A-LEVEL MATHEMATICS PAPER 3 EXAM-STYLE
PRACTICE SET 2026
◉ (draw the graph). Answer: Show that x² + 8x + 16 ≥ 0 for all values
of x
◉ (delta). Answer: What is the symbol for the discriminant?
◉ (delta) = b² - 4ac. Answer: What is the equation for the
discriminant?
◉ (delta) = -23
There are no real roots. Answer: Work out the discriminant and
determine how many roots the equation has.
2x² + 4 = 3x
◉ (delta) = 0
There are equal roots (one repeated root). Answer: Work out the
discriminant and determine how many roots the equation has.
x² + 12x + 36 = 0
◉ (delta) = 12
,There are two distinct real roots. Answer: Work out the discriminant
and determine how many roots the equation has.
6x² = 6x - 1
◉ x = -0.966 or 0.901. Answer: Solve. Write answer correct to 3
significant figures.
4x¹⁰ + x⁵ = 2
◉ p = 2/5. Answer: Find the value of p for which the quadratic
equation px² - 4px + 2 - p = 0 has equal roots
◉ (delta) = -55/4
There are no real roots. Answer: Show that there are no real
solutions to the simultaneous equations
y = 1 + 2x - x² and y = (1/2)x + 5
◉ When x = 1, y = 2
and when x = 2, y = 4. Answer: Solve the simultaneous equation
finding values for y and x.
y = 2x
y = x² - x + 2
◉ When x = 1, y = 0
,When x = -3, y = -4. Answer: Solve the simultaneous equation finding
values for y and x.
y = 0.5(1 - x²)
y=x-1
◉ When x = -1/2, y = 0
When x = -1, y = -1. Answer: Solve the simultaneous equation finding
values for y and x.
y = 1 + 2x
y² = 2x² + x
◉ Minimum at (-1, -1)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = x² + 2x
◉ Maximum at (0.5, 4)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = 3 + 4x - 4x²
, ◉ Minimum at (0.4, -2.8)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = 5x² - 4x - 2
◉ 24.5 cm². Answer: A right angled triangle has a width of x cm. The
length of the hypotenuse is 10 cm. The perimeter of the triangle is
24 cm. Find the maximum area of the triangle.
◉ 64 cm². Answer: A rectangle has a width of x cm. The perimeter of
the rectangle is 32 cm. Find the maximum area of the rectangle
◉ The y values. Answer: In functions, what is the range?
◉ The x values. Answer: In functions, what is the domain?
◉ f(x) ≤ 1. Answer: Find the range of the function f(x) = x -2, x ≤ 3
◉ h(x) > -4. Answer: Find the range of the function h(x) = x³ + 4, x > -
2
◉ 0 > f(x) ≥ 1. Answer: Find the range of the function f(x) = 1/x, x ≥ 1
PRACTICE SET 2026
◉ (draw the graph). Answer: Show that x² + 8x + 16 ≥ 0 for all values
of x
◉ (delta). Answer: What is the symbol for the discriminant?
◉ (delta) = b² - 4ac. Answer: What is the equation for the
discriminant?
◉ (delta) = -23
There are no real roots. Answer: Work out the discriminant and
determine how many roots the equation has.
2x² + 4 = 3x
◉ (delta) = 0
There are equal roots (one repeated root). Answer: Work out the
discriminant and determine how many roots the equation has.
x² + 12x + 36 = 0
◉ (delta) = 12
,There are two distinct real roots. Answer: Work out the discriminant
and determine how many roots the equation has.
6x² = 6x - 1
◉ x = -0.966 or 0.901. Answer: Solve. Write answer correct to 3
significant figures.
4x¹⁰ + x⁵ = 2
◉ p = 2/5. Answer: Find the value of p for which the quadratic
equation px² - 4px + 2 - p = 0 has equal roots
◉ (delta) = -55/4
There are no real roots. Answer: Show that there are no real
solutions to the simultaneous equations
y = 1 + 2x - x² and y = (1/2)x + 5
◉ When x = 1, y = 2
and when x = 2, y = 4. Answer: Solve the simultaneous equation
finding values for y and x.
y = 2x
y = x² - x + 2
◉ When x = 1, y = 0
,When x = -3, y = -4. Answer: Solve the simultaneous equation finding
values for y and x.
y = 0.5(1 - x²)
y=x-1
◉ When x = -1/2, y = 0
When x = -1, y = -1. Answer: Solve the simultaneous equation finding
values for y and x.
y = 1 + 2x
y² = 2x² + x
◉ Minimum at (-1, -1)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = x² + 2x
◉ Maximum at (0.5, 4)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = 3 + 4x - 4x²
, ◉ Minimum at (0.4, -2.8)
( (x + p)² + q and max/min coords at (-p, q) ). Answer: Sketch and
state the coordinates of the vertex and wheather it is a maximum or
a minimum for the following equation.
y = 5x² - 4x - 2
◉ 24.5 cm². Answer: A right angled triangle has a width of x cm. The
length of the hypotenuse is 10 cm. The perimeter of the triangle is
24 cm. Find the maximum area of the triangle.
◉ 64 cm². Answer: A rectangle has a width of x cm. The perimeter of
the rectangle is 32 cm. Find the maximum area of the rectangle
◉ The y values. Answer: In functions, what is the range?
◉ The x values. Answer: In functions, what is the domain?
◉ f(x) ≤ 1. Answer: Find the range of the function f(x) = x -2, x ≤ 3
◉ h(x) > -4. Answer: Find the range of the function h(x) = x³ + 4, x > -
2
◉ 0 > f(x) ≥ 1. Answer: Find the range of the function f(x) = 1/x, x ≥ 1
