Skill Align QCE Specialist Mathematics Paper 1 - Free Online Pack 0
Full-length Units 3&4 external-assessment-style showcase paper
- Paper
- Paper 1 Technology-free Question and Response Book Showcase
- Reading
- 5 minutes perusal
- Writing
- 90 minutes
- Assessment
- 60 marks
QCAA formula book provided; calculators, technology, notes and other resources are not permitted for Paper 1. No public PDF or formula-book download is supplied with Pack 0.
Section 1
Questions 1-10 are multiple choice. Select the best answer. Calculators are not permitted.
Question 1
1 mark- -1+sqrt3i
- 1+sqrt3i
- -1-sqrt3i
- 2
Question 2
1 mark- (-2,3)
- (2,3)
- (3,2)
- (-3,-2)
Question 3
1 mark- mathbf r=(1,-1,2)+t(4,1,3)
- mathbf r=(3,2,1)+t(2,3,1)
- mathbf r=(1,-1,2)+t(2,3,-1)
- mathbf r=(2,3,-1)+t(1,-1,2)
Question 4
1 mark- 3ln|x+1|+5ln|x+2|+C
- 2ln|x+2|+ln|x+1|+C
- ln|(x+1)(x+2)|+C
- 2ln|x+1|+ln|x+2|+C
Question 5
1 mark- x=1 or y=-2
- x=-1 or y=2
- y=x-3
- y=-2x+1
Question 6
1 mark- sqrt2
- 2sqrt2
- 4
- 8
Question 7
1 mark- half as large
- twice as large
- four times as large
- eight times as large
Question 8
1 mark- 5operatorname(cis)(fracpi6)
- 6operatorname(cis)(fracpi2)
- frac23operatorname(cis)(fracpi2)
- 6operatorname(cis)(fracpi6)
Question 9
1 mark- the vertical line x=2
- the horizontal line y=2
- the circle x²+y²=4
- the ray from the origin with argument π / 2
Question 10
1 mark- mathbf u × mathbf v=0
- (mathbf b-mathbf a) × (mathbf u × mathbf v)=0
- (mathbf b-mathbf a) × (mathbf u × mathbf v)=mathbf0
- lvertmathbf urvert=lvertmathbf vrvert
Section 2
Questions 11-19 are short response. Show exact working and mathematical reasoning.
Question 11
5 marksQuestion 12
6 marksQuestion 13
5 marksQuestion 14
6 marksQuestion 15
5 marksQuestion 16
6 marksQuestion 17
5 marksQuestion 18
6 marksQuestion 19
6 marksWorked Solutions And Marking Guide
Section 1 Question 1
Answer: -1+sqrt3i
The roots are 2operatorname(cis)(2kpi / 3), so the required root is 2operatorname(cis)(2π / 3)=-1+sqrt3i.
Section 1 Question 2
Answer: (2,3)
Multiply Abinom3(-2)=binom23, which is a quarter-turn anticlockwise.
Section 1 Question 3
Answer: mathbf r=(1,-1,2)+t(2,3,-1)
Use point A and direction overrightarrow(AB)=B-A=(2,3,-1).
Section 1 Question 4
Answer: 2ln|x+1|+ln|x+2|+C
Since ((3x+5) / ((x+1)(x+2)))=frac2(x+1)+frac1(x+2), integrate each linear fraction.
Section 1 Question 5
Answer: x=1 or y=-2
The product (x-1)(y+2) is zero when either factor is zero.
Section 1 Question 6
Answer: 2sqrt2
The cross product is mathbf a × mathbf b=(-2,0,2), whose magnitude is 2sqrt2.
Section 1 Question 7
Answer: four times as large
The margin of error is proportional to 1 / sqrt n, so halving it requires multiplying n by 4.
Section 1 Question 8
Answer: 6operatorname(cis)(fracpi6)
Multiply the moduli and add the arguments: zw=6operatorname(cis)(π / 3-π / 6).
Section 1 Question 9
Answer: the vertical line x=2
Since z+overline z=(x+iy)+(x-iy)=2x, the equation reduces to x=2.
Section 1 Question 10
Answer: (mathbf b-mathbf a) × (mathbf u × mathbf v)=0
The zero scalar triple product makes the displacement and both directions coplanar. Since the directions are not parallel, the coplanar lines intersect.
Section 2 Question 11
(a) z=sqrt2operatorname(cis)(fracpi4) and w=2operatorname(cis)(-fracpi6), so frac zw=frac1(sqrt2)operatorname(cis)(((5π) / (12))).
Divide the moduli and subtract the arguments.
(b) (frac1(sqrt2))^6operatorname(cis)(((5π) / (2)))=frac18operatorname(cis)(fracpi2)=frac i8.
Apply de Moivre's theorem and reduce the argument modulo two π.
Detailed marking criteria
Part Part (a) (3 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) (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.
Section 2 Question 12
(a) The directions (1,1,0) and (0,1,1) are not parallel. Equating coordinates gives s=-1, then t=-3 from y but t=1 from z, so the lines do not intersect. Hence they are skew.
Establish both non-parallel directions and the absence of an intersection.
(b) With mathbf d_1 × mathbf d_2=(1,-1,1) and mathbf q-mathbf p=(-1,2,-1), the distance is ((|(-1,2,-1) × (1,-1,1)|) / (sqrt3))=frac4(sqrt3).
Project the vector between points on the lines onto their common normal.
(c) A normal is (1,-1,1). Through (1,0,1), the plane is (x-1)-y+(z-1)=0, or x-y+z=2.
Use both direction vectors in the plane to construct its normal.
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) (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 (c) (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.
Section 2 Question 13
(a) v(t)=3t²-4t+1 and x(t)=t³-2t²+t.
Integrate twice and apply the two initial conditions.
(b) 3t²-4t+1=(3t-1)(t-1)=0, so t=frac13 and t=1.
Set velocity equal to zero.
(c) The velocity is negative between the roots. Since x(frac13)=frac4(27) and x(1)=0, the distance is frac4(27).
Use the sign of velocity so that displacement is converted to distance.
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)
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 (c) (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.
Section 2 Question 14
(a) The margin is 3, so 3=((2(9)) / (sqrt n)). Hence sqrt n=6 and n=36.
Use half the total width as the margin of error.
(b) The interval is (69,75). No; 70 lies inside the interval, so this interval does not rule out a population mean of 70.
Construct the endpoints before interpreting the claim.
(c) The width is halved to 3.
Interval width is inversely proportional to the square root of sample size.
Detailed marking criteria
Part Part (a) (3 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) (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 (c) (1 mark)
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.
Section 2 Question 15
(a) Take u=ln x and dv=x,dx. Then du=frac1x,dx, v=((x²) / (2)), and int xln x,dx=((x²) / (2))ln x-frac12int x,dx=((x²) / (2))ln x-((x²) / (4))+C.
State the parts and complete the remaining elementary integral.
(b) [((x²) / (2))ln x-((x²) / (4))]_1^e=((e²) / (4))-(-frac14)=((e²+1) / (4)).
Evaluate the antiderivative at both bounds.
Detailed marking criteria
Part Part (a) (3 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) (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.
Section 2 Question 16
(a) On mathbf r=P+tmathbf d, the plane expression is 3+2t. Thus 3+2t=9, so t=3 and the point is H=(7,2,-1).
Form and solve the line-plane intersection without assuming a supplied parameter value.
(b) A plane normal is mathbf n=(1,2,2). The normal component is operatorname(proj)_(mathbf n)mathbf d=((mathbf d × mathbf n) / (mathbf n × mathbf n))mathbf n=frac29(1,2,2). Hence mathbf d_(ref)=mathbf d-2operatorname(proj)_(mathbf n)mathbf d=frac19(14,1,-17), so (14,1,-17) is a direction vector.
Decompose the incident direction into normal and parallel components, then reverse only the normal component.
(c) |mathbf d|²=6, while |frac19(14,1,-17)|²=((196+1+289) / (81))=6, so the magnitudes are equal.
Compare squared magnitudes to avoid unnecessary radicals.
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) (3 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 (c) (1 mark)
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.
Section 2 Question 17
(a) Abinom2(-1)=binom13, so the image is (1,3).
Multiply the matrix by the column vector.
(b) det A=-2, so A⁻¹=begin(pmatrix)frac12½frac12&-frac12end(pmatrix).
Use the two-by-two inverse formula and the non-zero determinant.
(c) A⁻¹binom42=binom31, so the preimage is (3,1).
Apply the inverse transformation.
Detailed marking criteria
Part Part (a) (1 mark)
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) (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 (c) (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.
Section 2 Question 18
(a) (y+2),dy=(x+1),dx, so frac12(y+2)²=frac12(x+1)²+C. Using (0,0) gives C=frac32, hence (y+2)²=(x+1)²+3. Since y+2=2>0 at x=0, y=-2+√((x+1)²+3).
Separate, integrate, apply the initial condition and justify the square-root branch.
(b) At x=1, y=-2+sqrt7 and ((dy) / (dx))=frac2(sqrt7). Thus y+2-sqrt7=frac2(sqrt7)(x-1).
Use the differential equation for the gradient after finding the point.
Detailed marking criteria
Part Part (a) (4 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) (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.
Section 2 Question 19
(a) Squaring the distances gives (x-1)²+(y-1)²=(x-5)²+(y-1)^2. Cancelling common terms gives 8x=24, so x=3.
Translate the equal-modulus condition into Cartesian distances and simplify.
(b) Substituting x=3 into x²+y²=25 gives y²=16, so z=3+4i or z=3-4i.
Intersect the line from part (a) with the circle of radius five.
(c) The vertical base Z_1Z_2 has length 8, and its perpendicular distance from O is 3. Thus the area is frac12(8)(3)=12.
Use the two Argand-plane points as a base and the origin's perpendicular distance from their line.
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) (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 (c) (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.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Complex Numbers and Polar Form | Q1, Q8-Q9, Q11, Q19 | 14 | ___ | Review complex roots, polar form, loci and Argand-plane geometry. |
| Vectors, Matrices and Geometry | Q2-Q3, Q6, Q10, Q12, Q16-Q17 | 21 | ___ | Review matrix transformations, vector lines and products, skew lines, planes and reflection. |
| Calculus, Differential Equations and Inference | Q4-Q5, Q7, Q13-Q15, Q18 | 25 | ___ | Review partial fractions, slope fields, motion, confidence intervals and separable equations. |