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

## 1. | IGCSE Core |

Nora and Dora are trying to solve the following simultaneous equations by finding the intersection of their graphs.

$$y+x=6$$ $$y-x=1$$Here are their graphs:

(a) Who has plotted the graphs most correctly?

(b) What is the correct solution?

## 2. | GCSE Higher |

(a) Use the red graphs to solve the simultaneous equations:

$$4x-7y=-47$$ $$3y=-5x$$(b) Use the blue graph to find estimates for the solutions of the quadratic equation:

$$x^2-5x+3=0$$## 4. | GCSE Higher |

Estimate the solutions of the following simultaneous equations using their graphs as drawn on the grid below.

$$3x-5y=17$$ $$y=5-\frac{7x}{5}$$## 6. | GCSE Higher |

A rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways, either by cutting along line A or by cutting along line B.

When the original sheet of paper is cut along line A, the perimeter of each of the two pieces is 56 cm.

When the original sheet of paper is cut along line B, the perimeter of each of the two pieces is 64 cm.

What is the perimeter of the original sheet of paper?

## 7. | GCSE Higher |

Show that you understand equations and inequalities by answering the following:

(a) Solve \(5x^2=80\)

(b) Solve \(8x + 2 \gt x + 7\)

(c) Write down the largest integer that satisfies \(8x - 2 \lt 25\)

(d) Solve the following pair of equations

$$3x + 5y = 21$$ $$8x - 5y = 1$$## 8. | GCSE Higher |

Draw the graph of \(y = 2x^2 + 3x - 7\) for \( -3.5 \le x \le 2\). Draw suitable straight lines to find approximate solutions (to one decimal place) of the equations:

(a) \(2x^2 +3x - 9 = 0\)

(b) \(2x^2 +4x - 7 = 0\)

(c) \(2x^2 - x -5 = 3 - x\)

## 9. | GCSE Higher |

Solve algebraically the simultaneous equations to find the solution where \(-10 \le x \le 10\).

$$ 3x^2 - y^2 = 11 $$ $$ 5x + 3y = 27 $$## 10. | GCSE Higher |

Two numbers are chosen so that the sum of their squares is 25.

If those numbers are represented by \(x\) and \(y\) they will also satisfy the equation:

$$y-3x=13$$Use an algebraic method to find two possible values of \(x\) and \(y\) .

## 11. | GCSE Higher |

The prices of two watches are in the ratio \(a:b\).

When the prices are both increased by £10, the ratio becomes \(5 : 7\).

When the prices are both reduced by £10, the ratio becomes \(1 : 3\).

Express the ratio \(a:b\) in its lowest terms.

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