Skill Align

Skill Align QCE Specialist Mathematics Paper 2 - Free Online Pack 0

Full-length Units 3&4 external-assessment-style showcase paper

Paper: Paper 2 Technology-active Question and Response Book Showcase
Reading: 5 minutes perusal
Writing: 90 minutes
Assessment: 60 marks

QCAA formula book provided; approved graphics or scientific calculator permitted for Paper 2; CAS calculators are not permitted. No public PDF or formula-book download is supplied with Pack 0.

Section 1

Questions 1-10 are multiple choice. Select the best answer. Technology may be used where appropriate.

Question 1

1 mark

A confidence interval for a population mean is (72.4,77.6). Its centre and margin of error are

75.0 and 2.6
72.4 and 5.2
77.6 and 2.6
75.0 and 5.2

Question 2

1 mark

Forces of 40 N east, 15 N south and 10 N west act on a particle. The resultant is

Diagram Preview

50mathbf i+15mathbf j
30mathbf i-15mathbf j
30mathbf i+15mathbf j
15mathbf i-30mathbf j

Question 3

1 mark

Simpson's rule with h=0.5 and values 1,4,5 gives

Graph Preview

3
5
((11) / (3))
((22) / (3))

Question 4

1 mark

For mathbf a=(2,3,1) and mathbf b=(1,-1,2), the projection of a onto b is

6(1,-1,2)
(2,-3,2)
frac13(1,-1,2)
frac16(1,-1,2)

Question 5

1 mark

The polar point (5,π) is

(-5,0)
(5,0)
(0,-5)
(0,5)

Question 6

1 mark

If sample size is quadrupled with all else unchanged, confidence-interval width is

doubled
halved
quartered
unchanged

Question 7

1 mark

For dP / dt=0.5P(1-P / 30), the non-zero equilibrium is

Question 8

1 mark

Under the mapping w=(1+i)z, the point z=2-i maps to

1+3i
2
1+i
3+i

Question 9

1 mark

For mathbf a=(1,2,0) and mathbf b=(0,1,3), |mathbf a × mathbf b|=

√46
√14
6
46

Question 10

1 mark

For z=1.645, sigma=6 and n=144, the margin of error is

0.5000
0.8225
1.6450
9.8700

Section 2

Questions 11-18 are short response. Show working, modelling decisions and interpretation.

Question 11

5 marks

A process has population standard deviation sigma=6. Independent samples of sizes 25 and 75 from the same process have means 102 and 98 respectively. Use z=1.96.

(a) 2 marks

Find the mean of all 100 observations.

(b) 1 mark

Find the standard error of the combined mean.

(c) 2 marks

Construct the 95% confidence interval and state whether it establishes that the population mean is below 100.

Question 12

7 marks

A particle has speed v(t)=√(16+t³) metres per second for 0le tle4.

(a) 2 marks

Calculate v(1) and v(3), to three decimal places.

(b) 3 marks

Use Simpson's rule with four equal subintervals to estimate the distance travelled.

(c) 2 marks

Technology gives the integral as 21.826807 m. Find the percentage error in the Simpson estimate and explain why exactness was not guaranteed.

Question 13

6 marks

The mapping w=(1-2i)z+3+i is applied to the circle |z-(1+i)|=2.

(a) 2 marks

Describe the dilation and rotation caused by multiplication by 1-2i.

(b) 3 marks

Find an equation for the image circle in the w-plane.

(c) 1 mark

Find the area of the image circle.

Question 14

6 marks

A straight cable occupies the segment from A=(1,2,0) to B=(5,-1,4). A sensor is at C=(3,2,5). Safety clearance requires the sensor to be at least 4 units from the cable segment.

(a) 3 marks

Determine whether the point on line AB closest to C lies on the cable segment, and give its coordinates.

(b) 3 marks

Determine whether the safety requirement is met.

Question 15

6 marks

A metal component cools in a room held at 20^circC. Its temperature satisfies ((dT) / (dt))=-k(T-20), where k>0, T(0)=80 and T(10)=50. Time t is measured in minutes.

(a) 4 marks

Solve the differential equation and determine k.

(b) 2 marks

Find when the component first reaches 25^circC.

Question 16

7 marks

Consider the polar curve r=2(1+costheta), for -pilethetalepi.

(a) 2 marks

State the cusp, the point furthest from the pole and the axis of symmetry.

(b) 2 marks

Find the exact area enclosed by the curve.

(c) 3 marks

Find the exact arc length of the curve.

Question 17

6 marks

A 95% confidence interval for a process mean uses z=1.96, known sigma=12 and a current sample of 64 observations. The target total interval width is at most 3. Each additional observation costs $8. Alternatively, a $900 calibration reduces sigma to 6 while retaining the sample size 64.

(a) 3 marks

Without calibration, find the minimum total sample size that meets the target width.

(b) 2 marks

Compare the two costs and recommend a plan that meets the target.

(c) 1 mark

State one assumption needed for the comparison.

Question 18

7 marks

A 2 kg probe starts at mathbf r(0)=(0,1,0) with velocity mathbf v(0)=(1,-1,2). Constant forces mathbf F_1=(6,0,4) N and mathbf F_2=(-4,4,-2) N act on it. A planar sensor is described by x+y+z=5.

(a) 3 marks

Find the position vector of the probe for t greater than or equal to 0.

(b) 2 marks

Find when and where the probe first crosses the sensor plane.

(c) 2 marks

Find the speed at the crossing and the acute angle between the velocity and the plane.

Worked Solutions And Marking Guide

Section 1 Question 1

Answer: 75.0 and 2.6

Average the endpoints to find the centre, then subtract the centre from either endpoint.

Section 1 Question 2

Answer: 30mathbf i-15mathbf j

Combine signed horizontal and vertical components.

Section 1 Question 3

Answer: ((11) / (3))

Use ((h) / (3))[1+4(4)+5].

Section 1 Question 4

Answer: frac16(1,-1,2)

The dot product is 1 and b dot b is 6.

Section 1 Question 5

Answer: (-5,0)

Use cosine π=-1 and sine π=0.

Section 1 Question 6

Answer: halved

Width is proportional to one over square root n.

Section 1 Question 7

Answer: 30

Set the rate equal to zero.

Section 1 Question 8

Answer: 3+i

Expand (1+i)(2-i)=2-i+2i-i²=3+i.

Section 1 Question 9

Answer: √46

The cross product is (6,-3,1).

Section 1 Question 10

Answer: 0.8225

Compute 1.645 times 6 / 12.

Section 2 Question 11

(a) bar x=((25(102)+75(98)) / (100))=99.

Weight each sample mean by its sample size before dividing by the combined sample size.

(b) ((6) / (√100))=0.6.

The combined sample contains 100 independent observations from the same process.

(c) 99pm1.96(0.6)=(97.824,100.176). No; the interval contains 100.

Use the combined mean and standard error. A value inside the interval cannot be ruled out by this interval.

Detailed marking criteria

Part Part (a) (2 marks)

Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (1 mark)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 12

(a) v(1)=√17approx4.123 and v(3)=√43approx6.557.

Substitute the two times into the non-polynomial speed function.

(b) frac13[4+4(4.1231056)+2(4.8989795)+4(6.5574385)+8.9442719]approx21.821 m.

With four subintervals, h=1 and the Simpson weights are 1, 4, 2, 4, 1.

(c) ((|21.821469-21.826807|) / (21.826807)) × 100%approx0.0245%. Exactness is not guaranteed because √(16+t³) is not a polynomial of degree at most 3.

Compare unrounded values and identify the polynomial exactness condition for Simpson's rule.

Detailed marking criteria

Part Part (a) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 13

(a) A dilation by sqrt5 and a clockwise rotation through tan⁻¹(2).

The modulus of 1-2i is sqrt5, and its argument is -tan⁻¹(2).

(b) The centre maps to (1-2i)(1+i)+3+i=6, and the radius becomes 2sqrt5, so |w-6|=2sqrt5.

Map the original centre and scale the radius by the modulus of the multiplier.

(c) π(2sqrt5)²=20π.

Use the mapped radius, or multiply the original area by the area scale factor 5.

Detailed marking criteria

Part Part (a) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (1 mark)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 14

(a) With overrightarrow(AB)=(4,-3,4) and overrightarrow(AC)=(2,0,5), t=((overrightarrow(AC) × overrightarrow(AB)) / (|overrightarrow(AB)|²))=((28) / (41)). Since 0<t<1, the closest point lies on the segment and is H=A+toverrightarrow(AB)=(((153) / (41)),-((2) / (41)),((112) / (41))).

Infer the projection parameter rather than assuming that the closest point is an endpoint.

(b) CH=√(|overrightarrow(AC)|²-(((overrightarrow(AC) × overrightarrow(AB))²) / (|overrightarrow(AB)|²)))=√(29-((784) / (41)))=√(((405) / (41)))approx3.143. The clearance is less than 4, so the requirement is not met.

Use the perpendicular component of AC and interpret the result against the stated constraint.

Detailed marking criteria

Part Part (a) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 15

(a) ((dT) / (T-20))=-k,dt, so ln(T-20)=-kt+C and T=20+Ce^(-kt). From T(0)=80, C=60. Then 50=20+60e^(-10k), so e^(-10k)=frac12 and k=((ln2) / (10)). Thus T=20+60e^(-(ln2)t / 10).

Separate and integrate before applying both temperature observations.

(b) 25=20+60e^(-kt) gives e^(-kt)=frac1(12), so t=((ln12) / (k))=((10ln12) / (ln2))approx35.85 minutes.

Solve the calibrated model for time.

Detailed marking criteria

Part Part (a) (4 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 16

(a) The cusp is the pole (0,0) at theta=pmpi; the furthest point is (4,0) at theta=0; the curve is symmetric about the polar axis.

Evaluate r at the limiting and central angles, and use the even cosine function for symmetry.

(b) A=frac12int_(-π)^(π)4(1+costheta)²,dtheta=6π.

Expand the squared radius and integrate over one complete trace.

(c) Since ((dr) / (dtheta))=-2sintheta, √(r²+(dr / dtheta)²)=√(8(1+costheta))=4cos(theta / 2) on the interval. Hence L=int_(-π)^(π)4cos(theta / 2),dtheta=16.

Simplify the polar arc-length integrand and use the sign of cosine on the specified interval.

Detailed marking criteria

Part Part (a) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 17

(a) 2(1.96)((12) / (sqrt n))le3, so nge(((1.96(12)) / (1.5)))²=245.8624. Therefore the minimum total sample size is 246.

Convert total width to a margin of error and round the required sample size up.

(b) Adding 246-64=182 observations costs 182(8)=$1456. Calibration gives width 2(1.96)(6 / 8)=2.94 and costs $900. Calibrate, because it meets the target at the lower stated cost.

Verify that each option meets the target before comparing its cost.

(c) The observations must be independent and representative of the same stable process, and the stated population standard deviations must be valid.

State one relevant modelling or sampling assumption.

Detailed marking criteria

Part Part (a) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (1 mark)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Section 2 Question 18

(a) The resultant force is (2,4,2), so mathbf a=(1,2,1). Hence mathbf v(t)=(1+t,-1+2t,2+t) and mathbf r(t)=(t+frac12t²,1-t+t²,2t+frac12t²).

Use Newton's second law, then integrate while applying both initial vectors.

(b) Substitution gives 1+2t+2t²=5, so (t-1)(t+2)=0. For tge0, the first crossing is at t=1, at (frac32,1,frac52).

Reject the negative-time root after solving the plane-intersection condition.

(c) At t=1, mathbf v=(2,1,3), so the speed is √14 m s⁻¹. With plane normal mathbf n=(1,1,1), sinalpha=((|mathbf v × mathbf n|) / (|mathbf v||mathbf n|))=frac6(√42), giving alphaapprox67.8^circ.

Use the complementary relationship between the angle to a plane and the angle to its normal.

Detailed marking criteria

Part Part (a) (3 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (b) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Part Part (c) (2 marks)

Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.

Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.

Diagnostic Checklist

Topic	Questions	Marks	Marks Lost	Action
Statistical Inference and Normal Models	Q1, Q6, Q10-Q11, Q17	14	___	Review combined samples, interval width and sample-size decisions.
Forces, Vectors and Geometry	Q2, Q4, Q9, Q14, Q18	16	___	Review projection, constrained distance, resultant force and three-dimensional motion.
Numerical Methods, Complex Numbers and Differential Equations	Q3, Q5, Q7-Q8, Q12-Q13, Q15-Q16	30	___	Review Simpson's rule, complex mappings, non-logistic differential equations and polar curves.

QCE Specialist Mathematics Units 3&4 Free Online Pack 0 — Paper 2 Technology-active Question and Response Book Showcase

Exam-pack paper structure

Read Paper 2 Technology-active Question and Response Book Showcase online

Section 1

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Section 2

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Worked Solutions And Marking Guide

Section 1 Question 1

Section 1 Question 2

Section 1 Question 3

Section 1 Question 4

Section 1 Question 5

Section 1 Question 6

Section 1 Question 7

Section 1 Question 8

Section 1 Question 9

Section 1 Question 10

Section 2 Question 11

Detailed marking criteria

Section 2 Question 12

Detailed marking criteria

Section 2 Question 13

Detailed marking criteria

Section 2 Question 14

Detailed marking criteria

Section 2 Question 15

Detailed marking criteria

Section 2 Question 16

Detailed marking criteria

Section 2 Question 17

Detailed marking criteria

Section 2 Question 18

Detailed marking criteria

Diagnostic Checklist

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