QCE Specialist Maths Practice Questions & Past Papers

Q148 · Practice questionTechnology-freeSimple familiar1 mark

QUESTION 148

The four roots of

z^4=-16

are shown on the Argand plane. For each root, a new complex number is defined by

w=z+\frac{4}{z}.

Which statement describes the set of distinct values of

w

Argand diagram with four roots of z to the fourth equals minus sixteen, lying on a radius-two circle.

(A)

Four points on a circle of radius 4 centred at the origin.

(B)

Exactly two points on the real axis.

(C)

Exactly two points on the imaginary axis.

(D)

Only the origin.

ABCD

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Q149 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 149 (6 marks)

Two particles move in the plane for

0\leq t\leq4

, where

t

is measured in seconds. Their position vectors, in metres, are

\mathbf r_A(t)=t\mathbf i+(t-2)^2\mathbf j,\qquad \mathbf r_B(t)=(t+1)\mathbf i+\mathbf j.

Their paths are shown. The points

A_0

and

B_0

indicate their positions at

t=0

Equal-scale paths: A follows y=(x-2)^2 for 0<=x<=4; B follows y=1 for 1<=x<=5. Start positions A0=(0,4), B0=(1,1).

Determine the two points where the paths cross. Explain why the particles never meet during the given time interval.

[3 marks]

Determine the minimum distance between the particles and all times at which this minimum occurs.

[3 marks]

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Q150 · Practice questionTechnology-activeComplex familiar1 mark

QUESTION 150

An instrument records

Y=1.2X+0.6

, where

X

is the true measurement and

Y

is the recorded measurement, both in millimetres. This calibration relationship is exact.

A random sample of 64 independent measurements has recorded sample mean

\overline y=18.6

and recorded sample standard deviation

s_Y=2.4

. Assume the conditions for an approximate normal confidence interval are satisfied.

Using

z=1.96

, which interval is the approximate 95% confidence interval for the population mean of the true measurement

X

(A)

(18.012,\ 19.188)\ \mathrm{mm}

(B)

(14.510,\ 15.490)\ \mathrm{mm}

(C)

(14.412,\ 15.588)\ \mathrm{mm}

(D)

(14.93875,\ 15.06125)\ \mathrm{mm}

ABCD

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Q151 · Practice questionTechnology-activeComplex unfamiliar8 marks

QUESTION 151 (8 marks)

The region

R

is bounded by

y=\ln x

, the

x

-axis and

x=e

, as shown. A vertical line

x=c

, where

1<c<e

, divides

R

into two regions of equal area.

The region under y=ln x from x=1 to x=e, above the x axis, is shaded.

Determine the exact area of R.

[2 marks]

Determine c, correct to three decimal places.

[2 marks]

The whole region R is rotated about the x-axis. Determine the percentage of the resulting solid's volume produced by the part of R with

1\leq x\leq c

, correct to one decimal place. Use the radii of the circular cross-sections to explain, without relying only on rounded calculations, why this percentage is less than 50%.

[4 marks]

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Q152 · Practice questionTechnology-activeComplex unfamiliar8 marks

QUESTION 152 (8 marks)

A population is modelled as a continuous quantity

N(t)

, where

t

is measured in days. It satisfies

\frac{dN}{dt}=kN(K-N),\qquad k>0,\quad K>0.

The graph shows the growth rate as a function of population. The axis intercepts and the labelled point P are exact. Initially,

N(0)=50

Exact growth-rate curve with intercepts N=0 and N=400 and labelled point P=(100,30).

Use the graph to determine K and k.

[2 marks]

Determine the first time after

t=0

when the growth rate returns to its initial value. Give an exact expression and an answer in days, correct to two decimal places.

[6 marks]

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Q153 · Practice questionTechnology-freeComplex familiar1 mark

QUESTION 153

The points

A

B

and

C

represent the three roots of

z^3=-8i

on the Argand plane. The point

P

represents

z=2

. What is the exact value of

PA\times PB\times PC

Equal-scale Argand plane: A=(sqrt(3),-1), B=(0,2), C=(-sqrt(3),-1), P=(2,0). Dashed segments join P to the three roots.

(A)

8

(B)

8\sqrt2

(C)

16

(D)

128

ABCD

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Q154 · Practice questionTechnology-freeComplex familiar7 marks

QUESTION 154 (7 marks)

A sequence of partial sums is defined by

S_n=\sum_{k=1}^{n}\frac{k}{(k+1)!},\qquad n\in\mathbb Z^+.

Here

m!

denotes the factorial of the positive integer

m

Calculate

S_3

exactly.

[1 mark]

Prove by mathematical induction that

S_n=1-1/(n+1)!

for every positive integer

n

[4 marks]

Determine the least

n

for which

S_n>0.999

. Justify your answer.

[2 marks]

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Q155 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 155 (6 marks)

A point

P

has coordinates

(4,1,3)

. A plane

\Pi

has equation

x+2y+2z=3.

The point

H

is the perpendicular projection of

P

onto

\Pi

. The point

P'

is the reflection of

P

\Pi

, so

H

is the midpoint of

PP'

. The diagram is schematic.

Schematic plane Pi with P and P-prime on opposite sides, H on the plane, and a perpendicular dashed line through all three points. H is their midpoint.

Determine the coordinates of

H

using a normal vector to

\Pi

[3 marks]

Determine the coordinates of

P\prime

and the length

PP\prime

[3 marks]

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