satisfies this condition.
Which of the following is a function from A = {1, 2} to B = {x, y}?
- {(1,x), (1,y)}
- {(1,y), (2,y)} Correct answer.
- {(1,x)}
- {(1,x), (2,x), (2,y)}
Explanation
A function from A to B must assign exactly one element of B to each element of A. The set {(1,y), (2,y)} gives one image for 1 and one image for 2.
The number of one-one functions from a set with 3 elements to a set with 4 elements is
- 24 Correct answer.
- 12
- 81
- 64
Explanation
A one-one function must use distinct images for distinct elements. The first element has 4 choices, the second 3 and the third 2, giving 4 × 3 × 2 = 24.
The number of onto functions from a set with 4 elements to a set with 3 elements is
- 24
- 81
- 45
- 36 Correct answer.
Explanation
By inclusion-exclusion, onto functions = 3^4 − C(3,1)2^4 + C(3,2)1^4 = 81 − 48 + 3 = 36. Each codomain element must be used at least once.
If f(x) = 2x + 1 and g(x) = x^2, then (f ∘ g)(x) equals
- 4x^2 + 4x + 1
- 2x^2 + 1 Correct answer.
- 2x^2 + 2x + 1
- x^2 + 2x + 1
Explanation
(f ∘ g)(x) means f(g(x)). Substituting g(x) = x^2 into f gives f(x^2) = 2x^2 + 1.
If f: R → R is defined by f(x) = 3x − 7, then f⁻¹(x) is
- (x + 7)/3 Correct answer.
- 1/(3x − 7)
- 3x + 7
- (x − 7)/3
Explanation
Write y = 3x − 7, then solve for x: x = (y + 7)/3. Interchanging x and y gives f⁻¹(x) = (x + 7)/3.
The domain of the function f(x) = √(x − 3)/(x − 5) is
- (3, ∞)
- R \ {5}
- [3, ∞) \ {5} Correct answer.
- [3, ∞)
Explanation
The square root requires x − 3 ≥ 0, so x ≥ 3. The denominator requires x ≠ 5, so the domain is [3, ∞) with 5 removed.
The number of equivalence relations on the set {1, 2, 3} is
- 5 Correct answer.
- 8
- 3
- 9
Explanation
Equivalence relations on a finite set correspond to partitions of the set. The partitions of {1,2,3} are 1+1+1, three of type 2+1, and 3, giving 5 in total.
The binary operation * on Z is defined by a * b = a + b − 1. The inverse of 5 under * is
- 3
- 1/5
- −4
- −3 Correct answer.
Explanation
The identity e satisfies a * e = a + e − 1 = a, so e = 1. The inverse b of 5 must satisfy 5 * b = 1, hence 5 + b − 1 = 1 and b = −3.
If set A has 2 elements and set B has 3 elements, then the number of elements in A × B is
- 9
- 5
- 6 Correct answer.
- 8
Explanation
The Cartesian product A × B consists of all ordered pairs (a, b) with a in A and b in B, so its size is n(A)n(B) = 2 × 3 = 6. Adding the sizes or raising one size to the other does not count ordered pairs.
The relation R = {(1, 2), (2, 3), (3, 1)} is defined on A = {1, 2, 3}. The inverse relation R⁻¹ is
- {(1, 1), (2, 2), (3, 3)}
- {(1, 3), (2, 1), (2, 3)}
- {(1, 2), (2, 3), (3, 1)}
- {(2, 1), (3, 2), (1, 3)} Correct answer.
Explanation
The inverse relation is obtained by reversing every ordered pair in R. Reversing (1,2), (2,3), and (3,1) gives (2,1), (3,2), and (1,3), respectively.
The total number of functions from A = {a, b, c} to B = {0, 1} is
- 9
- 6
- 3
- 8 Correct answer.
Explanation
A function from a three-element set to a two-element set is determined by choosing one of the two outputs independently for each of the three inputs. Hence the number of functions is 2 × 2 × 2 = 2^3 = 8.
On A = {1, 2, 3}, which relation is symmetric but neither reflexive nor transitive?
- {(1,1), (2,2), (3,3)}
- {(1,2), (2,1)} Correct answer.
- {(1,1), (2,2)}
- {(1,2), (2,3), (1,3)}
Explanation
The relation {(1,2),(2,1)} is symmetric because each pair has its reverse. It is not reflexive since (1,1), (2,2), and (3,3) are not all present, and it is not transitive because (1,2) and (2,1) would require (1,1).
If f = {(1,2), (2,3), (3,1)} is a function on A = {1,2,3}, then f ∘ f is
- {(1,3), (2,1), (3,2)} Correct answer.
- {(1,3), (2,2), (3,1)}
- {(1,2), (2,3), (3,1)}
- {(1,1), (2,2), (3,3)}
Explanation
Compute f ∘ f by applying f twice: (f ∘ f)(1)=f(2)=3, (f ∘ f)(2)=f(3)=1, and (f ∘ f)(3)=f(1)=2. Therefore the composite relation is {(1,3),(2,1),(3,2)}.
If f(x) = x² + 1 and g(x) = √(x − 1), then (g ∘ f)(x) is
- x² + 1
- x
- √(x − 1) + 1
- |x| Correct answer.
Explanation
By definition, (g ∘ f)(x)=g(f(x))=√((x²+1)−1)=√(x²). Since the principal square root of x² is |x|, the composite is |x| for all real x.
For the binary operation * on Q defined by a * b = ab/2, the identity element is
- 0
- 2 Correct answer.
- 4
- 1
Explanation
An identity element e must satisfy a * e = a and e * a = a for every a. Here a * e = ae/2, so ae/2 = a gives e = 2 for a ≠ 0, and e = 2 also works when a = 0.
The relation R on Z is defined by a R b if and only if a − b is divisible by 4. This is an equivalence relation. The number of distinct equivalence classes is
- 3
- 4 Correct answer.
- 2
- infinitely many
Explanation
Two integers are related exactly when their difference is a multiple of 4, so they have the same remainder modulo 4. The possible remainders are 0, 1, 2, and 3, giving four distinct equivalence classes.
The range of the function f: R → R defined by f(x) = x/(1 + x²) is
- [-1/2, 1/2] Correct answer.
- [-1, 1]
- (-1/2, 1/2)
- [0, 1/2]
Explanation
For y to be in the range, yx² − x + y = 0 must have a real solution x. Its discriminant is 1 − 4y², so 1 − 4y² ≥ 0, giving −1/2 ≤ y ≤ 1/2; the endpoints occur at x = 1 and x = −1.
The number of bijections f from A = {1, 2, 3, 4} onto itself such that f(1) ≠ 1 and f(2) ≠ 2 is
- 16
- 14 Correct answer.
- 12
- 10
Explanation
There are 4! = 24 bijections in total. The conditions fail for 3! = 6 bijections with f(1)=1 and for 3! = 6 with f(2)=2, while 2! = 2 bijections satisfy both failures; inclusion-exclusion gives 24 − 6 − 6 + 2 = 14.
The modulus of the complex number z = 3 + 4i is
- 7
- 25
- √7
- 5 Correct answer.
Explanation
For z = a + bi, the modulus is |z| = √(a² + b²), the distance of the point (a, b) from the origin. Here a = 3 and b = 4, so |z| = √(9 + 16) = √25 = 5. The modulus of any complex number is a non-negative real number.
The roots of the quadratic equation x² + 2x + 5 = 0 are
- two complex conjugate roots Correct answer.
- two equal real roots
- two distinct real roots
- exactly one real root
Explanation
The nature of the roots is decided by the discriminant Δ = b² − 4ac = 4 − 20 = −16. Since the coefficients are real and Δ < 0, the equation has a pair of non-real complex conjugate roots. A quadratic with real coefficients always has two roots counted with multiplicity.
The 10th term of the arithmetic progression 3, 7, 11, 15, ... is
- 43
- 36
- 39 Correct answer.
- 40
Explanation
In an arithmetic progression, aₙ = a + (n − 1)d, where a is the first term and d is the common difference. Here a = 3 and d = 4, so a₁₀ = 3 + 9 × 4 = 3 + 36 = 39. Checking the pattern, the terms continue 19, 23, 27, 31, 35, 39.
The coefficient of x⁵ in the expansion of (x + 2)⁸ is
- 1792
- 448 Correct answer.
- 56
- 224
Explanation
The general term of (x + 2)⁸ is C(8, r) x^(8−r) 2^r. The power of x is 5 when 8 − r = 5, that is r = 3, so the required coefficient is C(8, 3) × 2³ = 56 × 8 = 448.
The number of distinct arrangements of all the letters of the word ASSAM is
- 15
- 30 Correct answer.
- 60
- 120
Explanation
The word ASSAM has 5 letters, with A repeated twice and S repeated twice. When letters are repeated, the number of distinct arrangements is 5!/(2! 2!) = 120/4 = 30, because interchanging the two identical A's or the two identical S's does not give a new word.
If A is a 3 × 3 matrix with determinant 4, then the determinant of 2A is
- 8
- 64
- 12
- 32 Correct answer.
Explanation
For an n × n matrix A and a scalar k, every row is multiplied by k, so det(kA) = kⁿ det(A). Here n = 3 and k = 2, giving det(2A) = 2³ × 4 = 8 × 4 = 32.
The sum to infinity of the geometric series 1 + 1/3 + 1/9 + ... is
- 9/4
- 2
- 3/2 Correct answer.
- 4/3
Explanation
For a geometric series with first term a and common ratio r, the sum to infinity is a/(1 − r) provided |r| < 1. Here a = 1 and r = 1/3, so the sum is 1/(1 − 1/3) = 1/(2/3) = 3/2.
The inverse of the matrix A = [[2, 3], [1, 2]] is
- [[1/2, −3], [−1, 1/2]]
- [[2, 3], [1, 2]]
- [[2, −3], [−1, 2]] Correct answer.
- [[−2, 3], [1, −2]]
Explanation
For a 2 × 2 matrix [[a, b], [c, d]], the inverse is (1/det) [[d, −b], [−c, a]]. Here det A = 4 − 3 = 1, so the inverse is simply [[2, −3], [−1, 2]], which indeed multiplies with A to give the identity matrix.
The term independent of x in the expansion of (x + 1/x²)⁹ is
- 84 Correct answer.
- 9
- 126
- 36
Explanation
The general term is C(9, r) x^(9−r) (x⁻²)^r = C(9, r) x^(9−3r). For the term independent of x, the exponent must be zero, so 9 − 3r = 0 and r = 3. The constant term is therefore C(9, 3) = 84.
If α, β, γ are the roots of x³ − 6x² + 11x − 6 = 0, then α² + β² + γ² equals
- 14 Correct answer.
- 11
- 22
- 36
Explanation
By Vieta's formulas, α + β + γ = 6 and αβ + βγ + γα = 11. Using the identity α² + β² + γ² = (α + β + γ)² − 2(αβ + βγ + γα), we get 36 − 22 = 14; indeed the roots are 1, 2 and 3.
The value of i^2023, where i = √(−1), is
- i
- −i Correct answer.
- −1
- 1
Explanation
Since i^4 = 1, the powers of i repeat with period 4. Write 2023 = 4 × 505 + 3, so i^2023 = i^3 = −i.
The sum of the first 20 terms of the arithmetic progression 5, 8, 11, … is
- 680
- 660
- 650
- 670 Correct answer.
Explanation
Here a = 5, d = 3, and n = 20. Using S_n = n/2 [2a + (n − 1)d], we get S_20 = 10[10 + 57] = 670.
If 2 is a root of the quadratic equation x^2 − 5x + k = 0, then the value of k is
- 4
- 14
- −6
- 6 Correct answer.
Explanation
Substitute x = 2 into the equation: 4 − 10 + k = 0. Hence k = 6.
The coefficient of x^3 in the expansion of (1 − 2x)^6 is
- −160 Correct answer.
- −1600
- −80
- 160
Explanation
The general term is C(6, r)(−2x)^r. For x^3, take r = 3, giving C(6, 3)(−2)^3 = 20 × (−8) = −160.
How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if repetition of digits is not allowed?
- 125
- 10
- 20
- 60 Correct answer.
Explanation
The hundreds place can be filled in 5 ways, the tens in 4 ways, and the units in 3 ways. Hence the number is 5 × 4 × 3 = 60.
The matrix [[x, 2], [3, 6]] is singular when x equals
- 1 Correct answer.
- −1
- 2
- 0
Explanation
A matrix is singular when its determinant is zero. Here det = 6x − 6, so 6x − 6 = 0 gives x = 1.
The sum of an infinite geometric series is 8 and its first term is 4. The common ratio is
- 2
- −1/2
- 1/4
- 1/2 Correct answer.
Explanation
For an infinite geometric series, S = a/(1 − r) provided |r| < 1. Substituting S = 8 and a = 4 gives 8 = 4/(1 − r), so 1 − r = 1/2 and r = 1/2.
The complex number (1 + i)/(1 − i) expressed in the form a + ib is
- −i
- 1 − i
- 1 + i
- i Correct answer.
Explanation
Multiply numerator and denominator by the conjugate 1 + i: (1 + i)^2 / ((1 − i)(1 + i)) = 2i/2 = i. Thus a = 0 and b = 1.
If ω is a non-real cube root of unity, then (1 + ω − ω^2)^7 equals
- −128ω² Correct answer.
- −128ω
- 128ω²
- −128ω² + 1
Explanation
Since 1 + ω + ω^2 = 0, we have 1 + ω = −ω^2, so 1 + ω − ω^2 = −2ω^2. Raising to the seventh power gives (−2)^7 ω^14 = −128ω^2, because ω^14 = ω^2.
If A is a square matrix such that A^2 = A, then (I + A)^3 − 7A equals
- 2I
- 0
- A
- I Correct answer.
Explanation
Expand: (I + A)^3 = I + 3A + 3A^2 + A^3. Since A^2 = A, also A^3 = A, so this becomes I + 7A. Subtracting 7A leaves I.
The real part of the complex number (2 + 3i)(1 − i) is
- 5 Correct answer.
- 3
- 2
- −1
Explanation
Expand the product using i² = −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 5 + i. Therefore the real part is 5, while the imaginary part is 1.
In how many ways can a committee of 2 teachers be chosen from a group of 6 teachers?
- 36
- 12
- 30
- 15 Correct answer.
Explanation
Choosing a committee means order does not matter, so use combinations: C(6,2) = 6!/(2!4!) = (6 × 5)/2 = 15. Permuting the two selected teachers does not create a new committee.
If α and β are the roots of 2x² − 5x + 3 = 0, then 1/α + 1/β equals
- 5/2
- 3/5
- 5/3 Correct answer.
- 2/3
Explanation
For ax² + bx + c = 0, α + β = −b/a = 5/2 and αβ = c/a = 3/2. Then 1/α + 1/β = (α + β)/(αβ) = (5/2)/(3/2) = 5/3.
The coefficient of x² in the expansion of (1 + 3x)^5 is
- 30
- 90 Correct answer.
- 270
- 45
Explanation
The term containing x² is C(5,2)(3x)² = 10 · 9x² = 90x². Hence the coefficient is 90. The binomial coefficient C(5,2) counts the ways to choose the two x-factors.
If A is a 3 × 3 matrix with |A| = 5, then |adj A| equals
- 125
- 1/5
- 25 Correct answer.
- 5
Explanation
For an n × n matrix A, A · adj A = |A| I. Taking determinants gives |A| |adj A| = |A|^n, so |adj A| = |A|^(n−1). With n = 3 and |A| = 5, |adj A| = 5² = 25.
If a, b, c are positive numbers in geometric progression, a + b + c = 14 and abc = 64, then a² + b² + c² equals
- 84 Correct answer.
- 104
- 56
- 72
Explanation
In a geometric progression, b² = ac, so abc = b³. Given abc = 64, b³ = 64 and hence b = 4 because b is positive. Then ac = 16 and a + c = 10, giving the outer terms 2 and 8, so a² + b² + c² = 4 + 16 + 64 = 84.
The value of lim_{x→2} (x² − 4)/(x − 2) is
- 2
- 8
- 0
- 4 Correct answer.
Explanation
Factorise the numerator as (x − 2)(x + 2). For x ≠ 2, the expression cancels to x + 2. Substituting x = 2 gives 2 + 2 = 4, so the limit is 4.
The derivative of sin 3x with respect to x is
- −3 cos 3x
- cos 3x
- 3 sin 3x
- 3 cos 3x Correct answer.
Explanation
Use the chain rule: d/dx sin(3x) = cos(3x) · d/dx(3x). Since d/dx(3x) = 3, the derivative is 3 cos 3x.
For x ≠ 0, the indefinite integral ∫ (1/x) dx equals
- −1/x² + C
- x + C
- 1/x + C
- ln|x| + C Correct answer.
Explanation
The antiderivative of 1/x is ln|x| + C because differentiating ln|x| gives 1/x for x ≠ 0. The absolute value keeps the result valid on either side of zero.
The function f(x) = (x² − 9)/(x − 3) for x ≠ 3, and f(3) = k, is continuous at x = 3 when k equals
- 6 Correct answer.
- 9
- 3
- 0
Explanation
For continuity at x = 3, the limit as x → 3 must equal f(3). Factor x² − 9 = (x − 3)(x + 3), cancel to get x + 3, and the limit is 6; hence k = 6.