QCE Mathematical Methods: Exam practice Q18 (2025, Paper 1)

Mathematical Methods — Question 18

QCAA 2025, Paper 1 · 6 marks

Q18 · 2025 · Technology-freeComplex unfamiliar6 marks

QUESTION 18 (6 marks)

Two objects are launched simultaneously from different positions and travel along the same straight-line path. The objects are launched towards each other with the same initial speed. The first object’s displacement (m) from the origin is given by

d_1=\frac13t^3-\frac12t^2+kt,

where

t

is the time (s) since the objects were launched and

k

is a constant,

k\ne0

. The second object is moving with a constant acceleration of

4

m s

^{-2}

. The second object changes its direction, and at time

t=1

s the objects have equal velocities and continue to travel in the same direction. Compared to the first object, how much further does the second object travel between

t=1

s and the next time the objects have equal velocities?

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

4.5

Worked solution

For the first object,

v_1(t)=\frac{d}{dt}\left(\frac13t^3-\frac12t^2+kt\right)=t^2-t+k.

For the second object,

v_2(t)=\int4\,dt=4t+c.

Given

v_1(1)=v_2(1)

1^2-1+k=4+c,

c=k-4.

Equating the velocities,

t^2-t+k=4t+k-4,

t^2-5t+4=0,

(t-1)(t-4)=0,

so the second time is

t=4

As the objects travel in one direction between the given times, distance travelled equals displacement. The sample response then evaluates

d_2-d_1=\int_1^4(v_2-v_1)\,dt =\int_1^4\big((4t-2)-(t^2-t+2)\big)\,dt

=\int_1^4(-t^2+5t-4)\,dt=4.5.

correctly determines the equation for the first object’s velocity

[1 mark]

correctly determines the equation for the second object’s velocity including the constant

c

[1 mark]

determines the relationship between constants

k

and

c

[1 mark]

determines the second time when the two velocities are the same

[1 mark]

uses a suitable method for determining the difference between distances the objects have travelled in the given time interval

[1 mark]

determines the difference between distances of the two objects

[1 mark]

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Mathematical Methods — Question 18

QUESTION 18 (6 marks)

QCAA guide · typeset solution

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