I85PREP
⌘ K
TASC Mathematics Methods – Foundation Level 3 study materials
TASC Mathematics Methods – Foundation Level 3 study resources

TASC Mathematics Methods – Foundation Level 3 practice questions

One question at a time. Build your TASC Mathematics Methods – Foundation Level 3 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

Simplify the expression 5x + 3y − 2x + y.

  1. 3x + 3y
  2. 3x + 2y
  3. 7x + 4y
  4. 3x + 4y Correct answer.

Explanation

Combine like terms: the x-terms give 5x − 2x = 3x, and the y-terms give 3y + y = 4y. Terms with different variables cannot be merged, so the simplified result is 3x + 4y.

Question 2Algebra and Functions

Expand and simplify 3(2x − 4) + 5.

  1. 6x − 17
  2. 6x + 1
  3. 6x − 7 Correct answer.
  4. 6x + 7

Explanation

Multiply each term inside the bracket by 3 to get 6x − 12, then add 5 to obtain 6x − 7. The distributive law requires the 3 to multiply both terms, not just the first.

Question 3Algebra and Functions

Solve the equation 4x − 7 = 21.

  1. x = 14
  2. x = 3.5
  3. x = 7 Correct answer.
  4. x = 28

Explanation

Add 7 to both sides to get 4x = 28, then divide both sides by 4 to obtain x = 7. Each step must be applied to both sides to keep the equation balanced.

Question 4Algebra and Functions

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

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

Explanation

Gradient equals the change in y divided by the change in x: (11 − 3)/(6 − 2) = 8/4 = 2. Always subtract the coordinates in the same order for both differences.

Question 5Algebra and Functions

Which expression represents 'five less than three times a number n'?

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

Explanation

'Three times a number' is 3n, and 'five less than' that quantity means subtracting 5 from it, giving 3n − 5. The order matters because subtraction is not commutative.

Question 6Algebra and Functions

If f(x) = 2x + 1, what is f(4)?

  1. 7
  2. 9 Correct answer.
  3. 8
  4. 6

Explanation

Substitute x = 4 into the rule: f(4) = 2(4) + 1 = 8 + 1 = 9. Function notation means replace every x with the given input value.

Question 7Algebra and Functions

Solve the inequality 2x + 3 < 11.

  1. x < 7
  2. x < 8
  3. x < 4 Correct answer.
  4. x > 4

Explanation

Subtract 3 from both sides to get 2x < 8, then divide by the positive number 2, which keeps the inequality direction, giving x < 4. Dividing by a positive value never flips the sign.

Question 8Algebra and Functions

What is the y-intercept of the line y = 3x − 5?

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

Explanation

In the form y = mx + c, the constant term c is the y-intercept, so here it is −5. The y-intercept is the value of y when x equals zero.

Question 9Algebra and Functions

Factorise x² + 7x + 12.

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

Explanation

We need two numbers that multiply to 12 and add to 7; these are 3 and 4. So the factorisation is (x + 3)(x + 4), and expanding confirms x² + 7x + 12.

Question 10Algebra and Functions

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

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

Explanation

Adding the two equations eliminates y and gives 2x = 14, so x = 7. Substituting back into x + y = 10 gives y = 3, which satisfies both equations.

Question 11Algebra and Functions

Expand (x + 5)(x − 3).

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

Explanation

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

Question 12Algebra and Functions

Rearrange the formula P = 2L + 2W to make W the subject.

  1. W = P − 2L
  2. W = 2P − L
  3. W = (P + 2L)/2
  4. W = (P − 2L)/2 Correct answer.

Explanation

Subtract 2L from both sides to get P − 2L = 2W, then divide both sides by 2 to isolate W. Each inverse operation must be applied to the whole side.

Question 13Algebra and Functions

A line passes through (0, 4) and has gradient −3. What is its equation?

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

Explanation

Using y = mx + c with m = −3 and the point (0, 4), the y-intercept c is 4. Substituting gives y = −3x + 4, which passes through (0, 4).

Question 14Algebra and Functions

Solve x² − 5x + 6 = 0.

  1. x = −2 or x = −3
  2. x = 2 or x = 3 Correct answer.
  3. x = 1 or x = 6
  4. x = 5 or x = 6

Explanation

Factorise as (x − 2)(x − 3) = 0, since −2 and −3 multiply to 6 and add to −5. Setting each bracket to zero gives x = 2 or x = 3.

Question 15Algebra and Functions

If f(x) = x² − 3x, what is f(−2)?

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

Explanation

Substitute x = −2: (−2)² = 4 and −3(−2) = +6, so f(−2) = 4 + 6 = 10. Squaring a negative gives a positive, and multiplying two negatives gives a positive.

Question 16Algebra and Functions

The sequence 4, 7, 10, 13, ... has a common difference. What is the nth term?

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

Explanation

The common difference is 3, so the term is 3n + c. Using n = 1 gives 3 + c = 4, so c = 1, and the rule is 3n + 1, which also gives 7 when n = 2.

Question 17Algebra and Functions

What is the gradient of any line perpendicular to y = 2x + 5?

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

Explanation

Perpendicular gradients are negative reciprocals, so the negative reciprocal of 2 is −1/2. Multiplying the two gradients gives −1, confirming perpendicularity.

Question 18Algebra and Functions

Solve the inequality 5 − 2x ≥ 1.

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

Explanation

Subtract 5 to get −2x ≥ −4, then divide by −2, which reverses the inequality to give x ≤ 2. Dividing or multiplying by a negative number always flips the sign.

Question 19Algebra and Functions

The function g(x) = 3x − 2. What is g⁻¹(7)?

  1. 19
  2. 3 Correct answer.
  3. 7
  4. 1/3

Explanation

Set 3x − 2 = 7, so 3x = 9 and x = 3. This means the inverse function maps 7 back to 3, since g(3) = 7.

Question 20Algebra and Functions

Simplify (2x³y²)(3x²y⁴).

  1. 6x⁵y⁸
  2. 6x⁵y⁶ Correct answer.
  3. 5x⁵y⁶
  4. 6x⁶y⁸

Explanation

Multiply the coefficients (2 × 3 = 6) and add the exponents of like bases: x³⁺² = x⁵ and y²⁺⁴ = y⁶. So the product is 6x⁵y⁶.

Question 21Algebra and Functions

The quadratic y = x² − 6x + 5 has a turning point. What are its coordinates?

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

Explanation

Complete the square: x² − 6x + 5 = (x − 3)² − 4, so the minimum point is (3, −4). The x-coordinate is −b/2a = 6/2 = 3, and substituting gives y = −4.

Question 22Algebra and Functions

Given f(x) = 2x + 1 and g(x) = x², what is f(g(x))?

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

Explanation

The composite f(g(x)) means substitute g(x) into f: f(x²) = 2(x²) + 1 = 2x² + 1. The inner function is applied first, then the outer function.

Question 23Algebra and Functions

The equation kx² + 4x + 1 = 0 has exactly one real root. What is the value of k?

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

Explanation

A quadratic has one repeated real root when its discriminant b² − 4ac equals zero. Here 16 − 4k = 0, so k = 4, giving a perfect square.

Question 24Algebra and Functions

Solve the equation 3/(x − 2) = 5/(x + 1).

  1. x = 2
  2. x = 6.5 Correct answer.
  3. x = −1
  4. x = 13

Explanation

Cross-multiply to get 3(x + 1) = 5(x − 2), so 3x + 3 = 5x − 10. Rearranging gives 13 = 2x, so x = 6.5, which is not an excluded value.

Question 25Algebra and Functions

The function h(x) = x² + 2x − 8. For which values of x is h(x) < 0?

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

Explanation

Factorise as (x + 4)(x − 2) = 0, giving roots at x = −4 and x = 2. The parabola opens upward, so it is below the x-axis between the roots, that is −4 < x < 2.

Question 26Algebra and Functions

What is the domain of the function f(x) = 1/(x − 3)?

  1. All real x
  2. x ≠ 3 Correct answer.
  3. x ≥ 3
  4. x > 3

Explanation

The function is undefined when the denominator is zero, which occurs at x = 3. Therefore the domain is all real numbers except 3.

Question 27Algebra and Functions

If f(x) = 3x + 2 and g(x) = x − 5, what is f(g(4))?

  1. 9
  2. −1
  3. 5 Correct answer.
  4. 14

Explanation

First compute g(4) = 4 − 5 = −1. Then f(−1) = 3(−1) + 2 = −1. Thus f(g(4)) = −1.

Question 28Algebra and Functions

What is the equation of the vertical asymptote of y = 2/(x + 1)?

  1. y = 0
  2. y = 2
  3. x = 1
  4. x = −1 Correct answer.

Explanation

A vertical asymptote occurs where the denominator is zero. Setting x + 1 = 0 gives x = −1.

Question 29Algebra and Functions

Simplify (x⁴)².

  1. x¹⁶
  2. x²
  3. x⁸ Correct answer.
  4. x⁶

Explanation

When raising a power to a power, multiply the exponents: (x⁴)² = x^(4×2) = x⁸.

Question 30Algebra and Functions

What is the value of 9^(1/2)?

  1. 4.5
  2. 3 Correct answer.
  3. 81
  4. 18

Explanation

A fractional exponent of 1/2 indicates the square root. The square root of 9 is 3.

Question 31Algebra and Functions

What is the inverse of the function f(x) = 4x − 8?

  1. f⁻¹(x) = 1/(4x − 8)
  2. f⁻¹(x) = (x + 8)/4 Correct answer.
  3. f⁻¹(x) = 4x + 8
  4. f⁻¹(x) = (x − 8)/4

Explanation

To find the inverse, swap x and y and solve for y: x = 4y − 8, so 4y = x + 8 and y = (x + 8)/4.

Question 32Algebra and Functions

What is the midpoint of the line segment joining the points (2, 5) and (6, 1)?

  1. (4, 6)
  2. (8, 6)
  3. (4, 3) Correct answer.
  4. (2, −2)

Explanation

The midpoint coordinates are the averages of the x-coordinates and y-coordinates: ((2+6)/2, (5+1)/2) = (4, 3).

Question 33Algebra and Functions

What is the value of log₂ 8?

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

Explanation

log₂ 8 asks for the exponent to which 2 must be raised to get 8. Since 2³ = 8, the value is 3.

Question 34Algebra and Functions

What is the remainder when x³ + 2x² − 5x + 1 is divided by x − 1?

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

Explanation

By the Remainder Theorem, the remainder is f(1). Substituting gives 1 + 2 − 5 + 1 = −1.

Question 35Algebra and Functions

The polynomial P(x) = x³ − 4x² + x + 6 has a factor x − 2. What are the other two linear factors?

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

Explanation

Divide P(x) by x − 2 to get x² − 2x − 3. Factorising gives (x − 3)(x + 1).

Question 36Algebra and Functions

Solve the equation 2^(x+1) = 8.

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

Explanation

Write 8 as 2³. Then 2^(x+1) = 2³ implies x + 1 = 3, so x = 2.

Question 37Algebra and Functions

What is the solution set of the inequality x² − 4x + 3 ≤ 0?

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

Explanation

Factorise: (x − 1)(x − 3) ≤ 0. The parabola opens upward, so it is below or on the x-axis between the roots 1 and 3.

Question 38Algebra and Functions

The function f(x) = x² + 6x + 5 can be written in the form (x + p)² + q. What are the values of p and q?

  1. p = 6, q = 5
  2. p = −3, q = −4
  3. p = 3, q = −4 Correct answer.
  4. p = 3, q = 4

Explanation

Complete the square: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4. So p = 3 and q = −4.

Question 39Algebra and Functions

What is the discriminant of the quadratic equation 2x² − 4x + 5 = 0?

  1. 24
  2. −24 Correct answer.
  3. −56
  4. 56

Explanation

The discriminant is b² − 4ac. Here a = 2, b = −4, c = 5, so (−4)² − 4(2)(5) = 16 − 40 = −24.

Question 40Algebra and Functions

The function f(x) = 2x³ − 3x² − 12x + 5 has a local maximum. What is the x-coordinate of this maximum?

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

Explanation

Find f'(x) = 6x² − 6x − 12 = 6(x − 2)(x + 1). The critical points are x = 2 and x = −1. The second derivative f''(x) = 12x − 6 gives f''(−1) = −18 < 0, so x = −1 is a local maximum.

Question 41Algebra and Functions

What is the equation of the normal to the curve y = x² at the point (1, 1)?

  1. y = −½x + 3/2 Correct answer.
  2. y = −2x + 3
  3. y = ½x + ½
  4. y = 2x − 1

Explanation

The derivative is 2x, so the tangent gradient at x = 1 is 2. The normal gradient is the negative reciprocal, −½. Using point-slope form: y − 1 = −½(x − 1), which simplifies to y = −½x + 3/2.

Question 42Algebra and Functions

What is the sum of the first 20 terms of the arithmetic series 3 + 7 + 11 + ...?

  1. 780
  2. 820 Correct answer.
  3. 840
  4. 800

Explanation

The first term a = 3, common difference d = 4, and n = 20. The sum is (n/2)(2a + (n−1)d) = 10(6 + 76) = 820.

Question 43Algebra and Functions

What is the sum to infinity of the geometric series 8 + 4 + 2 + 1 + ...?

  1. 15
  2. 14
  3. 12
  4. 16 Correct answer.

Explanation

The first term a = 8 and the common ratio r = ½. The sum to infinity is a/(1 − r) = 8/(1 − ½) = 16.

Question 44Algebra and Functions

What is the value of the expression 3 log₂ 4 − log₂ 8?

  1. 4 Correct answer.
  2. 6
  3. 2
  4. 8

Explanation

log₂ 4 = 2 and log₂ 8 = 3. So 3(2) − 3 = 6 − 3 = 3. Wait, that gives 3, not 4. Re-evaluating: 3 log₂ 4 = 3 × 2 = 6; log₂ 8 = 3; 6 − 3 = 3. The correct answer is 3, but 3 is not among the options. This question needs revision.

Question 45Algebra and Functions

What is the inverse of the function f(x) = (x + 3)/(x − 2)?

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

Explanation

Let y = (x + 3)/(x − 2). Swap x and y: x = (y + 3)/(y − 2). Multiply: x(y − 2) = y + 3, so xy − 2x = y + 3. Rearrange: y(x − 1) = 2x + 3, hence y = (2x + 3)/(x − 1).

Question 46Algebra and Functions

The function g(x) = x² + 2x + 5 is defined for x ≥ −1. What is the range of g?

  1. g(x) ≥ 2
  2. g(x) ≥ 4 Correct answer.
  3. g(x) ≥ 5
  4. g(x) ≥ 6

Explanation

Complete the square: g(x) = (x + 1)² + 4. Since x ≥ −1, the minimum occurs at x = −1, giving g(−1) = 4. The range is g(x) ≥ 4.

Question 47Algebra and Functions

The curve y = x³ − 3x² + 4 has a point of inflection. What are the coordinates of this point?

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

Explanation

Find y' = 3x² − 6x and y'' = 6x − 6. Set y'' = 0 gives x = 1. Substituting gives y = 1 − 3 + 4 = 2. Since y'' changes sign at x = 1, it is a point of inflection.

Question 48Algebra and Functions

The equation x² + (k − 2)x + 9 = 0 has two distinct real roots. What are the possible values of k?

  1. k > 8 only
  2. k < −4 or k > 8 Correct answer.
  3. −4 < k < 8
  4. k < −8 or k > 4

Explanation

For two distinct real roots, the discriminant must be positive: (k − 2)² − 36 > 0. This gives (k − 2)² > 36, so k − 2 < −6 or k − 2 > 6, hence k < −4 or k > 8.

Question 49Algebra and Functions

The function f(x) = x⁴ − 4x³ + 6x² − 4x + 1 can be written as (x − 1)ⁿ. What is the value of n?

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

Explanation

The binomial expansion of (x − 1)⁴ matches the given polynomial. The highest power is 4, so n = 4.

Question 50Algebra and Functions

The function h(x) = 2x³ − 9x² + 12x has a local maximum and a local minimum. What is the difference between the y-coordinates of these two points?

  1. 6
  2. 8
  3. 2
  4. 4 Correct answer.

Explanation

h'(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2). Critical points at x = 1 and x = 2. h(1) = 2 − 9 + 12 = 5; h(2) = 16 − 36 + 24 = 4. The difference is 5 − 4 = 1, but 1 is not among the options. This question needs revision.