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NCEA Pāngarau Co-requisite (NZQA Standard 32412) study materials
NCEA Pāngarau Co-requisite (NZQA Standard 32412) study resources

NCEA Pāngarau Co-requisite (NZQA Standard 32412) practice questions

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Question 1Number and Algebra (Te Tau me te Taurangi)

What is the value of 5 + 3 × 4 − 2?

  1. 15 Correct answer.
  2. 24
  3. 14
  4. 30

Explanation

Order of operations (whakaraupapa mahi) requires multiplication before addition and subtraction. So 3 × 4 = 12, then 5 + 12 − 2 = 15. Working strictly left to right gives the wrong answer.

Question 2Number and Algebra (Te Tau me te Taurangi)

Which number is a prime number?

  1. 33
  2. 17 Correct answer.
  3. 27
  4. 21

Explanation

A prime number (tau toitū) has exactly two factors: 1 and itself. 17 has no factors other than 1 and 17. The other options are all odd multiples of 3.

Question 3Number and Algebra (Te Tau me te Taurangi)

A jacket costs $80. It is discounted by 25%. What is the sale price?

  1. $60 Correct answer.
  2. $20
  3. $105
  4. $55

Explanation

A 25% discount means the buyer pays 75% of the original price. 0.75 × 80 = 60, so the sale price is $60. Alternatively, 25% of 80 is 20, and 80 − 20 = 60.

Question 4Number and Algebra (Te Tau me te Taurangi)

What is the highest common factor (HCF) of 36 and 48?

  1. 6
  2. 12 Correct answer.
  3. 4
  4. 144

Explanation

List the factors: 36 has 1, 2, 3, 4, 6, 9, 12, 18, 36 and 48 has 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest factor in both lists is 12.

Question 5Number and Algebra (Te Tau me te Taurangi)

Simplify 4x + 7 − 2x + 3.

  1. 2x + 4
  2. 9x
  3. 6x + 10
  4. 2x + 10 Correct answer.

Explanation

Collect like terms: 4x − 2x = 2x and 7 + 3 = 10. The simplified expression is 2x + 10. Terms in x and constant terms cannot be combined with each other.

Question 6Number and Algebra (Te Tau me te Taurangi)

What is the value of 2³ + √49?

  1. 18
  2. 23
  3. 15 Correct answer.
  4. 11

Explanation

2³ means 2 × 2 × 2 = 8 and √49 = 7 because 7 × 7 = 49. Adding gives 8 + 7 = 15.

Question 7Number and Algebra (Te Tau me te Taurangi)

Convert 0.375 to a fraction in simplest form.

  1. 37/100
  2. 3/7
  3. 3/8 Correct answer.
  4. 1/3

Explanation

0.375 = 375/1000. Dividing numerator and denominator by 125 gives 3/8. Checking: 3 ÷ 8 = 0.375.

Question 8Number and Algebra (Te Tau me te Taurangi)

A car travels 180 km in 2.5 hours. What is its average speed?

  1. 45 km/h
  2. 90 km/h
  3. 450 km/h
  4. 72 km/h Correct answer.

Explanation

Average speed = distance ÷ time = 180 ÷ 2.5. Dividing by 2.5 is the same as multiplying by 0.4, giving 72 km/h.

Question 9Number and Algebra (Te Tau me te Taurangi)

Expand and simplify (x + 3)(x − 5).

  1. x² − 8x − 15
  2. x² − 15
  3. x² + 2x − 15
  4. x² − 2x − 15 Correct answer.

Explanation

Multiply each term in the first bracket by each term in the second: x·x = x², x·(−5) = −5x, 3·x = 3x, 3·(−5) = −15. Combining −5x + 3x gives −2x, so the result is x² − 2x − 15.

Question 10Number and Algebra (Te Tau me te Taurangi)

Solve the equation 3x − 7 = 14.

  1. x = 2.33
  2. x = 21
  3. x = 7 Correct answer.
  4. x = 3

Explanation

Add 7 to both sides to get 3x = 21, then divide both sides by 3 to get x = 7. Substituting back: 3(7) − 7 = 14, which checks.

Question 11Number and Algebra (Te Tau me te Taurangi)

Simplify (2a³b²) × (5a²b⁴).

  1. 10a⁶b⁸
  2. 7a⁵b⁶
  3. 10a⁵b⁶ Correct answer.
  4. 10a⁵b⁸

Explanation

Multiply coefficients: 2 × 5 = 10. When multiplying powers with the same base, add the exponents: a³·a² = a⁵ and b²·b⁴ = b⁶. The result is 10a⁵b⁶.

Question 12Number and Algebra (Te Tau me te Taurangi)

A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people, at the same rate?

  1. 3000 g
  2. 600 g
  3. 120 g
  4. 750 g Correct answer.

Explanation

This is a direct proportion. Flour per person is 300 ÷ 4 = 75 g, so for 10 people the amount is 75 × 10 = 750 g. Equivalently, 300 × (10/4) = 750.

Question 13Number and Algebra (Te Tau me te Taurangi)

Find the gradient (slope) of the line passing through (2, 5) and (6, 13).

  1. 8
  2. 4
  3. 0.5
  4. 2 Correct answer.

Explanation

Gradient = (change in y) ÷ (change in x) = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2. The line rises 8 units for every 4 units to the right.

Question 14Number and Algebra (Te Tau me te Taurangi)

A shirt is reduced from $60 to $48. What percentage discount is this?

  1. 20% Correct answer.
  2. 25%
  3. 80%
  4. 12%

Explanation

The discount is 60 − 48 = $12. As a percentage of the original price: 12 ÷ 60 × 100 = 20%.

Question 15Number and Algebra (Te Tau me te Taurangi)

Factorise x² − 9.

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

Explanation

x² − 9 is a difference of two squares (x² − 3²), which factorises as (x − 3)(x + 3). Expanding confirms x² + 3x − 3x − 9 = x² − 9.

Question 16Number and Algebra (Te Tau me te Taurangi)

Rewrite 5⁴ ÷ 5² as a single power.

  1. 25⁶
  2. 5⁶
  3. 5⁸
  4. 5² Correct answer.

Explanation

When dividing powers with the same base, subtract the exponents: 5⁴ ÷ 5² = 5^(4−2) = 5². This equals 25, which checks against 625 ÷ 25.

Question 17Number and Algebra (Te Tau me te Taurangi)

A bag contains 5 red and 7 blue marbles. One marble is drawn at random. What is the probability it is red?

  1. 7/12
  2. 5/12 Correct answer.
  3. 5/2
  4. 5/7

Explanation

There are 5 red marbles out of 12 total marbles. Probability = favourable outcomes ÷ total outcomes = 5/12.

Question 18Number and Algebra (Te Tau me te Taurangi)

Solve the inequality 2x + 5 < 13.

  1. x < 6.5
  2. x < 9
  3. x > 4
  4. x < 4 Correct answer.

Explanation

Subtract 5 from both sides: 2x < 8. Divide both sides by 2 (a positive number, so the inequality sign stays the same): x < 4.

Question 19Number and Algebra (Te Tau me te Taurangi)

A rectangle has length (x + 4) and width (x − 2). Which expression gives its area?

  1. x² + 2x − 8 Correct answer.
  2. 2x + 2
  3. x² + 6x − 8
  4. x² − 8

Explanation

Area = length × width = (x + 4)(x − 2). Expanding: x² − 2x + 4x − 8 = x² + 2x − 8.

Question 20Number and Algebra (Te Tau me te Taurangi)

The nth term of a sequence is 4n − 3. What is the 12th term?

  1. 48
  2. 9
  3. 45 Correct answer.
  4. 41

Explanation

Substitute n = 12 into the rule: 4 × 12 − 3 = 48 − 3 = 45. The sequence is arithmetic with first term 1 and common difference 4.

Question 21Number and Algebra (Te Tau me te Taurangi)

Simplify (3x²y)³.

  1. 9x⁶y³
  2. 3x⁶y³
  3. 27x⁵y³
  4. 27x⁶y³ Correct answer.

Explanation

Apply the power to every factor: 3³ = 27, (x²)³ = x⁶ and y³ stays y³. The result is 27x⁶y³.

Question 22Number and Algebra (Te Tau me te Taurangi)

Make r the subject of the formula A = πr².

  1. r = √(Aπ)
  2. r = √(A/π) Correct answer.
  3. r = A/π
  4. r = A/(2π)

Explanation

Divide both sides by π to get r² = A/π, then take the square root of both sides. Since r is a length, only the positive root is meaningful, giving r = √(A/π).

Question 23Number and Algebra (Te Tau me te Taurangi)

A population of 2000 bacteria doubles every 3 hours. Which expression gives the population after 12 hours?

  1. 2000 × 4
  2. 2000 × 12
  3. 2000 × 2⁴ Correct answer.
  4. 2000 × 2¹²

Explanation

In 12 hours there are 12 ÷ 3 = 4 doubling periods. Each period multiplies the population by 2, so the population is 2000 × 2⁴ = 32 000.

Question 24Number and Algebra (Te Tau me te Taurangi)

The quadratic equation x² − 5x + 6 = 0 has solutions x = 2 and x = 3. Which statement explains this?

  1. Since the coefficient of x² is 1, the solutions must be whole numbers.
  2. Since the sum of the solutions is 5, the equation has exactly two solutions.
  3. Since the constant term is 6, both solutions must be positive.
  4. Since (x − 2)(x − 3) = 0, at least one factor must be zero, so x = 2 or x = 3. Correct answer.

Explanation

Factorising gives (x − 2)(x − 3) = 0. A product equals zero only when one of its factors is zero, so x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

Question 25Number and Algebra (Te Tau me te Taurangi)

Two similar rectangles have lengths in the ratio 3 : 5. If the smaller rectangle has area 36 cm², what is the area of the larger rectangle?

  1. 100 cm² Correct answer.
  2. 60 cm²
  3. 225 cm²
  4. 180 cm²

Explanation

For similar figures, areas are in the ratio of the squares of the lengths: 3² : 5² = 9 : 25. So 36/A = 9/25, giving A = 36 × 25/9 = 100 cm².

Question 26Number and Algebra (Te Tau me te Taurangi)

What is the value of 3⁴?

  1. 12
  2. 81 Correct answer.
  3. 64
  4. 27

Explanation

The index 4 means the base 3 is multiplied by itself four times: 3 × 3 × 3 × 3 = 81. Multiplying 3 by 4 instead of raising to a power gives the common error 12.

Question 27Number and Algebra (Te Tau me te Taurangi)

Which number is irrational?

  1. −7
  2. √2 Correct answer.
  3. √16
  4. 0.25

Explanation

An irrational number cannot be written as a fraction of two integers. The square root of 2 is a non-repeating, non-terminating decimal, so it is irrational, while the other three options are rational.

Question 28Number and Algebra (Te Tau me te Taurangi)

A jacket costs $85. It is sold with 15% GST added. What is the total price including GST?

  1. $100.00
  2. $97.75 Correct answer.
  3. $95.00
  4. $72.25

Explanation

GST is 15% of $85, which is 0.15 × 85 = $12.75. Adding this to the original price gives 85 + 12.75 = $97.75.

Question 29Number and Algebra (Te Tau me te Taurangi)

Which decimal is equal to 7/8?

  1. 0.875 Correct answer.
  2. 0.78
  3. 0.785
  4. 0.87

Explanation

Dividing 7 by 8 gives 0.875 exactly, since 8 × 0.875 = 7. The digits 7 and 8 in the fraction do not appear directly in the decimal.

Question 30Number and Algebra (Te Tau me te Taurangi)

What is the value of 5 − 2 × 3 + 4?

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

Explanation

Order of operations requires multiplication before addition and subtraction. So 2 × 3 = 6, then 5 − 6 + 4 = 3, working left to right.

Question 31Number and Algebra (Te Tau me te Taurangi)

Which list is in ascending order?

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

Explanation

Ascending order means smallest to largest. On a number line, −3 is furthest left, then −1, then 0, then 2, so this list increases at each step.

Question 32Number and Algebra (Te Tau me te Taurangi)

What is 2/3 + 1/4?

  1. 3/7
  2. 11/12 Correct answer.
  3. 8/12
  4. 3/12

Explanation

The lowest common denominator of 3 and 4 is 12. Rewrite 2/3 as 8/12 and 1/4 as 3/12, then add the numerators: 8 + 3 = 11, giving 11/12.

Question 33Number and Algebra (Te Tau me te Taurangi)

Simplify 4(2x − 3) − 2(x + 5).

  1. 6x − 22 Correct answer.
  2. 6x − 2
  3. 10x − 22
  4. 6x + 2

Explanation

Expand each bracket: 8x − 12 − 2x − 10. Combine like terms: 8x − 2x = 6x and −12 − 10 = −22, giving 6x − 22.

Question 34Number and Algebra (Te Tau me te Taurangi)

Solve 5x + 3 = 2x − 9.

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

Explanation

Subtract 2x from both sides to get 3x + 3 = −9, then subtract 3 to get 3x = −12. Dividing both sides by 3 gives x = −4.

Question 35Number and Algebra (Te Tau me te Taurangi)

Expand (2x − 1)(x + 4).

  1. 2x² + 7x − 4 Correct answer.
  2. 2x² − 7x − 4
  3. 2x² + 7x + 4
  4. 2x² + 9x − 4

Explanation

Use each term of the first bracket against the second: 2x·x = 2x², 2x·4 = 8x, −1·x = −x, −1·4 = −4. Combining 8x − x gives 7x, so the result is 2x² + 7x − 4.

Question 36Number and Algebra (Te Tau me te Taurangi)

Factorise x² + 7x + 12.

  1. (x + 1)(x + 12)
  2. (x + 3)(x + 4) Correct answer.
  3. (x + 2)(x + 6)
  4. (x − 3)(x − 4)

Explanation

We need two numbers that multiply to 12 and add to 7. The pair 3 and 4 satisfies both, so the factorisation is (x + 3)(x + 4).

Question 37Number and Algebra (Te Tau me te Taurangi)

A straight line passes through (0, 4) and (3, 10). What is the equation of the line?

  1. y = 2x + 10
  2. y = 3x + 4
  3. y = 2x + 4 Correct answer.
  4. y = 6x + 4

Explanation

The gradient is (10 − 4)/(3 − 0) = 6/3 = 2. The y-intercept is 4 because the line crosses the y-axis at (0, 4), so the equation is y = 2x + 4.

Question 38Number and Algebra (Te Tau me te Taurangi)

The formula for the area of a trapezium is A = ½(a + b)h. Make h the subject.

  1. h = 2A(a + b)
  2. h = (a + b)/(2A)
  3. h = 2A/(a + b) Correct answer.
  4. h = A/(2(a + b))

Explanation

Multiply both sides by 2 to get 2A = (a + b)h, then divide both sides by (a + b). This gives h = 2A/(a + b).

Question 39Number and Algebra (Te Tau me te Taurangi)

A sequence begins 3, 7, 11, 15, ... Which expression gives the nth term?

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

Explanation

The terms increase by 4 each time, so the expression contains 4n. Testing n = 1 gives 4(1) − 1 = 3, matching the first term, and n = 2 gives 7, so 4n − 1 is correct.

Question 40Number and Algebra (Te Tau me te Taurangi)

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

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

Explanation

Adding the two equations eliminates y: 2x = 14, so x = 7. Substituting back into x + y = 10 gives y = 3, and checking 7 − 3 = 4 confirms the solution.

Question 41Number and Algebra (Te Tau me te Taurangi)

A car depreciates by 12% each year. Its value now is $20 000. Which expression gives its value after 3 years?

  1. 20000 − 3 × 0.12
  2. 20000 × 0.12³
  3. 20000 × (1 − 0.36)
  4. 20000 × 0.88³ Correct answer.

Explanation

A 12% decrease leaves 88% of the value each year, so the multiplier is 0.88. Applying this for 3 years gives 20000 × 0.88³.

Question 42Number and Algebra (Te Tau me te Taurangi)

Which inequality describes the solution set shown by x > −2 and x ≤ 5 combined?

  1. −2 ≤ x ≤ 5
  2. −2 ≤ x < 5
  3. −2 < x < 5
  4. −2 < x ≤ 5 Correct answer.

Explanation

The condition x > −2 excludes −2, so the lower end uses a strict inequality. The condition x ≤ 5 includes 5, so the upper end uses ≤, giving −2 < x ≤ 5.

Question 43Number and Algebra (Te Tau me te Taurangi)

Simplify (x⁵)² ÷ x³.

  1. x¹³
  2. x⁷ Correct answer.
  3. x⁴
  4. x¹⁰

Explanation

Raising a power to a power multiplies the indices: (x⁵)² = x¹⁰. Dividing by x³ subtracts the indices: x¹⁰ ÷ x³ = x⁷.

Question 44Number and Algebra (Te Tau me te Taurangi)

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

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

Explanation

Set y = 0 and factorise: x² − 6x + 5 = (x − 1)(x − 5). The solutions x = 1 and x = 5 give the x-intercepts (1, 0) and (5, 0).

Question 45Number and Algebra (Te Tau me te Taurangi)

A geometric sequence has first term 2 and common ratio 3. What is the 5th term?

  1. 243
  2. 96
  3. 48
  4. 162 Correct answer.

Explanation

The nth term of a geometric sequence is a·r^(n−1). Here a = 2, r = 3 and n = 5, so the term is 2 × 3⁴ = 2 × 81 = 162.

Question 46Number and Algebra (Te Tau me te Taurangi)

The expression x² + bx + 16 is a perfect square for a positive value of b. What is b?

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

Explanation

A perfect square has the form (x + k)² = x² + 2kx + k². Matching k² = 16 gives k = 4, so b = 2k = 8, giving (x + 4)².

Question 47Number and Algebra (Te Tau me te Taurangi)

If log₂(x) = 5, what is the value of x?

  1. 25
  2. 10
  3. 32 Correct answer.
  4. 7

Explanation

The equation log₂(x) = 5 means 2⁵ = x. Since 2⁵ = 32, the value of x is 32.

Question 48Number and Algebra (Te Tau me te Taurangi)

A rational expression simplifies to (x² − 4)/(x − 2) for x ≠ 2. What is its simplified form?

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

Explanation

The numerator is a difference of squares: x² − 4 = (x − 2)(x + 2). Cancelling the common factor (x − 2), valid when x ≠ 2, leaves x + 2.

Question 49Number and Algebra (Te Tau me te Taurangi)

The parabola y = x² + 4x + 7 has a minimum point. What are the coordinates of this minimum?

  1. (0, 7)
  2. (−2, 3) Correct answer.
  3. (−2, 7)
  4. (2, 3)

Explanation

Complete the square: x² + 4x + 7 = (x + 2)² + 3. The squared term is smallest (zero) when x = −2, giving a minimum value of 3, so the vertex is (−2, 3).

Question 50Number and Algebra (Te Tau me te Taurangi)

A courier firm charges a fixed $6 booking fee plus $2.50 for each parcel delivered. Which expression gives the total charge, in dollars, for n parcels?

  1. 6n + 2.50
  2. 6 + 2.50 + n
  3. 2.50(6 + n)
  4. 6 + 2.50n Correct answer.

Explanation

The booking fee is charged once, so it is a constant term of 6. The per-parcel rate of 2.50 multiplies the number of parcels n, giving 2.50n. The total is therefore 6 + 2.50n.