Further Pure Math (Vector)

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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. $\begin{aligned}&\overrightarrow{A B}=-\mathbf{a}+\mathbf{b} \\&\overrightarrow{P Q}=\frac{\mathbf{a}}{4}+\frac{1}{2}(-\mathbf{a}+\mathbf{b}),=\frac{1}{4}(2 \mathbf{b}-\mathbf{a}) \text { oe } \\&\overrightarrow{Q R}=\frac{\mu}{4}(2 \mathbf{b}-\mathbf{a}) \\&\overrightarrow{Q R}=\frac{1}{2}(\mathbf{b}-\mathbf{a})+\lambda \mathbf{b} \\&\frac{2 \mu}{4} \mathbf{b}-\frac{\mu}{4} \mathbf{a}=\frac{1}{2} \mathbf{b}-\frac{1}{2} \mathbf{a}+\lambda \mathbf{b} \Rightarrow-\frac{\mu}{4}=-\frac{1}{2} \Rightarrow \mu=2 \\&\frac{2 \mu}{4}=\frac{1}{2}+\lambda \Rightarrow \lambda=\frac{1}{2} \end{aligned}$


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