2022 VCAA Mathematical Methods Exam 2 Solutions - Use CAS to save time - Section A and B

Q: What kinds of topics are covered in Section A?

Section A contains multiple-choice questions spanning calculus (derivatives, gradients, turning points), function properties (including inverses), geometry distance, and statistics (e.g., confidence intervals).

Q: How are results obtained quickly in these solutions?

The solutions explicitly note using CAS to save time and also reference CAS for values such as maxima and combined rates, indicating CAS-assisted computation throughout.

Q: What does Section B include beyond the multiple-choice section?

Section B provides multi-part written solutions, including steps for tangents and perpendiculars, enclosed area calculations, optimisation ranges, population model analysis with periodicity, and detailed probability/distribution work.

Sep 20, 2026

Uploaded by Elsa

2022 VCAA Mathematical Methods Exam 2 solutions provide worked answers across multiple-choice and extended response problems. Section A lists selections for a sequence of calculus, algebra, functions, distance, and statistics questions, including confidence intervals, derivatives, turning points, and inverse-function reasoning. Section B continues with stepwise solution parts covering tangents, perpendicular lines, areas, maximum/minimum analysis, periodicity in population models, and probability computations, with frequent use of CAS to streamline calculations and verify results.

LanguageEnglish

CategoryExam

FormatPDF

Pages2

Reading Time5 min

AccessFree to download

Section A – Multiple-choice questions

Q1–Q20

Section B

Q1a–Q1e

Q2ai–Q2h

Q3ai–Q3ciii

Q4a–Q5b

2022 VCAA Mathematical Methods Exam 2 Solutions © itute 2022 Use CAS to save time SECTION A – Multiple-choice questions | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | B | A | E | D | C | C | E | E | D | D | | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | C | A | C | B | E | B | A | B | D | A | Q1 Period = ~~ ~~2π2 = π B Q2 As x → ∞ , y → 4 A Q3 When x = 0 , dydx = 3e3x = 3 E Q4 f (x) = tan ~~ ~~x3 is undefined at x3 = ~~ ~~π2 , i.e. x = ~~ ~~3π2 < 5 D Q5 f ′(x) = 2x + 3 = 0 when x = − ~~ ~~32 , i.e. turning point at x = − ~~ ~~32 .: a = −1 .5 C Q6 Inverse of y = x 2 for x < 0 is x = y 2 for y < 0 i.e. y = − ~~ ~~x C Q7 Locate the points of zero gradient, after the second point the gradient is positive. b a b Q8 ∫ f (x)dx = ∫ f (x)dx + ∫ f (x)dx , 0 0 a b 10 = −4 + ∫ f (x)dx a b.: ∫ f (x)dx = 14 a Q9 Shortest distance is straight line distance. d = x 2 + y 2 = x 2 + 2x + 1 = (x + 1)2 = x + 1 E E D Q10 n = 1000 , ˆp = 0.55 , approx 95% confidence interval for ˆP 􀀇􀀅 0.55 × 0.45 0.55 × 0.45 􀀄􀀂 is 􀀆􀀅 0.55 −1.96 1000 , 0.55 + 1.96 1000 􀀃􀀂 ≈ (0.519, 0.581) or (51.9%, 58.1%) D Q11 xcos x = ~~ ~~ddx (x sin x) − sin x , 1k~~ ~~ ∫ x cos xdx = ~~ ~~1k~~ ~~ 􀀇􀀆􀀅 ∫ ~~ ~~ddx (xsin x)dx − ∫ sin xdx􀀄􀀃􀀂 = ~~ ~~1k~~ ~~ (xsin x − ∫ sin xdx)+ c CQ12 Pr (RB ) + Pr(BR ) = ~~ ~~3~~ ~~x~~ ~~ × ~~ ~~2~~ ~~~~ ~~x + ~~ ~~3~~ ~~~~ ~~x~~ ~~ × ~~ ~~2~~ ~~~~ ~~x = ~~ (~~3+~~ ~~x6~~)~~x~~(~~2~~ ~~+~~ ~~x~~) ~~A Q13 xx~~ ~~~~ ~~aa > 0 , either both x + a and x − a are both positive, i.e. x > a or both are negative, i.e. x < −a .: x ∈ R \ [− a, a] C Q14 Q15 Q16 100 ∫ ~~ ~~29 x2e − 19x2 dx ≈ 2.659 0 x2 − 2x − 3 ≥ 0 , (x + 1)(x − 3) ≥ 0 , Solve f ′(− 3) = 9 − 6m + n = 0 , f ′(1) = 9 + 2m + n = 0 .: m = 1 and n = −3 , .: f (x) = ~~ ~~13 x3 + x2 − 3x + p , (3, 4) .: p = −5 B E B Q17 Property 1 gives (a, L) is below (b, L) . Property 2 shows is a cubic shape function with turning points. Q18 Pr (X ≥ 16 | X ≥ a) = 0.9175 and a ≤ 15 Pr (X ≥ 16 ∩ X ≥ a) Pr (X ≥ 16) 0.91728 Pr (X ≥ a) = Pr (X ≥ a) = Pr (X ≥ a) = 0.9175 .: Pr (X ≥ a) ≈ 0.99976 , a = 12 Q19 V = x(a − 2x)(b − 2x) = abx − 2(a + b)x2 + 4x3 ddVx~~ ~~ = ab − 4(a + b)x + 12x2 = 0 , x = ~~ ~~a~~ ~~+~~ ~~b~~ ~~−~~ ~~a26~~ ~~−~~ ~~ab~~ ~~+~~ ~~b2~~ ~~ Q20 50 sin 2θ > 40 , 26.565 < θ < 63.435 , Pr (26 .565 < θ < 63 .435) ≈ 0.96947 SECTION B Q1a x = 0 Q1b f ′(x) = ~~ ~~16 x Q1c f ′(x) = ~~ ~~16 x = −2 , x = −12 , y = f (−12) = 12 Tangent at M : y −12 = −2(x − (−12)) .: y = −2x −12 g (x) A B D A Q1di Perpendicular at M : y −12 = ~~ ~~12 (x − (−12)) , y = ~~ ~~12 x + 18 Q1dii Point N : y = ~~ ~~12 x + 18 and y = ~~ ~~x212 .: x = 18 812􀀇􀀆􀀅 12 x + 18 − ~~ ~~x212~~ ~~􀀄􀀃􀀂 dx = 375 Q1e All parabolas are similar. To simplify calculations, consider y = x2 . M (− p, p 2 ) and the perpendicular at M is y = ~~ ~~21p~~ ~~ x + p 2 + ~~ ~~12 which intersects y = x 2 at x = p + ~~ ~~21p . p+2~~1~~p Enclosed area A = ∫ 21p~~ ~~ x + p 2 + ~~ ~~12 − x 2 dx , Amin when p = ~~ ~~12 and − p .: M 􀀅􀀇􀀆 − ~~ ~~12 , ~~ ~~14~~ ~~ 􀀄􀀃􀀂 . For y = ~~ ~~x4a22~~ ~~ , M 􀀅􀀇􀀆− b, ~~ ~~b4a22~~ ~~􀀄􀀃􀀂 . Q parabolas are similar,∴ same dilation in both directions .: − ~~ ~~12 : −b = ~~ ~~14 : ~~ ~~b4a22 .: b = 2a2 1 2022 VCAA Mathematical Methods Exam 2 Solutions © itute 2022 Q2ai r (0) = 2500 Q2aii Max = 1700 + 2500 = 4200 , min = −1700 + 2500 = 800 Q2aiii 200 − 40 = 160 Q2b (20, 700) and (100, 2500) , a = ~~ ~~2500~~ ~~700 = 900 π Same period as rabbit population .: b = 80 Q2c By CAS , max value of r (t)+ f (t)≈ 5339 Q2d By CAS, 160 Q2e 􀀍t′ 􀀊 􀀍 90 t + 60 􀀊 􀀍t 􀀊 􀀍 π (t′ − 60) 􀀊 For fox population, 􀀌􀀋 y′􀀉􀀈 = 􀀌􀀋 90π0y + 1600 􀀉􀀈 , 􀀌􀀋 y 􀀉􀀈 = 􀀌􀀋 9~~1~~0900(y′ − 1600)􀀉􀀈 9100 (y′ −1600) = sin ~~ ~~π90 (t′ − 60) .: f (t) = 900sin ~~ ~~π90 (t − 60) + 1600 300 ∫ ( f (t) + r(t

FAQs

What kinds of topics are covered in Section A?

Section A contains multiple-choice questions spanning calculus (derivatives, gradients, turning points), function properties (including inverses), geometry distance, and statistics (e.g., confidence intervals).

How are results obtained quickly in these solutions?

The solutions explicitly note using CAS to save time and also reference CAS for values such as maxima and combined rates, indicating CAS-assisted computation throughout.

What does Section B include beyond the multiple-choice section?

Section B provides multi-part written solutions, including steps for tangents and perpendiculars, enclosed area calculations, optimisation ranges, population model analysis with periodicity, and detailed probability/distribution work.

Sep 20, 2026

Uploaded by Elsa