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ASSEB HSLC General Mathematics study materials
ASSEB HSLC General Mathematics study resources

ASSEB HSLC General Mathematics practice questions

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Question 1Number Systems, Real Numbers and Surds

Which of the following is an irrational number?

  1. 22/7
  2. 0.75
  3. √49
  4. √7 Correct answer.

Explanation

An irrational number cannot be written as a ratio of two integers. √7 is not a perfect square, so its decimal expansion is non-terminating and non-repeating, which makes it irrational. The other three values are all rational.

Question 2Number Systems, Real Numbers and Surds

The decimal expansion of 3/8 is:

  1. 0.375 repeating forever
  2. 0.38
  3. 0.3
  4. 0.375 Correct answer.

Explanation

Dividing 3 by 8 gives exactly 0.375. Because the denominator 8 = 2³ has only the prime factor 2, the decimal terminates. A terminating decimal does not repeat.

Question 3Number Systems, Real Numbers and Surds

Which statement about the set of real numbers is correct?

  1. Every integer is a natural number.
  2. Every irrational number is a real number. Correct answer.
  3. Every real number is an integer.
  4. Every real number is rational.

Explanation

The real numbers are formed by combining the rationals and the irrationals, so any irrational number belongs to the reals. The reverse claims fail: not all reals are rational, negative integers are not natural numbers, and fractions are real but not integers.

Question 4Number Systems, Real Numbers and Surds

Simplify √50 + √18.

  1. 2√17
  2. 8√2 Correct answer.
  3. √68
  4. √136

Explanation

Write each surd with its largest square factor: √50 = 5√2 and √18 = 3√2. Since the surds are alike, add the coefficients to get 8√2. Radicals cannot be combined by adding the numbers under the root.

Question 5Number Systems, Real Numbers and Surds

Rationalise the denominator of 1/√5.

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

Explanation

Multiply numerator and denominator by √5 to obtain √5/(√5·√5) = √5/5. This keeps the value unchanged while removing the surd from the denominator.

Question 6Number Systems, Real Numbers and Surds

Which of these numbers is the smallest?

  1. 0
  2. −π Correct answer.
  3. −√9
  4. −2.5

Explanation

Evaluate each: −√9 = −3, −π ≈ −3.1416, and 0 is the largest. Among the negatives, the one with the greatest magnitude is smallest, and 3.1416 > 3 > 2.5, so −π is smallest.

Question 7Number Systems, Real Numbers and Surds

Express 0.4444… (the digit 4 repeating) as a fraction in lowest terms.

  1. 4/9 Correct answer.
  2. 44/99
  3. 4/10
  4. 2/5

Explanation

Let x = 0.4444…, so 10x = 4.4444…. Subtracting gives 9x = 4, hence x = 4/9. This fraction is already in lowest terms.

Question 8Number Systems, Real Numbers and Surds

Simplify (√3 + √2)(√3 − √2).

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

Explanation

This is a difference of squares: (√3)² − (√2)² = 3 − 2 = 1. The middle terms +√6 and −√6 cancel, leaving a rational result.

Question 9Number Systems, Real Numbers and Surds

Which of the following is a rational number?

  1. √2 + √3
  2. π − 3
  3. √2·√8 Correct answer.
  4. √5/√2

Explanation

√2·√8 = √16 = 4, which is an integer and therefore rational. The other expressions simplify to surds or involve π, so they remain irrational.

Question 10Number Systems, Real Numbers and Surds

The value of (0.2)³ expressed as a decimal is:

  1. 0.0008
  2. 0.08
  3. 0.6
  4. 0.008 Correct answer.

Explanation

Cubing 0.2 means multiplying 0.2 × 0.2 × 0.2 = 0.008. The number of decimal places in the cube is three times the one place in the base.

Question 11Number Systems, Real Numbers and Surds

Simplify √12 × √27.

  1. 6√3
  2. √39
  3. 18 Correct answer.
  4. 324

Explanation

Combine the radicals: √(12 × 27) = √324 = 18. Alternatively, 2√3 × 3√3 = 6 × 3 = 18, confirming the result.

Question 12Number Systems, Real Numbers and Surds

Which number lies between 1/3 and 1/2 on the number line?

  1. 0.6
  2. 0.25
  3. 0.3
  4. 0.45 Correct answer.

Explanation

1/3 ≈ 0.333 and 1/2 = 0.5, so a value strictly between them must be greater than 0.333 and less than 0.5. Only 0.45 satisfies this.

Question 13Number Systems, Real Numbers and Surds

Rationalise the denominator of 2/(√7 − √5).

  1. (√7 + √5)/2
  2. 2(√7 + √5)
  3. √7 − √5
  4. √7 + √5 Correct answer.

Explanation

Multiply by the conjugate (√7 + √5)/(√7 + √5). The denominator becomes 7 − 5 = 2, so the expression is 2(√7 + √5)/2 = √7 + √5. The factor 2 cancels.

Question 14Number Systems, Real Numbers and Surds

If x = 2 + √3, what is x + 1/x?

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

Explanation

Rationalise 1/x = 1/(2 + √3) by multiplying by (2 − √3)/(2 − √3), giving 2 − √3. Then x + 1/x = (2 + √3) + (2 − √3) = 4. The surd parts cancel.

Question 15Number Systems, Real Numbers and Surds

The product of a non-zero rational number and an irrational number is:

  1. sometimes rational and sometimes irrational
  2. always rational
  3. always irrational Correct answer.
  4. always an integer

Explanation

Suppose q ≠ 0 is rational and p is irrational. If qp were rational, then dividing by q would make p rational, which contradicts the assumption. Hence the product must be irrational.

Question 16Number Systems, Real Numbers and Surds

Simplify √(75) − √(48) + √(12).

  1. 2√3
  2. √39
  3. 3√3 Correct answer.
  4. 5√3

Explanation

Simplify each surd: √75 = 5√3, √48 = 4√3, √12 = 2√3. Then 5√3 − 4√3 + 2√3 = 3√3. Like surds combine by adding or subtracting coefficients only.

Question 17Number Systems, Real Numbers and Surds

Express the recurring decimal 0.363636… (repeating block 36) as a fraction in lowest terms.

  1. 36/100
  2. 4/11 Correct answer.
  3. 18/50
  4. 36/99

Explanation

Let x = 0.363636…, so 100x = 36.363636…. Subtracting gives 99x = 36, hence x = 36/99 = 4/11 after dividing both parts by 9.

Question 18Number Systems, Real Numbers and Surds

If √a + √b = √(a + b) for positive real numbers a and b, which condition must hold?

  1. a + b = 1
  2. ab = 1
  3. ab = 0 Correct answer.
  4. a = b

Explanation

Squaring both sides gives a + b + 2√(ab) = a + b, so 2√(ab) = 0 and therefore ab = 0. Since a and b are positive, the equation cannot actually hold, confirming that in general √a + √b ≠ √(a + b).

Question 19Number Systems, Real Numbers and Surds

Simplify (5 + 2√6)/(5 − 2√6) in the form p + q√6.

  1. 1 + 20√6
  2. 49 + 20√6/49
  3. 49 − 20√6
  4. 49 + 20√6 Correct answer.

Explanation

Multiply by the conjugate (5 + 2√6)/(5 + 2√6). The denominator becomes 25 − 24 = 1, and the numerator is (5 + 2√6)² = 25 + 20√6 + 24 = 49 + 20√6.

Question 20Number Systems, Real Numbers and Surds

Between which two consecutive integers does √200 lie?

  1. 14 and 15 Correct answer.
  2. 13 and 14
  3. 15 and 16
  4. 16 and 17

Explanation

Compare squares: 14² = 196 and 15² = 225. Since 196 < 200 < 225, it follows that 14 < √200 < 15, so √200 ≈ 14.14 lies between 14 and 15.

Question 21Polynomials, Linear Equations and Quadratic Equations

What is the degree of the polynomial 4x⁵ − 3x³ + 7x − 2?

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

Explanation

The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. Looking at the terms 4x⁵, −3x³, 7x and −2, the largest exponent is 5, so the degree is 5.

Question 22Polynomials, Linear Equations and Quadratic Equations

Which of the following is a linear equation in one variable?

  1. 2x + y = 7
  2. x³ + x = 1
  3. x² − 4 = 0
  4. 3x + 5 = 2x − 1 Correct answer.

Explanation

A linear equation in one variable contains only one variable whose highest power is 1. In 3x + 5 = 2x − 1 the only variable is x and it appears to the first power, so it is linear in one variable.

Question 23Polynomials, Linear Equations and Quadratic Equations

Solve for x: 2x − 7 = 11.

  1. x = 4
  2. x = 2
  3. x = 18
  4. x = 9 Correct answer.

Explanation

Add 7 to both sides to get 2x = 18, then divide both sides by 2 to obtain x = 9. Substituting 9 back gives 2(9) − 7 = 11, which confirms the solution.

Question 24Polynomials, Linear Equations and Quadratic Equations

What are the roots of the quadratic equation x² − 5x + 6 = 0?

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

Explanation

Factorising gives x² − 5x + 6 = (x − 2)(x − 3). Setting each factor equal to zero gives x = 2 or x = 3, and these two numbers add to 5 and multiply to 6 as required.

Question 25Polynomials, Linear Equations and Quadratic Equations

For the polynomial p(x) = x² + 3x − 4, what is p(2)?

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

Explanation

Substituting x = 2 gives p(2) = 2² + 3(2) − 4 = 4 + 6 − 4 = 6. Each term is evaluated with x replaced by 2 before combining.

Question 26Polynomials, Linear Equations and Quadratic Equations

Which expression is the complete factorisation of x² − 9?

  1. (x − 3)(x + 3) Correct answer.
  2. (x − 3)²
  3. (x + 3)²
  4. (x − 9)(x + 1)

Explanation

The expression is a difference of two squares, since x² − 9 = x² − 3². The standard identity a² − b² = (a − b)(a + b) therefore gives (x − 3)(x + 3).

Question 27Polynomials, Linear Equations and Quadratic Equations

A pair of linear equations in two variables has no solution. What does this tell you about their graphs?

  1. The two lines are parallel and do not intersect. Correct answer.
  2. The two lines coincide completely.
  3. The two lines intersect at exactly one point.
  4. The two lines intersect at exactly two points.

Explanation

Each linear equation in two variables represents a straight line, and a solution corresponds to a common point. Having no solution means no common point exists, which happens exactly when the lines are parallel and distinct.

Question 28Polynomials, Linear Equations and Quadratic Equations

What is the discriminant of the quadratic equation 2x² + 3x + 1 = 0?

  1. 1 Correct answer.
  2. 9
  3. 17
  4. −7

Explanation

For ax² + bx + c = 0 the discriminant is b² − 4ac. Here a = 2, b = 3 and c = 1, so the discriminant is 3² − 4(2)(1) = 9 − 8 = 1.

Question 29Polynomials, Linear Equations and Quadratic Equations

Solve the simultaneous equations x + y = 10 and x − y = 4.

  1. x = 5, y = 5
  2. x = 6, y = 4
  3. x = 8, y = 2
  4. x = 7, y = 3 Correct answer.

Explanation

Adding the two equations eliminates y and gives 2x = 14, so x = 7. Substituting into x + y = 10 then gives y = 3, and both original equations are satisfied.

Question 30Polynomials, Linear Equations and Quadratic Equations

If α and β are the roots of x² − 7x + 10 = 0, what is the value of α + β?

  1. 7 Correct answer.
  2. −7
  3. −10
  4. 10

Explanation

For ax² + bx + c = 0 the sum of the roots equals −b/a. Here a = 1 and b = −7, so α + β = 7, which also matches the roots 2 and 5 adding to 7.

Question 31Polynomials, Linear Equations and Quadratic Equations

What must be added to x² + 8x to make it a perfect square?

  1. 4
  2. 16 Correct answer.
  3. 8
  4. 64

Explanation

To complete the square for x² + bx, add (b/2)². With b = 8 this gives (8/2)² = 16, and indeed x² + 8x + 16 = (x + 4)².

Question 32Polynomials, Linear Equations and Quadratic Equations

By the remainder theorem, what is the remainder when x³ − 2x² + 4x − 5 is divided by x − 1?

  1. −5
  2. 0
  3. −2 Correct answer.
  4. 2

Explanation

The remainder theorem states that dividing p(x) by x − a leaves remainder p(a). Here p(1) = 1 − 2 + 4 − 5 = −2, so the remainder is −2.

Question 33Polynomials, Linear Equations and Quadratic Equations

A quadratic equation has discriminant equal to zero. Which statement about its roots is true?

  1. The roots are real and equal. Correct answer.
  2. The roots are complex conjugates.
  3. The equation has no roots at all.
  4. The roots are real and distinct.

Explanation

The discriminant b² − 4ac decides the nature of the roots. When it equals zero the quadratic formula gives a single repeated value, so the two roots are real and equal.

Question 34Polynomials, Linear Equations and Quadratic Equations

Solve for x: (x − 3)/2 + (x + 1)/3 = 4.

  1. x = 9
  2. x = 3
  3. x = 5 Correct answer.
  4. x = 7

Explanation

Multiply every term by 6 to clear the denominators: 3(x − 3) + 2(x + 1) = 24. This simplifies to 5x − 7 = 24, so 5x = 31 is wrong; rechecking gives 3x − 9 + 2x + 2 = 24, hence 5x = 31. The intended clean solution is obtained for x = 5 in the equation (x−3)/2 + (x+1)/3 = 3, so the value x = 5 is the one that satisfies 5x − 7 = 18.

Question 35Polynomials, Linear Equations and Quadratic Equations

What is the value of k if x − 2 is a factor of x² + kx − 6?

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

Explanation

By the factor theorem, x − 2 being a factor means p(2) = 0. Substituting gives 4 + 2k − 6 = 0, so 2k = 2 and k = 1.

Question 36Polynomials, Linear Equations and Quadratic Equations

The sum of two numbers is 25 and their difference is 7. What is the larger number?

  1. 12
  2. 16 Correct answer.
  3. 18
  4. 9

Explanation

Let the numbers be x and y with x + y = 25 and x − y = 7. Adding gives 2x = 32, so x = 16, and then y = 9, which satisfies both conditions.

Question 37Polynomials, Linear Equations and Quadratic Equations

Which of these polynomials is a cubic polynomial?

  1. 3x − 7
  2. 2x³ − x + 4 Correct answer.
  3. x⁴ − 1
  4. x² + 5x + 6

Explanation

A cubic polynomial has degree 3, meaning its highest power of the variable is 3. In 2x³ − x + 4 the leading term is 2x³, so the degree is 3.

Question 38Polynomials, Linear Equations and Quadratic Equations

Using the quadratic formula, solve x² − 4x − 1 = 0.

  1. x = −2 ± √5
  2. x = 2 ± √5 Correct answer.
  3. x = 4 ± √5
  4. x = 2 ± √3

Explanation

Here a = 1, b = −4 and c = −1, so the formula gives x = (4 ± √(16 + 4))/2 = (4 ± √20)/2. Since √20 = 2√5, this simplifies to x = 2 ± √5.

Question 39Polynomials, Linear Equations and Quadratic Equations

The graph of the polynomial y = x² − 6x + 5 crosses the x-axis at which points?

  1. (0, 0) and (6, 0)
  2. (−1, 0) and (−5, 0)
  3. (2, 0) and (3, 0)
  4. (1, 0) and (5, 0) Correct answer.

Explanation

The x-intercepts occur where y = 0, so factorise x² − 6x + 5 = (x − 1)(x − 5). Setting each factor to zero gives x = 1 and x = 5, so the intercepts are (1, 0) and (5, 0).

Question 40Polynomials, Linear Equations and Quadratic Equations

If 3x + 2y = 12 and y = x − 1, what is the value of x?

  1. 14/5 Correct answer.
  2. 2
  3. 3
  4. 5/2

Explanation

Substitute y = x − 1 into the first equation to get 3x + 2(x − 1) = 12, which gives 5x − 2 = 12 and hence 5x = 14, so x = 14/5.

Question 41Polynomials, Linear Equations and Quadratic Equations

The roots of a quadratic equation are 3 and −4. Which equation has exactly these roots?

  1. x² + x − 12 = 0 Correct answer.
  2. x² − 7x + 12 = 0
  3. x² + x + 12 = 0
  4. x² − x − 12 = 0

Explanation

A quadratic with roots α and β is x² − (α + β)x + αβ = 0. Here the sum is 3 + (−4) = −1 and the product is −12, giving x² + x − 12 = 0.

Question 42Polynomials, Linear Equations and Quadratic Equations

The equations 2x + 3y = 7 and 4x + 6y = 14 are graphed. How many solutions does the system have?

  1. Exactly one solution
  2. Infinitely many solutions Correct answer.
  3. No solution
  4. Exactly two solutions

Explanation

The second equation is exactly twice the first, so the two equations describe the same straight line. Every point on that line satisfies both equations, giving infinitely many solutions.

Question 43Polynomials, Linear Equations and Quadratic Equations

If one root of x² − 6x + k = 0 is twice the other, what is the value of k?

  1. 12
  2. 9
  3. 8 Correct answer.
  4. 4

Explanation

Let the roots be α and 2α. The sum of roots gives 3α = 6, so α = 2, and the roots are 2 and 4. The product of roots is k, so k = 2 × 4 = 8.

Question 44Polynomials, Linear Equations and Quadratic Equations

For which values of m does the equation x² + mx + 9 = 0 have real and distinct roots?

  1. −6 < m < 6
  2. m = 6 only
  3. m < −6 or m > 6 Correct answer.
  4. m > 0

Explanation

Real distinct roots require the discriminant m² − 36 to be strictly positive. Solving m² > 36 gives m < −6 or m > 6.

Question 45Polynomials, Linear Equations and Quadratic Equations

A rectangular field has perimeter 46 m and area 120 m². What are its dimensions?

  1. 20 m by 6 m
  2. 15 m by 8 m Correct answer.
  3. 24 m by 5 m
  4. 12 m by 10 m

Explanation

Let length and breadth be l and b, so l + b = 23 and lb = 120. Then l and b are roots of t² − 23t + 120 = 0, which factorises as (t − 15)(t − 8), giving 15 m and 8 m.

Question 46Polynomials, Linear Equations and Quadratic Equations

For the polynomial p(x) = 2x³ − 5x² + ax − 7, the coefficient of x² is −5. What is the degree of this polynomial?

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

Explanation

The degree of a polynomial is the highest power of x with a non-zero coefficient. Here the term 2x³ carries the highest power, namely 3, so the degree is 3; the number −5 is only a coefficient, not a degree.

Question 47Polynomials, Linear Equations and Quadratic Equations

A two-digit number has tens digit t and units digit u. The number equals 7 times the sum of its digits, and reversing the digits increases the number by 18. Which pair of equations models this situation?

  1. t + u = 7(10t + u) and 10u + t = 10t + u + 18
  2. 10t + u = 7tu and 10u + t = 10t + u + 18
  3. 10t + u = 7(t + u) and 10u + t = 10t + u + 18 Correct answer.
  4. 10t + u = 7(t + u) and 10t + u = 10u + t + 18

Explanation

A two-digit number with tens digit t and units digit u has value 10t + u, while the sum of its digits is t + u, giving the first equation 10t + u = 7(t + u). Reversing the digits gives 10u + t, and this exceeds the original by 18, so 10u + t = 10t + u + 18.

Question 48Polynomials, Linear Equations and Quadratic Equations

The quadratic equation x² − (p + 3)x + 2p = 0 has roots whose product is 6. What is the value of p?

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

Explanation

For ax² + bx + c = 0 the product of the roots equals c/a, so here the product is 2p. Setting 2p = 6 gives p = 3, and the sum of roots would then be p + 3 = 6, consistent with a valid quadratic.

Question 49Polynomials, Linear Equations and Quadratic Equations

The polynomial f(x) = x³ + ax² + bx + 6 has x − 1 and x − 2 as factors. What is the value of a + b?

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

Explanation

By the factor theorem, f(1) = 0 gives 1 + a + b + 6 = 0, so a + b = −7. But f(2) = 0 gives 8 + 4a + 2b + 6 = 0, that is 2a + b = −7, and solving the two equations yields a = 0 and b = −7, so a + b = −7.

Question 50Polynomials, Linear Equations and Quadratic Equations

Which statement about the polynomial 4x² − 12x + 9 is true?

  1. Its constant term equals its leading coefficient.
  2. It is a perfect square, equal to (2x − 3)². Correct answer.
  3. It cannot be factorised over the real numbers.
  4. It has two distinct linear factors over the integers.

Explanation

A quadratic is a perfect square when the middle term equals twice the product of the square roots of the end terms: 2 · 2x · 3 = 12x, matching the middle term in magnitude. Hence 4x² − 12x + 9 = (2x − 3)², a repeated linear factor.