Question 15 - A-Level Maths

AQA Paper 2 2019 June — Question 15 9 marks

Exam Board	AQA
Module	Paper 2 (Paper 2)
Year	2019
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Vectors Introduction & 2D
Type	Geometric properties using vectors
Difficulty	Moderate -0.3 This is a straightforward vectors question requiring standard techniques: computing direction vectors to show parallel sides for part (a), and calculating distance then speed for part (b). While the numbers are large, the conceptual demands are routine—no novel insight or complex multi-step reasoning is needed, making it slightly easier than average.
Spec	1.10a Vectors in 2D: i,j notation and column vectors 1.10d Vector operations: addition and scalar multiplication 1.10f Distance between points: using position vectors

Four buoys on the surface of a large, calm lake are located at $A$, $B$, $C$ and $D$ with position vectors given by $$\overrightarrow{OA} = \begin{bmatrix} 410 \\ 710 \end{bmatrix}, \overrightarrow{OB} = \begin{bmatrix} -210 \\ 530 \end{bmatrix}, \overrightarrow{OC} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} \text{ and } \overrightarrow{OD} = \begin{bmatrix} 590 \\ -40 \end{bmatrix}$$ All values are in metres.

Prove that the quadrilateral $ABCD$ is a trapezium but not a parallelogram. [5 marks]
A speed boat travels directly from $B$ to $C$ at a constant speed in 50 seconds. Find the speed of the boat between $B$ and $C$. [4 marks]

Show mark scheme Show mark scheme source

Question 15:

Answer	Marks
15(a)	Finds or or or

correctly

Condo𝐴𝐴�� 𝐴𝐴� ⃗ d𝐶𝐶�� 𝐶𝐶� ⃗ ctio𝐴𝐴� ��𝐶𝐶� ⃗ erro𝐶𝐶� � �� i𝐴𝐴� ⃗

n e a i r e n r n the

label

OE

or −130 −180

𝐴𝐴��𝐶𝐶�⃗ = � � �𝐶𝐶��𝐴𝐴�⃗ = � �

Finds g−ra8d4i0ent of or 750 or or

correctly

𝐴𝐴𝐴𝐴 9 𝐶𝐶𝐶𝐶 𝐴𝐴𝐶𝐶

𝐶𝐶G𝐴𝐴radient = = OE

31 𝐴𝐴𝐴𝐴 𝐶𝐶8𝐶𝐶4

Gradient = OE

13 𝐴𝐴𝐶𝐶 25

Gradient = − OE

6 𝐶𝐶𝐴𝐴 31 13 6

Accept ratios , ,− OE

9 84 25

Ignore any incorrect labelling of

Answer	Marks	Guidance
ratios here	3.1a	M1

𝐴𝐴��𝐴𝐴�⃗ = � � 𝐶𝐶��𝐶𝐶�⃗ = � �

−180 270

�𝐶𝐶��𝐶𝐶�⃗ = −1.5 ×�𝐴𝐴��𝐴𝐴�⃗

Thus and are parallel but

not equal in length

𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶

is a trapezium but not a

parallelogram

𝐴𝐴 𝐴𝐴𝐶𝐶𝐶𝐶

Finds and correctly

OE

𝐴𝐴��𝐴𝐴�⃗ 𝐶𝐶��𝐶𝐶�⃗

or

Finds gradients of and

correctly

or 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶

Finds a corresponding pair of ratios

correctly – Do not award if

reciprocals of gradients are

Answer	Marks	Guidance
labelled as gradients or vectors	1.1b	A1

Shows/states OE

or

𝐶𝐶��𝐶𝐶�⃗ = ±1.5 ×𝐴𝐴��𝐴𝐴�⃗

Shows/states that

or

�𝐴𝐴��𝐶𝐶�⃗ ≠ 𝑘𝑘×�𝐶𝐶��𝐴𝐴�⃗

Finds and correctly

or

g𝐴𝐴�� r𝐶𝐶�⃗ �𝐶𝐶s �� 𝐴𝐴� ⃗

Finds adient o f and

correctly

or 𝐴𝐴𝐶𝐶 𝐶𝐶𝐴𝐴

Finds a second corresponding pair

of ratios correctly– Do not award if

reciprocals of gradients are

labelled as gradients or vectors

If incorrect labelling used for ratios

then maximum mark is

Answer	Marks	Guidance
M1 A0 A0 E1 R0	1.1b	A1

Deduces that and are

parallel - implied by reference to

gradients𝐴𝐴�� 𝐴𝐴�⃗ 𝐶𝐶��𝐶𝐶�⃗

equal

or

Deduces correctly that and

are not parallel

�𝐴𝐴�� 𝐶𝐶 �⃗ �𝐶𝐶h ��𝐴𝐴e �⃗

NB E1 is Independent o f any ot r

Answer	Marks	Guidance
marks	3.2a	E1

Completes rigorous proof by

deducing correctly that the scalar

multiple of OE means the

parallel sides are not equal in

length ±1.5

or

Completes rigorous proof by

deducing correctly that and

are parallel giving justification and

𝐴𝐴��𝐴𝐴�⃗ �𝐶𝐶��𝐶𝐶�⃗

that and are not parallel

giving justification

�𝐴𝐴��𝐶𝐶�⃗ �𝐶𝐶��𝐴𝐴�⃗

Must include a statement that

is not a parallelogram at

some point

𝐴𝐴 𝐴𝐴𝐶𝐶𝐶𝐶

NB R1 can be awarded even if E1

was not awarded

Answer	Marks	Guidance
CSO	2.1	R1

Answer	Marks
15(b)	Uses velocity/displacement/time

relationship

Evidenced by dividing any vector

Answer	Marks	Guidance
/distance from part 15(a) by 50	3.1b	M1

𝒗𝒗= ×� �

50 −840

− 2.6

𝒗𝒗 = � �

−16.8

2 2

Answer	Marks	Guidance
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 =	𝒗𝒗	= �2.6 +16.8

m s-1

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 17

Finds the magnitude of their

or v

Answer	Marks	Guidance
𝐴𝐴��𝐶𝐶�⃗	1.1a	M1
Obtains 17	1.1a	A1

States correct speed with correct

Answer	Marks	Guidance
units	3.2a	A1
Total	9
Q	Marking Instructions	AO

Question 15:
--- 15(a) ---
15(a) | Finds or or or
correctly
Condo𝐴𝐴�� ��𝐴𝐴� ⃗ d𝐶𝐶�� � �𝐶𝐶� ⃗ ctio𝐴𝐴� ���𝐶𝐶� ⃗ erro𝐶𝐶� � �� i𝐴𝐴� ⃗
n e a i r e n r n the
label
OE
or −130 −180
𝐴𝐴����𝐶𝐶�⃗ = � � �𝐶𝐶���𝐴𝐴�⃗ = � �
Finds g−ra8d4i0ent of or 750 or or
correctly
𝐴𝐴𝐴𝐴 9 𝐶𝐶𝐶𝐶 𝐴𝐴𝐶𝐶
𝐶𝐶G𝐴𝐴radient = = OE
31
𝐴𝐴𝐴𝐴 𝐶𝐶8𝐶𝐶4
Gradient = OE
13
𝐴𝐴𝐶𝐶 25
Gradient = − OE
6
𝐶𝐶𝐴𝐴 31 13 6
Accept ratios , ,− OE
9 84 25
Ignore any incorrect labelling of
ratios here | 3.1a | M1 | −620 930
𝐴𝐴����𝐴𝐴�⃗ = � � 𝐶𝐶����𝐶𝐶�⃗ = � �
−180 270
�𝐶𝐶���𝐶𝐶�⃗ = −1.5 ×�𝐴𝐴���𝐴𝐴�⃗
Thus and are parallel but
not equal in length
𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶
is a trapezium but not a
parallelogram
𝐴𝐴 𝐴𝐴𝐶𝐶𝐶𝐶
Finds and correctly
OE
𝐴𝐴����𝐴𝐴�⃗ 𝐶𝐶����𝐶𝐶�⃗
or
Finds gradients of and
correctly
or 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶
Finds a corresponding pair of ratios
correctly – Do not award if
reciprocals of gradients are
labelled as gradients or vectors | 1.1b | A1
Shows/states OE
or
𝐶𝐶����𝐶𝐶�⃗ = ±1.5 ×𝐴𝐴����𝐴𝐴�⃗
Shows/states that
or
�𝐴𝐴���𝐶𝐶�⃗ ≠ 𝑘𝑘×�𝐶𝐶���𝐴𝐴�⃗
Finds and correctly
or
g𝐴𝐴���� r𝐶𝐶�⃗ �𝐶𝐶s ��� 𝐴𝐴� ⃗
Finds adient o f and
correctly
or 𝐴𝐴𝐶𝐶 𝐶𝐶𝐴𝐴
Finds a second corresponding pair
of ratios correctly– Do not award if
reciprocals of gradients are
labelled as gradients or vectors
If incorrect labelling used for ratios
then maximum mark is
M1 A0 A0 E1 R0 | 1.1b | A1
Deduces that and are
parallel - implied by reference to
gradients𝐴𝐴��� �𝐴𝐴�⃗ 𝐶𝐶����𝐶𝐶�⃗
equal
or
Deduces correctly that and
are not parallel
�𝐴𝐴�� �𝐶𝐶 �⃗ �𝐶𝐶h ���𝐴𝐴e �⃗
NB E1 is Independent o f any ot r
marks | 3.2a | E1
Completes rigorous proof by
deducing correctly that the scalar
multiple of OE means the
parallel sides are not equal in
length ±1.5
or
Completes rigorous proof by
deducing correctly that and
are parallel giving justification and
𝐴𝐴����𝐴𝐴�⃗ �𝐶𝐶���𝐶𝐶�⃗
that and are not parallel
giving justification
�𝐴𝐴���𝐶𝐶�⃗ �𝐶𝐶���𝐴𝐴�⃗
Must include a statement that
is not a parallelogram at
some point
𝐴𝐴 𝐴𝐴𝐶𝐶𝐶𝐶
NB R1 can be awarded even if E1
was not awarded
CSO | 2.1 | R1
--- 15(b) ---
15(b) | Uses velocity/displacement/time
relationship
Evidenced by dividing any vector
/distance from part 15(a) by 50 | 3.1b | M1 | 1 −130
𝒗𝒗= ×� �
50 −840
− 2.6
𝒗𝒗 = � �
−16.8
2 2
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = |𝒗𝒗|= �2.6 +16.8
m s-1
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 17
Finds the magnitude of their
or v
𝐴𝐴����𝐶𝐶�⃗ | 1.1a | M1
Obtains 17 | 1.1a | A1
States correct speed with correct
units | 3.2a | A1
Total | 9
Q | Marking Instructions | AO | Marks | Typical Solution

Show LaTeX source

Four buoys on the surface of a large, calm lake are located at $A$, $B$, $C$ and $D$ with position vectors given by
$$\overrightarrow{OA} = \begin{bmatrix} 410 \\ 710 \end{bmatrix}, \overrightarrow{OB} = \begin{bmatrix} -210 \\ 530 \end{bmatrix}, \overrightarrow{OC} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} \text{ and } \overrightarrow{OD} = \begin{bmatrix} 590 \\ -40 \end{bmatrix}$$

All values are in metres.

\begin{enumerate}[label=(\alph*)]
\item Prove that the quadrilateral $ABCD$ is a trapezium but not a parallelogram. [5 marks]

\item A speed boat travels directly from $B$ to $C$ at a constant speed in 50 seconds.

Find the speed of the boat between $B$ and $C$. [4 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 2 2019 Q15 [9]}}

This paper (17 questions)

View full paper

Q1 1 Q2 1 Q3 1 Q4 4 Q5 7 Q6 6 Q7 10 Q8 11 Q9 9 Q10 1 Q11 1 Q12 1 Q13 5 Q14 8 Q15 9 Q16 16 Q17 9

AQA Paper 2 2019 June — Question 15 9 marks

Question 15:

correctly

Condo𝐴𝐴�� ��𝐴𝐴� ⃗ d𝐶𝐶�� � �𝐶𝐶� ⃗ ctio𝐴𝐴� ���𝐶𝐶� ⃗ erro𝐶𝐶� � �� i𝐴𝐴� ⃗

n e a i r e n r n the

label

OE

or −130 −180

𝐴𝐴����𝐶𝐶�⃗ = � � �𝐶𝐶���𝐴𝐴�⃗ = � �

Finds g−ra8d4i0ent of or 750 or or

correctly

𝐴𝐴𝐴𝐴 9 𝐶𝐶𝐶𝐶 𝐴𝐴𝐶𝐶

𝐶𝐶G𝐴𝐴radient = = OE

31

𝐴𝐴𝐴𝐴 𝐶𝐶8𝐶𝐶4

Gradient = OE

13

𝐴𝐴𝐶𝐶 25

Gradient = − OE

6

𝐶𝐶𝐴𝐴 31 13 6

Accept ratios , ,− OE

9 84 25

Ignore any incorrect labelling of

𝐴𝐴����𝐴𝐴�⃗ = � � 𝐶𝐶����𝐶𝐶�⃗ = � �

−180 270

�𝐶𝐶���𝐶𝐶�⃗ = −1.5 ×�𝐴𝐴���𝐴𝐴�⃗

Thus and are parallel but

not equal in length

𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶

is a trapezium but not a

parallelogram

𝐴𝐴 𝐴𝐴𝐶𝐶𝐶𝐶

Finds and correctly

OE

𝐴𝐴����𝐴𝐴�⃗ 𝐶𝐶����𝐶𝐶�⃗

or

Finds gradients of and

correctly

or 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶

Finds a corresponding pair of ratios

correctly – Do not award if

reciprocals of gradients are

Shows/states OE

or

𝐶𝐶����𝐶𝐶�⃗ = ±1.5 ×𝐴𝐴����𝐴𝐴�⃗

Shows/states that

or

�𝐴𝐴���𝐶𝐶�⃗ ≠ 𝑘𝑘×�𝐶𝐶���𝐴𝐴�⃗

Finds and correctly

or

g𝐴𝐴���� r𝐶𝐶�⃗ �𝐶𝐶s ��� 𝐴𝐴� ⃗

Finds adient o f and

correctly

or 𝐴𝐴𝐶𝐶 𝐶𝐶𝐴𝐴

Finds a second corresponding pair

of ratios correctly– Do not award if

reciprocals of gradients are

labelled as gradients or vectors

If incorrect labelling used for ratios

then maximum mark is

Deduces that and are

parallel - implied by reference to

gradients𝐴𝐴��� �𝐴𝐴�⃗ 𝐶𝐶����𝐶𝐶�⃗

equal

or

Deduces correctly that and

are not parallel

�𝐴𝐴�� �𝐶𝐶 �⃗ �𝐶𝐶h ���𝐴𝐴e �⃗

NB E1 is Independent o f any ot r

Completes rigorous proof by

deducing correctly that the scalar

multiple of OE means the

parallel sides are not equal in

length ±1.5

or

Completes rigorous proof by

deducing correctly that and

are parallel giving justification and

𝐴𝐴����𝐴𝐴�⃗ �𝐶𝐶���𝐶𝐶�⃗

Condo𝐴𝐴�� 𝐴𝐴� ⃗ d𝐶𝐶�� 𝐶𝐶� ⃗ ctio𝐴𝐴� ��𝐶𝐶� ⃗ erro𝐶𝐶� � �� i𝐴𝐴� ⃗

𝐴𝐴��𝐶𝐶�⃗ = � � �𝐶𝐶��𝐴𝐴�⃗ = � �

𝐴𝐴��𝐴𝐴�⃗ = � � 𝐶𝐶��𝐶𝐶�⃗ = � �

�𝐶𝐶��𝐶𝐶�⃗ = −1.5 ×�𝐴𝐴��𝐴𝐴�⃗

𝐴𝐴��𝐴𝐴�⃗ 𝐶𝐶��𝐶𝐶�⃗

𝐶𝐶��𝐶𝐶�⃗ = ±1.5 ×𝐴𝐴��𝐴𝐴�⃗

�𝐴𝐴��𝐶𝐶�⃗ ≠ 𝑘𝑘×�𝐶𝐶��𝐴𝐴�⃗

g𝐴𝐴�� r𝐶𝐶�⃗ �𝐶𝐶s �� 𝐴𝐴� ⃗

gradients𝐴𝐴�� 𝐴𝐴�⃗ 𝐶𝐶��𝐶𝐶�⃗

�𝐴𝐴�� 𝐶𝐶 �⃗ �𝐶𝐶h ��𝐴𝐴e �⃗

𝐴𝐴��𝐴𝐴�⃗ �𝐶𝐶��𝐶𝐶�⃗

�𝐴𝐴��𝐶𝐶�⃗ �𝐶𝐶��𝐴𝐴�⃗