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Grade 11 math review pack

30 mixed questions from every topic in Grade 11, 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 11 Math Review Pack

Name: ____________________Date: ____________Score: ____ / 30
  1. Functions
    Solve $|2x - 9| = 9$. What is the larger solution?
    Answer: ______________________
  2. f(x) = $2x + 2$ and g(x) = $2x - 3$. Solve f(g(x)) = 12.
    Answer: ______________________
  3. Is $f(x) = -4x^4 - 4x^2 - 5$ odd, even or neither?
    A. OddB. EvenC. Neither
  4. Quadratic Functions
    How many times does the line $y = 2x - 3$ meet the parabola $y = x^2 + 1$?
    A. TwiceB. Once (tangent)C. Never
  5. Work out the maximum value of $y = -2x^2 - 24x - 79$.
    Answer: ______________________
  6. The quadratic with zeros −6 and −2 that passes through (−3, 3): what is the y-coordinate of its vertex?
    Answer: ______________________
  7. Polynomial and Rational Functions
    Factor fully: $x^3 + 7x^2 + 14x + 8$
    A. $(x + 2)(x^2 - 5x + 4)$B. $(x + 2)(x + 4)(x + 1)$C. $(x - 2)(x - 4)(x - 1)$D. $(x + 2)(x + 4)(x - 1)$
  8. For $x^2 - x - 10 = 0$, what is the sum of the roots? (fraction if needed)
    Answer: ______________________
  9. Exponential and Logarithmic Functions
    Solve $1.5^x > 47$. Give the boundary value of x to two decimal places (the answer is x > this value).
    Answer: ______________________
  10. Find the y-intercept of $y = 0.5(3)^{x - 2} - 1$. (decimal if needed)
    Answer: ______________________
  11. Sequences and Series
    $t_1 = 0$, $t_n = -2t_{n-1} - 5$. Find $t_5$.
    Answer: ______________________
  12. A school uses a phone tree: the principal calls 4 parents, each of them calls 4 more, and so on. How many parents are called after 8 rounds?
    Answer: ______________________
  13. Financial Mathematics
    A rectangle is measured as 5.2 cm by 1.3 cm, each to the nearest 0.1 cm. What is the largest its area could actually be? (cm², to two decimal places)
    Answer: ______________________
  14. Noah borrows $19,000 for a car at 8.4% per year, compounded monthly, paid back monthly over 4 years. What is the monthly payment? (dollars, to the nearest cent)
    Answer: ______________________
  15. Trigonometry
    A right triangle has legs 5 and 12 and hypotenuse 13. θ is opposite the side of length 5. Find csc θ. (fraction)
    Answer: ______________________
  16. Evaluate $\cos^{-1}(-0.1)$ in radians, to two decimal places.
    Answer: ______________________
  17. Periodic and Sinusoidal Functions
    Convert (-12, 35) to polar form (r, θ). What is r?
    Answer: ______________________
  18. What is the maximum value of $y = -2\sin\left[4(x - 45^\circ)\right] + 5$?
    Answer: ______________________
  19. Complex Numbers, Counting and Proof
    How many different "words" can be made by rearranging all the letters of TORONTO?
    Answer: ______________________
  20. The formula 1 + 2 + 3 + … + n = n(n + 1)/2 can be proved by induction. Use it to find the sum when n = 10.
    Answer: ______________________
  21. Vectors, Matrices and Conics
    Find the larger eigenvalue of $\begin{pmatrix} 0 & -1 \\ 12 & 8 \end{pmatrix}$.
    Answer: ______________________
  22. Two cafés sell coffees and pancakes. Café A sold 20 coffees and 7 pancakes; Café B sold 17 coffees and 12 pancakes. A coffee is $4 and a muffin $3. Using matrix multiplication, what is Café B's total revenue? (dollars)
    Answer: ______________________
  23. 3D Geometry, Voronoi Diagrams and Graphs
    A cone has radius 3 cm and height 8 cm. What is its total surface area? (cm², to one decimal place)
    Answer: ______________________
  24. Find the distance between (-5, -1, -6) and (7, 8, 1), to one decimal place.
    Answer: ______________________
  25. Statistics
    A 95% confidence interval for the mean commute time is 22 to 28 minutes. Which interpretation is correct?
    A. 95% of people commute 22 to 28 minutesB. We are 95% confident the true mean commute is between 22 and 28 minutesC. The sample mean is 95% likely to be 25D. There is a 95% chance the interval is wrong
  26. Is this an experiment or an observational study? "A doctor randomly assigns patients to a new drug or a placebo."
    A. ExperimentB. Observational study
  27. Probability
    5% of emails are spam. A filter flags 90% of spam emails and 5% of other emails. What is the probability a random email is flagged? (to four decimal places)
    Answer: ______________________
  28. The distance a brand of car tire lasts is normally distributed with mean 90,000 km and SD 5,000 km. What is the probability that it is less than 90,500 km? (to three decimal places)
    Answer: ______________________
  29. Introduction to Calculus
    f(x) = $-6x^6 + 9x + 2$. Find f′(-1).
    Answer: ______________________
  30. For f(x) = $x^2$, find [f(2 + h) − f(2)]/h when h = 0.001. (to three decimal places)
    Answer: ______________________

Answer key

  1. 9 — 2x − 9 = 9 gives x = 9; 2x − 9 = −9 gives x = 0.
  2. 4 — f(g(x)) = 2(2x − 3) + 2 = 12. So g(x) = 5, and 2x − 3 = 5 gives x = 4.
  3. B (Even) — Only even powers (and a constant): f(−x) = f(x).
  4. C (Never) — x² + 1 = 2x - 3 → x^2 - 2x + 4 = 0. D = 4 − 16 = -12.
  5. -7 — Vertex at x = −b/(2a) = 24/-4 = −6. y = −7. Since a < 0 it opens down: a maximum.
  6. 4 — First a: 3 = a(3)(−1), so a = -1. The vertex is at x = -4; y = -1(2)(−2) = 4.
  7. B ($(x + 2)(x + 4)(x + 1)$) — Test divisors of 8: P(−2) = 0, so (x + 2) is a factor. The remaining quadratic x^2 + 5x + 4 factors as (x + 4)(x + 1).
  8. 1 — Sum = −b/a = 1/1 = 1.
  9. 9.50 — x log 1.5 > log 47. log 1.5 is positive, so x > log 47 ÷ log 1.5 ≈ 9.50.
  10. -0.94 — x = 0: 0.5 × 3^-2 − 1 = -0.94.
  11. 25 — Work forward: 0, -5, 5, -15, 25.
  12. 87380 — 4 + 16 + … for 8 rounds: S = 4(4^8 − 1)/(4 − 1) = 87380.
  13. 7.09 — Upper bounds: 5.25 and 1.35. Largest area = 5.25 × 1.35 ≈ 7.09 cm².
  14. 467.42 — i = 0.70%, n = 48. R = 19,000 × 0.0070 ÷ [1 − (1 + 0.0070)^(−48)] ≈ $467.42.
  15. 13/5 — Opposite 5, adjacent 12, hypotenuse 13. csc θ = hyp/opp = 13/5.
  16. 1.67 — cos⁻¹ gives an angle between 0 and π: cos⁻¹(−0.1) ≈ 1.67 rad.
  17. 37 — r = √(12² + 35²) = 37.
  18. 7 — Max = axis + amplitude = 5 + 2 = 7.
  19. 420 — 7! ÷ (2! × 3!) = 5040 ÷ 12 = 420 (divide out the repeated letters).
  20. 55 — 10(11)/2 = 55.
  21. 6 — λ² − 8λ + 12 = 0 → λ = 2 or 6.
  22. 104 — [20 7; 17 12] × [4; 3]: Café B = 17(4) + 12(3) = $104.
  23. 108.8 — Slant height l = √(3² + 8²) ≈ 8.54. SA = πr² + πrl ≈ 108.8 cm².
  24. 16.6 — √(144 + 81 + 49) ≈ 16.6.
  25. B (We are 95% confident the true mean commute is between 22 and 28 minutes) — The confidence is about the method capturing the true population mean, not about individual commutes.
  26. A (Experiment) — The researchers assign the treatment, so it's an experiment (it can show cause and effect).
  27. 0.0925 — 0.05(0.9) + 0.95(0.05) = 0.0925.
  28. 0.540 — z = (90,500 − 90,000) ÷ 5,000 ≈ 0.10. P(Z < 0.10) ≈ 0.540.
  29. 45 — f′(x) = -36x^5 + 9. f′(−1) = 45.
  30. 4.001 — 4.001: as h shrinks this approaches f′(2) = 4.

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