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HSC Mathematics Extension 1 Mastery Pack
Proof, vectors, further calculus and the binomial distribution — full 70-mark HSC papers with worked solutions.
HSC Mathematics Extension 1 exam: Fri 23 Oct, 1:50pm — 13 days away
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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:
- Identify a and b from a sin x + b cos x.
- Compute R = √(a² + b²) — this is always positive.
- 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.
- 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 ✓.
- 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
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HSC Mathematics Extension 1 exam: Fri 23 Oct, 1:50pm — 13 days away
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All 20 practice exams
- 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)
- 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)
- 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
- Exam 4 — Further Calculus Skills (integration by substitution, inverse trig differentiation & integration); Vectors (dot product, projection, angle); Trigonometric Equations (double-angle & t-formula)
- 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)
- 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)
- 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
- Exam 8 — Vectors (28/60 marks — emphasis): projection, collision modelling, Varignon proof, perpendicularity/parallelism, extended projectile-by-vectors; Proof by Mathematical Induction; Trigonometric Equations
- Exam 9 — Trigonometric Equations (auxiliary-angle, quadratic-in-sin, modelling); Proof by Mathematical Induction; Vectors (projection and perpendicular component)
- 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)
- Exam 11 — Applications of Calculus (related rates, exponential growth/decay DEs, projectile motion, volumes of revolution); Proof by Mathematical Induction; Vectors (projection, angle, perpendicularity)
- 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
- 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
- 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)
- Exam 15 — Trigonometric Equations (emphasis); Proof by Mathematical Induction; Vectors
- 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)
- 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)
- 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
- Exam 19 — Proof by Mathematical Induction (series identity and divisibility); Vectors (projection, angle, geometric proof); Trigonometric equations and Further Calculus Skills
- 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
Common questions about HSC Mathematics Extension 1
Which syllabus does the HSC Mathematics Extension 1 exam follow?
HSC papers from 2020 to 2026 are set on the Mathematics Extension 1 Stage 6 Syllabus published in 2017. A new Mathematics Extension 1 11-12 Syllabus applies from 2027. Papers before 2020 come from the earlier 3 Unit era course, which shared some content but organised it differently.
Are pre-2020 Extension 1 past papers still worth doing?
Selectively. Induction, combinatorics, related rates and differential equation questions transfer well. Skip the old simple harmonic motion, circle geometry proof and parabola locus questions that appear in papers up to 2019, since those strands are not part of the current syllabus and no equivalent question can appear in your exam.
What topics are in HSC Mathematics Extension 1?
Seven: Functions, Trigonometric Functions, Calculus, Combinatorics, Proof, Vectors and Statistical Analysis. Vectors and formal proof by induction carry a lot of exam weight, and projectile motion is examined as an application of calculus and vectors rather than as a topic of its own.
Does the Extension 1 exam assume Mathematics Advanced content?
Yes. Extension 1 is studied alongside Advanced, not instead of it, and Extension 1 questions freely use Advanced techniques such as differentiation rules, logarithms, integration and the normal distribution. A question can be set entirely on Extension 1 content while still requiring fluent Advanced algebra to reach the answer.
Does HSC Mathematics Extension 1 scale up or down?
Mathematics Extension 1 scales up strongly, sitting just below Extension 2. Its cohort performs well across their other courses, which is what drives the adjustment. Scaling is recalculated every year, so this describes a past cohort rather than the year you are sitting.
What is included in the HSC Mathematics Extension 1 Mastery Pack?
Original practice exams with answer guides, worked questions, digital flashcards and revision notes for Mathematics Extension 1. Complete revision notes are also available free. Official past papers are free external links, not material we sell. Preview the sample note, worked question and contents here. Paid resources unlock with a one-time purchase from $20, with access while the platform operates.
Where can I buy HSC Mathematics Extension 1 notes and practice exams?
You can buy the Mathematics Extension 1 Mastery Pack here as a one-time purchase: original practice exams with answer guides, revision notes, worked questions and flashcards. Printed study guides, trial-exam packs and student note marketplaces are other options, and official NESA past papers are free — see the past-paper index for this subject.
Is the HSC Mathematics Extension 1 Mastery Pack a subscription?
No. It is a single payment per subject with no renewal, and access continues while the platform operates. You can preview a sample note, a worked question and the full contents before paying.
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