Exam-Style Questions.

1.	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 \( x \ge 0.5\)
  
  

Worked Solution

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2.	GCSE Higher

The volume of the larger red cube is 386cm³ greater than the volume of the smaller blue cube.

The lengths of the edges of the blue cube can be expressed as \(x\) cm.

The lengths of the edges of the red cube can be expressed as \(x+2\) cm.

(a) Show that \( x^2 + 2x - 63 = 0 \).

(b) Find the volume of the blue cube.

Worked Solution

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3.	IGCSE Extended

(a) Show that the equation \(\frac{3}{x+1}+\frac{3x-9}{2}=1\) can be simplified to \(3x^2-8x-5=0\).

(b) Solve the equation \(3x^2-8x-5=0\) showing all of your working and giving answers to three significant figures.

(c) The total surface area of a cone with radius \(x\) and slant height \(8x\) is equal to the area of a circle with radius r. Show that \(r = 3x\).

[The curved surface area, \(A\), of a cone with radius \(r\) and slant height \(l\) is \(A=\pi rl\).]

Worked Solution

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4.	IB Studies

A red rug has a width of \(x-3\) cm and a length of \(4x\) cm.

(a) Write down an expression for the area, A, in cm², of the rug.

The area of the rug is 3240 cm².

(b) Calculate the value of \(x\).

(c) Hence, write down the value of the length and of the width of the rug in centimetres.

Worked Solution

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5.	GCSE Higher

In the diagram below, which is not drawn to scale, all dimensions are in centimetres and all angles are multiples of 90^o. If the shaded area is 698cm², work out the value of \(x\).

Worked Solution

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6.	GCSE Higher

Given that:

$$ x^2 : (5x + 3) = 1 : 3 $$

find the possible values of \(x\).

Worked Solution

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7.	GCSE Higher

The area of triangle ABC (not drawn to scale) is

$$ \frac{35 \sqrt{3}}{4} m^2$$

If AB = \(x+2\) metres and AC = \(2x+1 \) metres, find the value of \(x\).

Worked Solution

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