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Grade 10 Prep Guide

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Adaptive Math Test — Grade 10 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 10 adaptive math assessment. None of the practice problems below are taken from the test bank — they are fresh examples designed to teach the same skills.

For each topic you'll find:

  • A short explanation of the key idea
  • Multiple worked examples (problems solved step by step)
  • Common mistakes to watch out for
  • Practice problems with answers at the end of each section

Table of Contents

Trigonometry

  1. Right Triangle Trig (SOH-CAH-TOA)
  2. Solving Right Triangles
  3. The Unit Circle
  4. Law of Sines and Law of Cosines

Polynomial Functions 5. Polynomial Operations 6. End Behaviour and Degree 7. Zeros and Multiplicities

Logarithms 8. Logarithm Basics 9. Properties of Logarithms 10. Solving Exponential and Logarithmic Equations

Radicals & Exponents 11. Simplifying Radicals 12. Rational Exponents 13. Solving Radical Equations

Complex Numbers 14. Powers of i 15. Operations with Complex Numbers

Rational Functions 16. Simplifying Rational Expressions 17. Vertical and Slant Asymptotes 18. Solving Rational Equations

Sequences & Series 19. Arithmetic Sequences 20. Geometric Sequences 21. Sigma Notation and Series

Statistics 22. Standard Deviation 23. Probability and Combinations

1. Right Triangle Trig (SOH-CAH-TOA)

Key Concepts

For a right triangle, with respect to angle θ:

  • sin θ = Opposite / Hypotenuse
  • cos θ = Adjacent / Hypotenuse
  • tan θ = Opposite / Adjacent

The acronym: SOH-CAH-TOA.

Worked Examples

Example 1. Right triangle: opp = 3, adj = 4, hyp = 5. Find sin, cos, tan of the acute angle opposite the side of length 3. Solution. sin = 3/5, cos = 4/5, tan = 3/4.

Example 2. sin θ = 5/13. Find cos θ (assume θ acute). Solution. opp = 5, hyp = 13 → adj = 12. cos θ = 12/13.

Example 3. A ramp rises 4 ft over a 9 ft horizontal distance. Angle? Solution. tan θ = 4/9 → θ ≈ 24°.

Example 4. From the top of a 20 m tree, the angle of depression to a person is 35°. Distance from base? Solution. tan 35° = 20/d → d = 20/tan 35° ≈ 28.6 m.

Common Mistakes

  • Using the wrong side (e.g., adjacent for sine).
  • Forgetting the calculator must be in degree mode (not radians) for degree problems.

Practice

  1. opp = 7, hyp = 25. sin θ?
  2. adj = 9, hyp = 15. cos θ?
  3. opp = 6, adj = 8. tan θ?
  4. A 10 m pole leans against a wall at 60° from horizontal. Height up the wall?

Answers: 1. 7/25 | 2. 3/5 | 3. 3/4 | 4. 10 sin 60° ≈ 8.66 m

2. Solving Right Triangles

Key Concepts

To solve a right triangle means finding all sides and angles. With one angle (other than 90°) and one side, you can find everything using SOH-CAH-TOA, Pythagoras, and the fact angles sum to 180°.

Worked Examples

Example 1. Right triangle: hyp 10, one acute angle 30°. Find the legs. Solution. opp = 10 sin 30° = 5; adj = 10 cos 30° = 5√3 ≈ 8.66.

Example 2. Right triangle: opp 7, angle 40°. Hypotenuse? Solution. sin 40° = 7/hyp → hyp = 7/sin 40° ≈ 10.89.

Example 3. Legs 5 and 12. Find both acute angles. Solution. tan A = 5/12 → A ≈ 22.6°. Other ≈ 67.4°.

Common Mistakes

  • Using sin when cos was needed (or vice versa).
  • Round too early (final answer should be more precise).

Practice

  1. Right triangle: hyp 20, angle 25°. Legs?
  2. Right triangle: opp 8, hyp 17. Find adj.
  3. Legs 9 and 9. Angles?

Answers: 1. opp ≈ 8.45, adj ≈ 18.13 | 2. 15 | 3. 45° and 45°

3. The Unit Circle

Key Concepts

The unit circle has radius 1 centred at the origin. A point on it at angle θ is (cos θ, sin θ).

Key values:

  • sin 0° = 0; sin 30° = 1/2; sin 45° = √2/2; sin 60° = √3/2; sin 90° = 1.
  • cos values mirror: cos 0° = 1; cos 30° = √3/2; cos 45° = √2/2; cos 60° = 1/2; cos 90° = 0.

Signs by quadrant:

  • Q1: all positive.
  • Q2: sin +, cos −.
  • Q3: both −.
  • Q4: sin −, cos +.

Worked Examples

Example 1. cos 180° = −1.

Example 2. sin 270° = −1.

Example 3. sin 150° (Q2, sin positive). Reference angle 30° → sin 150° = 1/2.

Example 4. cos 225° (Q3). Reference 45° → cos 225° = −√2/2.

Common Mistakes

  • Forgetting signs in different quadrants.
  • Mixing up reference angle with actual angle.

Practice

  1. cos 60°
  2. sin 90°
  3. cos 300°
  4. sin 210°

Answers: 1. 1/2 | 2. 1 | 3. 1/2 | 4. −1/2

4. Law of Sines and Law of Cosines

Key Concepts

For any triangle with angles A, B, C opposite sides a, b, c:

  • Law of Sines: a/sin A = b/sin B = c/sin C.
  • Law of Cosines: c² = a² + b² − 2ab cos C.

Use Law of Sines when you have a pair (angle + opposite side). Law of Cosines is useful for SSS or SAS.

Worked Examples

Example 1. a = 7, A = 40°, B = 65°. Find b. Solution. b = sin 65° × 7 / sin 40° ≈ 9.87.

Example 2. Find c if a = 6, b = 8, C = 60°. Solution. c² = 36 + 64 − 2(6)(8)(0.5) = 100 − 48 = 52. c = √52 ≈ 7.21.

Example 3. Find angle A if a = 10, b = 14, B = 50°. Solution. sin A / 10 = sin 50° / 14 → sin A ≈ 0.547 → A ≈ 33.2°.

Common Mistakes

  • Picking the wrong formula (SSA can have 0, 1, or 2 triangles — ambiguous case).
  • Forgetting to use the included angle for Law of Cosines.

Practice

  1. A = 30°, a = 5, B = 45°. Find b.
  2. a = 9, b = 12, C = 70°. Find c.
  3. a = 8, b = 15, c = 17. Find C.

Answers: 1. b ≈ 7.07 | 2. c ≈ 12.36 | 3. 90° (Pythagorean triple)

5. Polynomial Operations

Key Concepts

  • Add / subtract: combine like terms.
  • Multiply: distribute (or use FOIL for binomials × binomials).
  • Divide: long division or synthetic division.

Worked Examples

Example 1. (3x² + 5x − 2) + (x² − 4x + 7) = 4x² + x + 5.

Example 2. (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12.

Example 3. (x + 5)(x² − 3x + 1) = x³ − 3x² + x + 5x² − 15x + 5 = x³ + 2x² − 14x + 5.

Example 4. (x² + 5x + 6) ÷ (x + 2) = (x + 2)(x + 3) ÷ (x + 2) = x + 3.

Common Mistakes

  • Forgetting to distribute across every term.
  • Sign errors when subtracting.

Practice

  1. (4x² − 2x + 1) − (x² + 3x − 5)
  2. (x − 7)(x + 7)
  3. (2x + 1)²
  4. (x³ − 8) ÷ (x − 2)

Answers: 1. 3x² − 5x + 6 | 2. x² − 49 | 3. 4x² + 4x + 1 | 4. x² + 2x + 4

6. End Behaviour and Degree

Key Concepts

For a polynomial f(x) = aₙxⁿ + ... :

  • If n is even, both ends go the same direction.
    • Positive leading coefficient → up–up.
    • Negative → down–down.
  • If n is odd, ends go opposite directions.
    • Positive → down–up.
    • Negative → up–down.

Worked Examples

Example 1. End behaviour of f(x) = 2x³ − 5x? Solution. Odd degree, positive lead → as x → ∞, f → ∞; as x → −∞, f → −∞.

Example 2. End behaviour of f(x) = −x⁴ + 3x²? Solution. Even degree, negative lead → both ends → −∞.

Example 3. Max real zeros of a degree-5 polynomial? Solution. 5.

Common Mistakes

  • Confusing leading coefficient sign with the constant term.
  • Counting turning points incorrectly (max = degree − 1).

Practice

  1. End behaviour of f(x) = −2x⁵ + x³?
  2. End behaviour of f(x) = 4x⁶ − 7?
  3. Max number of real zeros of degree-7 polynomial?

Answers: 1. As x → ∞, f → −∞; as x → −∞, f → ∞ | 2. Both → ∞ | 3. 7

7. Zeros and Multiplicities

Key Concepts

A zero of f(x) is an x with f(x) = 0. From a factored form (x − r) appearing n times, r is a zero of multiplicity n:

  • Odd multiplicity → graph crosses the x-axis at r.
  • Even multiplicity → graph touches (bounces off) the x-axis at r.

Worked Examples

Example 1. f(x) = (x − 2)(x + 1)². Zeros and behaviour? Solution. Zeros at 2 (mult 1, crosses) and −1 (mult 2, bounces).

Example 2. f(x) = (x + 3)²(x − 4)³. Behaviour at zeros? Solution. −3 bounces (even); 4 crosses (odd).

Example 3. Build a degree-3 polynomial with roots 1, 2, −5. Solution. f(x) = (x − 1)(x − 2)(x + 5).

Common Mistakes

  • Reading roots as opposite signs (zero of (x − 3) is +3, not −3).
  • Saying every zero crosses the axis.

Practice

  1. Zeros of (x − 5)(x + 1)(x + 3)?
  2. Behaviour at zero in (x + 2)⁴?
  3. Build f(x) with roots 0, 3, −2.

Answers: 1. 5, −1, −3 | 2. Bounces (even mult) | 3. x(x − 3)(x + 2)

8. Logarithm Basics

Key Concepts

A logarithm is the inverse of an exponential: log_b(x) = y ⟺ bʸ = x.

Common bases:

  • log (no subscript) = log₁₀.
  • ln = log_e (natural log).

log_b(b) = 1; log_b(1) = 0.

Worked Examples

Example 1. log₂ 8 = 3 (since 2³ = 8).

Example 2. log 100 = 2 (since 10² = 100).

Example 3. log₃(1/9) = −2.

Example 4. ln(e²) = 2.

Common Mistakes

  • Treating log as a coefficient (log 2x ≠ 2 log x without rules).
  • Forgetting that you can't take log of zero or a negative.

Practice

  1. log₅ 25
  2. log 1000
  3. log₂(1/4)
  4. ln 1

Answers: 1. 2 | 2. 3 | 3. −2 | 4. 0

9. Properties of Logarithms

Key Concepts

  • Product: log_b(xy) = log_b(x) + log_b(y).
  • Quotient: log_b(x/y) = log_b(x) − log_b(y).
  • Power: log_b(xⁿ) = n log_b(x).
  • Change of base: log_b(x) = log(x) / log(b) = ln(x)/ln(b).

Worked Examples

Example 1. log(2) + log(50) = log(100) = 2.

Example 2. log_2(32) − log_2(4) = log_2(8) = 3.

Example 3. Expand log(x²y/z). Solution. 2 log x + log y − log z.

Example 4. Condense: 3 log x − log y. Solution. log(x³/y).

Common Mistakes

  • Distributing log over a sum/difference (log(a + b) ≠ log a + log b).
  • Misapplying the power rule (only over a single factor inside the log).

Practice

  1. log(4) + log(25)
  2. log_3(27/9)
  3. Expand log(a²b)
  4. Condense 2 log m + 3 log n

Answers: 1. 2 | 2. 1 | 3. 2 log a + log b | 4. log(m²n³)

10. Solving Exponential and Logarithmic Equations

Key Concepts

  • Same base exponentials: bˣ = bʸ → x = y.
  • Different bases: take a log of both sides.
  • Logarithmic equations: rewrite as an exponential, OR combine logs first.

Worked Examples

Example 1. Solve 2ˣ = 16. Solution. 2ˣ = 2⁴ → x = 4.

Example 2. Solve 3ˣ = 20. Solution. x = log₃ 20 = ln 20 / ln 3 ≈ 2.73.

Example 3. Solve log₂(x) = 5. Solution. x = 2⁵ = 32.

Example 4. Solve log(x) + log(x − 3) = 1. Solution. log(x(x − 3)) = 1 → x² − 3x = 10 → (x − 5)(x + 2) = 0 → x = 5 (reject −2, makes log undefined).

Common Mistakes

  • Forgetting to check for extraneous solutions (log of negatives).
  • Mixing up the base when changing.

Practice

  1. Solve 5ˣ = 125.
  2. Solve 4ˣ = 30. (≈)
  3. Solve log₃ x = 4.
  4. Solve log(x + 1) = 2.

Answers: 1. x = 3 | 2. x ≈ 2.45 | 3. 81 | 4. 99

11. Simplifying Radicals

Key Concepts

  • Pull out perfect-square factors: √50 = √(25·2) = 5√2.
  • Rationalize denominators: multiply by the conjugate or by the radical itself.
  • Combine like radicals (same index, same radicand): 3√2 + 4√2 = 7√2.

Worked Examples

Example 1. Simplify √72. Solution. √(36·2) = 6√2.

Example 2. Rationalize 3/√5. Solution. 3/√5 × √5/√5 = 3√5/5.

Example 3. Simplify 2√3 + 5√3 − √12. Solution. √12 = 2√3. 2√3 + 5√3 − 2√3 = 5√3.

Example 4. Rationalize 1/(2 + √3). Solution. × (2 − √3)/(2 − √3) → (2 − √3)/(4 − 3) = 2 − √3.

Common Mistakes

  • Adding radicands instead of leaving roots (√2 + √3 ≠ √5).
  • Missing a perfect-square factor.

Practice

  1. √98
  2. Rationalize 6/√3.
  3. 4√5 − √45
  4. Rationalize 2/(1 + √2).

Answers: 1. 7√2 | 2. 2√3 | 3. √5 | 4. 2(√2 − 1)

12. Rational Exponents

Key Concepts

xᵃ/ᵇ = (ᵇ√x)ᵃ = ᵇ√(xᵃ).

x^(−n) = 1/xⁿ.

(xᵐ)(xⁿ) = xᵐ⁺ⁿ etc.

Worked Examples

Example 1. 8^(2/3) = (³√8)² = 2² = 4.

Example 2. 16^(3/4) = (⁴√16)³ = 2³ = 8.

Example 3. 27^(−2/3) = 1/(27^(2/3)) = 1/9.

Example 4. (x^(1/2))(x^(1/3)) = x^(5/6).

Common Mistakes

  • Mixing the fraction (m/n: nth root then mth power).
  • Forgetting the sign on negative exponents.

Practice

  1. 4^(3/2)
  2. 32^(2/5)
  3. 81^(−1/4)
  4. (x^(2/3))³

Answers: 1. 8 | 2. 4 | 3. 1/3 | 4. x²

13. Solving Radical Equations

Key Concepts

To solve, isolate the radical then raise both sides to the appropriate power. Check for extraneous solutions (squaring can introduce false roots).

Worked Examples

Example 1. Solve √(x + 4) = 6. Solution. x + 4 = 36 → x = 32.

Example 2. Solve √(x − 1) = x − 7. Solution. x − 1 = (x − 7)² → x² − 15x + 50 = 0 → (x − 5)(x − 10). Check: x = 5 gives √4 = 2 ≠ −2 (extraneous). x = 10 gives √9 = 3 = 3 ✓. So x = 10.

Example 3. Solve ³√(2x − 1) = 3. Solution. 2x − 1 = 27 → x = 14.

Example 4. Solve √x + 2 = x. Solution. √x = x − 2 → x = (x − 2)² → x² − 5x + 4 = 0 → (x − 1)(x − 4). Check x = 1: √1 + 2 = 3 ≠ 1 (extraneous). x = 4: √4 + 2 = 4 ✓. x = 4.

Common Mistakes

  • Not isolating the radical first.
  • Forgetting to check solutions in the original equation.

Practice

  1. √(x − 2) = 5
  2. √(3x + 1) = x − 1
  3. ³√(x + 2) = 2

Answers: 1. 27 | 2. x = 8 (check) | 3. 6

14. Powers of i

Key Concepts

i = √(−1), so i² = −1. The cycle of i:

  • i¹ = i
  • i² = −1
  • i³ = −i
  • i⁴ = 1
  • (then repeats)

Take the exponent mod 4 to simplify.

Worked Examples

Example 1. i¹⁰: 10 mod 4 = 2 → −1.

Example 2. i²⁵: 25 mod 4 = 1 → i.

Example 3. i⁸⁷: 87 mod 4 = 3 → −i.

Example 4. i¹²⁰: 120 mod 4 = 0 → 1.

Common Mistakes

  • Forgetting that i² = −1.
  • Computing exponent mod 2 instead of mod 4.

Practice

  1. i¹⁵
  2. i²²
  3. i¹⁰⁰
  4. i³⁷

Answers: 1. −i | 2. −1 | 3. 1 | 4. i

15. Operations with Complex Numbers

Key Concepts

A complex number z = a + bi, where a = real part, b = imaginary part.

  • Add/sub: combine real and imaginary separately.
  • Multiply: distribute (FOIL), use i² = −1.
  • Divide: multiply numerator and denominator by the conjugate.

The conjugate of a + bi is a − bi.

Worked Examples

Example 1. (3 + 2i) + (4 − 5i) = 7 − 3i.

Example 2. (2 + i)(3 − 4i) = 6 − 8i + 3i − 4i² = 6 − 5i + 4 = 10 − 5i.

Example 3. (1 + 2i)² = 1 + 4i + 4i² = 1 + 4i − 4 = −3 + 4i.

Example 4. Divide (4 + 3i) / (2 − i). Solution. × (2 + i)/(2 + i) = (8 + 4i + 6i + 3i²) / (4 − i²) = (5 + 10i)/5 = 1 + 2i.

Common Mistakes

  • Forgetting i² = −1 during multiplication.
  • Not multiplying by the conjugate when dividing.

Practice

  1. (5 − 2i) − (3 + 4i)
  2. (1 + i)(2 + 3i)
  3. (2 − i)²
  4. (3 + i)/(1 − i)

Answers: 1. 2 − 6i | 2. −1 + 5i | 3. 3 − 4i | 4. 1 + 2i

16. Simplifying Rational Expressions

Key Concepts

  • Simplify by factoring numerator and denominator, then cancelling common factors.
  • Multiply: multiply tops, multiply bottoms, then simplify.
  • Divide: multiply by reciprocal.
  • Add/Sub: get a common denominator first.

Worked Examples

Example 1. Simplify (x² − 9)/(x² + 5x + 6). Solution. (x − 3)(x + 3) / (x + 2)(x + 3) = (x − 3)/(x + 2).

Example 2. (x/2) × (4/x²) = 4x/(2x²) = 2/x.

Example 3. (x + 1)/(x − 2) ÷ (x² − 1)/(x − 2). Solution. (x + 1)/(x − 2) × (x − 2)/((x + 1)(x − 1)) = 1/(x − 1).

Example 4. 1/x + 1/y = (y + x)/(xy).

Common Mistakes

  • Cancelling terms instead of factors (you can't cancel x in (x + 2)/x).
  • Forgetting domain restrictions (cancelling doesn't remove them).

Practice

  1. (x² − 4)/(x + 2)
  2. (3x/4) × (8/x²)
  3. (x + 5)/(x − 3) ÷ (x + 5)/(x²)
  4. 2/x − 3/(x + 1)

Answers: 1. x − 2 | 2. 6/x | 3. x²/(x − 3) | 4. (−x + 2)/(x(x + 1)) = (2 − x)/(x(x + 1))

17. Vertical and Slant Asymptotes

Key Concepts

For f(x) = N(x)/D(x) (in lowest terms):

  • Vertical asymptote: where D(x) = 0 (after cancelling).
  • Horizontal asymptote:
    • degree N < degree D → y = 0.
    • degree N = degree D → y = ratio of leading coefficients.
    • degree N > degree D → no horizontal; possibly slant asymptote.
  • Slant (when deg N = deg D + 1): do polynomial long division; the quotient (linear part) is the slant.

Worked Examples

Example 1. f(x) = (x + 1)/(x² − 4). Vertical asymptotes? Solution. x² − 4 = 0 → x = ±2.

Example 2. Horizontal asymptote of (2x + 1)/(x − 3)? Solution. Same degree → ratio of leading coeffs → y = 2.

Example 3. Horizontal asymptote of 3/(x² + 1)? Solution. Smaller numerator degree → y = 0.

Example 4. Slant asymptote of (x² + 3x + 2)/(x + 1)? Solution. Division gives x + 2 (with remainder 0) → y = x + 2.

Common Mistakes

  • Confusing vertical (x = a) with horizontal (y = a).
  • Saying y = 0 when there's no horizontal asymptote.

Practice

  1. Vertical asymptote of f(x) = 1/(x − 5).
  2. Horizontal asymptote of f(x) = (4x² + 1)/(2x²).
  3. Slant asymptote of f(x) = (x² − 1)/x.

Answers: 1. x = 5 | 2. y = 2 | 3. y = x

18. Solving Rational Equations

Key Concepts

Find a common denominator and multiply through (or cross-multiply for two-fraction equations). Always check for extraneous solutions that make the denominator zero.

Worked Examples

Example 1. Solve (x − 1)/3 = (x + 2)/4. Solution. 4(x − 1) = 3(x + 2) → 4x − 4 = 3x + 6 → x = 10.

Example 2. Solve 2/(x − 1) = 6/x. Solution. 2x = 6(x − 1) → 2x = 6x − 6 → x = 3/2.

Example 3. Solve 1/x + 1/(x − 2) = 1/2. Solution. LCD 2x(x − 2). Multiply: 2(x − 2) + 2x = x(x − 2) → 4x − 4 = x² − 2x → x² − 6x + 4 = 0 → x = 3 ± √5.

Example 4. Solve x/(x − 1) = 3/(x − 1). Solution. x = 3 (check: x = 1 makes denom zero, so 1 is excluded; x = 3 ✓).

Common Mistakes

  • Skipping the check — extraneous solutions are common.
  • Forgetting to multiply every term by the common denominator.

Practice

  1. (x + 2)/5 = (x − 1)/3
  2. 1/x = 4/(x + 6)
  3. 3/(x + 1) + 2 = 1

Answers: 1. x = 11/2 | 2. x = 2 | 3. x = −4

19. Arithmetic Sequences

Key Concepts

  • An arithmetic sequence has a constant difference d.
  • nth term: aₙ = a₁ + (n − 1)d.
  • Sum of first n terms: Sₙ = n(a₁ + aₙ)/2.

Worked Examples

Example 1. Sequence 3, 7, 11, 15, ... Find the 10th term. Solution. d = 4; a₁ₐ = 3 + 9(4) = 39.

Example 2. Sum of first 20 terms of 5, 8, 11, ... Solution. a₂₀ = 5 + 19(3) = 62. S = 20(5 + 62)/2 = 670.

Example 3. Find d if a₁ = 7 and a₆ = 27. Solution. 27 = 7 + 5d → d = 4.

Common Mistakes

  • Using (n) instead of (n − 1) in the formula.
  • Confusing arithmetic with geometric.

Practice

  1. 15th term of 2, 5, 8, 11, ...
  2. Sum of first 30 terms of 1, 4, 7, ...
  3. d in a sequence where a₁ = 10 and a₈ = 38?

Answers: 1. 44 | 2. 1335 | 3. 4

20. Geometric Sequences

Key Concepts

  • A geometric sequence has a constant ratio r.
  • nth term: aₙ = a₁ · r^(n−1).
  • Sum of first n terms (r ≠ 1): Sₙ = a₁(1 − rⁿ)/(1 − r).
  • Infinite sum (|r| < 1): S = a₁/(1 − r).

Worked Examples

Example 1. Sequence 3, 6, 12, 24, ... 8th term? Solution. r = 2; a₈ = 3 · 2⁷ = 384.

Example 2. Sum of first 5 terms of 2, 6, 18, ... Solution. r = 3; S = 2(1 − 3⁵)/(1 − 3) = 2(−242)/(−2) = 242.

Example 3. Infinite sum of 1, 1/2, 1/4, 1/8, ... Solution. r = 1/2; S = 1/(1 − 1/2) = 2.

Example 4. Find r if a₁ = 5 and a₄ = 40. Solution. 40 = 5r³ → r³ = 8 → r = 2.

Common Mistakes

  • Using n instead of n − 1 in the exponent.
  • Applying the infinite-sum formula when |r| ≥ 1.

Practice

  1. 6th term of 4, 8, 16, ...
  2. Sum of first 6 terms of 1, 3, 9, ...
  3. Infinite sum of 8, 4, 2, 1, ...
  4. r if a₁ = 2 and a₅ = 162?

Answers: 1. 128 | 2. 364 | 3. 16 | 4. 3 (since r⁴ = 81)

21. Sigma Notation and Series

Key Concepts

Sigma notation: Σ (from k = 1 to n) of f(k) = sum of f at each integer from 1 to n.

Common sums:

  • Σ k = n(n + 1)/2.
  • Σ k² = n(n + 1)(2n + 1)/6.

Worked Examples

Example 1. Σ (k = 1 to 5) k = 1 + 2 + 3 + 4 + 5 = 15.

Example 2. Σ (k = 1 to 10) k = 10 · 11 / 2 = 55.

Example 3. Σ (k = 1 to 4) k² = 1 + 4 + 9 + 16 = 30.

Example 4. Σ (k = 1 to 100) k = 100 · 101/2 = 5,050.

Common Mistakes

  • Including k = 0 when the lower bound is 1.
  • Squaring after summing instead of summing the squares.

Practice

  1. Σ (k = 1 to 6) k
  2. Σ (k = 1 to 20) k
  3. Σ (k = 1 to 3) k²

Answers: 1. 21 | 2. 210 | 3. 14

22. Standard Deviation

Key Concepts

Standard deviation σ measures spread around the mean.

For a dataset {x₁, ..., xₙ}:

  • mean μ = (Σxᵢ)/n.
  • variance = (Σ(xᵢ − μ)²)/n.
  • σ = √variance.

A smaller σ means data clusters closer to the mean.

Worked Examples

Example 1. Find σ of {2, 4, 4, 4, 5, 5, 7, 9}. Solution. μ = 40/8 = 5. Deviations²: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32. Variance = 32/8 = 4. σ = 2.

Example 2. Find σ of {1, 1, 1, 1} (all same value). Solution. σ = 0 (no spread).

Example 3. Two datasets: A = {10, 20, 30}, B = {18, 20, 22}. Which has greater σ? Solution. A — wider spread.

Common Mistakes

  • Forgetting to square the deviations before averaging.
  • Mixing up sample vs population formulas (population uses n, sample uses n − 1).

Practice

  1. σ of {3, 7, 8, 5, 12} (population).
  2. σ of {10, 10, 10}.
  3. Which has greater σ: {1, 5, 9} or {4, 5, 6}?

Answers: 1. ≈ 3.04 (variance ≈ 9.2) | 2. 0 | 3. {1, 5, 9}

23. Probability and Combinations

Key Concepts

  • Permutations P(n, r) = n!/(n − r)! count arrangements (order matters).
  • Combinations C(n, r) = n!/(r!(n − r)!) count selections (order doesn't).
  • Conditional: P(A | B) = P(A and B)/P(B).
  • Independent events: P(A and B) = P(A) × P(B).

Worked Examples

Example 1. How many 3-letter arrangements of {A, B, C, D, E}? Solution. P(5, 3) = 5·4·3 = 60.

Example 2. Choose 3 from 8 students. Solution. C(8, 3) = 8!/(3!5!) = 56.

Example 3. Draw 2 cards without replacement from a standard deck. P(both hearts)? Solution. (13/52)(12/51) = 156/2652 ≈ 0.059.

Example 4. P(at least one head in 3 coin flips)? Solution. 1 − P(no heads) = 1 − (1/2)³ = 7/8.

Common Mistakes

  • Using P (order) when you should use C (no order).
  • Forgetting the without-replacement adjustment.

Practice

  1. Arrangements of {1, 2, 3, 4} taken 2 at a time.
  2. Choose 2 from 6.
  3. Draw 2 from a deck (no replacement). P(both kings)?
  4. P(at least one tail in 2 flips)?

Answers: 1. 12 | 2. 15 | 3. (4/52)(3/51) ≈ 0.0045 | 4. 3/4

That's Grade 10. By now you've covered everything from trig to logs to complex numbers — the foundation for precalculus, calculus, and SAT-level math. The single biggest accelerator is identifying patterns across topics (e.g., quadratics ↔ logs ↔ exponentials all link). Practice mixed problems daily!