$\def\frac{\dfrac}$
1. (2011/june/Paper02/q5)
Relative to a fixed origin $O$, the position vector of the point $A$ is $5 \mathbf{i}+p \mathbf{j}$ and the position vector of the point $B$ is $q \mathbf{i}+12 \mathbf{j} .$ The point $D$ with position vector $13 \mathbf{i}+10 \mathbf{j}$ divides the line $A B$ in the ratio $2: 1$
(a) Find the value of $p$ and the value of $q$. (3)
The points $A, B$ and $E$ lie in that order on a straight line and $A E: B E$ is $5: 2$
(b) Find, in terms of $\mathbf{i}$ and $\mathbf{j}$, the position vector of the point $E$. (3)
2. (2012/jan/paper02/q1)
Referred to a fixed origin $O$, the position vectors of the points $P$ and $Q$ are $(10 \mathbf{i}-3 \mathbf{j})$ and $(4 \mathbf{i}+6 \mathbf{j})$ respectively. The point $R$ divides $P Q$ internally in the ratio $2: 1$
(a) Find the position vector of $R$
The point $S$ divides $O Q$ internally in the ratio $5: 4$ and area $\triangle O P Q=\lambda$ area $\Delta S R Q$
(b) Find the exact value of $\lambda$ (4)
3. (2012/june/paper02/q10)
The points $A, B, C$ and $D$ are the vertices of a quadrilateral and $\overrightarrow{A B}=3 \mathbf{i}+5 \mathbf{j}, \quad \overrightarrow{A C}=6 \mathbf{i}+6 \mathbf{j} \quad$ and $\quad \overrightarrow{A D}=9 \mathbf{i}+3 \mathbf{j}$
(a) (i) Find $\overrightarrow{B C}$
(ii) Hence show that $A B C D$ is a trapezium.
(b) (i) Find the exact value of $|\overrightarrow{B D}|$
(ii) Find a unit vector parallel to $\overrightarrow{B D}$
The point $F$ is on the line $B D$ and $B F: F D=1: 2$
(c) Find $\overrightarrow{A F}$ (2)
The point $E$ is on the line $A D$ such that $A B C E$ is a parallelogram.
(d) (i) Show that $F$ lies on the line $C E$
(ii) Find the ratio $E F: F C$
4. (2013/june/paper01/q11)
$O, A, B$ and $C$ are fixed points such that
$$\overrightarrow{O A}=\mathbf{p}+\mathbf{q} \quad \overrightarrow{O B}=3 \mathbf{p}-\mathbf{q} \quad \overrightarrow{O C}=6 \mathbf{p}-4 \mathbf{q}$$
(a) Find $\overrightarrow{A B}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.
(b) Show that the points $A, B$ and $C$ are collinear.
(c) Find the ratio $A B: B C$ (1)
The point $D$ lies on $A C$ produced such that $A C=2 C D$
(d) Find $\overrightarrow{O D}$ in terms of $\mathbf{p}$ and $\mathbf{q}$, simplifying your answer. (4)
5. (2014/june/paper02/q3)
Relative to a fixed origin $O$, the point $A$ has position vector $3 \mathbf{i}-4 \mathbf{j}$
The point $B$ is such that $\overrightarrow{A B}=\mathbf{i}+7 \mathbf{j}$
(a) Show that the triangle $O A B$ is isosceles.
(b) Find a unit vector parallel to $\overrightarrow{O B}$ (1)
6. (2015/june/paper02/q4)
Referred to a fixed origin $O$, the position vectors of the points $P$ and $Q$ are $(3 \mathrm{i}+6 \mathbf{j})$ and $(4 i-2 j)$ respectively.
(a) Find, as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}, \overrightarrow{P Q}$
(b) Find a unit vector which is parallel to $\overrightarrow{P Q}$
(c) Show that $\overrightarrow{O P}$ is perpendicular to $\overrightarrow{O Q}$
7. (2016/june/рареr02/q2)
Relative to a fixed origin $O$, the point $A$ has position vector $6 \mathbf{i}+5 \mathbf{j}$ and the point $B$ has position vector $3 \mathbf{i}+9 \mathbf{j}$
(a) Find $\overrightarrow{A B}$ as a simplified vector in terms of i and $\mathbf{j}$
The line $P Q$ is parallel to $A B .$ Given that $\overrightarrow{P Q}=12 \mathbf{i}+\lambda \mathbf{j}$
(b) find the value of $\lambda$
(c) Find a unit vector parallel to $A B .$
8. (2017/jan/paper02/q8)
[In this question, $\mathbf{p}$ and $\mathbf{q}$ are non-zero and non-parallel vectors.]
$O, A, B$ and $C$ are fixed points such that
$$\overrightarrow{O A}=5 \mathbf{p}-3 \mathbf{q} \quad \overrightarrow{O B}=11 \mathbf{p} \quad \overrightarrow{O C}=13 \mathbf{p}+\mathbf{q}$$
(a) (i) Show that the points $A, B$ and $C$ are collinear.
(ii) Write down the ratio $A B: B C$. (4)
The midpoint of $O A$ is $M$ and the midpoint of $O B$ is $N$.
(b) Show that the ratio of the area of the quadrilateral $A B N M$ to the area of the triangle $O A C$ is $9: 16$ $(7)$
9. (2018/june/paper01/q10)
The points $A, B, C$ and $D$ are such that
$$\overrightarrow{A B}=5 \mathbf{i}+5 \mathbf{j} \quad \overrightarrow{A C}=-2 \mathbf{i}+15 \mathbf{j} \quad \overrightarrow{A D}=-7 \mathbf{i} +10 \mathbf{j}$$
(a) (i) Find $\overrightarrow{D C}$ as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}$.
(ii) Hence show that $A B C D$ is a parallelogram. (4)
(b) Find a unit vector parallel to $\overrightarrow{B D}$ as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}$.
The point $E$ lics on $B D$ and $B E: E D-3: 10$
(c) Find $\overrightarrow{A E}$ as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}$. (2)
The point $F$ is such that $D C F$ and $A E F$ are both straight lines.
(d) Find $D C: C F$
10. $(2019 /$ june/paper02/q1)
Referred to a fixed origin $O$, the point $A$ has position vector $(4 \mathbf{i}+3 \mathbf{j})$ and the point $B$ has position vector $(\mathbf{i}+7 \mathbf{j})$
(a) Find $\overrightarrow{A B}$ as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}$
(b) Find a unit vector that is parallel to $\overrightarrow{A B}$
11. (2019/juneR/paper01/q7)
$O, A, B$ and $C$ are fixed points such that
$$\overrightarrow{O A}=8 \mathbf{i}-6 \mathbf{j} \quad \overrightarrow{O B}=15 \mathbf{i}-6 \mathbf{j} \quad \overrightarrow{O C}=8 \mathbf{i}+\mathbf{j}$$
(a) Find $\overrightarrow{B C}$ as a simplified expression in terms of $\mathbf{i}$ and $\mathbf{j}$
(b) Find a unit vector parallel to $\overrightarrow{B C}$
The point $M$ is the midpoint of $O A$ and the point $N$ lies on $O B$ such that $O N: N B=1: 2$
(c) Show that the points $M, N$ and $C$ are collinear.
12. (2012/jan/paper01/q10)
Figure 2 shows a trapezium $O A B C$ in which $A B$ is parallel to $O C$ and $A B=\frac{1}{2} O C$. The point $P$ divides $O A$ in the ratio $1: 3$ and the point $Q$ divides $B C$ in the ratio $1: 2$
The line $A C$ intersects the line $P Q$ at the point $T$.
$\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O C}=\mathbf{c}$
(a) Find, as simplified expressions in terms of a and $\mathbf{c}$
(i) $\overrightarrow{B C}$
(ii) $\overrightarrow{P Q}$ (5)
(b) (i) Given that $\overrightarrow{P T}=\lambda \overrightarrow{P Q}$, find an expression for $\overrightarrow{A T}$ in terms of $\lambda, \mathbf{a}$ and $\mathbf{c}$
(ii) Given also that $\overrightarrow{A T}=\mu \overrightarrow{A C}$, find an expression for $\overrightarrow{A T}$ in terms of $\mu, \mathbf{a}$ and $\mathbf{c}$
(c) Use your answers from part (b) to find the value of $\lambda$ and hence write down the ratio $P T: T Q$
13. (2013/jan/paper02/q8)
In Figure $3, \overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $M$ is the mid-point of $A B$
The point $P$ is on $O A$ such that $O P: P A=3: 2$
The point $X$ lies on $O B$ produced.
(a) Find, as simplified expressions in terms of $\mathbf{a}$ and $\mathbf{b}$,
(i) $\overrightarrow{A B}$
(ii) $\overrightarrow{O M}$
(iii) $\overrightarrow{P M}$ (6)
Given that $P, M$ and $X$ are collinear
(b) find, in terms of $\mathbf{b}, \overrightarrow{O X}$
(c) Find the ratio (area $\triangle O A M):($ area $\triangle O A X$ ).
14. (2014/jan/paper02/q8)
Figure 4 shows a hexagon $O A B C D E$. Each internal angle of the hexagon is $120^{\circ}$.
$$O A=O E, A B=E D=2 \times O A \text { and } O C=3 \times O A$$
$\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O E}=\mathbf{e}$
Find as simplified expressions in terms of a and e
(a) $\overrightarrow{A B}$
(b) $\overrightarrow{B E}$
The point $P$ divides $A B$ internally in the ratio $2: 3$
(c) Find $\overrightarrow{P C}$ as a simplified expression in terms of $\mathbf{a}$ and $\mathrm{e} .$
The point $Q$ lies on $E D$ produced so that the points $P, C$ and $Q$ are collinear.
(d) Find $\overrightarrow{O Q}$ in the form $\lambda \mathbf{a}+\mu \mathbf{e}$, stating the value of $\lambda$ and the value of $\mu$.
15. (2015/jan/paper01/q3)
Figure 1 shows the quadrilateral $O A B C$.
$$\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b} \text { and } \overrightarrow{O C}=\mathbf{c}$$
(a) Find, in terms of $\mathbf{a}$ and $\mathbf{b}, \overrightarrow{A B}$
The midpoint of $O A$ is $P$ and the midpoint of $A B$ is $Q$.
(b) Show that $\overrightarrow{P Q}=\mu \mathbf{b}$, where $\mu$ is a scalar, stating the value of $\mu$.
The point $S$ lies on $O C$ and the point $R$ lies on $B C$ such that $\overrightarrow{O S}=\lambda \overrightarrow{O C}$ and $\overrightarrow{B R}=\lambda \overrightarrow{B C}$
(c) Show that $P Q$ is parallel to $S R$.
Given that $\overrightarrow{P Q}=\frac{3}{2} \overrightarrow{S R}$
(d) find the value of $\lambda$ (2)
16. (2016/jan/paper02/q9)
Figure 2 shows a quadrilateral $O A B C$
$$\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b} \text { and } \overrightarrow{B C}=\mathbf{b}-2 \mathbf{a}$$
(a) (i) Prove that $\overrightarrow{A B}$ is parallel to $\overrightarrow{O C}$
(ii) Show that $A B: O C=1: 2$ (4)
The point $D$ lies on $O B$ such that $O D: D B=2: 3$
(b) Find the ratio of the area of $\triangle O D C$ :the area of $\triangle O A B$. (6)
17. (2016/june/paper01/q8)
In Figure $1, \overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $\overrightarrow{O D}=\frac{2}{3} \mathbf{b}$
The point $E$ divides $A D$ in the ratio $2: 3$
(a) Find as simplified expressions in terms of $\mathbf{a}$ and $\mathbf{b}$
(i) $\overrightarrow{A D}$
(ii) $\overrightarrow{O E}$
(iii) $\overrightarrow{B E}$
The point $F$ lies on $O A$ such that $\overrightarrow{O F}=\lambda \overrightarrow{O A}$ and $F, E$ and $B$ are collinear.
(b) Find the value of $\lambda$.
The area of triangle $O F B$ is 5 square units.
(c) Find the area of triangle $O A D$.
Give your answer in the form $\frac{p}{q}$, where $p$ and $q$ are integers. (3)
18. (2017/june/paper01/q3)
In Figure $1, \overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O B}=\mathbf{b}$
The point $C$ is the midpoint of $O A$ and the point $D$ divides $O B$ in the ratio $2: 1$
(a) Find $\overrightarrow{C D}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ (2)
The point $E$ lies on $A B$ produced such that $\overrightarrow{O E}=2 \mathbf{b}-\mathbf{a}$
(b) Find $\overrightarrow{C E}$ in terms of $\mathbf{a}$ and $\mathbf{b}$
(c) Hence show that $C, D$ and $E$ are collinear.
19. (2018/jan/paper02/q6)
Figure 3 shows the triangle $O A B$ with $\overrightarrow{O A}=a$ and $\overrightarrow{O B}=\mathbf{b}$.
(a) Find $\overrightarrow{A B}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
The point $P$ is such that $\overrightarrow{O P}=\frac{3}{4} \overrightarrow{O A}$, and the point $Q$ is the midpoint of $A B$
(b) Find $\overrightarrow{P Q}$ as a simplified expression in terms of a and $\mathbf{b}$.
The point $R$ is such that $P Q R$ and $O B R$ are straight lines where
$$\overrightarrow{Q R}=\mu \overrightarrow{P Q} \text { and } \overrightarrow{B R}=\lambda \overrightarrow{O B}$$
(c) Express $\overrightarrow{Q R}$ in terms of
(i) $\mathbf{a}, \mathbf{b}$ and $\mu$
(ii) $\mathbf{a}, \mathbf{b}$ and $\lambda$
(d) Hence find the value of
(i) $\mu$
(ii) $\lambda$
Answer
1. (a) $\quad p=6, q=17$ (b) $25 \hat{i}+16 \hat{j}$
2. (a) $\overrightarrow{O R}=6 \hat{i}+3 \hat{j}$ (b) $\lambda=\frac{27}{4}$
3. (a)(i) $\overrightarrow{B C}=3 \hat{i}+\hat{j}$ (ii) Show (b) (i) $2 \sqrt{10}$ (ii) $\frac{1}{2 \sqrt{10}}(6 \hat{i}-2 \hat{j})$ (c) $\vec{AF}=5 \hat{i}+4 \frac{1}{3} \hat{\jmath} \quad d(i)$ show $(i i) 2: 1$
4. (a) $\overrightarrow{A B}=2 \vec{p}-2 \vec{q}$ (b) Show (c) $2: 3$ (d) $\overrightarrow{O D}=\frac{17}{2} \vec{p}-\frac{13}{2} \vec{q}$
5. $(a)$ show $(b) \pm \frac{1}{5}(4 \hat{i}+3 \hat{j})$
6. (a) $\overrightarrow{P Q}=\hat{i}-8 \hat{j}$ (b) $\pm \frac{1}{\sqrt{65}}(\hat{i}-8 \hat{j})$ (c) Show
7. (a) $\overrightarrow{A B}=-3 \hat{i}+4 \hat{j}$ (b) $\lambda=-16$ (c) $\pm \frac{1}{5}(3 \hat{i}-4 \hat{j})$
8. $(a)$ (i) Show (ii) $3: 1$ (b) Show
9. (a)(i) $\overrightarrow{DC}=5i+5j$ (ii) Prove (b) $\frac{1}{13}(-12i+5j)$ (c) $\overrightarrow{AE}=\frac{29}{13}i+\frac{80}{13}j$ (d) $3:7$
10. (a)$-3 \hat{i}+4 \hat{j}$ (b) $\frac{1}{5}(-3 \hat{i}+4 \hat{j})$
11. (a) $-7i+7j$ (b) $\frac{1}{\sqrt{98}}(-7i+7j)$ (c) Show
12. (a)(i) $\overrightarrow{B C}=\frac{1}{2} \vec{c}-\vec{a}$ (ii) $\overrightarrow{P Q}=\frac{5}{12} \vec{a}+\frac{2}{3} \vec{c}$ (b)(i) $\overrightarrow{A T}=-\frac{3}{4} \vec{a}+\lambda\left(\frac{5}{12} \vec{a}+\frac{2}{3} \vec{c}\right)$ (ii) $\vec{A} T=\mu(\vec{c}-\vec{a})$ (c) $9: 4$
13. (a)(i) $\quad \overrightarrow{A B}=\vec{b}-\vec{a}$ (ii) $\overrightarrow{O M}=\frac{1}{2}(\vec{a}+\vec{b})$ (iii) $\overrightarrow{P M}=\frac{1}{2} \vec{b}-\frac{1}{10} \vec{a}$ (b) $\overrightarrow{O X}=3 \vec{b}$ (c) $1: 6$
14. (a) $\overrightarrow{A B}=2(\vec{a}+\vec{e})$ (b) $\overrightarrow{B E}=-3 \vec{a}-\vec{e}$ (c) $\overrightarrow{P C}=\frac{6}{5} \vec{a}+\frac{11}{5} \vec{e}$ (d) $\overrightarrow{O Q}=\vec{e}+p(\vec{a}+\vec{e}), \lambda=\frac{21}{5}, \mu=\frac{26}{5}$
15. (a) $\overrightarrow{A B}=\vec{b}-\vec{a}$ (b) Show (c) Show (d) $\lambda=\frac{2}{3}$
16. (a)(i) Prove (ii) Show (b) $\frac{4}{5}$
17. (a)(i) $\frac{2}{3} \vec{b}-\vec{a}$ (ii) $\frac{3}{5} \vec{a}+\frac{4}{15} \vec{b}$ (iii) $\frac{3}{5} \vec{a}-\frac{11}{15} \vec{b}$ (b) $\lambda=\frac{9}{11}$ (c) $\frac{110}{27}$
18. $(a) \overrightarrow{C D}=\frac{2}{3} \vec{b}-\frac{1}{2} \vec{a}$ (b) $\overrightarrow{\mathrm{CE}}=2 \vec{b}-\frac{3}{2} \vec{a}$ (c) Show
19. $\begin{aligned}&\overrightarrow{A B}=-\mathbf{a}+\mathbf{b} \\&\overrightarrow{P Q}=\frac{\mathbf{a}}{4}+\frac{1}{2}(-\mathbf{a}+\mathbf{b}),=\frac{1}{4}(2 \mathbf{b}-\mathbf{a}) \text { oe } \\&\overrightarrow{Q R}=\frac{\mu}{4}(2 \mathbf{b}-\mathbf{a}) \\&\overrightarrow{Q R}=\frac{1}{2}(\mathbf{b}-\mathbf{a})+\lambda \mathbf{b} \\&\frac{2 \mu}{4} \mathbf{b}-\frac{\mu}{4} \mathbf{a}=\frac{1}{2} \mathbf{b}-\frac{1}{2} \mathbf{a}+\lambda \mathbf{b} \Rightarrow-\frac{\mu}{4}=-\frac{1}{2} \Rightarrow \mu=2 \\&\frac{2 \mu}{4}=\frac{1}{2}+\lambda \Rightarrow \lambda=\frac{1}{2} \end{aligned}$
Post a Comment