iGCSE Mathematics (0580) :E7.1 Describe a translation by using a vector.iGCSE Style Questions Paper 2

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Question

$M = \begin{pmatrix}5 &-4 \\2 &3 \end{pmatrix}$

Find
(a) M²,

Answer/Explanation

Ans: $\begin{pmatrix}17 &-32 \\16 &1 \end{pmatrix}$

(b) 2M,

Answer/Explanation

Ans: $\begin{pmatrix}10 &-8 \\4 &6 \end{pmatrix}$

Answer/Explanation

Ans: 23 cao

(d) M^-1.

Answer/Explanation

Ans: $\frac{1}{23}\begin{pmatrix}3 &4 \\-2 &5 \end{pmatrix}$

Question

O is the origin and OPQRST is a regular hexagon.

$\overrightarrow{OP}=p and \overrightarrow{OT}=t.$

Find, in terms of p and t, in their simplest forms,

(a) $\overrightarrow{PT}$,

Answer/Explanation

Ans: -p+t

(b) $\overrightarrow{PR}$,

Answer/Explanation

Ans: p+2t

Answer/Explanation

Ans: 2(p + t) or 2p + 2t

Question

(a) Work out $\begin{pmatrix}1 &-2 \\3 &4 \end{pmatrix}\begin{pmatrix}-5 &-3 \\2 &1 \end{pmatrix}.$

Answer/Explanation

Ans: $\begin{pmatrix}-9 &-5 \\-7 &-5 \end{pmatrix}$

(b) Find the inverse of $\begin{pmatrix}1 &-2 \\3 &4 \end{pmatrix}$.

Answer/Explanation

Ans: $\frac{1}{10}\begin{pmatrix}4 &2 \\-3 &1 \end{pmatrix} oe$

Answer/Explanation

Ans: Not the same order oe

Question

$M = \begin{pmatrix}5 &2 \\-3 &4 \end{pmatrix}$ $N = \begin{pmatrix}-1 &-2 \\2 &6 \end{pmatrix}$

Calculate
(a) MN,

Answer/Explanation

Ans: $\begin{pmatrix}-1 &2 \\11 &30 \end{pmatrix}$

(b) M⁻¹, the inverse of M.

Answer/Explanation

Ans: $\frac{1}{26}\begin{pmatrix}4 &-2 \\3 &5 \end{pmatrix}oe$

Question

A is the point (–1, 1) and B is the point (8, 7).
(a) Write $\overrightarrow{AB}$ as a column vector.
(b) Find $|\overrightarrow{AB}|$.
(c) $\overrightarrow{AC} = 2 \overrightarrow{AB}$.
Write down the co-ordinates of C.

Answer/Explanation

Ans:

(a) $\binom{9}{6}$

(b) 10.8 or 10.81 to 10.82

Question

$A = \begin{pmatrix}
1 & 2\\
3 & 4
\end{pmatrix}$ $B =\begin{pmatrix}
4 & 3\\
1 & 2
\end{pmatrix}$
Find
(a) AB
(b) $B^{-1}$, the inverse of B,

Answer/Explanation

Ans:

(a) AB = $\begin{pmatrix}
6 & 7\\
16 & 17
\end{pmatrix}$
(b) $B^{-1}$ = $\frac{1}{5} \begin{pmatrix}
2 & -3\\
-1 & 4
\end{pmatrix}$

Question

OPQR is a parallelogram, with O the origin.
M is the midpoint of PQ.
OM and RQ are extended to meet at S.
$\overrightarrow{OP}=p$ and $\overrightarrow{OR}=r$.
(a) Find, in terms of p and r, in its simplest form,
(i) $\overrightarrow{OM}$,
(ii) the position vector of S.
(b) When $\overrightarrow{PT} = -\frac{1}{2}p + r$, what can you write down about the position of T?

Answer/Explanation

Ans:

(a) (i) $\overrightarrow{OM} = p + \frac{1}{2} r$
(ii) 2p + r

(b) Midpoint of RQ

Question

y varies directly with $\sqrt{x+5}.$
y = 4 when x = –1.

Find y when x = 11.

Answer/Explanation

Ans:

Question

(a) Calculate $\begin{pmatrix}3 &7 \\-1 &4 \end{pmatrix}\begin{pmatrix}-2 &1 \\4 &2 \end{pmatrix}.$

Answer/Explanation

Ans: $\begin{pmatrix}22 &17 \\18 &7 \end{pmatrix}$

(b) Calculate the inverse of $\begin{pmatrix}5 &3 \\6&4\end{pmatrix}.$

Answer/Explanation

Ans: $\frac{1}{2}\begin{pmatrix}4 &-3 \\-6 &5 \end{pmatrix}$

Question

Work out.
(a) $2\binom{3}{5}-\binom{1}{2}$
(b) (1 2)$\binom{2}{3}$

Answer/Explanation

Ans:

(a) $\binom{5}{8}$
(b) (8) final answer

Question

PQRS is a quadrilateral and M is the midpoint of PS.

$\overrightarrow{PQ}=a, \overrightarrow{QR}=b and \overrightarrow{SQ}=a-2b.$

(a) Show that $\overrightarrow{PS}=2b.$

Answer/Explanation

Ans: a + 2b − a or a − ( a − 2b) oe

(b) Write down the mathematical name for the quadrilateral PQRM, giving reasons for your answer.

Answer/Explanation

Ans: Parallelogram

PM equal and parallel to QR

PM or PS parallel to QR
and MR found = a so 2 pairs of
parallel sides

Question

$M=\begin{pmatrix}
5 &3 \\
3&-2
\end{pmatrix}$ $N = \begin{pmatrix}
3 & -6 \\
4& 2
\end{pmatrix}$
Calulate
(a) MN,
(b) $M^{-1}$.

Answer/Explanation

Ans:

(a) $\begin{pmatrix}
27 & -24\\
-5 & -10
\end{pmatrix}$
(b) $-\frac{1}{13}\begin{pmatrix}
-2 & -3\\
-1 & 5
\end{pmatrix}$ oe isw

Question

Find the inverse of the matrix $\begin{pmatrix}
3 & -2\\
-8 & 7
\end{pmatrix}$

Answer/Explanation

Ans:

$\frac{1}{5} \begin{pmatrix}
7 & 2\\
8 & 3
\end{pmatrix}$ oe isw

Question

Points A and B are marked on the grid.
$\overheadarrow{BC}$ = \(\begin{pmatrix}
-4\\0

\end{pmatrix}\)
(a) On the grid, plot the point C.
(b) Write $\overheadarrow{AC}$ as a column vector.
(c) $\overheadarrow{DE}$ is a vector that is perpendicular to $\overheadarrow{BC}$.
The magnitude of $\overheadarrow{DE}$ is equal to the magnitude of $\overheadarrow{BC}$.
Write down a possible column vector for $\overheadarrow{DE}$.

Answer/Explanation

Ans:

(a) Point at (3, 5)
(b) \(\begin{pmatrix}
1\\-3

\end{pmatrix}\)
(c) \(\begin{pmatrix}
0\\4

\end{pmatrix}
or
\begin{pmatrix}
0\\-4

\end{pmatrix}\)

FT their $\overheadarrow{AC}$

Question

(a) D is the point (2, ‒5) and \(\overheadarrow{DE} =\begin{pmatrix}
7\\1

\end{pmatrix}\)
Find the co-ordinates of the point E.
(………………… , …………………)
(b) v= \(\begin{pmatrix}
t\\ 12

\end{pmatrix}\) and |v| = 13.

Work out the value of t, where t is negative.
t = …………………………………………..

Answer/Explanation

Ans:

(a) (9, −4)
(b) -5

Question

The diagram shows a parallelogram OCEG.

O is the origin, $\overheadarrow{OA} = a$ and $\overheadarrow{OB} = b$
BHF and AHD are straight lines parallel to the sides of the parallelogram.
$\overheadarrow{OG} = 3\overheadarrow{OA}$ and $\overheadarrow{OC}= 2 \overheadarrow{OB}$.
(a) Write the vector $\overheadarrow{HE}$ in terms of a and b.
$\overheadarrow{HE}$ = …………………..
(b) Complete this statement.
a + 2b is the position vector of point …………………….
(c) Write down two vectors that can be written as 3a – b.
…………………………. and ………………………….

Answer/Explanation

Ans:

(a) 2a + b
(b) D
(c) $\overheadarrow{CF}$ and $\overheadarrow{BG}$

Question

$\begin{pmatrix}
8 &3 \\4
&2
\end{pmatrix}$

Find
(a) A²,

Answer/Explanation

Ans: $\begin{pmatrix}
76 &30 \\40
&16
\end{pmatrix}$

(b) A^–1.

Answer/Explanation

Ans: $\frac{1}{4}\begin{pmatrix}
2 &-3 \\-4
&8
\end{pmatrix}oe$

Question

Simplify.
(a) $(m^5)^2$
(b) $4x^3y \times 5x^2y$

Answer/Explanation

Ans:

(a) $m^{10}$ final answer
(b) $20x^5y^2$ final answer

Question

M=$\begin{pmatrix}8 & 2\\ 7 & 3\end{pmatrix}$ $N=\begin{pmatrix}4 &-1 \\ -3 & 5\end{pmatrix}$
(a) Find MN
(b) Find $M^{-1}$.

Answer/Explanation

(a)$M=\begin{pmatrix}26 & 2\\ 19 & 8\end{pmatrix}$
(b)$\frac{1}{10}\begin{pmatrix}3 & -2 \\ -7& 8\end{pmatrix}$

Question

The diagram shows a regular hexagon ABCDEF.
$\vec{CD}=p$ and$ \vec{CB}=q$
Find CA, in terms of p and q, giving your answer in its simplest form.

Answer/Explanation

2q+p

Question

$M=\bigl(\begin{smallmatrix}1 & & 2\\3 & & 4\end{smallmatrix}\bigr)$ $P=\bigl(\begin{smallmatrix}5 & & 6\\7 & & 8\end{smallmatrix}\bigr)$

(a) Find MP.

$\begin{pmatrix}& & \\& &\end{pmatrix}$[2]

(b) Find $\left | M \right |$ .[1]

Answer/Explanation

Ans:

19(a) $\begin{pmatrix}19 & 22\\43 & 50\end{pmatrix}$

19(b) –2 final answer

Question

$P=\begin{pmatrix}3 & & 1\\2 & & 4\end{pmatrix}$

(a) Find P².[2]

$\begin{pmatrix}& & \\& &\end{pmatrix}$ [2]

(b) Find P^–1.

$\begin{pmatrix}& & \\& &\end{pmatrix}$ [2]

Answer/Explanation

Ans:

23(a) $\begin{pmatrix}11 & & 7\\14 & & 18\end{pmatrix}$

23(b) $\frac{1}{10}\begin{pmatrix}4 & & -1\\-2 & & 3\end{pmatrix}$ oe isw

Question

(a) Work out $\bigl(\begin{smallmatrix}2 & &-1 \\4& & 3\end{smallmatrix}\bigr)$ $\bigl(\begin{smallmatrix}1 & &6 \\-5& & 4\end{smallmatrix}\bigr)$.[2]

(b) Find the value of x when the determinant o f$\bigl(\begin{smallmatrix}3 & &-1 \\-7& & x\end{smallmatrix}\bigr)$ is 5.

x = __ [2]

Answer/Explanation

Ans:

20(a) $\bigl(\begin{smallmatrix}7 & &8 \\-11& & 36\end{smallmatrix}\bigr)$

20(b) 4

Question

(a)
The diagram shows a parallelogram CDEF.
$\vec{FE}=m $and $\vec{CE}=n.$
B is the midpoint of CD.
FA=2AC
Find an expression ,in terms of m and n ,for $\vec{AB}.$
Give your answer in simplest form.
$\vec{AB}=………………………..$
(b)$\vec{GH}=\frac{2}{6}(2p+q)$ $ \vec{JK}=\frac{5}{18}(2p+q)$
Write down two factors about vectors $\vec{GH}$ and $\vec{JK}.$
…………………………………………………………………………
…………………………………………………………………………

Answer/Explanation

(a)$\frac{5}{6}m-\frac{1}{3}n$
(b)$\bar{GH}=3\bar{JK} $ or $\bar{GH}$ has a greater magnitude
$\bar{GH} $and $\bar{JK}$ are parallel.

Question

(a)(i)$m=\begin{pmatrix}5\\ 7\end{pmatrix}$.Find 3m
(ii)$\vec{VW}=\begin{pmatrix}10\\ -24\end{pmatrix}$.Find$ \left | \vec{\vec{VW}} \right |$
……………………

(b)

OABC is a parallelogram.
$\vec{OA}= p$ and $\vec{OC} = q.$
E is the point on AB such that AE : EB = 3 : 1.
Find $\vec{OE}$, in terms of p and q, in its simplest form.
$\vec{OE}= …………………………………………..$

Answer/Explanation

(a)(i)$\begin{pmatrix}15\\ 21\end{pmatrix}$
(ii)26
(b)$p+\frac{3}{4}q$

Question

Point A has coordinates (6, 4) and point B has coordinates (2, 7).
Write $\vec{AB}$ as a column vector.
$\vec{AB}=\left ( \right )$.

Answer/Explanation

$\begin{pmatrix}-4\\ 3\end{pmatrix}$

Question

The diagram shows a trapezium OPQR.

O is the origin, $\overrightarrow{OR}=a and \overrightarrow{OP}=b.
\left | \overrightarrow{RQ} \right |=\frac{3}{5}|\overrightarrow{OP}|$

(a) Find $\overrightarrow{PQ}$ in terms of a and b in its simplest form.

(b) When PQ and OR are extended, they intersect at W.

Find the position vector of W.

Answer/Explanation

(a) $a-\frac{2}{5}b $oe simplified
(b) $ \frac{5}{2}$a oe

Question

In the diagram, PQS, PMR, MXS and QXR are straight lines.

PQ = 2 QS.

M is the midpoint of P

QX : XR = 1 : 3.

$\underset{PQ}{\rightarrow})$=q and $\underset{PR}{\rightarrow})$=r.

(a) Find, in terms of q and r,

(i) $\underset{RQ}{\rightarrow})$,

(ii) $\underset{MS}{\rightarrow})$

(b) By finding , show that $\underset{MX}{\rightarrow})$ is the midpoint of MS.

Answer/Explanation

(a) (i) -r + q or q – r (ii) $\frac{1}{2}$(3q – r)

Question

$
\mathbf{A}=\left(\begin{array}{rr}
2 & 2 \\
2 & -2
\end{array}\right)
$
Work out
(a) $\mathbf{A}^2$,

(b) $\mathbf{A}^{-1}$, the inverse of $\mathbf{A}$.

▶️Answer/Explanation

(a) $\left(\begin{array}{ll}8 & 0 \\ 0 & 8\end{array}\right)$ oe
(b) $\left(\begin{array}{rr}\frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & -\frac{1}{4}\end{array}\right)$ oe

Question

5
$
\mathbf{A}=\left(\begin{array}{ll}
-2 & 3 \\
-4 & 5
\end{array}\right)
$
Find $\mathbf{A}^{-1}$, the inverse of the matrix $\mathbf{A}$.

▶️Answer/Explanation

$\frac{1}{2}\left(\begin{array}{ll}5 & -3 \\ 4 & -2\end{array}\right)$ or $\left(\begin{array}{cc}2.5 & -1.5 \\ 2 & -1\end{array}\right)$

Question

The diagram is made from three identical parallelograms.
$O$ is the origin. $\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O G}=\mathbf{g}$.
Write down in terms of $\mathbf{a}$ and $\mathbf{g}$
(a) $\overrightarrow{G B}$

(b) the position vector of the centre of the parallelogram BCDE.

▶️Answer/Explanation

(a) $2 \mathbf{a}-\mathbf{g}$ cao
(b) $2 \frac{1}{2} \mathbf{a}+\frac{1}{2} \mathbf{g}$ oe cao

Question

$
\mathbf{A}=\left(\begin{array}{ll}
x & 6 \\
4 & 3
\end{array}\right)
$
$
\mathbf{B}=\left(\begin{array}{ll}
2 & 3 \\
2 & 1
\end{array}\right)
$
(a) Find $\mathbf{A B}$.

(b) When $\mathbf{A B}=\mathbf{B A}$, find the value of $x$.

▶️Answer/Explanation

(a) $\left(\begin{array}{cc}2 x+12 & 3 x+6 \\ 14 & 15\end{array}\right)$
(b) 5

Question

$
\mathbf{A}=\left(\begin{array}{rr}
0 & 1 \\
-8 & -4
\end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rr}
7 & 1 \\
0 & -5
\end{array}\right)
$
Calculate the value of $5|\mathbf{A}|+|\mathbf{B}|$, where $|\mathbf{A}|$ and $|\mathbf{B}|$ are the determinants of $\mathbf{A}$ and $\mathbf{B}$.

▶️Answer/Explanation

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