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HSC · HSC Year 12

Mathematics Extension 1

Proof, vectors, further calculus and the binomial distribution — full 70-mark HSC papers with worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
7official past papers

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Sample revision note

Auxiliary-Angle Method and Products to Sums

The Auxiliary-Angle Form: Converting a sin(x) + b cos(x)

Any expression of the form a sin(x) + b cos(x), where a and b are real constants (not both zero), can be rewritten as a single sinusoidal function with amplitude R and a phase shift. This is called the auxiliary-angle method (also called the harmonic form). The two most useful forms are:

  • R sin(x + α)
  • R cos(x − α)

where R > 0 and α is the auxiliary angle, typically chosen so that 0 < α < π/2 (keeping α in the first quadrant simplifies most problems).

Derivation of R sin(x + α):

Expand R sin(x + α) using the compound-angle identity:

R sin(x + α) = R [sin x cos α + cos x sin α] = (R cos α) sin x + (R sin α) cos x

Matching coefficients with a sin x + b cos x gives the system:

  • R cos α = a
  • R sin α = b

Squaring and adding: R² cos²α + R² sin²α = a² + b², so R = √(a² + b²).

Dividing: R sin α / R cos α = b/a, so tan α = b/a (with α chosen to match the signs of a and b).

Derivation of R cos(x − α):

Expand R cos(x − α) = R [cos x cos α + sin x sin α] = (R cos α) cos x + (R sin α) sin x.

Matching: R sin α = a, R cos α = b, giving the same R = √(a² + b²) but now tan α = a/b.

The choice of form (sin or cos) usually depends on the question. Both give identical R; only the formula for α differs.

Step-by-Step Method and a Fully Worked Example

Follow this systematic procedure every time:

  1. Identify a and b from a sin x + b cos x.
  2. Compute R = √(a² + b²) — this is always positive.
  3. Determine α from tan α = b/a (for R sin form) or tan α = a/b (for R cos form). Check the quadrant using the signs of both a and b.
  4. Write the final auxiliary form and verify by expanding back.

Worked Example: Express f(x) = 3 sin x + 4 cos x in the form R sin(x + α), giving α correct to 2 decimal places.

Step 1: a = 3, b = 4.

Step 2: R = √(3² + 4²) = √(9 + 16) = √25 = 5. (Arithmetic check: 9 + 16 = 25 ✓, √25 = 5 ✓.)

Step 3: tan α = b/a = 4/3. Since a = 3 > 0 and b = 4 > 0, both R cos α = 3 and R sin α = 4 are positive, so α is in the first quadrant.

α = arctan(4/3) ≈ 0.9273 rad ≈ 0.93 rad (2 d.p.) [equivalently ≈ 53.13°].

Step 4: f(x) = 5 sin(x + 0.93).

Verification: 5 sin(x + 0.93) = 5[sin x cos(0.93) + cos x sin(0.93)]. Now cos(0.93) ≈ 0.6006 and sin(0.93) ≈ 0.7994 (check: 0.6006² + 0.7994² ≈ 0.3607 + 0.6390 ≈ 0.9997 ≈ 1 ✓). So 5 × 0.6006 ≈ 3.003 ≈ 3 and 5 × 0.7994 ≈ 3.997 ≈ 4 ✓. The coefficients recover correctly.

Second example — negative coefficients: Express g(x) = −√3 sin x + cos x in the form R sin(x + α).

a = −√3, b = 1. R = √(3 + 1) = 2. tan α = b/a = 1/(−√3). But now R cos α = −√3 < 0 and R sin α = 1 > 0, so α is in the second quadrant. Reference angle = arctan(1/√3) = π/6 = 30°, so α = π − π/6 = 5π/6. Therefore g(x) = 2 sin(x + 5π/6). Verify: 2[sin x cos(5π/6) + cos x sin(5π/6)] = 2[sin x(−√3/2) + cos x(1/2)] = −√3 sin x + cos x ✓.

Sample exam question

A student is proving by mathematical induction that 1(2) + 2(3) + 3(4) + ... + n(n+1) = n(n+1)(n+2)/3 for all integers n >= 1. Having assumed the statement is true for n = k, which expression must be added to the assumed sum k(k+1)(k+2)/3 to complete the inductive step for n = k+1?

  • A. (k+1)(k+2)
  • B. (k+1)(k+2)(k+3)/3
  • C. k(k+1)
  • D. (k+1)(k+2)/3
Show the worked answer

Answer: A

The (k+1)th term of the series is n(n+1) evaluated at n = k+1, which is (k+1)(k+2). This single term is added to the assumed sum. Indeed k(k+1)(k+2)/3 + (k+1)(k+2) = (k+1)(k+2)(k+3)/3, the required result.

All 20 practice exams

  1. Exam 1 — Proof by Mathematical Induction (series sum and divisibility — emphasis topic); Vectors (2D projection and 3D magnitude/perpendicularity); Trigonometric Equations (auxiliary-angle R-method and double-angle)
  2. Exam 2 — Vectors (projection, geometry proof, relative motion/collision, projectile modelling) — emphasised, ~21/60 marks; Proof by Mathematical Induction (series summation); Trigonometric Equations (quadratic-in-sin and double-angle reduction)
  3. Exam 3 — Trigonometric Equations (emphasis): double-angle reduction, auxiliary-angle R-method, factoring trig equations over a restricted domain; Calculus: inverse-trig differentiation, integration by reduction identity and u-substitution, volumes of revolution; Vectors: projection, angle between vectors, internal division of an interval
  4. Exam 4 — Further Calculus Skills (integration by substitution, inverse trig differentiation & integration); Vectors (dot product, projection, angle); Trigonometric Equations (double-angle & t-formula)
  5. Exam 5 — Applications of Calculus (Newton's law of cooling modelling, related rates, areas/volumes, SHM) — the paper's emphasis; Proof by Mathematical Induction (divisibility); Vectors (dot product, angle, scalar/vector projection)
  6. Exam 6 — The Binomial Distribution (Bin(n,p), exact probabilities, mean/variance, normal approximation to a sample proportion) — the flagged emphasis, ~35/60 marks; Proof by Mathematical Induction (series identity and divisibility); Vectors (dot product, angle between vectors, scalar projection)
  7. Exam 7 — Proof by Mathematical Induction (series, divisibility and inequality forms); Projectile motion and exponential modelling applications of calculus; The Binomial Distribution including normal approximation and sample proportions
  8. Exam 8 — Vectors (28/60 marks — emphasis): projection, collision modelling, Varignon proof, perpendicularity/parallelism, extended projectile-by-vectors; Proof by Mathematical Induction; Trigonometric Equations
  9. Exam 9 — Trigonometric Equations (auxiliary-angle, quadratic-in-sin, modelling); Proof by Mathematical Induction; Vectors (projection and perpendicular component)
  10. Exam 10 — Further Calculus Skills (substitution, by parts, inverse-trig & double-angle integration) — dominant weighting; Applications of Calculus (Newton's Law of Cooling model, SHM, related rates); Proof by Mathematical Induction (series result)
  11. Exam 11 — Applications of Calculus (related rates, exponential growth/decay DEs, projectile motion, volumes of revolution); Proof by Mathematical Induction; Vectors (projection, angle, perpendicularity)
  12. Exam 12 — The Binomial Distribution (emphasis): exact/cumulative probabilities, mean & variance, sample proportions with normal approximation, multi-part modelling; Full Year 12 coverage: Mathematical Induction (series + divisibility), Vectors (dot product, projection, projectile, geometric proof), Trigonometric Equations, Further Calculus Skills, Applications of Calculus; Realistic HSC Ext 1 weighting and graduated difficulty, with every numerical answer recomputed and verified in Python/SymPy
  13. Exam 13 — Proof by Mathematical Induction (series, divisibility and inequality forms) — the emphasis of this paper; Vectors — projections, perpendicularity and 3D angle/dot product; Trigonometric equations — quadratic-in-sin, t-formula and auxiliary-angle methods
  14. Exam 14 — Vectors (projection, geometry proofs, projectile motion as vectors) — emphasised per exam flavour; Proof by Mathematical Induction; Trigonometric Equations (quadratic-in-sin, auxiliary-angle method)
  15. Exam 15 — Trigonometric Equations (emphasis); Proof by Mathematical Induction; Vectors
  16. Exam 16 — Further Calculus Skills (integration by substitution, inverse-trig derivatives/integrals); Applications of Calculus (related rates, volumes of revolution, exponential models); Proof by Mathematical Induction (divisibility)
  17. Exam 17 — Applications of Calculus (SHM, projectile modelling, area/volume of revolution); Further Calculus Skills (inverse-trig derivatives, integration by substitution and reverse chain); The Binomial Distribution (probabilities, mean/SD, recurrence and mode)
  18. Exam 18 — The Binomial Distribution (emphasis - 21/60 marks): Bernoulli trials, P(X=k)=C(n,k)p^k q^(n-k), mean np and sd sqrt(npq), most likely value, normal approximation and sample proportions; Proof by Mathematical Induction: divisibility and series-sum proofs with rigorous inductive step; Vectors: scalar (dot) product, projection, perpendicularity, geometry proofs and projectile motion
  19. Exam 19 — Proof by Mathematical Induction (series identity and divisibility); Vectors (projection, angle, geometric proof); Trigonometric equations and Further Calculus Skills
  20. Exam 20 — Vectors (35%): components, projection, angle, geometric proofs, projectile motion in vector form; Proof by Mathematical Induction (series sum and divisibility); Trigonometric Equations (quadratic-in-sin and auxiliary-angle method)

All 20 revision notes

  • Auxiliary-Angle Method and Products to Sums
  • Sum, Difference and Double-Angle Identities
  • Inequalities
  • Parametric Form of a Function or Relation
  • Reciprocal, Absolute-Value and Square-Root Graphs
  • Inverse Trigonometric Functions and Their Graphs
  • Division, Remainder and Factor Theorems
  • Roots and Coefficients; Multiple Roots
  • Related Rates and Exponential Growth and Decay
  • Permutations and Combinations
  • The Binomial Theorem and Pascal's Triangle
  • Differential Equations
  • Growth, Decay and Related Rates Models
  • Integrals of sin-squared, cos-squared and Inverse-Trig Forms
  • Integration by Substitution
  • Proof by Mathematical Induction
  • The Binomial Distribution and Normal Approximation
  • Solving Trigonometric Equations
  • Introduction to Vectors: Operations and Geometry
  • Projectile Motion