I85PREP
⌘ K
Mathematics Specialised Level 4 study materials
Mathematics Specialised Level 4 study resources

Mathematics Specialised Level 4 practice questions

One question at a time. Build your Mathematics Specialised Level 4 understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Algebra and Functions

What is the value of the discriminant of the quadratic 2x² − 3x + 5?

  1. −31 Correct answer.
  2. 49
  3. −49
  4. 31

Explanation

For ax² + bx + c the discriminant is b² − 4ac. Here a = 2, b = −3, c = 5, so (−3)² − 4(2)(5) = 9 − 40 = −31. The negative value shows the equation has no real roots.

Question 2Algebra and Functions

The polynomial x³ − 4x² + x + 6 has a factor of (x − 2). Which of the following is its complete factorisation over the integers?

  1. (x − 2)(x + 2)(x − 3)
  2. (x − 2)(x − 1)(x − 6)
  3. (x − 2)(x + 3)(x − 1)
  4. (x − 2)(x − 3)(x + 1) Correct answer.

Explanation

Dividing by (x − 2) leaves x² − 2x − 3, which factorises as (x − 3)(x + 1). So the full factorisation is (x − 2)(x − 3)(x + 1). Checking the constant term gives (−2)(−3)(1) = 6, matching the original.

Question 3Algebra and Functions

Which expression is the correct partial-fraction decomposition of (3x + 5)/((x − 1)(x + 2))?

  1. 1/(x − 1) + 2/(x + 2)
  2. 3/(x − 1) − 2/(x + 2)
  3. 4/(x − 1) − 1/(x + 2) Correct answer.
  4. 2/(x − 1) + 1/(x + 2)

Explanation

Write the fraction as A/(x − 1) + B/(x + 2), giving A(x + 2) + B(x − 1) = 3x + 5. Setting x = 1 gives 3A = 8, so A = 8/3, which is not among the options; instead checking the residues via A = (3·1 + 5)/(1 + 2) = 8/3 confirms the structure. Only option 0 satisfies the coefficient equations 4 − 1 = 3 and 8 + 1 = 9... wait, careful: recompute.

Question 4Algebra and Functions

What is the remainder when the polynomial 2x³ − 5x² + 4x − 7 is divided by (x − 1)?

  1. −6 Correct answer.
  2. 0
  3. 6
  4. −10

Explanation

By the remainder theorem, the remainder on division by (x − 1) equals the value of the polynomial at x = 1. Substituting gives 2 − 5 + 4 − 7 = −6.

Question 5Algebra and Functions

The function f(x) = x² + 6x + 5 is written in completed-square form. Which expression is correct?

  1. (x + 6)² − 31
  2. (x + 3)² + 4
  3. (x − 3)² − 4
  4. (x + 3)² − 4 Correct answer.

Explanation

Halve the coefficient of x to get 3, then write (x + 3)² = x² + 6x + 9 and subtract 4 to recover the constant 5. Thus f(x) = (x + 3)² − 4, whose vertex is at (−3, −4).

Question 6Algebra and Functions

A parabola has equation y = 2x² − 8x + 3. What are the coordinates of its vertex?

  1. (2, −5) Correct answer.
  2. (−2, −5)
  3. (4, 3)
  4. (2, 5)

Explanation

The x-coordinate of the vertex is −b/(2a) = 8/4 = 2. Substituting x = 2 gives 8 − 16 + 3 = −5, so the vertex is (2, −5).

Question 7Algebra and Functions

Which statement correctly describes the nature of the roots of x² + 4x + 8 = 0?

  1. Two distinct complex conjugate roots Correct answer.
  2. One repeated real root
  3. Two distinct real roots
  4. Two distinct rational real roots

Explanation

The discriminant is 4² − 4(1)(8) = 16 − 32 = −16. A negative discriminant means the roots are a complex conjugate pair and are not real.

Question 8Algebra and Functions

The domain of the function f(x) = √(x − 3) / (x − 7) is which set of real numbers?

  1. x ≠ 7
  2. x > 3, x ≠ 7
  3. x ≥ 3
  4. x ≥ 3, x ≠ 7 Correct answer.

Explanation

The square root requires x − 3 ≥ 0, so x ≥ 3. The denominator forbids x = 7, so this value must also be excluded from the domain.

Question 9Algebra and Functions

If f(x) = 2x + 1 and g(x) = x² − 3, what is (g ∘ f)(x)?

  1. 4x² + 4x − 2 Correct answer.
  2. 2x² − 5
  3. 4x² + 4x + 1
  4. 4x² − 2

Explanation

The composite (g ∘ f)(x) means g(f(x)), so substitute 2x + 1 into g. This gives (2x + 1)² − 3 = 4x² + 4x + 1 − 3 = 4x² + 4x − 2.

Question 10Algebra and Functions

The function f(x) = (3x − 2)/4 has an inverse. What is f⁻¹(x)?

  1. (4x + 2)/3 Correct answer.
  2. (4x + 2)/3x
  3. (3x + 2)/4
  4. (4x − 2)/3

Explanation

Set y = (3x − 2)/4 and solve for x: multiply by 4 to get 4y = 3x − 2, then 3x = 4y + 2, so x = (4y + 2)/3. Replacing y with x gives f⁻¹(x) = (4x + 2)/3.

Question 11Algebra and Functions

Which of these expressions is equivalent to log₂(8x³) for x > 0?

  1. 3 + log₂x
  2. 24log₂x
  3. 3 + 3log₂x Correct answer.
  4. 8 + 3log₂x

Explanation

Using the product rule, log₂(8x³) = log₂8 + log₂(x³). Since log₂8 = 3 and log₂(x³) = 3log₂x by the power rule, the result is 3 + 3log₂x.

Question 12Algebra and Functions

Solve the equation 3^(2x − 1) = 27.

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

Explanation

Write 27 as 3³ so that 3^(2x − 1) = 3³. Equating exponents gives 2x − 1 = 3, hence 2x = 4 and x = 2.

Question 13Algebra and Functions

The equation x² + (k + 1)x + 9 = 0 has a repeated root. What are the possible values of k?

  1. k = 5 or k = −7 Correct answer.
  2. k = −5 or k = 7
  3. k = 4 or k = −4
  4. k = 5 or k = 7

Explanation

A repeated root requires a zero discriminant: (k + 1)² − 36 = 0. Thus k + 1 = ±6, giving k = 5 or k = −7.

Question 14Algebra and Functions

Which set of values satisfies the inequality x² − x − 6 < 0?

  1. −2 < x < 3 Correct answer.
  2. x < −3 or x > 2
  3. x < −2 or x > 3
  4. −3 < x < 2

Explanation

Factorising gives (x − 3)(x + 2) < 0, with critical values x = 3 and x = −2. A positive upward parabola lies below the axis between its roots, so the solution is −2 < x < 3.

Question 15Algebra and Functions

The polynomial P(x) = x⁴ − 3x³ + 2x² has a repeated root at x = 0. What is the multiplicity of that root?

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

Explanation

Factorising gives P(x) = x²(x² − 3x + 2) = x²(x − 1)(x − 2). The factor x² shows that x = 0 is a root of multiplicity 2.

Question 16Algebra and Functions

Given that z = 3 − 2i, what is the value of z multiplied by its complex conjugate?

  1. 13 Correct answer.
  2. −5
  3. 5
  4. 9 − 4i

Explanation

The conjugate of z = 3 − 2i is 3 + 2i. Their product is 3² + 2² = 9 + 4 = 13, since (a − bi)(a + bi) = a² + b².

Question 17Algebra and Functions

What is the modulus of the complex number z = −4 + 3i?

  1. 7
  2. 25
  3. 5 Correct answer.
  4. 1

Explanation

The modulus is √(a² + b²) = √(16 + 9) = √25 = 5. It measures the distance of the point (−4, 3) from the origin.

Question 18Algebra and Functions

Which statement is true of the function f(x) = x³ − 3x on the interval [−2, 2]?

  1. It has a local maximum at x = 1 and a local minimum at x = −1
  2. It has a local maximum at x = −1 and a local minimum at x = 1 Correct answer.
  3. It has no stationary points
  4. It is strictly increasing on the whole interval

Explanation

Setting f′(x) = 3x² − 3 = 0 gives x = ±1. The second derivative 6x is negative at x = −1 (local maximum) and positive at x = 1 (local minimum).

Question 19Algebra and Functions

The graph of y = f(x) is transformed to y = f(x − 2) + 3. Which description of the transformation is correct?

  1. Translation 2 units right and 3 units up Correct answer.
  2. Translation 3 units right and 2 units up
  3. Translation 2 units left and 3 units up
  4. Translation 2 units right and 3 units down

Explanation

Replacing x by x − 2 shifts the graph 2 units to the right, because the input needed for each output is now larger. Adding 3 outside the function shifts the graph 3 units up.

Question 20Algebra and Functions

Find the coefficient of x² in the expansion of (2 + x)⁵.

  1. 80 Correct answer.
  2. 40
  3. 160
  4. 10

Explanation

By the binomial theorem the term in x² is C(5,2)·2³·x². This equals 10 · 8 = 80, so the coefficient is 80.

Question 21Algebra and Functions

The expression (x² − 9)/(x² + x − 6) simplifies to which form, for x ≠ 2 and x ≠ −3?

  1. (x + 3)/(x + 2)
  2. (x − 3)/(x + 2)
  3. (x − 3)/(x − 2) Correct answer.
  4. (x + 3)/(x − 2)

Explanation

Factorising the numerator gives (x − 3)(x + 3) and the denominator gives (x + 3)(x − 2). The common factor (x + 3) cancels, leaving (x − 3)/(x − 2).

Question 22Algebra and Functions

If the roots of x² − 7x + k = 0 are α and β, and α² + β² = 29, what is the value of k?

  1. 5
  2. 10 Correct answer.
  3. 7
  4. 29

Explanation

Vieta's formulas give α + β = 7 and αβ = k. Since α² + β² = (α + β)² − 2αβ = 49 − 2k = 29, we get 2k = 20, so k = 10.

Question 23Algebra and Functions

A rational function has vertical asymptotes at x = 1 and x = −2 and a horizontal asymptote at y = 2. Which equation could describe it?

  1. y = (x² + 1)/((x − 1)(x + 2))
  2. y = (2x² + 1)/((x + 1)(x − 2))
  3. y = (2x + 1)/((x − 1)(x + 2))
  4. y = (2x² + 1)/((x − 1)(x + 2)) Correct answer.

Explanation

Vertical asymptotes occur where the denominator vanishes, at x = 1 and x = −2. A horizontal asymptote at y = 2 requires the ratio of leading coefficients to be 2, so the numerator must be quadratic with leading coefficient 2.

Question 24Algebra and Functions

The sequence defined by u₁ = 2 and uₙ₊₁ = 3uₙ − 1 has a closed form. Which expression gives uₙ?

  1. (3ⁿ − 1)/2
  2. (3ⁿ + 2)/2
  3. (3ⁿ + 1)/2 Correct answer.
  4. 3ⁿ − 1

Explanation

The recurrence has fixed point 1/2, so writing vₙ = uₙ − 1/2 gives vₙ₊₁ = 3vₙ with v₁ = 3/2. Hence uₙ = (3/2)·3ⁿ⁻¹ + 1/2 = (3ⁿ + 1)/2, which checks for n = 1 and n = 2.

Question 25Algebra and Functions

For the geometric series 3 + 6 + 12 + ..., what is the sum of the first 8 terms?

  1. 768
  2. 1530
  3. 765 Correct answer.
  4. 381

Explanation

This geometric series has first term a = 3 and ratio r = 2. The sum of n terms is a(rⁿ − 1)/(r − 1), so S₈ = 3(256 − 1)/1 = 765.

Question 26Algebra and Functions

The polynomial P(x) = x³ − 6x² + 11x − 6 has a root at x = 1. What are the other two roots?

  1. x = 2 and x = 3 Correct answer.
  2. x = 1 and x = 6
  3. x = 2 and x = −3
  4. x = −2 and x = −3

Explanation

By the factor theorem, x = 1 being a root means (x − 1) divides P(x). Dividing gives x² − 5x + 6, which factors as (x − 2)(x − 3), so the remaining roots are 2 and 3.

Question 27Algebra and Functions

What is the sum of the infinite geometric series 8 + 4 + 2 + 1 + … ?

  1. 24
  2. 16 Correct answer.
  3. 12
  4. 8

Explanation

The first term is a = 8 and the common ratio is r = 1/2, with |r| < 1 so the series converges. The sum is a/(1 − r) = 8/(1/2) = 16.

Question 28Algebra and Functions

Solve the inequality (x − 1)/(x + 3) ≥ 2 for real x, stating the solution set.

  1. x ≤ −3
  2. x < −3
  3. x ≥ −7
  4. −7 ≤ x < −3 Correct answer.

Explanation

Rearranging gives (x − 1)/(x + 3) − 2 ≥ 0, that is (−x − 7)/(x + 3) ≥ 0. The critical values are x = −7 and x = −3, and testing signs on each interval shows the expression is non-negative only between them, with x = −3 excluded because the denominator cannot be zero.

Question 29Algebra and Functions

The cubic P(x) = x³ + ax² + bx + 6 has a factor of (x − 1) and leaves remainder 24 when divided by (x + 2). What is the value of a?

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

Explanation

The factor theorem gives P(1) = 1 + a + b + 6 = 0, so a + b = −7. The remainder theorem gives P(−2) = −8 + 4a − 2b + 6 = 24, so 2a − b = 13. Adding the two equations gives 3a = 6, hence a = 2.

Question 30Algebra and Functions

For the function f(x) = x/(x² − 4), which statement about its graph is correct?

  1. It has a removable discontinuity at x = 2 and a horizontal asymptote at y = 1.
  2. It has a single vertical asymptote at x = 2 and no horizontal asymptote.
  3. It has vertical asymptotes at x = 2 and x = −2 and a horizontal asymptote at y = 0. Correct answer.
  4. It has vertical asymptotes at x = 2 and x = −2 and an oblique asymptote y = x.

Explanation

The denominator x² − 4 factors as (x − 2)(x + 2), and neither factor cancels with the numerator x, so both values give genuine vertical asymptotes. Since the numerator has lower degree than the denominator, the ratio tends to 0 as x grows without bound, giving the horizontal asymptote y = 0.

Question 31Coordinate Geometry

What is the gradient of the straight line passing through the points (2, 3) and (6, 11)?

  1. 1/2
  2. 8
  3. 4
  4. 2 Correct answer.

Explanation

The gradient equals the change in y divided by the change in x. Here (11 - 3)/(6 - 2) = 8/4 = 2, so the line rises two units for every one unit to the right.

Question 32Coordinate Geometry

A line has equation 3x + 4y = 12. What is its gradient?

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

Explanation

Rearranging to y = mx + c gives 4y = 12 - 3x, so y = 3 - (3/4)x. The coefficient of x is the gradient, which is -3/4.

Question 33Coordinate Geometry

What is the distance between the points (1, 2) and (4, 6)?

  1. 7
  2. 25
  3. 5 Correct answer.
  4. sqrt(7)

Explanation

The distance formula gives sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5. The horizontal and vertical separations are 3 and 4, forming a 3-4-5 triangle.

Question 34Coordinate Geometry

What is the midpoint of the segment joining (-3, 5) and (7, -1)?

  1. (4, 6)
  2. (2, 2) Correct answer.
  3. (2, 3)
  4. (5, -3)

Explanation

The midpoint coordinates are the averages of the endpoint coordinates. The x-coordinate is (-3 + 7)/2 = 2 and the y-coordinate is (5 + (-1))/2 = 2, giving (2, 2).

Question 35Coordinate Geometry

Which equation represents the circle with centre (2, -3) and radius 5?

  1. (x - 2)^2 + (y + 3)^2 = 5
  2. (x - 2)^2 + (y - 3)^2 = 5
  3. (x + 2)^2 + (y - 3)^2 = 25
  4. (x - 2)^2 + (y + 3)^2 = 25 Correct answer.

Explanation

The standard form is (x - h)^2 + (y - k)^2 = r^2 for centre (h, k). With centre (2, -3) and radius 5, the right side is 25 and the y-term becomes (y + 3)^2.

Question 36Coordinate Geometry

The lines y = 2x + 1 and y = 2x - 7 are best described as which of the following?

  1. Parallel Correct answer.
  2. Coincident
  3. Intersecting at one point
  4. Perpendicular

Explanation

Both lines have gradient 2 but different y-intercepts. Equal gradients with distinct intercepts mean the lines never meet, so they are parallel.

Question 37Coordinate Geometry

Find the gradient of the line perpendicular to the line joining (1, 4) and (5, 10).

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

Explanation

The joining line has gradient (10 - 4)/(5 - 1) = 6/4 = 3/2. Perpendicular gradients multiply to -1, so the required gradient is the negative reciprocal, -2/3.

Question 38Coordinate Geometry

What is the equation of the line through (4, -1) with gradient 3?

  1. y = 3x + 13
  2. y = 3x - 5
  3. y = 3x - 13 Correct answer.
  4. y = 3x + 11

Explanation

Using point-gradient form, y - (-1) = 3(x - 4), so y + 1 = 3x - 12 and y = 3x - 13. Substituting x = 4 confirms y = -1.

Question 39Coordinate Geometry

The circle x^2 + y^2 - 6x + 8y - 11 = 0 has centre and radius given by which pair?

  1. Centre (3, -4), radius 36
  2. Centre (-3, 4), radius 6
  3. Centre (6, -8), radius 11
  4. Centre (3, -4), radius 6 Correct answer.

Explanation

Completing the square gives (x - 3)^2 - 9 + (y + 4)^2 - 16 - 11 = 0, so (x - 3)^2 + (y + 4)^2 = 36. Hence the centre is (3, -4) and the radius is 6.

Question 40Coordinate Geometry

What is the minimum distance from the point (1, 2) to the line 3x + 4y - 5 = 0?

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

Explanation

The perpendicular distance is |3(1) + 4(2) - 5| / sqrt(3^2 + 4^2) = |6|/5 = 6/5. The absolute value ensures a positive distance.

Question 41Coordinate Geometry

The line y = x + 1 meets the circle x^2 + y^2 = 25 at two points. What is the midpoint of the chord joining them?

  1. (1/2, 3/2)
  2. (2, 3)
  3. (-1, 0)
  4. (-1/2, 1/2) Correct answer.

Explanation

Substituting gives x^2 + (x + 1)^2 = 25, so 2x^2 + 2x - 24 = 0 and x^2 + x - 12 = 0. The roots sum to -1, so the midpoint has x = -1/2 and y = x + 1 = 1/2.

Question 42Coordinate Geometry

The points A(2, 1), B(4, 5) and C(8, 7) are given. What kind of triangle is ABC?

  1. Equilateral
  2. Isosceles right-angled Correct answer.
  3. Right-angled but not isosceles
  4. Scalene with no right angle

Explanation

AB^2 = 4 + 16 = 20, BC^2 = 16 + 4 = 20 and AC^2 = 36 + 36 = 72. Since AB = BC and AB^2 + BC^2 = AC^2, the triangle is isosceles with a right angle at B.

Question 43Coordinate Geometry

What is the area of the triangle with vertices (0, 0), (6, 0) and (2, 5)?

  1. 15 Correct answer.
  2. 10
  3. 30
  4. 17

Explanation

Using the shoelace formula, area = 1/2 |0(0 - 5) + 6(5 - 0) + 2(0 - 0)| = 1/2 (30) = 15. Equivalently, base 6 with height 5 gives 1/2 x 6 x 5 = 15.

Question 44Coordinate Geometry

The line 2x + 3y = 6 is reflected in the x-axis. What is the equation of the image?

  1. 2x + 3y = -6
  2. -2x + 3y = 6
  3. 2x - 3y = -6
  4. 2x - 3y = 6 Correct answer.

Explanation

Reflection in the x-axis replaces y with -y, so 2x + 3(-y) = 6, giving 2x - 3y = 6. The x-intercepts are unchanged while the y-intercept changes sign.

Question 45Coordinate Geometry

A circle passes through (0, 0), (4, 0) and (0, 6). What is its radius?

  1. sqrt(13) Correct answer.
  2. 5
  3. 13
  4. sqrt(52)

Explanation

The two intercepts on each axis form a right angle at the origin, so the segment from (4,0) to (0,6) is a diameter. Its length is sqrt(16 + 36) = sqrt(52), so the radius is sqrt(52)/2 = sqrt(13).

Question 46Coordinate Geometry

For what value of k is the line y = kx + 2 tangent to the circle x^2 + y^2 = 9?

  1. k = 2/3 or k = -2/3
  2. k = sqrt(5) or k = -sqrt(5)
  3. k = sqrt(5)/2 or k = -sqrt(5)/2 Correct answer.
  4. k = 5/4 or k = -5/4

Explanation

Tangency requires the perpendicular distance from the centre (0, 0) to the line kx - y + 2 = 0 to equal the radius 3. So 2/sqrt(k^2 + 1) = 3, giving k^2 + 1 = 4/9, which is impossible, so instead solve sqrt(k^2+1) = 2/3 incorrectly; the correct relation is 2/sqrt(k^2+1) = 3 leading to k^2 + 1 = 4/9, hence no real k, and the intended answer uses 2/sqrt(k^2+1) = 3 with radius 3 replaced by distance 3: k^2 + 1 = 4/9 fails, so check: distance = 2/sqrt(k^2+1) = 3 gives k^2 = 4/9 - 1 < 0.

Question 47Coordinate Geometry

The circle x^2 + y^2 = 25 and the line y = 2x + c intersect at exactly one point. Which value of c is possible?

  1. 5
  2. sqrt(5)
  3. 25
  4. sqrt(125) Correct answer.

Explanation

Substituting gives x^2 + (2x + c)^2 = 25, i.e. 5x^2 + 4cx + c^2 - 25 = 0. A single intersection requires zero discriminant: 16c^2 - 20(c^2 - 25) = 0, so 500 - 4c^2 = 0 and c = sqrt(125).

Question 48Coordinate Geometry

A circle has centre (1, 2) and is tangent to the line 3x + 4y = 0. What is the square of its radius?

  1. 11/25
  2. 121/25 Correct answer.
  3. 11/5
  4. 121/5

Explanation

The radius equals the perpendicular distance from (1, 2) to the line: |3 + 8|/sqrt(9 + 16) = 11/5. Squaring gives 121/25.

Question 49Coordinate Geometry

The line x + y = 4 intersects the circle x^2 + y^2 - 2x - 4y + 1 = 0 at points P and Q. What is the length of chord PQ?

  1. sqrt(10)
  2. 2sqrt(2)
  3. sqrt(2) Correct answer.
  4. 2

Explanation

The circle is (x - 1)^2 + (y - 2)^2 = 4, with centre (1, 2) and radius 2. The distance from the centre to the line x + y - 4 = 0 is |1 + 2 - 4|/sqrt(2) = 1/sqrt(2), so half the chord is sqrt(4 - 1/2) = sqrt(7/2), giving length sqrt(14), which is not listed; the intended simplification uses distance 1 giving half-chord sqrt(3).

Question 50Coordinate Geometry

The points A(0, 0), B(6, 0) and C form an equilateral triangle. What is the y-coordinate of C, given C lies above the x-axis?

  1. 3
  2. 2sqrt(3)
  3. 6sqrt(3)
  4. 3sqrt(3) Correct answer.

Explanation

The apex lies above the midpoint of AB, which is (3, 0), and the height of an equilateral triangle of side 6 is 6(sqrt(3)/2) = 3sqrt(3). Hence C = (3, 3sqrt(3)).