## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | IGCSE Extended |

OABC is a parallelogram with O as origin. The position vector of A is \(a\) and the position vector of C is \(c\).

F is the mid-point of AB and the point E divides the line OC such that OE:EC = 2:1.

The point E also divides the line AD such that AE:ED = 3:2.

Find the following in terms of \(a\) and \(c\).

(a) \(\overrightarrow{OB}\)

(b) \(\overrightarrow{AC}\)

(c) \(\overrightarrow{AE}\)

(d) the position vector of F.

(e) \(\overrightarrow{AD}\)

(f) \(\overrightarrow{BD}\)

## 2. | GCSE Higher |

(a) Shape \(A\) is translated to shape \(B\) using the vector \( \begin{pmatrix}m\\n\\ \end{pmatrix}\). What are the values of \(m\) and \(n\)?

(b) Vectors \(a, b, c, d\) and \(e\) are drawn on an isometric grid. Write each of the vectors \(c, d\) and \(e\) in terms of \(a\) and/or \(b\).

## 3. | GCSE Higher |

ABCD is a quadrilateral. The points E, F, G and H are the midpoints of the sides of this quadrilateral.

$$\vec{CH} = a, \vec{DG} = b \text{ and } \vec{DF} = c$$Show that HG is parallel to EF.

## 4. | GCSE Higher |

In the parallelogram OABC two of the sides can be represented by vectors \(a\) and \(c\).

\( \overrightarrow{OA} = a \) and \( \overrightarrow{OC} = c \)

\( X \) is the midpoint of the line \( AC \).

\( OCD \) is a straight line such that \(OC:CD = k:1 \)

Given that \( \overrightarrow{XD} = 3c - \frac12 a \) find the value of \( k \).

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