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Level 5
Column vectors
16 questions
Lesson
Not started
Mark as done
Given that the vector
(
2
x
5
)
+
(
y
−
3
)
\begin{pmatrix} 2x \\ 5 \end{pmatrix} + \begin{pmatrix} y \\ -3 \end{pmatrix}
(
2
x
5
)
+
(
y
−
3
)
is parallel to the vector
(
x
2
)
+
(
3
y
1
)
\begin{pmatrix} x \\ 2 \end{pmatrix} + \begin{pmatrix} 3y \\ 1 \end{pmatrix}
(
x
2
)
+
(
3
y
1
)
find an expression for
y
y
y
in terms of
x
x
x
.
(3 marks)
●●●●●
Level 5
3 marks
Start
→
Mark as done
v
=
(
t
−
4
)
\mathbf{v} = \binom{t}{-4}
v
=
(
−
4
t
)
, where
t
t
t
is a positive number, and
∣
v
∣
=
5
|\mathbf{v}| = 5
∣
v
∣
=
5
.
(a)
Find the value of
t
t
t
.
(b)
Write down the value of
∣
−
5
v
∣
|-5\mathbf{v}|
∣
−
5
v
∣
.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A
A
A
is the point
(
−
3
,
5
)
(-3, 5)
(
−
3
,
5
)
and
B
B
B
is the point
(
7
,
−
1
)
(7, -1)
(
7
,
−
1
)
.
(a)
Write
A
B
→
\overrightarrow{AB}
A
B
as a column vector.
(b)
C
C
C
is the point such that
B
C
→
=
1
2
A
B
→
\overrightarrow{BC} = \tfrac12\overrightarrow{AB}
B
C
=
2
1
A
B
. Find the coordinates of
C
C
C
.
(c)
Work out
∣
A
C
→
∣
|\overrightarrow{AC}|
∣
A
C
∣
, giving your answer correct to
1
1
1
decimal place.
●●●●●
Level 5
7 marks
Start
→
Vectors on a shape
13 questions
Lesson
Not started
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
.
P
P
P
is the point on
A
B
AB
A
B
such that
A
P
:
P
B
=
1
:
2
AP : PB = 1 : 2
A
P
:
P
B
=
1
:
2
.
a
b
O
A
B
P
Diagram NOT accurately drawn
(a)
Write
A
B
→
\overrightarrow{AB}
A
B
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. (1 mark)
(b)
Show that
O
P
→
=
2
3
a
+
1
3
b
\overrightarrow{OP} = \dfrac{2}{3}\mathbf{a} + \dfrac{1}{3}\mathbf{b}
O
P
=
3
2
a
+
3
1
b
(3 marks)
●●●●●
Level 5
4 marks
Start
→
Mark as done
O
A
B
C
OABC
O
A
B
C
is a quadrilateral.
C
B
T
CBT
C
B
T
is a straight line.
6a
4b
3a
O
A
B
C
T
M
N
M
M
M
is the point on
O
A
OA
O
A
such that
O
M
:
M
A
=
1
:
2
OM : MA = 1 : 2
O
M
:
M
A
=
1
:
2
N
N
N
is the midpoint of
A
B
AB
A
B
.
O
A
→
=
6
a
\overrightarrow{OA} = 6\mathbf{a}
O
A
=
6
a
\qquad
O
C
→
=
4
b
\overrightarrow{OC} = 4\mathbf{b}
O
C
=
4
b
\qquad
C
B
→
=
3
a
\overrightarrow{CB} = 3\mathbf{a}
C
B
=
3
a
B
T
→
=
k
C
B
→
\overrightarrow{BT} = k\overrightarrow{CB}
B
T
=
k
C
B
where
k
k
k
is a scalar.
Given that
M
N
T
MNT
M
N
T
is a straight line, find the value of
k
k
k
.
You must show all your working.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
O
P
Q
R
OPQR
O
P
QR
is a quadrilateral.
a
b
kb
O
P
Q
R
X
Y
X
X
X
is the point on
O
P
OP
O
P
such that
O
X
:
X
P
=
1
:
3
OX : XP = 1 : 3
O
X
:
X
P
=
1
:
3
Y
Y
Y
is the point on
R
Q
RQ
R
Q
such that
R
Y
:
Y
Q
=
2
:
1
RY : YQ = 2 : 1
R
Y
:
Y
Q
=
2
:
1
O
P
→
=
a
\overrightarrow{OP} = \mathbf{a}
O
P
=
a
\qquad
O
R
→
=
b
\overrightarrow{OR} = \mathbf{b}
O
R
=
b
\qquad
P
Q
→
=
k
b
\overrightarrow{PQ} = k\mathbf{b}
P
Q
=
k
b
where
k
k
k
is a positive integer.
(a)
Find
X
Y
→
\overrightarrow{XY}
X
Y
in terms of
k
k
k
,
a
\mathbf{a}
a
and
b
\mathbf{b}
b
.
Give your answer in its simplest form.
(4 marks)
(b)
Is
X
Y
XY
X
Y
parallel to
O
P
OP
O
P
?
Give a reason for your answer.
(1 mark)
●●●●●
Level 5
5 marks
Start
→
Mark as done
O
A
B
C
OABC
O
A
B
C
is a parallelogram.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
C
→
=
b
\overrightarrow{OC} = \mathbf{b}
O
C
=
b
.
D
D
D
is the point on
A
B
AB
A
B
such that
A
D
:
D
B
=
3
:
1
AD:DB = 3:1
A
D
:
D
B
=
3
:
1
.
E
E
E
is the midpoint of
O
C
OC
O
C
.
a
b
O
A
B
C
D
E
The diagram is a sketch and is not accurately drawn.
(a)
Find
O
D
→
\overrightarrow{OD}
O
D
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
.
(b)
Find
E
D
→
\overrightarrow{ED}
E
D
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
, giving your answer in its simplest form.
●●●●●
Level 5
6 marks
Start
→
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
.
M
M
M
is the midpoint of
O
A
OA
O
A
.
N
N
N
is the point on
A
B
AB
A
B
such that
B
N
:
N
A
=
1
:
3
BN:NA = 1:3
B
N
:
N
A
=
1
:
3
.
a
b
O
A
B
M
N
The diagram is a sketch and is not accurately drawn.
(a)
Find
O
N
→
\overrightarrow{ON}
O
N
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
, giving your answer in its simplest form.
(b)
Find
M
N
→
\overrightarrow{MN}
M
N
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
, giving your answer in its simplest form.
●●●●●
Level 5
6 marks
Start
→
Vector proofs
12 questions
Lesson
Not started
Mark as done
O
A
C
B
OACB
O
A
C
B
is a parallelogram.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
M
M
M
is the midpoint of
A
C
AC
A
C
and
N
N
N
is the midpoint of
B
C
BC
B
C
.
a
b
O
A
C
B
M
N
Diagram NOT accurately drawn
Prove that
M
N
MN
M
N
is parallel to
A
B
AB
A
B
. (4 marks)
●●●●●
Level 5
4 marks
Start
→
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
2
a
\overrightarrow{OA} = 2\mathbf{a}
O
A
=
2
a
and
O
B
→
=
2
b
\overrightarrow{OB} = 2\mathbf{b}
O
B
=
2
b
.
P
P
P
is the point on
A
B
AB
A
B
such that
A
P
:
P
B
=
1
:
3
AP : PB = 1 : 3
A
P
:
P
B
=
1
:
3
.
Q
Q
Q
is the point on
O
B
OB
O
B
such that
O
Q
:
Q
B
=
1
:
3
OQ : QB = 1 : 3
O
Q
:
QB
=
1
:
3
.
Prove that
P
Q
PQ
P
Q
is parallel to
O
A
OA
O
A
. [5]
●●●●●
Level 5
5 marks
Start
→
Mark as done
O
A
B
C
OABC
O
A
B
C
is a quadrilateral.
C
B
T
CBT
C
B
T
is a straight line.
6a
4b
3a
O
A
B
C
T
M
N
M
M
M
is the point on
O
A
OA
O
A
such that
O
M
:
M
A
=
1
:
2
OM : MA = 1 : 2
O
M
:
M
A
=
1
:
2
N
N
N
is the midpoint of
A
B
AB
A
B
.
O
A
→
=
6
a
\overrightarrow{OA} = 6\mathbf{a}
O
A
=
6
a
\qquad
O
C
→
=
4
b
\overrightarrow{OC} = 4\mathbf{b}
O
C
=
4
b
\qquad
C
B
→
=
3
a
\overrightarrow{CB} = 3\mathbf{a}
C
B
=
3
a
B
T
→
=
k
C
B
→
\overrightarrow{BT} = k\overrightarrow{CB}
B
T
=
k
C
B
where
k
k
k
is a scalar.
Given that
M
N
T
MNT
M
N
T
is a straight line, find the value of
k
k
k
.
You must show all your working.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
O
P
Q
R
OPQR
O
P
QR
is a quadrilateral.
a
b
kb
O
P
Q
R
X
Y
X
X
X
is the point on
O
P
OP
O
P
such that
O
X
:
X
P
=
1
:
3
OX : XP = 1 : 3
O
X
:
X
P
=
1
:
3
Y
Y
Y
is the point on
R
Q
RQ
R
Q
such that
R
Y
:
Y
Q
=
2
:
1
RY : YQ = 2 : 1
R
Y
:
Y
Q
=
2
:
1
O
P
→
=
a
\overrightarrow{OP} = \mathbf{a}
O
P
=
a
\qquad
O
R
→
=
b
\overrightarrow{OR} = \mathbf{b}
O
R
=
b
\qquad
P
Q
→
=
k
b
\overrightarrow{PQ} = k\mathbf{b}
P
Q
=
k
b
where
k
k
k
is a positive integer.
(a)
Find
X
Y
→
\overrightarrow{XY}
X
Y
in terms of
k
k
k
,
a
\mathbf{a}
a
and
b
\mathbf{b}
b
.
Give your answer in its simplest form.
(4 marks)
(b)
Is
X
Y
XY
X
Y
parallel to
O
P
OP
O
P
?
Give a reason for your answer.
(1 mark)
●●●●●
Level 5
5 marks
Start
→
Mark as done
O
A
B
C
OABC
O
A
B
C
is a parallelogram, with
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
C
→
=
b
\overrightarrow{OC} = \mathbf{b}
O
C
=
b
.
M
M
M
is the midpoint of
A
B
AB
A
B
.
N
N
N
is the point on the diagonal
O
B
OB
O
B
such that
O
N
→
=
2
3
O
B
→
\overrightarrow{ON} = \tfrac{2}{3}\overrightarrow{OB}
O
N
=
3
2
O
B
.
a
b
O
A
B
C
M
N
Diagram not accurately drawn.
(a)
Find
C
N
→
\overrightarrow{CN}
C
N
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form.
(b)
Prove that
C
C
C
,
N
N
N
and
M
M
M
lie on the same straight line.
●●●●●
Level 5
7 marks
Start
→
Mark as done
O
A
B
OAB
O
A
B
is a triangle, with
O
A
→
=
6
a
\overrightarrow{OA} = 6\mathbf{a}
O
A
=
6
a
and
O
B
→
=
6
b
\overrightarrow{OB} = 6\mathbf{b}
O
B
=
6
b
.
M
M
M
is the midpoint of
O
A
OA
O
A
.
N
N
N
is the point on
A
B
AB
A
B
such that
A
N
:
N
B
=
2
:
1
AN : NB = 2 : 1
A
N
:
N
B
=
2
:
1
.
P
P
P
is the point on
O
B
OB
O
B
extended such that
O
P
→
=
p
b
\overrightarrow{OP} = p\,\mathbf{b}
O
P
=
p
b
, where
p
p
p
is a number.
(a)
Find
M
N
→
\overrightarrow{MN}
M
N
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form.
(b)
M
M
M
,
N
N
N
and
P
P
P
lie on the same straight line.
Work out the value of
p
p
p
.
●●●●●
Level 5
6 marks
Start
→