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

## 2. | GCSE Higher |

(a) Simplify:

$$ p^6 \times p^3$$(b) Simplify:

$$ \dfrac{a^5b^7}{a^4b} $$(c) Solve:

$$ \dfrac{2w}{9} \gt 10$$## 3. | GCSE Higher |

Find the highest common factor of the following two expressions:

$$ 8x^5y^3 $$ $$ 6x^2y $$## 4. | GCSE Higher |

Simplify then find the square root of this expression:

$$\frac{y}{(1-y)^2} - \frac{y}{1-y}$$## 5. | 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:

$$(x-2)^2 \lt \frac{16}{49}$$## 6. | GCSE Higher |

(a) Simplify \( \left(\dfrac{3a}{a^3 - 3}\right)^0 \)

(b) Simplify \( \dfrac{9(2b-1)}{(2b-1)^2}\)

(c) Simplify \( (2c^3d^4)^5 \)

## 7. | GCSE Higher |

Work out the exact value of \(n\).

## 8. | GCSE Higher |

(a) Express the following as a single fraction in its simplest form.

$$ \dfrac{6}{x-3} - \dfrac{2}{x-1} $$(b) Expand and simplify the following:

$$ (x+ 2)(3x-5)(5x+1) $$## 9. | GCSE Higher |

(a) Simplify \( 6(3a-2)-2(a-3) \)

(b) Simplify \( \frac{3x}{4} - \frac{2x}{3} \)

(c) Rearrange the formula to make m the subject:

$$ w = \frac{m+n}{3-m} + 7 $$(d) Solve \( x^8 = 5600 \)

(e) Solve \( |x+5| = 9 \)

(f) Solve by factorising:

$$ 9z^2 - 27z + 20 = 0 $$## 10. | GCSE Higher |

(a) The \(n\)th term of a sequence is \(2^n+2^{n+1}\)

Work out the 8th term of the sequence.

(b) The \(n\)th term of a different sequence is \(9(3^n + 3^{n+1})\)

Expand and express this expression as the sum of two powers of three.

## 11. | GCSE Higher |

(a) Simplify the following expression.

$$ \frac{x^2 - 4}{3x^2 + 13x + 14}$$(b) Make b the subject of the following formula.

$$ a = \frac{7(3b-c)}{b}$$## 12. | GCSE Higher |

(a) Without using a calculator, show that \(\sqrt{28}=2\sqrt7\)

(b) The point \(X\) is shown on the unit grid below. The point \(Y\) is \(\sqrt17\) units from \(X\) and lies on the intersection of two grid lines. Mark one possible position for \(Y\).

## 13. | GCSE Higher |

If a, b and c are positive integers use the following statements to find the values of a, b and c.

$$ (ab^c)^3 = 27b^{21} $$ $$ b= 9a $$## 15. | GCSE Higher |

Show that:

$$3 - (4x^2+11x+7) \div (x^2+4x+3) $$simplifies to \( \frac{a-x}{x+b}\) where \(a\) and \(b\) are integers.

## 16. | GCSE Higher |

The expression below can be written as a single fraction in the form \( \dfrac{a-bx}{x^2-25} \) where \(a\) and \(b\) are integers.

$$ \frac{x-4}{x-5} - 2 + \frac{x+1}{x+5}$$Work out the value of \(a\) and the value of \(b\).

## 17. | GCSE Higher |

\(y = a \times b^{x – 2}\) where \(a\) and \(b\) are numbers.

\(y = 5\) when \(x = 2\)

\(y = 0.005\) when \(x = 5\)

Work out the value of \(y\) when \(x = 4\)

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