Past Exam Questions

The diagram shows the triangle \(OAB\), where \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

The point \(P\) lies on \(OA\) such that \(\overrightarrow{OP}=\frac34\overrightarrow{OA}\).

The point \(Q\) lies on \(AB\) such that \(\overrightarrow{AQ}=\frac13\overrightarrow{AB}\).

The straight line through \(P\) and \(Q\) meets the straight line through \(O\) and \(B\) at the point \(R\). It is given that \(\overrightarrow{OR}=\lambda\mathbf{b}\) and \(\overrightarrow{PR}=\mu\overrightarrow{PQ}\), where \(\lambda\) and \(\mu\) are constants.

(a) Find \(\overrightarrow{OR}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mu\).

(b) Hence find the values of \(\lambda\) and \(\mu\).

The diagram shows four points, \(O,A,B\) and \(C\).

\(A,B\) and \(C\) lie in a straight line and are such that \(\frac{AB}{AC}=\frac13\).

\(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

(a) Find \(\overrightarrow{OC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Simplify your answer.

(b) The line \(OA\) is extended to the point \(D\) such that \(OA:AD=2:7\). Point \(E\) lies on \(CD\) such that \(\overrightarrow{OE}=\lambda\mathbf{b}\). Find the value of \(\lambda\).

In the diagram, \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\).

The point \(M\) is the midpoint of \(OB\).

The point \(N\) is such that \(\overrightarrow{ON}=3\overrightarrow{NA}\).

The lines \(BN\) and \(AM\) intersect at the point \(X\).

\(\overrightarrow{BX}=\lambda\overrightarrow{BN}\), where \(\lambda\) is a constant.

\(\overrightarrow{MX}=\mu\overrightarrow{MA}\), where \(\mu\) is a constant.

(a) Find \(\overrightarrow{OX}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\lambda\).

(b) Find \(\overrightarrow{OX}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\mu\).

(c) Hence find the values of \(\lambda\) and \(\mu\).

The diagram shows the trapezium \(O A B C\), where \(\overrightarrow{O A}=4 \mathbf{a}, \overrightarrow{O C}=\mathbf{c}\), and \(\overrightarrow{C B}=2 \mathbf{a}\). The point \(D\) lies on \(A B\) such that \(A D: D B=2: 1\). The point \(X\) is the point of intersection of the lines \(O D\) and \(A C\). It is given that \(\overrightarrow{A X}=\lambda \overrightarrow{A C}\) and \(\overrightarrow{O X}=\mu \overrightarrow{O D}\).

Find in terms of \(\mathbf{a}\) and \(\mathbf{c}\) (a) \(\overrightarrow{A B}\)

(b) \(\overrightarrow{O D}\).

(c) Find \(\overrightarrow{O X}\) in terms of \(\mathbf{a}, \mathbf{c}\) and \(\mu\).

(d) Find \(\overrightarrow{A X}\) in terms of \(\mathbf{a}, \mathbf{c}\) and \(\lambda\).

(e) Hence find the values of \(\lambda\) and \(\mu\).

The diagram shows the triangle \(O A C\). The point \(B\) lies on \(A C\) such that \(A B: B C=p: q\), where \(p\) and \(q\) are constants ( \(p \neq-q\) ). \(\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b} \text { and } \overrightarrow{O C}=\mathbf{c} .\)

Show that \(\mathbf{b}=\frac{q \mathbf{a}+p \mathbf{c}}{q+p}\).

(a)

The diagram shows a triangle \(O A B\). The point \(P\) lies on \(A B\). The ratio \(A P: P B\) is \(1: 3\). Given that \(\overrightarrow{O A}=\mathbf{a}\) and \(\overrightarrow{O B}=\mathbf{b}\), find an expression for \(\overrightarrow{O P}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Simplify your answer.

(b) Vector \(\mathbf{q}\) has magnitude \(12 \sqrt{5}\) and direction \(\binom{6}{-3}\).

Vector \(\mathbf{r}\) has magnitude \(15 \sqrt{2}\) and direction \(\binom{-5}{5}\). Find the unit vector in the direction of \(\mathbf{q}+\mathbf{r}\).

The diagram shows a triangle \(O B C\). \(O A: O B=4: 7\) and \(O D: O C=4: 7\). \(\overrightarrow{O B}=\mathbf{b} \text { and } \overrightarrow{O C}=\mathbf{c}\)

The point \(P\) is the point of intersection of \(A C\) and \(B D\) such that \(\overrightarrow{A P}=\lambda \overrightarrow{A C}\) and \(\overrightarrow{B P}=\mu \overrightarrow{B D}\) where \(\lambda\) and \(\mu\) are scalars. (a) Find two expressions for \(\overrightarrow{O P}\), each in terms of \(\mathbf{b}, \mathbf{c}\) and a scalar, and hence show that \(P\) divides both \(A C\) and \(D B\) in the ratio \(4: 7\).

(b) The point \(Q\) is such that \(\overrightarrow{O Q}=\frac{2}{7} \mathbf{b}+\frac{2}{7} \mathbf{c}\).

Use a vector method to show that \(O, Q\) and \(P\) are collinear. Justify your answer.

The diagram shows a parallelogram \(O A B C\). The point \(D\) divides the line \(O C\) in the ratio \(2: 3\). \(\overrightarrow{O A}=\mathbf{a} \text { and } \overrightarrow{O C}=\mathbf{c}\)

The point \(P\) lies on \(A D\) such that \(\overrightarrow{O P}=\lambda \overrightarrow{O B}\) and \(\overrightarrow{A P}=\mu \overrightarrow{A D}\), where \(\lambda\) and \(\mu\) are scalars. Find two expressions for \(\overrightarrow{O P}\), each in terms of \(\mathbf{a}\), \(\mathbf{c}\) and a scalar, and hence show that \(P\) divides both \(D A\) and \(O B\) in the ratio \(m: n\), where \(m\) and \(n\) are integers to be found.

The diagram shows the triangle \(OAB\) with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). The point \(X\) lies on the line \(OA\) such that \(\overrightarrow{OX}=\frac35\mathbf a\). The point \(Y\) is the midpoint of the line \(AB\). Find, in terms of \(\mathbf a\) and \(\mathbf b\),

(a) \(\overrightarrow{AB}\),

(b) \(\overrightarrow{XY}\).

The lines \(OB\) and \(XY\) are extended to meet at the point \(Z\). It is given that \(\overrightarrow{YZ}=\lambda\overrightarrow{XY}\) and \(\overrightarrow{BZ}=\mu\mathbf b\).

(c) Find \(\overrightarrow{XZ}\) in terms of \(\lambda\), \(\mathbf a\) and \(\mathbf b\).

(d) Find \(\overrightarrow{XZ}\) in terms of \(\mu\), \(\mathbf a\) and \(\mathbf b\).

(e) Hence find the values of \(\lambda\) and \(\mu\).

The diagram shows a triangle \(OAB\). The point \(C\) is the midpoint of \(OA\). The point \(D\) lies on \(CB\) such that \(CD:DB=2:3\).

\(\overrightarrow{OC}=\mathbf c,\qquad \overrightarrow{CB}=\mathbf b.\)

The point \(E\) lies on \(AB\) such that \(\overrightarrow{OE}=\lambda\overrightarrow{OD}\) and \(\overrightarrow{AE}=\mu\overrightarrow{AB}\), where \(\lambda\) and \(\mu\) are scalars. Find two expressions for \(\overrightarrow{OE}\), each in terms of \(\mathbf b\), \(\mathbf c\) and a scalar, and hence find \(AE:EB\).

In the triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). The mid-point of the line \(OB\) is \(X\), and the mid-point of the line \(AB\) is \(Y\). The lines \(OY\) and \(AX\) intersect at the point \(Z\). It is given that \(\overrightarrow{AZ}=\lambda\overrightarrow{AX}\) and \(\overrightarrow{OZ}=\mu\overrightarrow{OY}\), where \(\lambda\) and \(\mu\) are rational numbers.

(a) Find \(\overrightarrow{OZ}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\lambda\).

(b) Find \(\overrightarrow{OZ}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\mu\).

(c) Find the values of \(\lambda\) and \(\mu\).

(d) Hence find \(\overrightarrow{OZ}\) in terms of \(\mathbf a\) and \(\mathbf b\) only.

In the triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\).

The straight line \(XYZ\) is such that:

\(\overrightarrow{OX}=\frac45\mathbf b\)

\(\overrightarrow{AY}=\frac13\overrightarrow{AB}\)

\(\overrightarrow{AZ}=\mu\mathbf a\), where \(\mu\) is a constant

\(\overrightarrow{YZ}=\lambda\overrightarrow{XY}\), where \(\lambda\) is a constant.

(a) Show that \(\overrightarrow{XY}=\frac23\mathbf a-\frac7{15}\mathbf b\).

(b) Find \(\overrightarrow{YZ}\) in terms of \(\lambda\), \(\mathbf a\) and \(\mathbf b\).

(c) Find \(\overrightarrow{YZ}\) in terms of \(\mu\), \(\mathbf a\) and \(\mathbf b\).

(d) Hence find the values of \(\lambda\) and \(\mu\).

The diagram shows a triangle \(OAC\). The point \(B\) lies on \(AC\) such that \(AB:AC=2:5\). It is given that

\(\overrightarrow{OA}=\mathbf a,\qquad \overrightarrow{OB}=\mathbf b,\qquad \overrightarrow{OC}=\mathbf c.\)

(a) Show that

\(5\mathbf b-3\mathbf a=2\mathbf c.\)

The diagram now includes points \(X\) and \(Y\), such that

\(\overrightarrow{OX}=\frac34\overrightarrow{OA}\)

and

\(\overrightarrow{OY}=m\overrightarrow{OB},\)

where \(m\) is a constant. It is also given that \(XY:XC=\lambda:1\), where \(\lambda\) is a constant.

(b) Using part (a), find \(\overrightarrow{XC}\) in terms of \(\mathbf a\) and \(\mathbf b\).

(c) Hence find the values of \(m\) and \(\lambda\).

(a) Find the vector with magnitude \(200\) in the direction of

\(\begin{pmatrix}7\\-24\end{pmatrix}.\)

(b) The diagram shows triangle \(AOB\) such that \(\overrightarrow{OA}=\mathbf a\), and \(\overrightarrow{OB}=\mathbf b\). The point \(C\) lies on the line \(AB\) such that \(AC:AB=1:3\). Find the vector \(\overrightarrow{OC}\) in terms of \(\mathbf a\) and \(\mathbf b\), giving your answer in its simplest form.

(c) Given the vector equation

\(p\begin{pmatrix}2\\1\end{pmatrix}+q\begin{pmatrix}2\\4\end{pmatrix} =5\begin{pmatrix}-p+1\\p+q\end{pmatrix},\)

find the values of \(p\) and \(q\).

In the diagram, \(OP=2\mathbf a\), \(SR=5\mathbf a\), \(OS=3\mathbf b\) and \(QR=\mathbf b\).

The point \(X\) lies on \(PS\) and on \(OQ\). It is given that \(PX=\lambda PS\) and \(OX=\mu OQ\).

(a) Express \(\overrightarrow{OX}\) in terms of \(\lambda\), \(\mathbf a\) and \(\mathbf b\).

(b) Express \(\overrightarrow{OQ}\) in terms of \(\mathbf a\) and \(\mathbf b\).

(c) Find the values of \(\lambda\) and \(\mu\).

(d) Find \(OX:OQ\).

(e) Find \(PX:XS\).

The diagram shows a quadrilateral \(OABC\), where

\(\overrightarrow{OA}=\mathbf a,\qquad \overrightarrow{OB}=\mathbf b,\qquad \overrightarrow{OC}=\mathbf c.\)

The line \(AC\) intersects \(OB\) at \(P\), and \(AP:PC=3:2\).

(a) Find \(\overrightarrow{OP}\) in terms of \(\mathbf a\) and \(\mathbf c\).

(b) Given that \(OP:PB=2:3\), show that

\(2\mathbf b=3\mathbf c+2\mathbf a.\)

The diagram shows the parallelogram \(OABC\), such that \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). The point \(D\) lies on \(CB\) such that \(CD:DB=3:1\). When extended, the lines \(AB\) and \(OD\) meet at the point \(E\). It is given that

\(\overrightarrow{OE}=h\overrightarrow{OD} \quad\text{and}\quad \overrightarrow{BE}=k\overrightarrow{AB},\)

where \(h\) and \(k\) are constants.

(a) Find \(\overrightarrow{DE}\) in terms of \(\mathbf a\), \(\mathbf c\) and \(h\).

(b) Find \(\overrightarrow{DE}\) in terms of \(\mathbf a\), \(\mathbf c\) and \(k\).

(c) Hence find the value of \(h\) and of \(k\).

(a) The diagram shows triangle \(OAC\), where \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\) and \(\overrightarrow{OC}=\mathbf c\). The point \(B\) lies on the line \(AC\) such that \(AB:BC=m:n\), where \(m\) and \(n\) are constants.

(i) Write down \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf b\).

(ii) Write down \(\overrightarrow{BC}\) in terms of \(\mathbf b\) and \(\mathbf c\).

(iii) Hence show that \(n\mathbf a+m\mathbf c=(m+n)\mathbf b\).

(b) Given that

\(\lambda\begin{pmatrix}2\\1\end{pmatrix}+(\mu-1)\begin{pmatrix}-4\\7\end{pmatrix} =(\lambda+1)\begin{pmatrix}4\\-2\end{pmatrix},\)

find the value of each of the constants \(\lambda\) and \(\mu\).

The diagram shows the triangle \(OAC\). The point \(B\) is the midpoint of \(OC\). The point \(Y\) lies on \(AC\) such that \(OY\) intersects \(AB\) at the point \(X\), where \(AX:XB=3:1\). It is given that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

(a) Find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), giving your answer in its simplest form.

(b) Find \(\overrightarrow{AC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

(c) Given that \(\overrightarrow{OY}=h\overrightarrow{OX}\), find \(\overrightarrow{AY}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(h\).

(d) Given that \(\overrightarrow{AY}=m\overrightarrow{AC}\), find the value of \(h\) and of \(m\).

In the diagram, \(\overrightarrow{OP}=2\mathbf{b}\), \(\overrightarrow{OS}=3\mathbf{a}\), \(\overrightarrow{SR}=\mathbf{b}\), and \(\overrightarrow{PQ}=\mathbf{a}\). The lines \(OR\) and \(QS\) intersect at \(X\).

(a) Find \(\overrightarrow{OQ}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

(b) Find \(\overrightarrow{QS}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

(c) Given that \(\overrightarrow{QX}=\mu\overrightarrow{QS}\), find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\), and \(\mu\).

(d) Given that \(\overrightarrow{OX}=\lambda\overrightarrow{OR}\), find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\), and \(\lambda\).

(e) Find the values of \(\lambda\) and \(\mu\).

(f) Find the value of \(\dfrac{QX}{XS}\).

(g) Find the value of \(\dfrac{OR}{OX}\).