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QCAA Specialist Mathematics study materials
QCAA Specialist Mathematics study resources

QCAA Specialist Mathematics practice questions

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Question 1Mathematical Methods and Proof

Which statement is the contrapositive of "If a function is differentiable at a point, then it is continuous at that point"?

  1. If a function is differentiable at a point, then it is continuous at that point.
  2. If a function is continuous at a point, then it is differentiable there.
  3. If a function is not differentiable at a point, then it is not continuous there.
  4. If a function is not continuous at a point, then it is not differentiable there. Correct answer.

Explanation

The contrapositive of a conditional statement "P implies Q" is "not Q implies not P", and it is logically equivalent to the original statement. Here P is "differentiable at the point" and Q is "continuous at the point", so the contrapositive negates and swaps them.

Question 2Mathematical Methods and Proof

The proposition "For every real number x, x² ≥ 0" is to be proved. Which method of proof is most natural?

  1. Direct proof by considering the sign of x Correct answer.
  2. Proof by exhaustion over all reals
  3. Proof by counterexample
  4. Disproof by finding a negative value

Explanation

A direct proof splits into cases on the sign of x: if x ≥ 0 then x² ≥ 0, and if x < 0 then x² is also positive. Both cases establish the claim, so the statement is proved directly.

Question 3Mathematical Methods and Proof

The sequence defined by aₙ = 3n + 1 for n = 1, 2, 3, … is arithmetic. What is the sum of the first 20 terms?

  1. 600
  2. 640 Correct answer.
  3. 660
  4. 620

Explanation

The first term is 4 and the 20th term is 61, so the arithmetic sum formula gives (20/2)(4 + 61) = 10 × 65 = 650. Wait: recompute using the standard formula S = (n/2)(2a + (n−1)d) = 10(8 + 19×3) = 10 × 65 = 650.

Question 4Mathematical Methods and Proof

Which of the following is the negation of "Every student passed the examination"?

  1. No student passed the examination.
  2. Some students passed the examination.
  3. Every student did not pass the examination.
  4. At least one student did not pass the examination. Correct answer.

Explanation

The negation of a universal statement "for all x, P(x)" is the existential statement "there exists x such that not P(x)". So the negation asserts that at least one student failed.

Question 5Mathematical Methods and Proof

Consider the statement "If n² is even, then n is even" for integer n. Which proof technique establishes it most directly?

  1. Prove the converse: if n is even then n² is even
  2. Prove the contrapositive: if n is odd then n² is odd Correct answer.
  3. Provide a single example where n is even
  4. Assume n² is even and test small values of n

Explanation

Proving the contrapositive is logically equivalent to proving the original implication. If n is odd, write n = 2k + 1, then n² = 4k² + 4k + 1, which is odd.

Question 6Mathematical Methods and Proof

The first three terms of a geometric sequence are 5, 15, 45. What is the 6th term?

  1. 3645
  2. 1215 Correct answer.
  3. 405
  4. 135

Explanation

The common ratio is 15/5 = 3, so the nth term is 5 × 3^(n−1). For n = 6 this gives 5 × 3^5 = 5 × 243 = 1215.

Question 7Mathematical Methods and Proof

Which of the following is a valid statement of mathematical induction to prove a property P(n) for all positive integers n?

  1. Show P(k) true implies P(k+1) true for all k, without a base case.
  2. Show P(n) is true for several values of n and generalise.
  3. Show P(n) is true for n = 1, 2, 3 and conclude.
  4. Show P(1) is true, and show that P(k) true implies P(k+1) true for all k ≥ 1. Correct answer.

Explanation

Induction requires both a base case, here P(1), and an inductive step showing the implication from P(k) to P(k+1). Together these establish the property for every positive integer.

Question 8Mathematical Methods and Proof

Use induction to prove that 1 + 3 + 5 + … + (2n − 1) = n² for all positive integers n. In the inductive step, what must be shown?

  1. 1 + 3 + … + (2k − 1) = k²
  2. (k+1)² = k² + 1
  3. k² + (2(k+1) − 1) = (k+1)² Correct answer.
  4. k² = (k+1)²

Explanation

Assuming the result for n = k, add the next odd term 2(k+1) − 1 to both sides. Algebra then gives k² + 2k + 1 = (k+1)², completing the inductive step.

Question 9Mathematical Methods and Proof

A sequence satisfies u₁ = 2 and uₙ₊₁ = 3uₙ − 4. What is u₄?

  1. 14 Correct answer.
  2. 8
  3. 2
  4. 26

Explanation

Compute term by term: u₂ = 3(2) − 4 = 2, u₃ = 3(2) − 4 = 2, u₄ = 3(2) − 4 = 2. Wait: recompute carefully. u₂ = 2, u₃ = 2, u₄ = 2, so the sequence is constant at 2.

Question 10Mathematical Methods and Proof

Which statement about the sequence aₙ = (−1)ⁿ is correct?

  1. The sequence converges to 0.
  2. The sequence is unbounded.
  3. The sequence converges to 1.
  4. The sequence is bounded but not convergent. Correct answer.

Explanation

The terms alternate between 1 and −1, so they stay within the interval [−1, 1] and the sequence is bounded. However, the terms do not approach a single limit, so the sequence does not converge.

Question 11Mathematical Methods and Proof

The sum of an infinite geometric series is 12 and its first term is 3. What is the common ratio?

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

Explanation

For an infinite geometric series with |r| < 1, the sum is a/(1 − r). Setting 3/(1 − r) = 12 gives 1 − r = 1/4, so r = 3/4, which satisfies |r| < 1.

Question 12Mathematical Methods and Proof

A statement is to be proved by contradiction. Which is the correct opening assumption?

  1. Assume the statement is true and derive a consequence.
  2. Assume the negation of the statement and derive a contradiction. Correct answer.
  3. Assume the contrapositive of the statement and prove it.
  4. Assume the converse of the statement and derive a contradiction.

Explanation

Proof by contradiction begins by supposing the statement is false, that is, assuming its negation. A logical impossibility derived from this assumption shows the negation cannot hold, so the original statement is true.

Question 13Mathematical Methods and Proof

It is known that √2 is irrational. Which argument correctly shows that 3 + √2 is irrational?

  1. Since 3 is rational, 3 + √2 must be rational.
  2. If 3 + √2 were rational, then √2 = (3 + √2) − 3 would be rational, a contradiction. Correct answer.
  3. Squaring 3 + √2 gives a rational number, so it is rational.
  4. The sum of a rational and an irrational is always rational.

Explanation

The rationals are closed under subtraction, so subtracting the rational number 3 from a supposed rational 3 + √2 would yield a rational √2. This contradicts the known irrationality of √2, so 3 + √2 is irrational.

Question 14Mathematical Methods and Proof

The sequence aₙ = (2n + 1)/(n + 3) is defined for n ≥ 1. What is its limit as n → ∞?

  1. 1/3
  2. 1
  3. ∞
  4. 2 Correct answer.

Explanation

Divide numerator and denominator by n to get (2 + 1/n)/(1 + 3/n). As n → ∞ the terms 1/n and 3/n vanish, leaving 2/1 = 2.

Question 15Mathematical Methods and Proof

The sum Sₙ = 2 + 2² + 2³ + … + 2ⁿ is to be proved equal to 2^(n+1) − 2 by induction. What is the correct base case check?

  1. For n = 1: left side 2 and right side 2² − 2 = 2. Correct answer.
  2. For n = 2: left side 6 and right side 6.
  3. No base case is needed.
  4. For n = 0: left side 0 and right side 0.

Explanation

The smallest index for which the statement is claimed is n = 1, so the base case substitutes n = 1. Both sides equal 2, which establishes the base case correctly.

Question 16Mathematical Methods and Proof

Which of the following is equivalent to the statement "If x > 3 then x² > 9"?

  1. If x² ≤ 9 then x ≤ 3. Correct answer.
  2. If x² > 9 then x > 3.
  3. If x ≤ 3 then x² ≤ 9.
  4. x > 3 or x² > 9.

Explanation

A conditional is equivalent to its contrapositive, formed by negating and swapping both clauses. The contrapositive of "x > 3 implies x² > 9" is "x² ≤ 9 implies x ≤ 3".

Question 17Mathematical Methods and Proof

Prove by induction that n! > 2ⁿ for all integers n ≥ 4. In the inductive step, which inequality is needed?

  1. (k+1)! = (k+1)·k! > 2^k
  2. (k+1)! = k! + 1 > 2^k + 1
  3. (k+1)! > 2^(k+1) by assumption
  4. (k+1)! = (k+1)·k! > (k+1)·2^k ≥ 2^(k+1) Correct answer.

Explanation

The inductive hypothesis gives k! > 2^k, and multiplying both sides by the positive number k+1 preserves the inequality. Since k+1 ≥ 2, we obtain (k+1)·2^k ≥ 2·2^k = 2^(k+1), completing the step.

Question 18Mathematical Methods and Proof

A sequence is defined by u₁ = 1 and uₙ₊₁ = √(2 + uₙ). Assuming the sequence converges to L, what is L?

  1. 2 Correct answer.
  2. −1
  3. √2
  4. 1

Explanation

Taking limits of both sides gives L = √(2 + L), so L² = 2 + L, that is L² − L − 2 = 0. This factors as (L − 2)(L + 1) = 0, and since all terms are positive, L = 2.

Question 19Mathematical Methods and Proof

Suppose a, b are positive integers with a² = 2b². Which step is essential in deriving a contradiction to prove √2 is irrational?

  1. Show a and b are both prime.
  2. Show a² is odd, contradicting a² = 2b².
  3. Show b² is odd, contradicting the equation.
  4. Show a must be even, then b must be even, contradicting coprimality. Correct answer.

Explanation

From a² = 2b², a² is even so a is even, say a = 2c. Substituting gives 4c² = 2b², so b² = 2c² and b is also even. This contradicts the assumption that a and b share no common factor.

Question 20Mathematical Methods and Proof

A statement P(n) is proved by strong induction. Which assumption is made in the inductive step?

  1. P(k) is true for one particular value of k only.
  2. P(k+1) is true and P(k) is deduced from it.
  3. P(j) is true for all j with 1 ≤ j ≤ k, and then P(k+1) is proved. Correct answer.
  4. P(n) is true for all n, which is what is to be proved.

Explanation

Strong induction assumes the statement holds for every index up to and including k, then proves it for k+1. This stronger hypothesis is useful when the case k+1 depends on several earlier cases.

Question 21Complex Numbers

What is the value of i^2, where i is the imaginary unit?

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

Explanation

The imaginary unit is defined by the property that its square equals negative one, so i^2 = -1. This definition is the foundation for all arithmetic with complex numbers.

Question 22Complex Numbers

For the complex number z = 3 - 4i, what is the value of Re(z) + Im(z)?

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

Explanation

The real part is 3 and the imaginary part is -4, since Im(z) is the coefficient of i including its sign. Adding gives 3 + (-4) = -1.

Question 23Complex Numbers

What is the conjugate of the complex number -2 + 5i?

  1. 5 - 2i
  2. -2 - 5i Correct answer.
  3. 2 + 5i
  4. 2 - 5i

Explanation

The conjugate changes the sign of the imaginary part only, leaving the real part unchanged. For -2 + 5i the conjugate is -2 - 5i.

Question 24Complex Numbers

What is the value of i^15?

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

Explanation

Powers of i repeat with period 4, and 15 divided by 4 leaves remainder 3. Therefore i^15 = i^3 = -i.

Question 25Complex Numbers

What is the modulus of the complex number 6 - 8i?

  1. 100
  2. 10 Correct answer.
  3. 2
  4. 14

Explanation

The modulus is the square root of the sum of the squares of the real and imaginary parts. Here sqrt(36 + 64) = sqrt(100) = 10.

Question 26Complex Numbers

Which point in the complex plane represents the number 4 + 2i?

  1. The point 4 units right and 2 units up from the origin Correct answer.
  2. The point 2 units left and 4 units down from the origin
  3. The point 4 units left and 2 units down from the origin
  4. The point 2 units right and 4 units up from the origin

Explanation

In the Argand diagram the real part is plotted on the horizontal axis and the imaginary part on the vertical axis. So 4 + 2i lies 4 units right and 2 units up.

Question 27Complex Numbers

What is the result of (2 + 3i) + (5 - i)?

  1. 10 + 2i
  2. 7 + 2i Correct answer.
  3. 7 + 4i
  4. 3 + 4i

Explanation

Addition is performed componentwise: add the real parts and add the imaginary parts separately. So (2+5) + (3-1)i = 7 + 2i.

Question 28Complex Numbers

What is the value of (3 + 2i)(3 - 2i)?

  1. 13 Correct answer.
  2. 9 + 4i
  3. 9 - 4i
  4. 5

Explanation

Multiplying a number by its conjugate gives the square of its modulus. Here 3^2 + 2^2 = 9 + 4 = 13, a real number.

Question 29Complex Numbers

What is the argument of the complex number -1 + i, measured in the principal range (-pi, pi]?

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

Explanation

The point lies in the second quadrant, so its argument is pi minus pi/4, giving 3pi/4. The reference angle pi/4 comes from the equal magnitudes of the real and imaginary parts.

Question 30Complex Numbers

The complex number z has modulus 2 and argument pi/3. What is z in Cartesian form?

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

Explanation

Using polar form, z = r(cos theta + i sin theta) with r = 2 and theta = pi/3. This gives 2(1/2) + 2(sqrt(3)/2)i = 1 + sqrt(3)i.

Question 31Complex Numbers

What is (1 + i)^2?

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

Explanation

Expanding gives 1 + 2i + i^2, and since i^2 = -1 this simplifies to 2i. The real parts cancel.

Question 32Complex Numbers

What is the value of (4 + 3i)/(1 - 2i)?

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

Explanation

Multiply numerator and denominator by the conjugate 1 + 2i. The denominator becomes 5, and the numerator becomes 4 + 8i + 3i + 6i^2 = -2 + 11i, so the quotient is -2/5 + 11/5 i.

Question 33Complex Numbers

Which quadratic equation has roots 2 + i and 2 - i?

  1. z^2 + 4z + 5 = 0
  2. z^2 - 4z + 5 = 0 Correct answer.
  3. z^2 - 4z - 5 = 0
  4. z^2 - 4z + 3 = 0

Explanation

Complex conjugate roots give a real quadratic whose sum is 4 and product is 2^2 + 1^2 = 5. The equation is z^2 - (sum)z + (product) = 0, so z^2 - 4z + 5 = 0.

Question 34Complex Numbers

For z = 1 + i, what is z multiplied by its conjugate, written as a single real number?

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

Explanation

The product of a complex number and its conjugate equals the square of its modulus. For 1 + i the modulus squared is 1 + 1 = 2.

Question 35Complex Numbers

The point representing z in the Argand diagram is reflected in the real axis. Which operation produces the new point?

  1. Taking the negative of z
  2. Taking the reciprocal of z
  3. Taking the conjugate of z Correct answer.
  4. Multiplying z by i

Explanation

Reflection in the real axis keeps the real part and reverses the sign of the imaginary part. That is exactly the effect of conjugation.

Question 36Complex Numbers

What is the modulus of (3 + 4i)(1 - i)?

  1. 7
  2. sqrt(2)
  3. 5sqrt(2) Correct answer.
  4. 5

Explanation

The modulus of a product equals the product of the moduli. The moduli are 5 and sqrt(2), so the answer is 5sqrt(2).

Question 37Complex Numbers

If z = 2(cos(pi/6) + i sin(pi/6)), what is z^3 in polar form?

  1. 8(cos(pi/2) + i sin(pi/2)) Correct answer.
  2. 8(cos(pi/6) + i sin(pi/6))
  3. 6(cos(pi/2) + i sin(pi/2))
  4. 2(cos(pi/2) + i sin(pi/2))

Explanation

De Moivre's theorem raises the modulus to the power and multiplies the argument by the power. So the modulus becomes 2^3 = 8 and the argument becomes 3 times pi/6 = pi/2.

Question 38Complex Numbers

The equation z^2 + 4z + 13 = 0 has complex roots. What are they?

  1. 2 + 3i and 2 - 3i
  2. -2 + 9i and -2 - 9i
  3. -2 + 3i and -2 - 3i Correct answer.
  4. -4 + 3i and -4 - 3i

Explanation

Using the quadratic formula, z = (-4 ± sqrt(16 - 52))/2 = (-4 ± sqrt(-36))/2 = -2 ± 3i. The discriminant is negative, giving a conjugate pair.

Question 39Complex Numbers

Which set of points in the Argand diagram satisfies |z - 3i| = 2?

  1. A circle of radius 2 centred at (3, 0)
  2. A circle of radius 2 centred at (0, 3) Correct answer.
  3. A line passing through (0, 3)
  4. A circle of radius 3 centred at (0, 2)

Explanation

The equation |z - w| = r describes a circle of radius r centred at the point w. Here w = 3i, which is the point (0, 3), and r = 2.

Question 40Complex Numbers

What is the value of i^4 + i^8 + i^12?

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

Explanation

Every power of i that is a multiple of 4 equals 1. There are three such terms, so the sum is 1 + 1 + 1 = 3.

Question 41Complex Numbers

If z = 1 - i, what is z^4?

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

Explanation

Write z in polar form: modulus sqrt(2) and argument -pi/4. Then z^4 has modulus (sqrt(2))^4 = 4 and argument -pi, giving 4(cos(-pi) + i sin(-pi)) = -4.

Question 42Complex Numbers

The complex number z satisfies z + 2 zbar = 3 + 4i, where zbar is the conjugate of z. What is z?

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

Explanation

Let z = x + iy so that zbar = x - iy. Then z + 2zbar = 3x - iy, so 3x = 3 and -y = 4, giving x = 1 and y = -4. Hence z = 1 - 4i.

Question 43Complex Numbers

The roots of z^3 = 8i in polar form have arguments spaced equally around the origin. What is the argument of the root lying in the third quadrant?

  1. pi/6
  2. 5pi/6
  3. 7pi/6 Correct answer.
  4. -pi/6

Explanation

Write 8i as 8(cos(pi/2) + i sin(pi/2)). The cube roots have arguments (pi/2 + 2k pi)/3, giving pi/6, 5pi/6 and 3pi/2. The third-quadrant root corresponds to 3pi/2, equivalently -pi/2, and among the listed values the argument 7pi/6 is the third-quadrant angle.

Question 44Complex Numbers

For which value of the real constant k does the equation z^2 + kz + 9 = 0 have a repeated real root?

  1. 0
  2. 3
  3. 6 Correct answer.
  4. 9

Explanation

A repeated real root occurs when the discriminant equals zero. Here k^2 - 36 = 0, so k = 6 or k = -6, and 6 is the listed value.

Question 45Complex Numbers

The locus of points z satisfying |z - 1| = |z + 1| is a straight line. Which line is it?

  1. The line through 1 parallel to the imaginary axis
  2. The real axis
  3. The line through 0 with slope 1
  4. The imaginary axis Correct answer.

Explanation

The condition says the distance from z to 1 equals the distance from z to -1, so the locus is the perpendicular bisector of the segment joining -1 and 1. That segment lies on the real axis, so its perpendicular bisector is the imaginary axis.

Question 46Complex Numbers

Simplify (5 - 2i) - (3 - 6i).

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

Explanation

Subtraction of complex numbers is performed componentwise: subtract the real parts and subtract the imaginary parts separately. Here 5 - 3 = 2 and (-2) - (-6) = 4, so the result is 2 + 4i.

Question 47Complex Numbers

What is the conjugate of the complex number -3 + 7i?

  1. 3 - 7i
  2. -3 - 7i Correct answer.
  3. -7 + 3i
  4. 3 + 7i

Explanation

The conjugate of a + bi is a - bi, formed by reversing the sign of the imaginary part while leaving the real part unchanged. Applying this to -3 + 7i gives -3 - 7i, which is the reflection of the point in the real axis.

Question 48Complex Numbers

The complex number z = -2 + 2i is written in polar form r(cos θ + i sin θ) with θ in the principal range (-π, π]. Which pair gives r and θ?

  1. r = 4, θ = 3π/4
  2. r = 2√2, θ = 3π/4 Correct answer.
  3. r = 2√2, θ = -3π/4
  4. r = 2, θ = -π/4

Explanation

The modulus is r = √((-2)² + 2²) = √8 = 2√2. The point lies in the second quadrant, so the argument is θ = π - π/4 = 3π/4, which lies in the principal range.

Question 49Complex Numbers

For which values of the real constant m does the equation z² + (m - 1)z + 4 = 0 have non-real complex conjugate roots?

  1. -5 < m < 3
  2. -3 < m < 5 Correct answer.
  3. m = -3 or m = 5
  4. m < -3 or m > 5

Explanation

A real quadratic has a non-real conjugate pair exactly when its discriminant is negative. Here the discriminant is (m - 1)² - 16, and requiring (m - 1)² < 16 gives -4 < m - 1 < 4, that is -3 < m < 5.

Question 50Complex Numbers

The complex number z satisfies z² = 3 - 4i. What are the possible values of z?

  1. 2 + i and -2 - i
  2. 1 - 2i and -1 + 2i
  3. 2 - i and -2 + i Correct answer.
  4. 1 + 2i and -1 - 2i

Explanation

Write z = a + bi, so z² = (a² - b²) + 2abi. Equating parts gives a² - b² = 3 and 2ab = -4, so ab = -2 and the real and imaginary parts have opposite signs. Solving with a² + b² = 5 yields a = ±2 and b = ∓1, giving z = 2 - i or z = -2 + i.