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NVO Grade 10 Mathematics study materials
NVO Grade 10 Mathematics study resources

NVO Grade 10 Mathematics practice questions

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Question 1Algebra and Functions

What is the value of the discriminant of the quadratic equation 2x² − 5x + 3 = 0?

  1. −1
  2. 49
  3. 24
  4. 1 Correct answer.

Explanation

For ax² + bx + c = 0, the discriminant is b² − 4ac. Here a = 2, b = −5, c = 3, so the discriminant equals 25 − 24 = 1. A positive value confirms two distinct real roots.

Question 2Algebra and Functions

Simplify (3x²y)(−2xy³).

  1. 6x³y⁴
  2. −6x³y⁴ Correct answer.
  3. −6x²y³
  4. −5x³y⁴

Explanation

Multiply the coefficients: 3 × (−2) = −6. For the variables, add the exponents of like bases: x²·x = x³ and y·y³ = y⁴. The product is therefore −6x³y⁴.

Question 3Algebra and Functions

If f(x) = 4x − 7, what is f(−2)?

  1. −1
  2. 1
  3. 15
  4. −15 Correct answer.

Explanation

Substitute x = −2 into the rule: 4(−2) − 7 = −8 − 7 = −15. Function notation means replace every x with the given input value.

Question 4Algebra and Functions

Solve the linear equation 5(x − 2) = 3x + 4.

  1. x = −7
  2. x = 7 Correct answer.
  3. x = 3
  4. x = 1

Explanation

Expand the left side to get 5x − 10 = 3x + 4. Subtract 3x from both sides to get 2x − 10 = 4, then add 10 to get 2x = 14, so x = 7.

Question 5Algebra and Functions

What is the slope of the line passing through the points (2, 5) and (6, 13)?

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

Explanation

Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. The slope measures vertical change per unit of horizontal change.

Question 6Algebra and Functions

Which expression is a perfect-square trinomial?

  1. x² + 9
  2. x² + 6x + 6
  3. x² − 9
  4. x² + 6x + 9 Correct answer.

Explanation

A perfect-square trinomial has the form a² + 2ab + b², which factors as (a + b)². Here x² + 6x + 9 = (x + 3)² because 2·x·3 = 6x and 3² = 9.

Question 7Algebra and Functions

The domain of the function f(x) = 1/(x − 4) is all real numbers except which value?

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

Explanation

A rational function is undefined where its denominator equals zero. Setting x − 4 = 0 gives x = 4, so this value must be excluded from the domain.

Question 8Algebra and Functions

Solve the inequality 3x + 5 < 20.

  1. x < 5 Correct answer.
  2. x < 15
  3. x > 15
  4. x > 5

Explanation

Subtract 5 from both sides to get 3x < 15, then divide by the positive number 3, which keeps the inequality direction: x < 5.

Question 9Algebra and Functions

What are the solutions of x² − 5x + 6 = 0?

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

Explanation

Factor the quadratic by finding two numbers that multiply to 6 and add to −5: these are −2 and −3. So (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

Question 10Algebra and Functions

The parabola y = −2(x − 1)² + 8 has its vertex at which point?

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

Explanation

Vertex form is y = a(x − h)² + k with vertex (h, k). Here h = 1 and k = 8, so the vertex is (1, 8). The negative a means the parabola opens downward.

Question 11Algebra and Functions

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

  1. 10
  2. 19 Correct answer.
  3. 7
  4. 49

Explanation

The composition (f ∘ g)(3) means f(g(3)). First g(3) = 9, then f(9) = 2(9) + 1 = 19.

Question 12Algebra and Functions

Which equation represents a line parallel to y = (2/3)x − 7 and passing through the point (3, 1)?

  1. y = (2/3)x + 1
  2. y = 3x − 7
  3. y = (2/3)x − 1 Correct answer.
  4. y = (−3/2)x + 1

Explanation

Parallel lines share the same slope, so the new slope is 2/3. Using point-slope form with (3, 1): y − 1 = (2/3)(x − 3), which simplifies to y = (2/3)x − 1.

Question 13Algebra and Functions

Simplify (x² − 9)/(x² + x − 12) for x ≠ 3 and x ≠ −4.

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

Explanation

Factor both parts: x² − 9 = (x − 3)(x + 3) and x² + x − 12 = (x + 4)(x − 3). Cancel the common factor (x − 3) to get (x + 3)/(x + 4).

Question 14Algebra and Functions

The system y = 2x + 1 and y = −x + 7 has its solution at which point?

  1. (0, 1)
  2. (2, 5) Correct answer.
  3. (5, 2)
  4. (3, 7)

Explanation

Set the expressions equal: 2x + 1 = −x + 7, so 3x = 6 and x = 2. Then y = 2(2) + 1 = 5, giving the intersection point (2, 5).

Question 15Algebra and Functions

For which values of k does x² + kx + 16 = 0 have exactly one real solution?

  1. k = 8 or k = −8 Correct answer.
  2. k = 64 or k = −64
  3. k = 16 only
  4. k = 4 or k = −4

Explanation

Exactly one real solution requires the discriminant to be zero: k² − 4(1)(16) = 0, so k² = 64 and k = ±8.

Question 16Algebra and Functions

If the graph of y = x² is shifted 3 units right and 2 units down, what is the new equation?

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

Explanation

A horizontal shift right by 3 replaces x with (x − 3), and a vertical shift down by 2 subtracts 2 from the whole function. The result is y = (x − 3)² − 2.

Question 17Algebra and Functions

What is the inverse of the function f(x) = 3x − 6?

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

Explanation

Write y = 3x − 6, then swap x and y to get x = 3y − 6. Solving for y gives y = (x + 6)/3, which is the inverse function.

Question 18Algebra and Functions

Which expression is equivalent to log₂(32)?

  1. 16
  2. 5 Correct answer.
  3. 6
  4. 2

Explanation

A logarithm answers the question of what exponent produces the argument. Since 2⁵ = 32, we have log₂(32) = 5.

Question 19Algebra and Functions

The expression 2x² − 8x + 5 can be written in the form a(x − h)² + k. What is the value of k?

  1. 5
  2. −8
  3. 3
  4. −3 Correct answer.

Explanation

Complete the square: 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3. Thus k = −3.

Question 20Algebra and Functions

If the roots of a quadratic are 4 and −1, which equation has these roots?

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

Explanation

A quadratic with roots r and s is (x − r)(x − s) = 0. Here (x − 4)(x + 1) = x² − 3x − 4, since the sum of roots 3 gives coefficient −3 and the product −4 gives the constant.

Question 21Algebra and Functions

A sequence is defined by a₁ = 3 and aₙ = 2aₙ₋₁ + 1. What is a₄?

  1. 31 Correct answer.
  2. 7
  3. 63
  4. 15

Explanation

Apply the recurrence step by step: a₂ = 2(3) + 1 = 7, a₃ = 2(7) + 1 = 15, and a₄ = 2(15) + 1 = 31.

Question 22Algebra and Functions

For the function f(x) = x² − 4x, on which interval is the function decreasing?

  1. x < 2 Correct answer.
  2. all real numbers
  3. x > 2
  4. x < 0

Explanation

The vertex of this upward parabola is at x = 2, found from x = −b/(2a) = 4/2. A parabola opening upward decreases to the left of its vertex, so f is decreasing for x < 2.

Question 23Algebra and Functions

The polynomial p(x) = x³ − 2x² − 5x + 6 has x = 1 as a root. What are all the roots of p(x)?

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

Explanation

Since x = 1 is a root, divide by (x − 1) to get x² − x − 6, which factors as (x − 3)(x + 2). The full factorization is (x − 1)(x − 3)(x + 2), giving roots 1, 3, and −2.

Question 24Algebra and Functions

A function is defined piecewise: f(x) = x² for x < 0, f(x) = 2x + 1 for 0 ≤ x ≤ 3, and f(x) = 10 for x > 3. What is f(f(−2))?

  1. 17
  2. 5
  3. 4
  4. 10 Correct answer.

Explanation

First f(−2) uses the x < 0 rule: (−2)² = 4. Then f(4) uses the x > 3 rule, so f(4) = 10.

Question 25Algebra and Functions

If 2^(x+1) = 8^(x−2), what is the value of x?

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

Explanation

Rewrite 8 as 2³, so 8^(x−2) = 2^(3x−6). Equating exponents gives x + 1 = 3x − 6, so 7 = 2x and x = 7/2.

Question 26Algebra and Functions

Which expression is equivalent to (3x²y)(4xy³)?

  1. 12x³y³
  2. 12x³y⁴ Correct answer.
  3. 12x²y³
  4. 7x³y⁴

Explanation

Multiply the coefficients 3 · 4 = 12, then add exponents of like bases: x² · x = x³ and y · y³ = y⁴. The product is therefore 12x³y⁴.

Question 27Algebra and Functions

What is the value of f(−2) when f(x) = x² + 3x − 1?

  1. −11
  2. −3 Correct answer.
  3. 1
  4. 9

Explanation

Substitute x = −2: (−2)² + 3(−2) − 1 = 4 − 6 − 1 = −3. Each term must be evaluated with the correct sign before combining.

Question 28Algebra and Functions

Which line has a y-intercept of −4 and a slope of 3?

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

Explanation

In slope-intercept form y = mx + b, m is the slope and b is the y-intercept. A slope of 3 and y-intercept of −4 give y = 3x − 4.

Question 29Algebra and Functions

Which expression is equivalent to (2x − 5)(x + 3)?

  1. 2x² + x + 15
  2. 2x² + 11x − 15
  3. 2x² − x − 15
  4. 2x² + x − 15 Correct answer.

Explanation

Using FOIL: 2x·x = 2x², 2x·3 = 6x, −5·x = −5x, and −5·3 = −15. Combining 6x − 5x gives x, so the product is 2x² + x − 15.

Question 30Algebra and Functions

What is the slope of any line perpendicular to y = (1/4)x + 9?

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

Explanation

Perpendicular lines have slopes whose product is −1, so the slope is the negative reciprocal of 1/4. The negative reciprocal of 1/4 is −4.

Question 31Algebra and Functions

Which statement describes the range of the function f(x) = x² + 2?

  1. All real numbers
  2. All real numbers y with y ≥ 2 Correct answer.
  3. All real numbers y with y ≤ 2
  4. All real numbers y with y ≥ 0

Explanation

Since x² is never negative, the smallest value of x² + 2 occurs at x = 0 and equals 2. The output can be any value at or above 2, so the range is y ≥ 2.

Question 32Algebra and Functions

What is the solution set of |2x − 6| = 10?

  1. {−8, 2}
  2. {2, 8}
  3. {−2, 8} Correct answer.
  4. {8}

Explanation

An absolute value equation splits into two cases: 2x − 6 = 10 gives x = 8, and 2x − 6 = −10 gives x = −2. Both values satisfy the original equation.

Question 33Algebra and Functions

The quadratic 3x² − 12x + 7 is written as 3(x − h)² + k. What is the value of h?

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

Explanation

Factor out 3: 3(x² − 4x) + 7, then complete the square to get 3(x − 2)² − 5. Comparing with 3(x − h)² + k shows h = 2.

Question 34Algebra and Functions

Which expression is equivalent to (x³ − 8)/(x − 2) for x ≠ 2?

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

Explanation

This is a difference of cubes: x³ − 8 = (x − 2)(x² + 2x + 4). Dividing by (x − 2) leaves x² + 2x + 4.

Question 35Algebra and Functions

If f(x) = x + 4 and g(x) = 2x − 1, what is (g ∘ f)(x)?

  1. x + 3
  2. 2x + 7 Correct answer.
  3. 2x + 3
  4. 2x − 5

Explanation

The composition (g ∘ f)(x) means g(f(x)). Substituting f(x) = x + 4 into g gives 2(x + 4) − 1 = 2x + 7.

Question 36Algebra and Functions

What is the solution set of the inequality −3x + 4 > 19?

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

Explanation

Subtract 4 to get −3x > 15, then divide by −3 and reverse the inequality sign, giving x < −5. Dividing by a negative number always flips the direction of the inequality.

Question 37Algebra and Functions

The system 2x + 3y = 12 and x − y = 1 has its solution at which point?

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

Explanation

From x − y = 1 we get x = y + 1. Substituting into 2x + 3y = 12 gives 2y + 2 + 3y = 12, so y = 2 and x = 3. Checking: 2(3) + 3(2) = 12 and 3 − 2 = 1.

Question 38Algebra and Functions

What is the value of log₃(81)?

  1. 4 Correct answer.
  2. 3
  3. 1/4
  4. 27

Explanation

The logarithm asks for the exponent to which the base must be raised to obtain 81. Since 3⁴ = 81, log₃(81) = 4.

Question 39Algebra and Functions

What is the sum of the first 20 terms of the arithmetic sequence 5, 9, 13, 17, …?

  1. 840
  2. 860 Correct answer.
  3. 820
  4. 900

Explanation

Here a₁ = 5 and d = 4, so a₂₀ = 5 + 19(4) = 81. The sum is n(a₁ + aₙ)/2 = 20(5 + 81)/2 = 860.

Question 40Algebra and Functions

A geometric sequence has a₁ = 3 and a₄ = 24. What is the common ratio?

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

Explanation

For a geometric sequence, a₄ = a₁r³, so 24 = 3r³ and r³ = 8. The real cube root of 8 is 2, giving r = 2.

Question 41Algebra and Functions

For the function f(x) = x² − 6x + 5, on which interval is the function increasing?

  1. x < 3
  2. x > 0
  3. All real numbers
  4. x > 3 Correct answer.

Explanation

The vertex occurs at x = −b/(2a) = 6/2 = 3, and the parabola opens upward. An upward parabola decreases to the left of its vertex and increases to the right, so it increases for x > 3.

Question 42Algebra and Functions

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

  1. 3
  2. −1
  3. 7 Correct answer.
  4. 11

Explanation

By the Remainder Theorem, the remainder equals p(2). Evaluating: 8 + 8 − 10 + 1 = 7.

Question 43Algebra and Functions

Which equation has a graph with no x-intercepts?

  1. y = x² − 4x + 3
  2. y = x² + 5x
  3. y = x² + 4x + 6 Correct answer.
  4. y = x² − 6x + 9

Explanation

A parabola has no x-intercepts when its discriminant b² − 4ac is negative. For y = x² + 4x + 6, the discriminant is 16 − 24 = −8, which is negative, so there are no real roots.

Question 44Algebra and Functions

If 4^(2x) = 8^(x+1), what is the value of x?

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

Explanation

Write both sides with base 2: 4^(2x) = 2^(4x) and 8^(x+1) = 2^(3x+3). Equating exponents gives 4x = 3x + 3, so x = 3. Wait-solving carefully, 4x = 3x + 3 yields x = 3; however the intended consistent check is 4x = 3x + 3, so x = 3.

Question 45Algebra and Functions

The polynomial p(x) = x³ − 4x² + x + 6 has x = −1 as a root. What are all the roots of p(x)?

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

Explanation

Dividing p(x) by (x + 1) gives x² − 5x + 6, which factors as (x − 2)(x − 3). Together with x = −1, the roots are −1, 2, and 3.

Question 46Algebra and Functions

The rational function f(x) = (x² − 1)/(x² − 3x + 2) has a hole at which x-value?

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

Explanation

Factor numerator and denominator: (x − 1)(x + 1) over (x − 1)(x − 2). The factor (x − 1) cancels, creating a hole at x = 1, while x = 2 remains a vertical asymptote.

Question 47Algebra and Functions

A function is defined by f(x) = x² for x < 1 and f(x) = 2x + 1 for x ≥ 1. What is f(f(−2))?

  1. 25
  2. 9 Correct answer.
  3. 5
  4. −3

Explanation

First, −2 < 1 so f(−2) = (−2)² = 4. Then 4 ≥ 1, so f(4) = 2(4) + 1 = 9.

Question 48Algebra and Functions

The function f(x) = 2x² − 8x + 3 is restricted to x ≥ 2. What is the range of this restricted function?

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

Explanation

The vertex is at x = 2, where f(2) = 8 − 16 + 3 = −5. Since the parabola opens upward and the domain x ≥ 2 begins at the vertex, the function takes all values y ≥ −5.

Question 49Algebra and Functions

If log₂(x) + log₂(x − 2) = 3, what is the value of x?

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

Explanation

Combine the logs: log₂(x(x − 2)) = 3, so x(x − 2) = 8 and x² − 2x − 8 = 0. This factors as (x − 4)(x + 2) = 0, giving x = 4 or x = −2; the domain requires x > 2, so x = 4.

Question 50Algebra and Functions

Which value is the slope of the line passing through the points (2, 5) and (6, 13)?

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

Explanation

The slope of a line through two points is the change in y divided by the change in x, so (13 − 5)/(6 − 2) equals 8/4, which simplifies to 2. This is the constant rate at which y increases as x increases by one.