CAIE IGCSE Additional Math 13 Vectors in…ensions

Covered syllabus points

13.1. Use vector notation 13.2. Use position and unit vectors 13.3. Calculate and combine vectors 13.4. Compose and resolve velocities

[Maximum number: 7]

The point O is the origin.
Two points P and Q are such that $\overrightarrow{PQ}$ is in the same direction as $-\mathbf{i}+5\mathbf{j}$ .

(a)

The point R is such that $\overrightarrow{OR}$ is in the same direction as $\overrightarrow{PQ}$ and the magnitude of $\overrightarrow{OR}$ is $3\sqrt{26}$ .
Find $\overrightarrow{OR}$ .

[ 3 ]

$[\overrightarrow{OR}=]k(-\mathbf{i}+5\mathbf{j})$ or $k\binom{-1}{5}$
B1 soi
$\sqrt{1^2+5^2}=\sqrt{26}$
B1 for finding the magnitude of $\overrightarrow{PQ}$ ;
allow unsimplified.
$-3\mathbf{i}+15\mathbf{j}$ or $\binom{-3}{15}$
B1
Allow $3(-\mathbf{i}+5\mathbf{j})$ or $3\binom{-1}{5}$ . Do not isw.

(b)

$\overrightarrow{OP}$ is in the same direction as $2\mathbf{i}-3\mathbf{j}$ and $\overrightarrow{OQ}=10\mathbf{i}+6\mathbf{j}$ .
Find $\overrightarrow{OP}$ .

[ 4 ]

$a(2\mathbf{i}-3\mathbf{j})+b(-\mathbf{i}+5\mathbf{j})=10\mathbf{i}+6\mathbf{j}$ oe
2a-b=10 oe, -3a+5b=6 oe
or $\frac{10-2a}{6+3a}=\frac{-1}{5}$ or $\frac{10+b}{6-5b}=\frac{-2}{3}$
M1 dep on previous B1 for equating like vectors.
a=8 or b=6 $16\mathbf{i}-24\mathbf{j}$ oe
Allow $8(2\mathbf{i}-3\mathbf{j})$ or $8\binom{2}{-3}$ .

[Maximum number: 8]

In this question lengths are in centimetres and time, t, is in seconds.
A particle P is moving in a straight line with a speed of 26 in the direction of the vector $\binom{5}{-12}$ .

All through this question allow correctly written vectors in $\mathbf{i},\mathbf{j}$ form;
incorrectly written column vectors do not earn full marks unless the correct form is recovered.

(a)

Find the velocity vector of P.

When t=0, P passes through a point A which has position vector $\binom{3}{6}$ .

[ 2 ]

$\binom{10}{-24}$ oe, isw
B1 for the magnitude of the direction vector is 13 soi, or M1 for $\sqrt{5^2+(-12)^2}$ oe.
A1

(b)

Write down the position vector of P at time t.

At the same time that P passes through A, a particle Q passes through a point B.
The position vector of Q at time t is given by $\binom{8 t-5}{2-25 t}$ .
The distance between P and Q at time t is d.

[ 2 ]

$\binom{3}{6}+t\binom{10}{-24}$ oe, isw
M1 FT for $\binom{3}{6}+t(\text{their }\binom{10}{-24})$ .
A1

(c)

Show that $d^{2}=m t^{2}+n t+r$ , where m, n and r are integers to be found.

[ 3 ]

$\binom{3+10t}{6-24t}-\binom{8t-5}{2-25t}=\binom{8+2t}{4+t}$ oe, soi
Finds $d^2$ or d for the difference of their position vectors.
M1 FT.
$d^2=5t^2+40t+80$ mark final answer

(d)

Hence show that P and Q do not collide.

[ 1 ]

Correct argument including d, e.g. when d=0, t=-4, but $t\geqslant0$ , so no collision;
or d>0 when $t\geqslant0$ , so $d\ne0$ .
B1 dep on part (c) being correct.

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CAIE IGCSE Additional Math 13 Vectors in Two Dimensions Topic Practice

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)