Vector (Myanmar Exam Board)

 

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Group (2015-2019)

1. (2015/Myanmar /q4 )
In $\triangle A B C, \overrightarrow{B P}=\overrightarrow{P C}$ and $\overrightarrow{C Q}=\frac{1}{3} \overrightarrow{C A}$. Prove that $2 \overrightarrow{B C}+\overrightarrow{C A}+\overrightarrow{B A}=6 \overrightarrow{P Q}$ (3 marks).

2. (2015/Myanmar /q13b )
Find the matrix which will rotate $30^{\circ}$. and then reftect in the line $O Y$. What is the map of the point $(1 ; 0) ? \quad \cdot(5$ marks $)$

3. (2015/FC /q4 )
The position vectors of $A, B$ and $C$ are $2 \vec{p}-\vec{q}, k \vec{p}+\vec{q}$ and $12 \vec{p}+4 \vec{q}$ respectively. Calculate the value of $k$ if $A, B$ and $C$ are collinear with $\vec{p} \neq \overrightarrow{0}, \vec{q} \neq \overrightarrow{0}, \vec{p}$ and $\vec{q}$ are not parallel.

4. (2015/FC /q13b )
In a qưadrilateral $O L N M, O M \| L N$, where $\overrightarrow{O L}=\vec{a}, \overrightarrow{O M}=\vec{b}$ and $\overrightarrow{L N}=k \vec{b}, O P$ is drawn parallel to $M N$ to meet the diagonal $M L$ at $P$. If $L P=\frac{1}{4} L M .$ Find the value of $k . \quad$ (5 marks)

5. (2016/Myanmar /q13b )
Given that $\alpha(\triangle E C D)=24 \mathrm{~cm}^{2}$, calculate $\alpha(\Delta E A B)$. $2 \hat{i}+3 \hat{j}, 10 \hat{i}+2 \hat{j}$ and $\lambda(-\hat{i}+5 \hat{j})$ respectively. Given that $|\overrightarrow{A B}|=|\overrightarrow{A C}|$ show that $\lambda^{2}-\lambda-2=0$ and hence find the two possible vectors $\overrightarrow{A C}$

6. (2016/FC /q13b )
The coordinates of points $P, Q$ and $R$ are $(1,2),(7,3)$ and $(4,7)$ respectively. If $P Q S R$ is a parallelogram, find the coordinates of $S$ by vector method. If $P S$ and $Q R$ meet at $T$, find the coordinate of $T$ by using vectors.

7. (2017/Myanmar /q13b )
The coordinates of $P, Q$ and $R$ are $(1,3),(5,4)$ and $(1,9)$ respectively. Find the corrdinates of $S$ if $P Q R S$ is a parallelogram.
Q13(b) Solution

8. (2017/FC /q13b )
The position vectors of $\mathrm{A}$ and $\mathrm{B}$ relative to an origin $\mathrm{O}$ are $\left(\begin{array}{c}5 \\ 15\end{array}\right)$ and $\left(\begin{array}{c}13 \\ 3\end{array}\right)$ respectively. Given that C lies on $\mathrm{AB}$ and has position vector $\left(\begin{array}{c}2 t+1 \\ t+1\end{array}\right)$, find the value of $t$ and the ratio AC : CB. $\quad$ (5 marks)

9. (2018/Myanmar /q13b )
Relative to an origin $O$ the position vectors of the points $P$ and $Q$ are $3 \hat{i}+\hat{j}$ and $7 \hat{i}-15 \hat{j}$ respectively. Given that $R$ is the point such that $3 \overrightarrow{P R}=\overrightarrow{R Q}$, find a unit vector in the direction $\overrightarrow{O R}$.
Click for Solution

10. (2018/FC /q13b )
Points $A$ and $B$ have position vectors $\left(\begin{array}{l}5 \\ 1\end{array}\right)$ and $\left(\begin{array}{l}3 \\ 4\end{array}\right)$ respectively, relative to an origin $O$. Given that point $C$ with position vector $\left(\begin{array}{l}0 \\ k\end{array}\right)$ lies on $A B$ produced, calculate the vlaue of $k$ and the value of $|2 \overrightarrow{A B}+\overrightarrow{O C}|$

11. (2019/Myanmar /q4b )
The coordinates of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $(1,0),(4,2)$ and $(5,4)$ respectively. Use vector method to determine the coordinates of $\mathrm{D}$ if $\mathrm{ABCD}$ is a parallelogram. (3 marks) Click for Solution

12. (2019/Myanmar /q14a )
In the quadrilateral $\mathrm{ABCD}, \mathrm{M}$ and $\mathrm{N}$ are the midpoints of $\mathrm{AC}$ and $\mathrm{BD}$ respectively. Prove that $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{CD}}=\overrightarrow{\mathrm{MN}} \quad$ (5 marks) Click for Solution

13. (2019/FC /q4b )
The coordinates of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $(1,0),(4,2)$ and $(5,4)$ respectively. Use vector method to determine the coordinates of D. If ACBD is a parallelogram. - (3 marks) Click for Solution 4(b)


14. (2019/FC /q14a )
By using geometric vectors, show that the diagonals of a parallelogram bisect each other. (5 marks) Click for Solution 14(a)

Answer (2015-2019)

1.  Prove
2.  $\left(-\frac{\sqrt 3}{2},\frac 12\right)$
3.  $k=6$
4.  $\frac 43$
5.  $\overrightarrow{A C}=-4 \hat{i}+7 \hat{j} \text { or } \overrightarrow{A C}=-\hat{i}-8 \hat{j}$
6.  $T=\left(\frac{11}{2}, 5\right)$
7.  $S=(-3, 8)$
8.  $t=5,AC:CB=3:1$
9.  $\frac{4}{5} \hat{i}-\frac{3}{5} \hat{j}$
10.  $k=\frac{17}{2}, \frac{\sqrt{905}}{2}$
11.  $D=(2,2)$
12.  Proof 
13.  $D=(0,-2)$
14.  Prove 

Group (2014)

1. In the diagram, $\overrightarrow{P Q}=3 \vec{a}, \overrightarrow{Q R}=\vec{b}, \overrightarrow{S R}=4 \vec{a}$, and $\overrightarrow{P X}=k \overrightarrow{P R}$. Find $\overrightarrow{S Q}$ and $\overrightarrow{S X}$ in terms of $\vec{a}, \vec{b}$ and $k$ and hence find the value of $k$. (3 marks) 


2. The points $P$ and $Q$ have position vectors $\vec{a}+\vec{b}$ and $3 \vec{a}-2 \vec{b}$ respectively, relative to an origin $O .$ Given that $O P Q R$ is a parallelogram, express the vectors $\overrightarrow{P Q}$ and $\overrightarrow{P R}$ in terms of $\vec{a}$ and $\vec{b}$. (3 marks)

3. The position vectors of the points $A$ and $B$, relative to an origin $O$ are $\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\left(\begin{array}{l}3 \\ 2\end{array}\right)$ respectively. Find the position vector of $C$ if $\overrightarrow{A C}=3 \overrightarrow{B A}$. (3 marks)

4. The three points $O, P$ and $Q$ are such that $\overrightarrow{O P}=\left(\begin{array}{l}2 \\ 3\end{array}\right)$ and $\overrightarrow{O Q}=\left(\begin{array}{c}q \\ 2 q\end{array}\right)$ Given that $\overrightarrow{P Q}$ is a unit vector, calculate the possible values of $q$. (3 marks)

5. Given that $P=(4,-9)$ and $Q=(-8,3)$. If $R$ is the middle point of $P Q$ and $S$ is the point such that $\overrightarrow{R S}=\left(\begin{array}{r}6 \\ -5\end{array}\right)$, find the coordinates of the point $S$. (3 marks)

6. Given that $\vec{p}=\left(\begin{array}{c}h \\ 12\end{array}\right)$, and $\vec{q}=\left(\begin{array}{c}3 \\ -9\end{array}\right)$ are parallel vectors, find the value of $h$. (3 marks)

7. Given that $\vec{a}=\left(\begin{array}{c}2 \\ 18\end{array}\right), \vec{b}=\left(\begin{array}{l}3 \\ 4\end{array}\right), \vec{c}=\left(\begin{array}{c}5 \\ 10\end{array}\right)$ and $\vec{a}+k \vec{b}$ is parallel to $\vec{c}$. Find $k$. (3 marks)

8. Find the map of the point $(-2,3)$ by the reflection matrix $F$. (3 marks)

9. In the figure $\overrightarrow{O A}$ and $\overrightarrow{O B}$ represent vectors $\vec{a}$ and $\vec{b}$ respectively. $P, Q$ and $R$ are points such that $\overrightarrow{O P}=\frac{2}{3} \overrightarrow{O A}, \overrightarrow{A Q}=\frac{3}{5} \overrightarrow{A B}$ and $\overrightarrow{O R}=\lambda \overrightarrow{O B}$. Given that $P, Q$ and $R$ are collinear, find the value of $\lambda$. (5 marks) 


10. The points $A, B$ and $C$ have position vectors $\vec{p}+\vec{q}, 3 \vec{p}-2 \vec{q}$ and $6 \vec{p}+k \vec{q}$ respectively relative to an origin $O$. Find $\overrightarrow{A B}$ and $\overrightarrow{A C}$. If $\overrightarrow{A B}=\lambda \overrightarrow{A C}$, find the value of $k$ and of $\lambda$. (5 marks)

11. Position vectors of points $P, Q$ and $R$ relative to an origin $O$ are $m \vec{a}-4 \vec{b}, 5 \vec{a}-3 \vec{b}$ and $-3 \vec{a}-\vec{b}$ respectively. If $P, Q$ and $R$ are collinear and $\vec{a}, \vec{b}$ are not parallel, $\vec{a} \neq 0, \vec{b} \neq 0$, find the value of $m$ and $P Q: Q R$. (5 marks)

12. The points $A$ and $B$ have position vectors $\vec{a}$ and $\vec{b}$ respectively, relative to an origin $O$. The point $P$ divides the line segment $O A$ in the ratio $1: 3$ and the point $R$ divides the line segment $A B$ in the ratio $1: 2 .$ Given that $P R B Q$ is a parallelogram, find the position vector of $Q$. (5 marks)

13. Position vectors of points $A, B$ and $C$ relative to an origin $O$ are $\left(\begin{array}{c}4 \\ 14\end{array}\right),\left(\begin{array}{c}10 \\ 5\end{array}\right)$ and $\left(\begin{array}{c}6 k \\ k\end{array}\right)$ respectively. If $C$ lies on $A B$ produced, find the value of k. If $D$ is a point on $B C$ such that $B D: D C=1: 2$, find the position vector of D relative to $O$. (5 marks)

14. The vector $\overrightarrow{O P}$ has a magnitude of 39 units and has the opposite direction as $\left(\begin{array}{c}5 \\ 12\end{array}\right)$. The vector $\overrightarrow{O Q}$ has a magnitude of 25 units and has the opposite direction as $\left(\begin{array}{l}3 \\ 4\end{array}\right)$. Express $\overrightarrow{O P}$ and $\overrightarrow{O Q}$ as column vectors and find the unit vector in the direction of $\overrightarrow{P Q}$.  (5 marks)

15. In quadrilateral $A B C D, P$ and $Q$ are the midpoints of $A D$ and $B C$ respectively. Prove that $\overrightarrow{A B}+\overrightarrow{D C}=2 \overrightarrow{P Q} .$ By using this result, show that if $A B$ is parallel to $D C$, then $P Q$ is also parallel to $D C$. (3 marks)

16. In $\Delta P Q R, X$ and $Y$ are points on the sides $P Q$ and $P R$ respectively such that $P X: X Q=P Y: Y R=3: 2 .$ Prove by a vector method that $X Q R Y$ is a trapezium. (3 marks)

17. Relative to the origin $O$, the position vectors of the points $A$ and $B$ are $\left(\begin{array}{l}p \\ q\end{array}\right)$ and $\left(\begin{array}{c}q \\ p+1\end{array}\right)$ respectively, where $q \neq 0 .$ If $O, A$ and $B$ are collinear, show that $q^{2}=p(p+1)$. (3 marks)

18. $O A B C$ is a parallelogram with the vertex $O$ at the origin and the vertices $A$ and $C$ at $(4,6)$ and $(8,2)$ respectively. If $P$ and $Q$ are the mid-points of $O A$ and $B C$ respectively, show that $O P B Q$ is a parallelogram by using vector method. (5 marks)

19. The position vectors of points $A, B$ and $C$ relative to an origin $O$ are $3 \vec{b}+5 \vec{c}-2 \vec{a}, 7 \vec{a}-\vec{c}$ and $\vec{a}+2 \vec{b}+3 \vec{c}$ respectively. Show that $A, B$ and $C$ are collinear and $A B=B C+A C$. (5 marks)

20. Find the matrix which will reflect in the line $O X$ followed by a rotating through $45^{\circ}$. What is the map of the point $(2,0)$ ? (5 marks)

21. Find the matrix which will translate through 3 units horizontally and 1 unit vertically followed by a rotation through $45^{\circ}$, and find the map of the point $(1,2)$. (5 marks)


Answer(2014)

1. $\overrightarrow{S Q}=4 \vec{a}-\vec{b}, \overrightarrow{S X}=(1+3 k) \vec{a}+(k-1) \vec{b}, k=\frac{3}{7}$ 
2. $\overrightarrow{P Q}=2 \vec{a}-3 \vec{b}, \overrightarrow{P R}=\vec{a}-4 \vec{b}$
3. $\overrightarrow{O C}=\left(\begin{array}{c}-5 \\ 6\end{array}\right) \quad $
4. $q=\frac{6}{5}($ or $) 2 \quad $
5. $S=(8,-8) \quad $
6. $h=-4$
7. $ k=7 \quad $
8. $(2,3)$
9. $\lambda=\frac{3}{2} \quad$ 
10. $\overrightarrow{A B}=2 \vec{p}-\vec{q}, \overrightarrow{A C}=5 \vec{p}+(k-1) \vec{q}, k=-\frac{3}{2}, \lambda=\frac{2}{5}$
11. $P Q: Q R=1: 2$
12. $\overrightarrow{O Q}=\frac{2}{3} \vec{b}-\frac{5}{12} \vec{a}$
13. $k=2, \overrightarrow{O D}=\left(\begin{array}{c}\frac{32}{3} \\ 4\end{array}\right)$ 
14. $\overrightarrow{O P}=\left(\begin{array}{c}-15 \\ -36\end{array}\right), \overrightarrow{O Q}=\left(\begin{array}{c}-15 \\ -20\end{array}\right), \overrightarrow{P Q}=\left(\begin{array}{l}0 \\ 1\end{array}\right)$ 
15. Prove
16. Prove
17. Show
18. Show
19. Show
20. $(\sqrt{2}, \sqrt{2}) \quad $
21. $\left(\begin{array}{lll}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & \sqrt{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 2 \sqrt{2} \\ 0 & 0 & 1\end{array}\right),\left(\frac{\sqrt{2}}{2}, \frac{7 \sqrt{2}}{2}\right)$


Group (2013)

1. The co-ordinates of $P, Q$ and $R$ are $(2,3),(8,4)$ and $(5,8)$ respectively. Find the co-ordinates of $S$ if $P Q R S$ is a parallelogram. (3 marks)

2. The three points $O, P$ and $Q$ are such that $\overrightarrow{O P}=\left(\begin{array}{l}2 \\ 3\end{array}\right)$ and $\overrightarrow{O Q}=\left(\begin{array}{c}q \\ 2 q\end{array}\right)$. Given that $\overrightarrow{P Q}$ is a unit vector, find the possible values of $q$. (3 marks)

3. The position vectors of $P$ and $Q$ relative to an origin $O$ are $\overrightarrow{O P}=\left(\begin{array}{c}k \\ 3 k\end{array}\right)$ and $\overrightarrow{O Q}=\left(\begin{array}{l}2 \\ 4\end{array}\right)$. If $|\overrightarrow{P Q}|=2$, calculate the possible values of $k$. (3 marks)

4. The vector $\overrightarrow{O A}$ has a magnitude of 51 units and has the opposite direction as the vector $\left(\begin{array}{c}-8 \\ 15\end{array}\right)$. Express the vector $\overrightarrow{O A}$ as a column vector. (3 marks)

5. Position vectors of points $A, B$ and $C$ relative to an origin $O$ are $\left(\begin{array}{l}2 \\ 7\end{array}\right),\left(\begin{array}{l}6 \\ 1\end{array}\right)$ and $\left(\begin{array}{l}2 t \\ t\end{array}\right)$ respectively. If $C$ lies on $A B$, find the value of $t$ and the ratio $A B: A C: C B$. (5 marks)

6. The position vectors of $A$ and $B$ relative to an origin $O$ are $\left(\begin{array}{c}5 \\ 15\end{array}\right)$ and $\left(\begin{array}{c}13 \\ 3\end{array}\right)$ respectively. Given that $C$ lies on $A B$ and has position vector $\left(\begin{array}{c}2 t+1 \\ t+1\end{array}\right)$, find the value of $t$ and ine ratio $A C: C B$. (5 marks)

7. The position vectors of $A$ and $B$ relative to an origin $O$ are $\overrightarrow{O A}=9 i+4 p \hat{j}$ and $\overrightarrow{O B}=p \hat{i}+4 \hat{j}$ respectively. Given that the magnitude of $\overrightarrow{A B}$ is 10, calculate the possible values of $p$. (3 marks)

8. Given that $\vec{p}=3 \hat{i}+15 \hat{j}$ and $\vec{q}=\hat{i}-6 \hat{j}$, find the unit vector whose direction is opposite to $2 \vec{p}+\vec{q}$. (3 marks)

9. Given that $\vec{p}=3 \hat{i}+15 \hat{j}$ and $\vec{q}=\hat{i}-6 \hat{j}$, find the unit vector which has the opposite direction as $2 \vec{p}+\vec{q}$. (3 marks)

10. $A, B$ and $C$ are points with position vectors $-\hat{i}+p \hat{j}, 5 \hat{i}+9 \hat{j}$ and $6 \hat{i}-8 \hat{j}$ respectively. Find the value of $p$ if $A, B$ and $C$ are collinear. Given $D$ is a point on $O C$ such that $O D$ is a unit vector, find the position vector of $D$ relative to $O$. (5 marks)

11. Find the map of the point $P(3,2)$ when it is reflected in the $X$-axis. (3 marks)

12. By a vector method prove that if the diagonals of a quadrilateral bisect one another then the quadrilateral is a parallelogram. (3 marks)

13. In the quadrilateral $O A B C, D$ is the midpoint of $B C$ and $G$ is a point on $A D$ such that $A G: G D=2: 3$. If $\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$ and $\overrightarrow{O C}=\vec{c}$, express $\overrightarrow{O D}$ and $\overrightarrow{O G}$ in terms of $\vec{a}, \vec{b}$ and $\vec{c}$. (3 marks)

14. The points $A, B$, and $C$ have position vectors $\vec{a}, \vec{b}$ and $m \vec{a}+n \vec{b}$ respectively. If $m+n=1$, show that $A, B$ and $C$ are collinear. (3 marks)

15. Let $P Q R S$ be a quadrilateral with $A, B, C, D$ the midpoints of the respective sides. Prove, by a vector method, that $A B C D$ is a parallelogram. (5 marks)

16. The median $A D$ of $\triangle A B C$ is produced to $E$ so that $\overrightarrow{D E}=\frac{1}{3} \overrightarrow{A D}$. If $\overrightarrow{A G}=\frac{2}{3} \overrightarrow{A D}$, prove that $B E C G$ is a parallelogram. (5 marks)

17. In $\triangle X Y Z, L$ and $M$ are mid-points of $Y Z$ and $X Z$ respectively. Prove that $\overrightarrow{X Y}+\overrightarrow{X Z}+\overrightarrow{Z Y}=4 \overrightarrow{M L}$. (5 marks)

18. Relative to an origin $O$, the position vectors of $X, Y$ and $Z$ are $\vec{a}-\vec{b}$, $(\lambda-1) \vec{a}+\lambda \vec{b}$ and $4 \vec{a}+(8+\lambda) \vec{b}$ where $\vec{a}$ and $\vec{b}$ are non-parallel vectors. Given that $X, Y$ and $Z$ are collinear, find the possible values of $\lambda$. (5 marks)

19. The position vectors of $A, B$ and $C$, relative to an origin $O$ are $3 \vec{p}+2 \vec{q},-5 \vec{p}-3 \vec{q}$ and $4 \vec{p}-\vec{q}$ respectively. The mid point of $A B$ is $M$ and the point $N$ is such that $\overrightarrow{A N}=\frac{1}{3} \overrightarrow{A C}$. Find $\overrightarrow{M N}$ in terms of $\vec{p}$ and $\vec{q}$. (5 marks)

20. Points $A, B, C, D$ have position vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively relative to the origin $O$. If $P$ divides $A B$ in the ratio $1: 2$ and $Q$ divides $C D$ in the ratio $1: 2$, obtain an expression for the position vector of $X$, where $X$ is the mid-point of $P Q$. (5 marks)

21. In the figure $\overrightarrow{O A}=\vec{a}$ and $\overrightarrow{O B}=\vec{b}$. Make the points $M$ and $N$ such that $\overrightarrow{O M}=\frac{1}{3} \vec{a}$ and $\overrightarrow{O N}=-\frac{2}{3} \vec{b}$. Find the vectors $\overrightarrow{A B}, \overrightarrow{M B}, \overrightarrow{M N}$ and $\overrightarrow{A N}$ in terms of $\vec{a}$ and $\vec{b}$. (5 marks)


Answer (2013)
1. $\cdot(-1,7) $
2. $\cdot \frac{6}{5}$ (or) $2 \quad$ 
3. $\frac{4}{5}($ or $) 2$
4. $\left(\begin{array}{r}24 \\ -45\end{array}\right)$ 
5. $t=\frac{5}{2}, A B: A C :C B=7: 5: 2$
6. $t=5, A C: C B=5: 1 \quad$ 
7. $-\frac{1}{17}($ or ) 3 
8. $-\frac{1}{25}(7 \hat{i}+24 \hat{j})$
9. $-\frac{1}{25}(7 \hat{i}+24 \hat{j})$ 
10. $p=111, \overrightarrow{O D}=\frac{3}{5} \hat{i}-\frac{4}{5} \hat{j}$
11. $(3,-2)$
12. Prove
13. $\overrightarrow{O D}=\frac{1}{2}(\vec{c}+\vec{b}), \quad \overrightarrow{O G}=\frac{1}{5}(3 \vec{a}+\vec{b}+\vec{c}) $
14. Show
15. Prove
16. Prove
17. Prove
18. $\lambda=-7$ (or) 3 
19. $\frac{13}{3} \vec{p}+\frac{3}{2} \vec{q} \quad$ 
20. $\overrightarrow{O X}=\frac{1}{3} \vec{a}+\frac{1}{6} \vec{b}+\frac{1}{3} \vec{c}+\frac{1}{6} \vec{d}$
21. $\overrightarrow{A B}=\vec{b}-\vec{a}, \overrightarrow{M B}=\vec{b}-\frac{1}{3} \vec{a}, \overrightarrow{M N}=-\frac{1}{3}(\vec{a}+2 \vec{b}), \overrightarrow{A N}=-\frac{1}{3}(3 \vec{a}+2 \vec{b})$




Group (2012)


$\qquad$$\,$
1.The position vectors of $A, B$ and $C$ are $3 \vec{p}+k \vec{q}, 4 \vec{p}-8 \vec{q}$ and $6 \vec{p}-4 \vec{q}$ respectively. Calculate the value of k if $A, B$ and $C$ are collinear. (3 marks)

2.The position vectors of $P, Q$ and $R$ with respect to an origin $O$ are $3 \vec{b}+5 \vec{c}-2 \vec{a}$, $\vec{a}+2 \vec{b}+3 \vec{c}$ and $7 \vec{a}-\vec{c}$. Prove that the points $P, Q$ and $R$ collinear. (3 marks)

3.Let $\overrightarrow{O P}=2 \vec{a}+\vec{b}, \overrightarrow{O Q}=-3 \vec{a}-\vec{b}$ and $\overrightarrow{O R}=\frac{2}{3} \vec{a}+5 \vec{b}$.

4.$A, B$ and $C$ are points with position vectors $\hat{i}+3 \hat{j}, 2 \hat{i}+5 \hat{j}$ and $k \hat{i}-4 \hat{j}$ respectively. Find the value of $k$ if $A, B$ and $C$ are collinear. (5 marks)

5. In the figure $\overrightarrow{O B}=\vec{b}$ and $\overrightarrow{O C}=\vec{c}$. Make the points $E$ and $F$ such that $\overrightarrow{O E}=\frac{1}{2} \vec{b}, \overrightarrow{O F}=-2 \vec{c}$. Find the vectors $\overrightarrow{E C}$ and $\overrightarrow{B F}$ in terms of $\vec{b}$ and $\vec{c}$. (3 marks)


6.The position vectors of $A$ and $B$ relative to an origin $O$ are $3 \vec{p}+2 \vec{q}$ and $-5 \vec{p}-3 \vec{q}$ respectively. $C$ is the point on $A B$ such that $A C=\frac{1}{3} A B$. Express $\overrightarrow{O C}$ and $\overrightarrow{A B}$ in terms of $\vec{p}$ and $\vec{q}$. (3 marks)

7.Given that $\vec{a}=3 \hat{i}+15 \hat{j}$ and $\vec{b}=\hat{i}-6 \hat{j}$, find the unit vector which has the opposite direction as $\vec{a}+2 \vec{b}$. (3 marks)

8.The coordinates of $A, B$ and $C$ are $(1,2),(7,1)$ and $(-3,7)$ respectively. If $O$ is the origin and $\overrightarrow{O C}=h \overrightarrow{O A}+k \overrightarrow{O B}$, where $h$ and $k$ are constants, find the values of $h$ and $\boldsymbol{k}$. (5 marks)

9.If $3 \overrightarrow{O P}=2 \vec{a}, \overrightarrow{O Q}=2 \vec{b}$, and $\overrightarrow{P R}=\lambda \overrightarrow{P Q}$, express $\overrightarrow{O R}$ in terms of $\lambda, \vec{a}$ and $\vec{b}$. (3 marks)

10.The position vectors of $A$ and $B$ relative to an origin $O$ are $4 \vec{a}$ and $4 \vec{b}$ respectively. The point $D$ on $\overrightarrow{O A}$ is such that $\overrightarrow{O D}=k \overrightarrow{O A}$ and the point $E$ on $A B$ is such that $\overrightarrow{A E}=m \overrightarrow{A B}$. The line segments $O E$ and $B D$ intersect at point $X$. If $\overrightarrow{O X}=\frac{2}{5} \overrightarrow{O E}$ and $\overrightarrow{X B}=\frac{4}{5} \overrightarrow{D B}$, express $\overrightarrow{O X}$ in two different forms and hence find the value of $k$ and of $m$. (5 marks)

11.Find the matrix which will translate a distance of 3 units horizontally and 1 unit vertically what is the map of $(1,2)$ ? (3 marks)

12.Find the matrix which rotates through $75^{\circ}$. (3 marks)

13.Find the matrix which will rotate $45^{\circ}$ and then reflect in the line $O Y$. What is the map of the point $(4,5)$ ? (5 marks)

14.The coordinates of points $P, Q$ and $R$ are $(1,2),(7,3)$ and $(4,7)$ respectively. If $P Q S R$ is a parallelogram, find the coordinates of $S$ by vector method. If $P S$ and $Q R$ meet at $T$, find the coordinates of $T$ by using vectors. (5 marks)

15.$A B C D E F$ is a regular hexagon. If $G$ is the common point of intersection of the diagonals, prove by a vector method that $\overrightarrow{A B}=\overrightarrow{E D}$. (3 marks)

16.$P$ is the midpoint of the side $C D$ of the parallelogram $A B C D$. If $B D$ and $A P$ intersect at $Q$, prove by vector method that $D Q=\frac{1}{3} D B$. (5 marks)

17.The median $A D$ of $\triangle A B C$ is produced to $K$ so that $D K=\frac{1}{3} A D$. If $A G=\frac{2}{3} A D$, prove by a vector method, $B K C G$ is a parallelogram. (5 marks)

18.$A B C D E F$ is a regular hexagon. If $G$ is the common point of intersection of the diagonals, prove by a vector method that $A B / / E D$ and $A D=2 B C$. (5 marks)

19.$P Q R S$ is a parallelogram, $X, Y$ are mid-points of $\overrightarrow{P Q}, \overrightarrow{P S}$. Show that $\overrightarrow{P X}+\overrightarrow{P Y}=\frac{1}{2} \overrightarrow{P R}$. (3 marks)

20.In quadrilateral $A B C D$, diagonals $A C$ and $B D$ meet at $O$. If $A O: O C=B O: O D=2: 1$, prove by a vector method that $A B C D$ is a trapezium. (3 marks)

21.In $\triangle A B C, B P=P C$ and $C Q=\frac{1}{3} C A$. Show that $2 \overrightarrow{B C}+\overrightarrow{C A}+\overrightarrow{B A}=6 \overrightarrow{P Q}$. (5 marks)



Answer (2012)


$\quad$$\,$
1.$-10$
2.Prove
3.Prove
4.$-2.5$
5.$\vec{c}-\frac{1}{2} \vec{b},-\vec{b}-2 \vec{c}$
6.$\frac{1}{3} \vec{p}+\frac{1}{3} \vec{q} ;-8 \vec{p}-5 \vec{q}$
7.$\frac{-5}{\sqrt{34}} \hat{i}-\frac{3}{\sqrt{34}} \hat{j}$
8.$h=4, k=-1$
9.$\left(\frac{2}{3}-\frac{2}{3} \lambda\right) \vec{a}+2 \lambda \vec{b}$
10.$\left(\frac{8}{5}-\frac{8}{5} m\right) \vec{a}+\frac{8}{5} m \vec{b} ;\frac{16}{5} k \vec{a}+\frac{4}{5} \vec{b} ; k=\frac{1}{4}, m=\frac{1}{2} \quad $
11.$\left(\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right),(4,3)$
12.$\left(\begin{array}{lc}\frac{\sqrt{6}-\sqrt{2}}{4} & \frac{-\sqrt{6}+\sqrt{2}}{4} \\ \frac{\sqrt{6}+\sqrt{2}}{4} & \frac{\sqrt{6}-\sqrt{2}}{4}\end{array}\right)$.
13.$ \left(\begin{array}{cc}\frac{-\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{array}\right),\left(\frac{\sqrt{2}}{2}, \frac{9 \sqrt{2}}{2}\right)$
14.$(10,8) ;\left(\frac{11}{2}, 5\right) \quad$
15.Prove
16.Prove
17.Prove
18.Prove
19.Prove
20.Prove
21.Prove


Group (2011)


$\quad\;\,$$\,$
1.In triangle $A B C, P$ and $Q$ are points on the sides $A B$ and $A C$ respectively such that $A P: P B=A Q: Q C=3: 1$.Prove by a vector method that $P Q \| B C$ and find the ratio $P Q: B C$.$\mbox{ (3 marks)}$

2.$P$ is the point $(-1,3) .\overrightarrow{P Q}=\left(\begin{array}{r}-2 \\ 5\end{array}\right)$ and $\overrightarrow{P R}=\left(\begin{array}{l}0 \\ 7\end{array}\right)$.Find the coordinates of $Q$ and $R$.$\mbox{ (3 marks)}$

3.$A$ is the point $(3,4)$.$\overrightarrow{A B}=\left(\begin{array}{r}2 \\ -1\end{array}\right)$ and $\overrightarrow{B C}=\left(\begin{array}{r}1 \\ -3\end{array}\right) .$ Find the coordinates of $B$ and $C$.$\mbox{ (3 marks)}$

4.The coordinates of the points $A$ and $B$ are $(3,2)$ and $(2 p, p)$ respectively.If $\overrightarrow{A B}$ is a unit vector, find the value of $p$.$\mbox{ (3 marks)}$

5.(Figure) If $\overrightarrow{O P}=3 \vec{p}, \overrightarrow{O Q}=6 \vec{q}, \overrightarrow{O M}=2 \vec{p}+2 \vec{q}$, find $P M: M Q$.$\mbox{ (3 marks)}$


6.Find the matrix which rotates through $45^{\circ}$ and find the map of the point $(1,1)$.$\mbox{ (3 marks)}$

7.$A, B$ and $C$ are points with position vectors $4 \vec{p}-\vec{q}, \lambda(\vec{p}+\vec{q})$ and $\vec{p}+2 \vec{q}$ respectively.Find $\overrightarrow{A B}$ and $\overrightarrow{A C}$.Given that $B$ lies on $A C$, find the value of $\lambda$.$\mbox{ (5 marks)}$

8.Find the matrix which will translate a distance of $-2$ units horizontally and 2 units vertically.What is the map of $(4,-1) ?$ $\mbox{ (3 marks)}$

9.Find the matrix which will reflect in the line $O Y$ followed by a translation through 3 units horizontally and $-2$ units vertically.What is the map of the point $(4,-1) ?$ $\mbox{ (5 marks)}$

10.$A B C D$ is a parallelogram.Let $O$ be the point of intersection of two diagonals and $M$ be the mid-point of $A B$.If $\overrightarrow{A O}=\vec{a}, \overrightarrow{B O}=\vec{b}$, find $\overrightarrow{B C}$ and $\overrightarrow{B M}$ in terms of $\vec{a}$ and $\vec{b}$.$\mbox{ (3 marks)}$

11.If $G$ is the centroid of triangle $A B C$, show that $\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=\overrightarrow{0}$.$\mbox{ (3 marks)}$

12.In $\triangle P Q R, P Q=Q R$.The line $R Q$ is produced to $S$ such that $R Q=Q S .X$ and $Y$ are points on $P R$ and $P Q$ such that $P X=X R$ and $Q Y=\frac{1}{3} Q P .$ Use a vector method to prove that $Y S=2 X Y$.$\mbox{ (5 marks)}$

13.Let $P, Q, R, S$ be the midpoints of the respective sides $A B, B C, C D, D A$ of quadrilateral $A B C D$.Show that $P Q R S$ is a parallelogram.$\mbox{ (5 marks)}$.$\mbox{ (5 marks)}$

14.In $\triangle A B C, P, Q$ and $R$ are points on the sides $B C, C A, A B$ respectively such that $B P=2 P C, Q A=2 C Q$ and $A R=2 R B$.Prove by a vector method that $P Q R B$ is a parallelogram.$\mbox{ (5 marks)}$

15.Show that the diagonals of a parallelogram bisect each other.$\mbox{ (5 marks)}$

16.In the quadrilateral $A B C D, M$ and $N$ are the midpoints of $A C$ and $B D$ respectively.Prove that $\overrightarrow{A B}+\overrightarrow{C B}+\overrightarrow{A D}+\overrightarrow{C D}=4 \overrightarrow{M N}$.$\mbox{ (5 marks)}$

17.The position vectors of points $P, Q$ and $R$ relative to an origin $O$ are $3 \vec{b}+5 \vec{c}-2 \vec{a}, 7 \vec{a}-\vec{c}$ and $\vec{a}+2 \vec{b}+3 \vec{c}$ respectively.Show that $P, Q$ and $R$ are collinear.Show also that $P Q=Q R+P R$.$\mbox{ (5 marks)}$

18.Given that $\overrightarrow{O P}=\vec{p}, \overrightarrow{O Q}=\vec{q}$ and $\overrightarrow{O R}=4 \overrightarrow{O Q} .$ If $\overrightarrow{P S}=\frac{2}{3} \overrightarrow{P Q}$ and $\overrightarrow{P T}=\frac{1}{3} \overrightarrow{P R}$, prove that $O, S, T$ are collinear.$\mbox{ (5 marks)}$



Answer (2011)


$\quad\;\,$$\,$
1.$P Q: B C=3: 4$
2.$(-3,8) ;(-1,10)$
3.$(5,3) ;(6,0)$
4.$P=\frac{6}{5}$ (or) 2
5.$P M: M Q=1: 2$
6.$R=\left(\begin{array}{cc}\frac{\sqrt{2} }{2 }& -\frac{\sqrt{2} }{2 }\\ \frac{\sqrt{2} }{2 }& \frac{\sqrt{2} }{2 }\end{array}\right) ;(0, \sqrt{2})$
7.$\overrightarrow{A B}=(\lambda-4) \vec{p}+(\lambda+1) \vec{q};\overrightarrow{A C}=3(\vec{q}-\vec{p}) ; \lambda=\frac{3}{2} \quad$
8.$\left(\begin{array}{llr}1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{array}\right) ;(2,1)$
9.$\left(\begin{array}{ccc}-1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 1\end{array}\right) ;(-1,-3)$
10.$\overrightarrow{B C}=\vec{a}+\vec{b}, \quad \overrightarrow{B M}=\frac{1}{2}(\vec{b}-\vec{a})$
11.Show
12.Prove
13.Prove
14.Prove
15.Prove
16.Prove
17.Prove
18.Prove


Group (2010)


$\quad\;\,$$\,$
1.$A$ is a point outside a quadrilateral $P Q R S.$ If $\overrightarrow{P A}+\overrightarrow{R A}=\overrightarrow{Q A}+\overrightarrow{S A}$, show that $P Q R S$ is a parallelogram.$\text{ (3 marks)}$

2.The coordinates of $A, B$ and $C$ are $(2,2),(4,5)$ and $(5,1)$ respectively.If $O$ is the origin and $\overrightarrow{O C}=h \overrightarrow{O A}+k \overrightarrow{O B}$, where $h$ and $k$ are constants, find the value of $h$ and of $k.$ $\text{ (3 marks)}$

3.The position vectors of $A$ and $B$ are $\overrightarrow{O A}=9 \hat{i}+4 p \hat{j}$ and $\overrightarrow{O B}=p \hat{i}+4 \hat{j}$ respectively.Given that the magnitude of $\overrightarrow{A B}$ is 10, calculate the possible values of $p.$ $\text{ (3 marks)}$

4.The position vectors of $P$ and $Q$ are $\overrightarrow{O P}=q \hat{i}+2 q \hat{j}$ and $\overrightarrow{O Q}=-3 \hat{i}+4 \hat{j}$ respectively.Given that the magnitude of $\overrightarrow{P Q}$ is 10, calculate the possible values of $q.$ $\text{ (3 marks)}$

5.The position vectors of the points $A, B$ and $C$ relative to an origin $O$ are $\vec{a}, \vec{b}$ and $\vec{c}$ respectively.If $\vec{b}=4 \vec{c}-3 \vec{a}$ and $C$ lies on the line $A B(C$ lies between $A$ and $B$ ), show that $B C=3 A C.$ $\text{ (3 marks)}$

6.The position vectors of $A, B, C$ are $\vec{a}, \vec{b}$ and $\vec{c}$ respectively.If $\overrightarrow{A C}=-2 \overrightarrow{C B}$, then find $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}.$ Find also the ratio $A B: B C.$ $\text{ (3 marks)}$

7.$A$ is the point $(3,4), \overrightarrow{A B}=\left(\begin{array}{c}-1 \\ 3\end{array}\right).$ Find the coordinate of $B$ and $|\overrightarrow{A B}|.$ $\text{ (3 marks)}$

8.$\overline{A B}$ is parallel to the vector $8 \hat{i}-15 \hat{j}$ and has a magnitude of 68.Find $\overrightarrow{A B}.$ $\text{ (3 marks)}$

9.Given that $\vec{p}=3 \hat{i}+15 \hat{j}$ and $\vec{q}=\hat{i}-6 \hat{j}$, find the unit vector whose direction is opposite to $2 \vec{p}+\vec{q}.$ $\text{ (3 marks)}$

10.Write the reflection matrix $S$ in the line $O X$ and find the map of the point $(20,10)$ by $S.$ $\text{ (3 marks)}$

11.Write the reflection matrix $F$ in the line $O Y$ and find the map of the point $(9,2)$ by $F.$ $\text{ (3 marks)}$

12.The coordinates of $A, B, C$ are $(1,2),(3,1)$ and $(4,3)$ respectively.If $A B C D$ is a parallelogram and $E$ is the mid-point of $B D$, find the coordinate of $E$ by vector method.$\text{ (5 marks)}$

13.Given that $\overrightarrow{O P}=2 \vec{a}-\vec{b}, \overrightarrow{O Q}=k \vec{a}+\vec{b}$, and $\overrightarrow{O R}=12 \vec{a}+4 \vec{b}$, find, in terms of $\vec{a}$ and $\vec{b}$, the vectors $\overrightarrow{P Q}$ and $\overrightarrow{P R}.$ If $P, Q$ and $R$ are collinear, find the value of $k.$ $\text{ (5 marks)}$

14.If $\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$ and $P$ is the point on $O A$ produced such thát $O A: A P=3: 2$ and $Q$ is the point on $O B$ produced such that $O B: B Q=3: 2.$ Find $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ in terms of $\vec{a}$ and $\vec{b}$ and hence prove that $\overrightarrow{P Q}$ is parallel to $\overrightarrow{A B}.$ $\text{ (5 marks)}$

15.If $\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$ and $P$ is the point on $O A$ produced such that $O A: A P=1: 2$ and $Q$ is the point on $O B$ produced such that $O B: B Q=1: 2.$ Find $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ in terms of $\vec{a}$ and $\vec{b}$ and hence prove that $\overrightarrow{P Q}$ is parallel to $\overrightarrow{AB}.$ $\text{ (5 marks)}$

16.Relative to an origin $O$, the position vectors of $X, Y$ and $Z$ are $\vec{a}-\vec{b},(\lambda-1) \vec{a}+\lambda \vec{b}$ and $4 \vec{a}+(8+\lambda) \vec{b}$ where $\vec{a}$ and $\vec{b}$ are non-parallel vectors.Given that $X, Y$ and $Z$ are collinear, find the possible values of $\lambda.$ $\text{ (5 marks)}$

17.Relative to an origin $O$, the position vectors of $X, Y$ and $Z$ are $\vec{a}-\vec{b}, \lambda \vec{a}+3 \vec{b}$ and $4 \vec{a}+(7+2 \lambda) \vec{b}$ where $\vec{a}$ and $\vec{b}$ are non-parallel vectors.Given that $X, Y$ and $Z$ are collinear, find the possible values of $\lambda.$ $\text{ (5 marks)}$

18.Position vectors of points $P, Q$ and $R$ relative to an origin $O$ are $m \vec{a}-4 \vec{b}, 5 \vec{a}-3 \vec{b}$ and $-3 \vec{a}-\vec{b}$ respectively.If $P, Q$ and $R$ are collinear and $\vec{a}, \vec{b}$ are not parallel, find the value of $m$ and the ratio $P Q: Q R.$ $\text{ (5 marks)}$

19.Points $A$ and $B$ have position vectors $\left(\begin{array}{l}5 \\ 1\end{array}\right)$ and $\left(\begin{array}{l}3 \\ 4\end{array}\right)$ respectively, relative to an origin $O.$ Given that $C$ with position vector $\left(\begin{array}{l}1 \\ k\end{array}\right)$ lies on $A B$ produced, calculate the value of $k$ and the value of $|2 \overrightarrow{A B}+\overrightarrow{\mathrm{OC}}|.$ $\text{ (5 marks)}$

20.Find the matrix which will rotate $135^{\circ}$ and then reflect in the line $O X.$ Find the map of the point $(1,1).$ $\text{ (5 marks)}$

21.Write down the matrix which rotates through an angle of $180^{\circ}$ anticlockwise about the origin.Find also the matrix which reflects in the $X$-axis and then in the $Y$-axis.Are these two matrices equal? $\text{ (5 marks)}$



Answer (2010)


$\quad\;\,$$\,$
1.$\frac{21}{2} ;-4$
2.Prove
3.$-\frac{1}{17}, 3$
4.$-3,5$
5.Prove
6.$\vec{c}=2 \vec{b}-\vec{a} ; 1: 1$
7.$(2,7) ;\sqrt{10}$
8.$32 \hat{i}-60 \hat{j}$
9.$-\frac{7}{25} \hat{i}-\frac{24}{25} \hat{j}$
10.$\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right),(20,-10)$
11.$\left(\begin{array}{ll}-1 & 0 \\ 0 & 1\end{array}\right),(-9,2)$
12.$\left(\frac{5}{2}, \frac{5}{2}\right)$
13.$\overrightarrow{P Q}=(k-2) \vec{a}+2 \vec{b}, \quad \overrightarrow{P R}=10 \vec{a}+5 \vec{b} ; k=6$
14.$\overrightarrow{A B}=\vec{b}-\vec{a}, \overrightarrow{P Q}=\frac 53 \vec{b}-\frac 53 \vec{a}$
15.$\overrightarrow{A B}=\vec{b}-\vec{a}, \overrightarrow{P Q}=3 \vec{b}-3 \vec{a}$
16.$-7,3$
17.$-5,2$
18.$9 ;1: 2$
19.$7 ; \sqrt{178}$
20.Prove
21.Prove

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