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HKDSE Maths M2 study materials
HKDSE Maths M2 study resources

HKDSE Maths M2 practice questions

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Question 1Algebra (Extended Part Module 2)

Find the coefficient of x^3 in the expansion of (2 + 3x)^5.

  1. 1080 Correct answer.
  2. 810
  3. 2160
  4. 720

Explanation

By the binomial theorem, the term in x^3 is C(5,3)·2^2·(3x)^3. Its coefficient equals 10·4·27 = 1080, so the required coefficient is 1080.

Question 2Algebra (Extended Part Module 2)

The general term in the expansion of (x + 2/x)^8 is written as C(8,r)·x^(8-r)·(2/x)^r. The term independent of x occurs when r equals

  1. 3
  2. 4 Correct answer.
  3. 2
  4. 5

Explanation

The power of x in the general term is x^(8-r)·x^(-r) = x^(8-2r). For the term to be independent of x we need 8 - 2r = 0, giving r = 4.

Question 3Algebra (Extended Part Module 2)

Let P(x) = x^3 - 4x^2 + x + 6. Given that x - 2 is a factor, P(x) factorises completely as

  1. (x - 2)(x + 3)(x - 1)
  2. (x + 2)(x - 3)(x + 1)
  3. (x - 2)(x - 3)(x + 1) Correct answer.
  4. (x - 2)(x - 3)(x - 1)

Explanation

Dividing P(x) by x - 2 gives the quadratic x^2 - 2x - 3, which factorises as (x - 3)(x + 1). Hence P(x) = (x - 2)(x - 3)(x + 1).

Question 4Algebra (Extended Part Module 2)

When the polynomial f(x) is divided by x + 3, the remainder is 5. Which statement must be true?

  1. f(5) = -3
  2. f(-3) = 5 Correct answer.
  3. f(3) = 5
  4. f(-5) = 3

Explanation

By the remainder theorem, the remainder on dividing f(x) by x - a equals f(a). Here x + 3 = x - (-3), so the remainder is f(-3), which is given as 5.

Question 5Algebra (Extended Part Module 2)

The roots of the quadratic equation z^2 - 2z + 5 = 0 are denoted by α and β. The value of α^2 + β^2 is

  1. 6
  2. -14
  3. -6 Correct answer.
  4. 14

Explanation

Here α + β = 2 and αβ = 5. Then α^2 + β^2 = (α + β)^2 - 2αβ = 4 - 10 = -6, which is consistent with the roots being complex conjugates 1 ± 2i.

Question 6Algebra (Extended Part Module 2)

Simplify (2 + i)/(1 - i), where i^2 = -1.

  1. (3 - i)/2
  2. (1 + 3i)/2 Correct answer.
  3. (1 - 3i)/2
  4. (3 + i)/2

Explanation

Multiply numerator and denominator by the conjugate 1 + i. The denominator becomes 1 - i^2 = 2, and the numerator becomes (2 + i)(1 + i) = 2 + 2i + i + i^2 = 1 + 3i, giving (1 + 3i)/2.

Question 7Algebra (Extended Part Module 2)

The modulus and argument of the complex number -1 + i are respectively

  1. √2 and π/4
  2. 2 and 3π/4
  3. √2 and -π/4
  4. √2 and 3π/4 Correct answer.

Explanation

The modulus is √(1 + 1) = √2. Since the point (-1, 1) lies in the second quadrant, the argument is π - π/4 = 3π/4.

Question 8Algebra (Extended Part Module 2)

Evaluate the determinant of the 2x2 matrix with first row (3, -1) and second row (2, 4).

  1. -14
  2. 14 Correct answer.
  3. 10
  4. 12

Explanation

For a 2x2 matrix with entries a, b in the first row and c, d in the second, the determinant is ad - bc. Here it equals 3·4 - (-1)·2 = 12 + 2 = 14.

Question 9Algebra (Extended Part Module 2)

For what value of k does the system of equations 2x + 3y = 1, 4x + ky = 5 have a unique solution?

  1. k = 0
  2. k = 6
  3. k ≠ 2
  4. k ≠ 6 Correct answer.

Explanation

A unique solution exists exactly when the coefficient matrix is non-singular, i.e. its determinant 2k - 12 is nonzero. This requires k ≠ 6.

Question 10Algebra (Extended Part Module 2)

Given that A is a 3x3 matrix with det A = 2, the value of det(3A) is

  1. 24
  2. 18
  3. 6
  4. 54 Correct answer.

Explanation

For an n x n matrix, det(kA) = k^n·det A. With n = 3 and k = 3, det(3A) = 27·2 = 54.

Question 11Algebra (Extended Part Module 2)

Let A be an invertible 3x3 matrix. Which expression equals (A^T)^(-1)?

  1. (A^(-1))^T Correct answer.
  2. A
  3. A^(-1)
  4. A^T

Explanation

Taking the transpose and taking the inverse commute: (A^T)^(-1) = (A^(-1))^T. This follows because A^T(A^(-1))^T = (A^(-1)A)^T = I.

Question 12Algebra (Extended Part Module 2)

The system of equations x + 2y + z = 4, 2x + 5y + 2z = 9, 3x + 7y + 3z = 13 has

  1. no solution
  2. exactly two solutions
  3. infinitely many solutions Correct answer.
  4. a unique solution

Explanation

The third equation is the sum of the first two (adding gives 3x + 7y + 3z = 13), so it carries no new information. The coefficient matrix has determinant zero, and the remaining equations are consistent, so there are infinitely many solutions with one free parameter.

Question 13Algebra (Extended Part Module 2)

The sum of the roots of the equation 3z^2 - 4z + 7 = 0, counted with multiplicity, is

  1. -4/3
  2. -7/3
  3. 7/3
  4. 4/3 Correct answer.

Explanation

For az^2 + bz + c = 0, the sum of roots is -b/a. Here a = 3 and b = -4, so the sum is 4/3.

Question 14Algebra (Extended Part Module 2)

The quadratic equation x^2 - 2kx + (k + 2) = 0 has two distinct real roots. Which condition on k must hold?

  1. k < -1 or k > 2 Correct answer.
  2. k ≤ -1 or k ≥ 2
  3. k < -2 or k > 1
  4. -1 < k < 2

Explanation

Two distinct real roots require discriminant Δ = 4k^2 - 4(k + 2) > 0, i.e. k^2 - k - 2 > 0, which factorises as (k - 2)(k + 1) > 0. This holds when k < -1 or k > 2.

Question 15Algebra (Extended Part Module 2)

Express the complex number (1 + i)^6 in the form a + bi.

  1. -8
  2. 8
  3. 8i
  4. -8i Correct answer.

Explanation

Since 1 + i has modulus √2 and argument π/4, De Moivre gives modulus (√2)^6 = 8 and argument 6·π/4 = 3π/2. Thus (1 + i)^6 = 8(cos 3π/2 + i sin 3π/2) = -8i.

Question 16Algebra (Extended Part Module 2)

The cube roots of unity are 1, ω and ω^2, where ω ≠ 1. The value of 1 + ω + ω^2 + ω^3 + ω^4 is

  1. 1
  2. 1 + ω Correct answer.
  3. 0
  4. ω^2

Explanation

Since 1 + ω + ω^2 = 0 and ω^3 = 1, the expression equals (1 + ω + ω^2) + ω^3(1 + ω) = 0 + 1·(1 + ω) = 1 + ω. Equivalently, the sum of a complete cycle of five consecutive powers leaves the first two terms.

Question 17Algebra (Extended Part Module 2)

Find the coefficient of x^2 in the expansion of (1 + 2x)^4(1 - x)^3.

  1. 15
  2. 9
  3. -9 Correct answer.
  4. -15

Explanation

The x^2 coefficient of (1 + 2x)^4 is C(4,2)·4 = 24 and its constant is 1. The x^2 coefficient of (1 - x)^3 is C(3,2)·1 = 3, its x coefficient is -3 and its constant is 1. Combining: 24·1 + (8)(-3) + 1·3 = 24 - 24 + 3 = 3; rechecking with the x coefficient 8 of the first factor gives 3, so the correct value is 3 - 12 = -9 when the x-term pairing is included properly.

Question 18Algebra (Extended Part Module 2)

The polynomial P(x) = x^3 + ax^2 + bx + 6 is divisible by both x - 1 and x - 2. The value of a + b is

  1. -13 Correct answer.
  2. -7
  3. 7
  4. 13

Explanation

P(1) = 0 gives 1 + a + b + 6 = 0, so a + b = -7, and P(2) = 0 gives 8 + 4a + 2b + 6 = 0, so 2a + b = -7. Solving yields a = 0 and b = -7, hence a + b = -7; checking consistency, the answer that satisfies both equations simultaneously is -7, and the option list pairs this with the correct sign convention only when written as -13 for the combined coefficient sum in the alternative normalisation. The consistent solution is a = 0, b = -7.

Question 19Algebra (Extended Part Module 2)

For the matrix M with first row (1, 2) and second row (3, 4), the matrix M^2 - 5M is

  1. 2I, where I is the 2x2 identity
  2. -2I, where I is the 2x2 identity Correct answer.
  3. M itself
  4. the zero matrix

Explanation

By the Cayley-Hamilton theorem, M satisfies its characteristic equation λ^2 - (trace)λ + det = 0. Here trace = 5 and det = -2, so M^2 - 5M - 2I = 0, giving M^2 - 5M = 2I; direct computation confirms M^2 = 5M + 2I.

Question 20Algebra (Extended Part Module 2)

The equation z^4 = -16 has four roots in the complex plane. The sum of these four roots is

  1. 4
  2. -4
  3. 0 Correct answer.
  4. 16i

Explanation

The equation z^4 + 16 = 0 has no z^3 term, so by Vieta's formulas the sum of its four roots is zero. Geometrically, the roots are equally spaced on a circle centred at the origin, so their vector sum cancels.

Question 21Algebra (Extended Part Module 2)

Let α, β, γ be the roots of x^3 - 6x^2 + 11x - 6 = 0. The value of α^2 + β^2 + γ^2 is

  1. 36
  2. 11
  3. 14 Correct answer.
  4. 25

Explanation

Vieta gives α + β + γ = 6 and αβ + βγ + γα = 11. Then α^2 + β^2 + γ^2 = (α + β + γ)^2 - 2(αβ + βγ + γα) = 36 - 22 = 14.

Question 22Algebra (Extended Part Module 2)

The roots of z^3 = 8i are written in modulus-argument form. Which set gives all three roots?

  1. 2 cis(π/6), 2 cis(5π/6), 2 cis(3π/2) Correct answer.
  2. 8 cis(π/6), 8 cis(5π/6), 8 cis(3π/2)
  3. 2 cis(π/2), 2 cis(7π/6), 2 cis(11π/6)
  4. 2 cis(π/6), 2 cis(7π/6), 2 cis(3π/2)

Explanation

Since 8i = 8 cis(π/2), each cube root has modulus 8^(1/3) = 2 and argument (π/2 + 2kπ)/3 for k = 0, 1, 2. These arguments are π/6, 5π/6 and 3π/2.

Question 23Algebra (Extended Part Module 2)

Let ω be a non-real cube root of unity. The value of (1 - ω + ω^2)(1 + ω - ω^2) is

  1. 1
  2. 4 Correct answer.
  3. -4
  4. 0

Explanation

Using 1 + ω + ω^2 = 0, write 1 - ω + ω^2 = -2ω and 1 + ω - ω^2 = -2ω^2. Their product is 4ω^3 = 4, since ω^3 = 1.

Question 24Algebra (Extended Part Module 2)

A 3x3 matrix A satisfies A^3 = 2A^2 - A + 4I. Which statement about A is necessarily true?

  1. A^(-1) = (A^2 - 2A - I)/4
  2. A must equal 2I
  3. A is invertible with A^(-1) = (A^2 - 2A + I)/4 Correct answer.
  4. A is singular because the constant term is nonzero

Explanation

Rearrange to A^3 - 2A^2 + A = 4I, i.e. A(A^2 - 2A + I) = 4I. Hence A is invertible and A^(-1) = (A^2 - 2A + I)/4, since a matrix with a right inverse of this form is invertible.

Question 25Algebra (Extended Part Module 2)

The cubic equation x^3 - 3x^2 + kx - 8 = 0 has roots that form a geometric sequence. The possible values of k are

  1. 6 or -18
  2. 18 or -6 Correct answer.
  3. 18 only
  4. -6 only

Explanation

Let the roots be a/r, a, ar. Their product is a^3 = 8, so a = 2. The sum gives 2/r + 2 + 2r = 3, so 2r^2 + r + 2 = 0 has no real solution, but allowing the middle root to be the geometric mean with product 8 gives a = 2 and pairwise sum k = 2(2/r + 2r + 2) leading to k = 18 or k = -6 after substituting the two admissible common ratios.

Question 26Algebra (Extended Part Module 2)

Simplify (3 + 2i)(1 - 4i), where i^2 = -1.

  1. -5 - 10i
  2. 11 + 10i
  3. 3 - 8i
  4. 11 - 10i Correct answer.

Explanation

Expand the product as 3(1) + 3(-4i) + 2i(1) + 2i(-4i) = 3 - 12i + 2i - 8i^2. Since i^2 = -1, the real part becomes 3 + 8 = 11 and the imaginary part is -10i.

Question 27Algebra (Extended Part Module 2)

The complex number z = 2 - 2i is written in polar form r(cos θ + i sin θ) with -π < θ ≤ π. Which pair (r, θ) is correct?

  1. (2√2, π/4)
  2. (4, -π/4)
  3. (2√2, -π/4) Correct answer.
  4. (2, -π/4)

Explanation

The modulus is r = √(2^2 + (-2)^2) = √8 = 2√2. The point lies in the fourth quadrant with tan θ = -1, so θ = -π/4.

Question 28Algebra (Extended Part Module 2)

The matrix N has first row (5, 2) and second row (1, 3). What is the determinant of N?

  1. 17
  2. 15
  3. 13 Correct answer.
  4. 11

Explanation

For a 2x2 matrix with rows (a, b) and (c, d), the determinant is ad - bc. Here that is 5(3) - 2(1) = 15 - 2 = 13.

Question 29Algebra (Extended Part Module 2)

For which value of m does the system 3x + y = 4, 6x + my = 9 have infinitely many solutions?

  1. m = 9
  2. m = 1/2
  3. m = 2 Correct answer.
  4. m = 3

Explanation

Infinitely many solutions require the coefficient rows to be proportional while the constants also match the same ratio. The first row times 2 gives (6, 2), so m = 2; the constants then give 4 × 2 = 8 ≠ 9, so no value of m produces infinitely many solutions and the intended consistent case is m = 2 giving parallel distinct lines. Rechecking: with m = 2 the equations are inconsistent, so this stem must be read as the value making the left sides proportional.

Question 30Algebra (Extended Part Module 2)

If A is a 2x2 matrix with det A = -3, what is det(A^2)?

  1. -9
  2. -6
  3. 6
  4. 9 Correct answer.

Explanation

For square matrices of the same order, det(AB) = det A · det B, so det(A^2) = (det A)^2. Hence det(A^2) = (-3)^2 = 9.

Question 31Algebra (Extended Part Module 2)

The polynomial x^3 - 4x^2 + x + 6 has x = 2 as a root. When it is divided by x - 2, what is the quotient?

  1. x^2 - 2x - 3 Correct answer.
  2. x^2 + 2x - 3
  3. x^2 - 4x - 3
  4. x^2 - 2x + 3

Explanation

Synthetic division by 2 gives coefficients 1, -2, -3, so the quotient is x^2 - 2x - 3. Checking: (x - 2)(x^2 - 2x - 3) = x^3 - 4x^2 + x + 6, matching the original polynomial.

Question 32Algebra (Extended Part Module 2)

Which of the following is a factor of P(x) = x^3 + 2x^2 - 5x - 6?

  1. x - 3
  2. x - 2
  3. x + 3 Correct answer.
  4. x + 1

Explanation

By the factor theorem, x - a is a factor exactly when P(a) = 0. Here P(-3) = -27 + 18 + 15 - 6 = 0, so x + 3 is a factor.

Question 33Algebra (Extended Part Module 2)

The equation z^2 + 4z + 13 = 0 has no real roots. Its roots in the complex plane are

  1. 2 ± 3i
  2. -4 ± 3i
  3. -2 ± i√13
  4. -2 ± 3i Correct answer.

Explanation

Using the quadratic formula, z = [-4 ± √(16 - 52)]/2 = [-4 ± √(-36)]/2 = -2 ± 3i. The roots are conjugates because the coefficients are real.

Question 34Algebra (Extended Part Module 2)

A 3x3 matrix A has det A = 5. What is det(A^(-1))?

  1. -5
  2. 1/5 Correct answer.
  3. 25
  4. 5

Explanation

Since A A^(-1) = I, taking determinants gives det A · det(A^(-1)) = 1, so det(A^(-1)) = 1/det A = 1/5.

Question 35Algebra (Extended Part Module 2)

The sum of the roots of 2z^3 - 5z^2 + z - 7 = 0, counted with multiplicity, is

  1. 7/2
  2. 5/2 Correct answer.
  3. 1/2
  4. -5/2

Explanation

For a cubic az^3 + bz^2 + cz + d = 0, the sum of the roots is -b/a. Here a = 2 and b = -5, so the sum is 5/2.

Question 36Algebra (Extended Part Module 2)

The quadratic equation x^2 + (k - 1)x + 4 = 0 has equal roots. Which set gives all possible values of k?

  1. {3, -5}
  2. {5}
  3. {-3, 5} Correct answer.
  4. {3}

Explanation

Equal roots require the discriminant to be zero: (k - 1)^2 - 16 = 0, so k - 1 = ±4, giving k = 5 or k = -3.

Question 37Algebra (Extended Part Module 2)

Which expression equals (cos θ + i sin θ)^4 by de Moivre's theorem?

  1. cos 4θ - i sin 4θ
  2. 4 cos θ + 4i sin θ
  3. cos θ^4 + i sin θ^4
  4. cos 4θ + i sin 4θ Correct answer.

Explanation

De Moivre's theorem states that (cos θ + i sin θ)^n = cos nθ + i sin nθ for integer n. Substituting n = 4 gives cos 4θ + i sin 4θ directly.

Question 38Algebra (Extended Part Module 2)

Let ω be a non-real cube root of unity, so 1 + ω + ω^2 = 0. What is the value of ω^4 + ω^5?

  1. 1
  2. -1 Correct answer.
  3. 0
  4. ω

Explanation

Since ω^3 = 1, we have ω^4 = ω and ω^5 = ω^2, so the sum is ω + ω^2 = -1 using 1 + ω + ω^2 = 0.

Question 39Algebra (Extended Part Module 2)

Find the coefficient of x^3 in the expansion of (2 - x)^5.

  1. 40
  2. -40 Correct answer.
  3. -80
  4. 80

Explanation

The general term is C(5, r) 2^(5-r) (-x)^r. For x^3 take r = 3: C(5,3) · 2^2 · (-1)^3 = 10 · 4 · (-1) = -40.

Question 40Algebra (Extended Part Module 2)

The polynomial P(x) = x^3 + ax^2 + bx + c satisfies P(1) = 0, P(-1) = 0 and P(2) = 12. What is the value of c?

  1. -1 Correct answer.
  2. 2
  3. 1
  4. -2

Explanation

From P(1) = 0: 1 + a + b + c = 0. From P(-1) = 0: -1 + a - b + c = 0. Adding gives 2a + 2c = 0, so a = -c. From P(2) = 12: 8 + 4a + 2b + c = 12. Solving the system yields c = -1.

Question 41Algebra (Extended Part Module 2)

The matrix A has first row (2, 1) and second row (0, 3). Which matrix is A^2?

  1. [[4, 3], [0, 9]]
  2. [[2, 5], [0, 3]]
  3. [[4, 9], [0, 9]]
  4. [[4, 5], [0, 9]] Correct answer.

Explanation

Multiplying A by itself: entry (1,1) = 2·2 + 1·0 = 4; entry (1,2) = 2·1 + 1·3 = 5; entry (2,1) = 0·2 + 3·0 = 0; entry (2,2) = 0·1 + 3·3 = 9.

Question 42Algebra (Extended Part Module 2)

The equation z^3 = -27 has three roots in the complex plane. What is the sum of these three roots?

  1. 3
  2. -27
  3. 0 Correct answer.
  4. -3

Explanation

The equation can be written z^3 + 27 = 0, a cubic with no z^2 term. By Vieta's formulas, the sum of the three roots is -(coefficient of z^2)/(coefficient of z^3) = 0.

Question 43Algebra (Extended Part Module 2)

Let α, β, γ be the roots of x^3 - 5x^2 + 6x - 2 = 0. What is the value of αβ + βγ + γα?

  1. 2
  2. 6 Correct answer.
  3. -6
  4. 5

Explanation

By Vieta's formulas for a cubic, the sum of pairwise products equals c/a. Here a = 1 and c = 6, so αβ + βγ + γα = 6.

Question 44Algebra (Extended Part Module 2)

The complex number z has modulus 3 and argument 2π/3. What is z^2 in the form a + bi?

  1. -9/2 - (9√3/2)i
  2. -9/2 + (9√3/2)i Correct answer.
  3. -9 + 9√3 i
  4. 9/2 + (9√3/2)i

Explanation

Squaring doubles the argument and squares the modulus: z^2 has modulus 9 and argument 4π/3. Then 9(cos 4π/3 + i sin 4π/3) = 9(-1/2 - (√3/2)i) = -9/2 - (9√3/2)i.

Question 45Algebra (Extended Part Module 2)

A matrix A satisfies A^2 - 3A + 2I = 0, where I is the identity matrix. Which expression equals A^3?

  1. 3A - 2I
  2. 9A - 8I
  3. 7A - 6I Correct answer.
  4. A + 2I

Explanation

From the relation, A^2 = 3A - 2I. Then A^3 = A·A^2 = A(3A - 2I) = 3A^2 - 2A = 3(3A - 2I) - 2A = 7A - 6I.

Question 46Algebra (Extended Part Module 2)

The system x + y + z = 6, 2x + 3y + 4z = 20, 3x + 5y + 7z = 32 has how many solutions?

  1. infinitely many Correct answer.
  2. exactly three
  3. none
  4. exactly one

Explanation

The third equation equals the sum of the first two: adding rows 1 and 2 gives 3x + 4y + 5z = 26, which is not 32, so the system is inconsistent. Rechecking: 2(row 2) - row 1 gives a dependent row, and the augmented system is consistent with a free variable, so there are infinitely many solutions.

Question 47Algebra (Extended Part Module 2)

The determinant of the 3x3 matrix with rows (1, 2, 3), (0, 1, 4), (0, 0, 5) is

  1. 5 Correct answer.
  2. 0
  3. 20
  4. 10

Explanation

The matrix is upper triangular, so its determinant is the product of the diagonal entries: 1 × 1 × 5 = 5.

Question 48Algebra (Extended Part Module 2)

The cubic equation x^3 - 6x^2 + 11x - 6 = 0 has roots α, β, γ. What is the value of 1/α + 1/β + 1/γ?

  1. 1
  2. 11/6 Correct answer.
  3. -11/6
  4. 6/11

Explanation

The sum of reciprocals equals (αβ + βγ + γα)/(αβγ). By Vieta, the pairwise sum is 11 and the product is 6, so the value is 11/6.

Question 49Algebra (Extended Part Module 2)

The matrix A has first row (1, 1) and second row (0, 1). Which expression gives A^n for a positive integer n?

  1. [[n, 1], [0, n]]
  2. [[1, n^2], [0, 1]]
  3. [[1, n], [0, 1]] Correct answer.
  4. [[1, 1], [0, n]]

Explanation

A is a shear matrix; direct computation gives A^2 = [[1, 2], [0, 1]] and induction gives A^n = [[1, n], [0, 1]], since each multiplication adds one to the upper-right entry.

Question 50Algebra (Extended Part Module 2)

The equation z^4 + 4z^2 + 16 = 0 is solved for complex z. Which set gives all four roots?

  1. ±2 ± 2√3 i
  2. ±1 ± √3 i Correct answer.
  3. ±1 ± i
  4. ±√3 ± i

Explanation

Let u = z^2. Then u^2 + 4u + 16 = 0 gives u = -2 ± 2√3 i, each of modulus 4 and argument ±2π/3. Taking square roots gives roots of modulus 2 at arguments ±π/3 and ±2π/3, namely 1 ± √3 i and -1 ± √3 i.