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

## 1. | GCSE Higher |

The red line in the diagram below shows an inequality. If the variable is \(x\), which inequality best describes \(x\)?

(a) \(-9 \le x \le 4\)

(b) \(-9 \lt x \le 4\)

(c) \(-9 \le x \lt 4\)

(d) \(-9 \lt x \lt 4\)

(e) \(-4 \le x \le 9\)

## 2. | GCSE Higher |

(a) Write down the integer values of \(x\) that satisfy the inequality \(–3 \le x \lt 1 \).

(b) Solve the compound inequality \( -4 \leq 2y + 1 \lt 6 \) and write down the integer solutions for \(y\).

(c) Find the single digit integers for \(z\) which satisfy the inequality \( -3z \leq 9 \).

## 3. | GCSE Higher |

(a) Solve 9m < 15m − 12

(b) On the number line below, show the set of values of \(x\) for which \( -4 \le x-1 \lt 5 \)

## 4. | GCSE Higher |

The graph of a quadratic function, \(y=f(x)\) is shown drawn accurately in the following diagram. Write down all the integer solutions of \(f(x) \le 0\).

## 5. | GCSE Higher |

The diagram below is a sketch of \(y = f(x)\) where \(f(x)\) is a quadratic function.

The graph intersects the x-axis where \(x=-2\) and \(x = 0.5\).

Which of the following is the solution of \(f(x) \le 0\) ?

- (a) \( x \ge -2 \) or \(x \ge 0.5\)
- (b) \( -2 \ge x \ge 0.5\)
- (c) \( x \ge -2 \) and \( x \le 0.5\)
- (d) \( x \le -2 \) and \( x \ge 0.5\)

## 6. | GCSE Higher |

Describe the unshaded (white) region by writing down three inequalities.

## 7. | GCSE Higher |

On the grid below indicate the region that satisfies all three of these inequalities.

$$y>-2$$ $$x+y<5$$ $$y-1 \le \frac{x}{2}$$## 8. | GCSE Higher |

(a) The expression \( (x+1)(2x-3)(3x+4) \) can be written in the form \(ax^3 + bx^2 + cx + d \) where \(a, b, c\) and \(d\) are integers. Find the values of \(a, b, c\) and \(d\).

(b) Solve the following inequality:

$$(x-2)^2 \lt \frac{16}{49}$$## 9. | GCSE Higher |

A region on a coordinate grid is described by the following three inequalities:

$$x>-2$$ $$x+y<7$$ $$y \ge \frac{x}{3}+2$$By shading the **unwanted** regions show the region on the grid below.

## 10. | GCSE Higher |

By shading the unwanted regions, show the region satisfied by these three inequalities.

$$ y \le x + 3 $$ $$ y> 4-x $$ $$ x < 2.5 $$## 11. | GCSE Higher |

Solve the following inequalities then explain how the whole number solutions to A and B different.

$$A: 5 \le 5x \lt 30$$ $$B: 5 \lt 5x \le 30$$## 12. | IGCSE Extended |

A large car park has an area of 1400m^{2} with space for \(x\) cars and \(y\) vans. Each car requires 14m^{2} of space and each van requires 35m^{2} of space.

(a) Show that \(2x+5y \le 200\)

(b) There must also be space for

(i) at least 50 vehicles,

(ii) at least 20 vans.

Write down two more inequalities to show this information.

(c) On the grid, show the three inequalities by drawing the three lines and shading the **unwanted** regions.

(d) Use your graph to find the largest possible number of vans.

(e) The company charges £6 for parking each car and £10 for parking each van. Find the number of cars and the number of vans which give the owners of the car park the greatest possible income and calculate this income.

## 13. | GCSE Higher |

A region on a coordinate grid is described by the following three inequalities:

$$-8 \le x \le 6$$ $$2y \ge x + 1$$ $$2y+x \le 12$$(a) By shading the **unwanted** regions show the region on the grid below.

(b) Mr Mushnik says that the point with coordinates (-8, 10) does not satisfy all the inequalities because it does not lie in the region. Is Mr Mushnik correct? You must give a reason for your answer.

## 14. | 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

$$3x + 5y = 21$$ $$8x - 5y = 1$$## 15. | GCSE Higher |

Here is a function machine that produces two outputs, A and B.

Work out the range of input values for which the output A is less than the output B.

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