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Grade 10 math review pack

30 mixed questions from every topic in Grade 10, with an answer key and explanations. Great for the start of the year, before a test, or over the holidays.

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Grade 10 Math Review Pack

Name: ____________________Date: ____________Score: ____ / 30
  1. Quadratic Expressions
    Four plots with areas $xy$, $7x$, $6y$ and 42 m² fit together to make one rectangular garden. Factor by grouping to find the garden's length and width.
    A. $(x + 7)(y + 6)$B. $(x + y)(7 + 6)$C. $(x + 6)(y + 7)$D. $(x + 42)(y + 1)$
  2. A rectangular deck has an area of $3x^2 + 29x + 40$ m². Factor the area to find its dimensions.
    A. $(3x + 5)(x + 8)$B. $(3x + 40)(x + 1)$C. $(3x - 5)(x - 8)$D. $(3x + 8)(x + 5)$
  3. A rectangular deck has an area of $3x^2 + 5x + 2$ m². Factor the area to find its dimensions.
    A. $(3x + 1)(x + 2)$B. $(3x + 2)(x + 1)$C. $(3x - 2)(x - 1)$
  4. Simplify $\dfrac{x^2 + 4x + 3}{x + 1}$.
    A. $x + 4$B. $x - 3$C. $x + 3$
  5. Quadratic Relations
    A food truck sells lunch combos for $17, and 775 combos are sold. For every $1 increase in price, 25 fewer are sold. What price gives the most revenue? (dollars)
    Answer: ______________________
  6. How many real roots does $-x^2 - 3x - 6 = 0$ have?
    A. TwoB. OneC. None
  7. Which describes how $y = 3(x + 4)^2 - 7$ is transformed from y = x²?
    A. vertical stretch by 3, 4 right, 7 downB. vertical stretch by 3, reflection in the x-axis, 4 left, 7 downC. vertical stretch by 3, 4 left, 7 downD. vertical stretch by 3, 7 left, 4 down
  8. A parabola has vertex (−3, 4) and passes through (−5, 12). What is the value of a in y = a(x − h)² + k?
    Answer: ______________________
  9. Linear Systems
    Solve for x: $0.5x + 0.4y = 173$ and $x + y = 360$. What is y?
    Answer: ______________________
  10. How many solutions does this system have? $y = 3x - 4$ and $-20x + 4y = -16$
    A. OneB. NoneC. Infinitely many
  11. Solve the system $4x + y = 2$ and $-x + 4y = 25$. What is x?
    Answer: ______________________
  12. Solve by elimination: $-3x - 5y = 6$ and $-4x + 4y = -24$. What is x?
    Answer: ______________________
  13. Analytic Geometry
    Triangle ABC has A(2, 2), B(1, 4) and C(0, 3). Is there a right angle at B?
    A. YesB. No
  14. A diameter of a circle has endpoints (3, −1) and (−2, −2). What is the centre of the circle? Write it like (−3, 5).
    Answer: ______________________
  15. What is the shortest distance from the point (2, −1) to the line $3x + 4y = -3$, to one decimal place?
    Answer: ______________________
  16. What is the equation of the perpendicular bisector of the segment from (6, 2) to (−2, 0)?
    A. $y = -4x - 9$B. $y = -4x + 9$C. $y = 4x - 7$
  17. Trigonometry
    A triangular sail has sides 23 m and 8 m with an angle of 82° between them. What is its area? Give the area to one decimal place. (m²)
    Answer: ______________________
  18. Chloe is 1.5 m tall and casts a 2 m shadow. At the same time, a tree casts a 9 m shadow. How tall is the tree, to one decimal place? (m)
    Answer: ______________________
  19. From a point 53 m from the base of a building, the angle of elevation to the top is 51°. How tall is the building, to one decimal place? (m)
    Answer: ______________________
  20. Two boats leave a harbour. One sails 6 km and the other sails 21 km, on courses that make an angle of 42°. How far apart are they, to one decimal place? (km)
    Answer: ______________________
  21. Functions, Exponentials and Logarithms
    The point (2, 2) is on y = f(x). What point is it moved to on $y = -2f(x - 2) + 2$? Write it like (3, −2).
    Answer: ______________________
  22. A medicine has a half-life of 8 hours. A patient takes 140 mg. How much is left after 24 hours? (mg, to one decimal place)
    Answer: ______________________
  23. Which is the logarithmic form of $10^{-2} = \dfrac{1}{100}$?
    A. $\log_{10} \dfrac{1}{100} = -2$B. $\log_{-2} \dfrac{1}{100} = 10$C. $\log_{10} -2 = \dfrac{1}{100}$D. $\log_{100} 10 = -2$
  24. $4,000 is invested at 8% a year, compounded yearly. How many years until it triples? Give the answer to one decimal place.
    Answer: ______________________
  25. Geometry and Measurement
    A sharpened crayon is a cylinder (radius 3 mm, length 110 mm) with a cone (height 4 mm) on the end. What is its volume, to the nearest cubic millimetre?
    Answer: ______________________
  26. For triangles ABC and DEF: ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Are they congruent, and why?
    A. SSSB. ASAC. Not enough informationD. SAS
  27. A circle has radius 15 cm. Find the length of an arc with a central angle of 60°, to one decimal place. (cm)
    Answer: ______________________
  28. Statistics and Probability
    In a class of 35, 12 like math, 13 like art and 7 like both. A student is chosen at random. What is the probability they like math or art? (fraction)
    Answer: ______________________
  29. Work out the population standard deviation of 15, 12, 2, 20, 13, 8, 12, to one decimal place.
    Answer: ______________________
  30. Ice cream sales and sunburns have a strong positive correlation. What is the best conclusion?
    A. Both rise in hot, sunny weather; one does not cause the otherB. The data must be wrongC. Eating ice cream causes sunburnD. Sunburns make people buy ice cream

Answer key

  1. C ($(x + 6)(y + 7)$) — Group in pairs: xy + 7x = x(y + 7), and 6y + 42 = 6(y + 7). The common bracket gives (x + 6)(y + 7).
  2. A ($(3x + 5)(x + 8)$) — Look for two numbers that multiply to 120 and add to 29: 24 and 5. Split the middle term and group. The dimensions are 3x + 5 m and x + 8 m.
  3. B ($(3x + 2)(x + 1)$) — Look for two numbers that multiply to 6 and add to 5: 3 and 2. Split the middle term and group. The dimensions are 3x + 2 m and x + 1 m.
  4. C ($x + 3$) — Top factors as (x + 1)(x + 3). Cancel (x + 1): x + 3.
  5. 24 — R(x) = (17 + x)(775 − 25x) has zeros at x = −17 and x = 31. The vertex is halfway: x = 7. Best price: 17 + 7 = $24.
  6. C (None) — D = 9 − 24 = -15. D < 0: no real roots.
  7. C (vertical stretch by 3, 4 left, 7 down) — a = 3; (x + 4) moves it 4 left; −7 moves it 7 down.
  8. 2 — y = a(x + 3)² + 4. Substitute (−5, 12): 12 = a(4) + 4, so 4a = 8 and a = 2.
  9. 70 — Multiply the first by 10: 5x + 4y = 1730. Use x = 360 − y: 5(360 − y) + 4y = 1730 → -y = -70 → y = 70.
  10. A (One) — Rewrite the second: y = 5x - 4. Different slopes: the lines cross once.
  11. -1 — Isolate y: y = -4x + 2. Substitute: −x + 4(−4x + 2) = 25 → −17x = 17 → x = −1. Then y = 6.
  12. 3 — Multiply to match a coefficient (e.g. first × 4, second × 3), then subtract to eliminate a variable. The solution is (3, −3).
  13. B (No) — Slope AB = 2/-1, slope BC = -1/-1. Their product is not −1, so AB and BC are not perpendicular.
  14. (0.5, −1.5) — The centre is the midpoint of any diameter: ((3 − 2) ÷ 2, (−1 − 2) ÷ 2) = (0.5, −1.5).
  15. 1.0 — Distance = |3(2) + 4(−1) + 3| ÷ √(9 + 16) = 5 ÷ 5 ≈ 1.
  16. B ($y = -4x + 9$) — Midpoint (2, 1). Slope of AB = 1/4, so the bisector's slope is -4. y − 1 = -4(x − 2) → y = -4x + 9.
  17. 91.1 — Area = ½ × 23 × 8 × sin 82° ≈ 91.1.
  18. 6.8 — Same sun angle, so similar triangles: 1.5/2 = h/9. h = 9 × 1.5 ÷ 2 ≈ 6.8 m.
  19. 65.4 — tan 51° = h/53, so h = 53 tan 51° ≈ 65.4 m.
  20. 17.0 — Two sides and the angle between them: cosine law. d² = 6² + 21² − 2(6)(21) cos 42° ≈ 289.73, d ≈ 17.0 km.
  21. (4, −2) — x: 2 + 2 = 4. y: -2 × 2 + 2 = −2.
  22. 17.5 — 24 ÷ 8 = 3 half-lives. 140 × (½)^3 = 17.5 mg.
  23. A ($\log_{10} \dfrac{1}{100} = -2$) — bʸ = x becomes log_b x = y: the base stays the base and the exponent is the answer.
  24. 14.3 — 4,000(1.08)ᵗ = 12,000 → 1.08ᵗ = 3. t = log 3 ÷ log 1.08 ≈ 14.3 years.
  25. 3148 — Cylinder: π(3)²(110) ≈ 3110.2. Cone: ⅓π(3)²(4) ≈ 37.7. Total ≈ 3148 mm³.
  26. C (Not enough information) — Three equal angles (AAA) only shows the triangles are similar: they could be different sizes.
  27. 15.7 — θ = 60° = 1.05 rad. s = rθ = 15 × 1.05 ≈ 15.7 cm.
  28. 18/35 — P(A or B) = (12 + 13 − 7) ÷ 35 = 18/35.
  29. 5.2 — Mean = 11.71. Variance = average squared distance ≈ 27.06. σ = √27.06 ≈ 5.2.
  30. A (Both rise in hot, sunny weather; one does not cause the other) — A third factor (hot, sunny weather) increases both. Correlation does not prove causation.

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