Part a (1 mark)

Substitute g(x) into f.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Solve 2x²-1=7.; Obtains the correct final result f(g(x))=2x²-1, with x=pm2.

Part b (1 mark)

Use x^2ge0.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Identify the minimum value 2(0)-1=-1.; Completes the required reasoning and reaches [-1,infty).

Part a (1 mark)

Set 800e^(-0.2t)=400.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Solve e^(-0.2t)=1 / 2 using logarithms.; Obtains the correct final result 5ln2 hours.

Part b (1 mark)

Set 800e^(-0.2t)=100.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Take logarithms and solve for t.; Obtains the correct final result 5ln8 hours.

Part a (1 mark)

Use 6k=1.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Calculate 0(k)+1(2k)+2(3k).; Obtains the correct final result k=frac16 and E(X)=frac43.

Part b (1 mark)

Calculate E(X²)=7 / 3.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Subtract [E(X)]²=16 / 9.; Obtains the correct final result frac59.

Part a (1 mark)

Apply the product rule.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Solve f'(x)=0.; Substitute x=1.; Obtains the correct final result f'(x)=e^(-x)(1-x), with stationary point (1,1 / e).

Part b (1 mark)

Show f' changes from positive to negative at 1.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Compare with f(0)=0 and f(x)to0.

Part b (1 mark)

Use the zero gradient to write the horizontal tangent.; Completes the required reasoning and reaches The absolute maximum is 1 / e at x=1, and the tangent is y=1 / e.

Part a (1 mark)

Identify first term 5000 and ratio 0.92.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Evaluate 5000(0.92)^5.; Obtains the correct final result V_n=5000(0.92)^(n-1), and V_6=3295.41.

Part b (1 mark)

Apply ten years of declining-balance depreciation: 5000(0.92)¹⁰approx2171.94.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Obtains the correct final result 2171.94.

Part a (1 mark)

Use the quadratic vertex at x=2.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Evaluate int_0⁴(4x-x²),dx.; Obtains the correct final result Maximum 4 at x=2; area ((32) / (3)) square units.

Part b (1 mark)

Divide the integral 32 / 3 by the interval length 4.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Solve x(4-x)=8 / 3.

Part b (1 mark)

Retain both solutions in [0,4].; Obtains the correct final result Average value frac83; x=2pm((2sqrt3) / (3)).

Part a (1 mark)

Standardise 78 to obtain z=1.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Calculate 1-Phi(1).; Obtains the correct final result 0.1587.

Part b (1 mark)

Use x=72+1.282(6).

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Evaluate the percentile.; Obtains the correct final result 79.692, approximately 79.7.

Part a (1 mark)

Interpret 2.4 using kilometres and minutes.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Substitute x=10.; Completes the required reasoning and reaches Each additional kilometre is associated with 2.4 additional predicted minutes; the prediction is 39 minutes.

Part b (1 mark)

Calculate observed minus predicted, 42-39=3.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Calculate r²=0.7396 and interpret in context.; Completes the required reasoning and reaches Residual 3 minutes; about 74% of delivery-time variation is explained by its linear association with route length.

Part a (1 mark)

At A, compare bearings 065^circ and 090^circ.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

At B, compare bearings 270^circ and 310^circ.; Subtract the two angles from 180^circ.; Obtains the correct final result angle A=25^circ, angle B=40^circ, and angle C=115^circ.

Part b (1 mark)

Identifies that AC is opposite 40^circ and AB is opposite 115^circ.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Applies the sine rule ((AC) / (sin 40^circ))=((18) / (sin 115^circ)).

Part b (1 mark)

Evaluates and rounds to ACapprox12.8 km.

Part a (1 mark)

Substitute B_0=20000.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Compare 0.005B_0=100 with the repayment.; Completes the required reasoning and reaches B_1=19,500; the 600 repayment exceeds the first month's 100 interest.

Part b (1 mark)

Iterate the recurrence to identify the accumulated repayment series.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Sum the finite geometric series.

Part b (1 mark)

Simplify using 600 / 0.005=120000.; Completes the required reasoning and reaches B_n=120000-100000(1.005)^n.

Part a (1 mark)

Require total probability 1.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Evaluate int_0^2x(2-x),dx=4 / 3.; Obtains the correct final result c=frac34.

Part b (1 mark)

Integrate the density from 0 to x.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Evaluate F(1)=1 / 2.

Part b (1 mark)

Use P(X>1)=1-F(1).; Obtains the correct final result F(x)=((3x²) / (4))-((x³) / (4)), and P(X>1)=frac12.

Part a (1 mark)

Read the multiplier -2.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Read the horizontal and vertical translations from vertex form.; Obtains the correct final result Reflect in the x-axis, stretch vertically by factor 2, then translate 3 units right and 5 units up.

Part b (1 mark)

Identify the vertex from the translated form.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Use the downward opening to state the maximum range.; Obtains the correct final result Vertex (3,5); range yle5.

Part a (1 mark)

Use h=128 / r^2.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Substitute into S=2pi r²+2pi rh.; Completes the required reasoning and reaches S(r)=2pi r²+256pi / r.

Part b (1 mark)

Solve S'(r)=4pi r-256pi / r²=0.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Obtain r=4 and hence h=8.

Part b (1 mark)

Use S''(4)>0 to justify the minimum.; Completes the required reasoning and reaches r=4 cm and h=8 cm.

Part a (1 mark)

Transform the mean using 1.5(42)-4.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Multiply the standard deviation by |1.5|.; Obtains the correct final result Mean 59; standard deviation 9.

Part b (1 mark)

Calculate y=1.5(50)-4=71.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Standardise using the transformed mean and standard deviation.; Completes the required reasoning and reaches The transformed value is 71, and (71-59) / 9=4 / 3.

Part a (1 mark)

Integrate f'(x).

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Use f(0)=5 to determine the constant.; Obtains the correct final result f(x)=2x³-6x²+2x+5.

Part b (1 mark)

Set 6x²-12x+2=0.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Apply the quadratic formula and simplify.; Obtains the correct final result x=1pm((sqrt6) / (3)).

Part a (1 mark)

Solve f'(x)=3x²-3=0.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Substitute the inputs and use f''(x)=6x.; Completes the required reasoning and reaches A local maximum at (-1,2) and a local minimum at (1,-2).

Part b (1 mark)

Factor to identify the zeros 0 and sqrt3.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Integrate 3x-x³ over the interval.; Obtains the correct final result frac94 square units.

Part a (1 mark)

Differentiate and factor v(t)=3(t-1)(t-3).

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Verify sign changes and substitute the two times into s.; Obtains the correct final result Direction changes at t=1,3; s(1)=4 and s(3)=0.

Part b (1 mark)

Evaluate s(0)=0 and s(4)=4.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Add the absolute changes 0to4, 4to0 and 0to4.; Obtains the correct final result 12 metres.

Part a (1 mark)

Recognise the deposits as a geometric series.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Use the future-value annuity sum.; Obtains the correct final result 300((1.004²⁴-1) / (0.004)).

Part b (1 mark)

Calculate total contributions 24(300)=7200.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Subtract them from the accumulated balance.; Obtains the correct final result 300((1.004²⁴-1) / (0.004))-7200.

Part a (1 mark)

Use 10=2(10)sin(theta / 2) to obtain theta=pi / 3.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Calculate sector area frac12r^2theta.

Part a (1 mark)

Calculate triangle area frac12r^2sintheta=25sqrt3.

Part a (1 mark)

Subtract triangle area from sector area.; Obtains the correct final result theta=fracpi3; sector area ((50pi) / (3))text( cm)²; segment area ((50pi) / (3))-25sqrt3text( cm)^2.

Part a (1 mark)

Use 1200a³=1800.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Solve a³=3 / 2 and substitute into the model.; Obtains the correct final result a=(3 / 2)^(1 / 3), so P(t)=1200(3 / 2)^(t / 3).

Part b (1 mark)

Set (3 / 2)^(t / 3)=2.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part b (1 mark)

Take logarithms and solve for t.; Obtains the correct final result ((3ln2) / (ln(3 / 2))).

Part a (1 mark)

Factor x(x-2)(x+2) to locate the three intercepts.

Acceptable alternatives: Accept an equivalent valid method or expression that demonstrates this mathematical decision.

Part a (1 mark)

Use odd symmetry to double the area on [0,2].

Part a (1 mark)

Integrate 4x-x³ from 0 to 2.

Part a (1 mark)

Evaluate 2(4)=8.; Obtains the correct final result 8 square units.