Cambridge IGCSE Mathematics Practice Paper Notes

Cambridge IGCSE Mathematics Practice Paper

General Information

• Duration: 1 hour 30 minutes

• Total Marks: 100

• Instructions:

  • Answer all questions.

  • Show workings in the space provided.

  • Write the final answer on the line labeled "Answer:"

  • Calculators are permitted only on questions marked "Calculator Allowed."

  • Marks are awarded for the method as well as the final answer.

• Topics Covered: Number, Algebra, Functions, Coordinate Geometry, Geometry.

Section 1: Number (20 marks)

Q1 (3 marks)

Express 72 as a product of its prime factors.

Q2 (3 marks)

Express 0.00045 in standard form.

Standard form is expressed as $a \times 10^b$, where 1 \leq a < 10 and b is an integer.

Q3 (3 marks)

Calculate the value of $5^3 \times 5^{-1}$.

Use the rule $a^m \times a^n = a^{m+n}$.

Q4 (3 marks)

Find the highest common factor (HCF) and the least common multiple (LCM) of 12 and 18.

• HCF: The largest number that divides both 12 and 18.

• LCM: The smallest number that is a multiple of both 12 and 18.

Q5 (4 marks)

A number is increased by 25% and then decreased by 20%. Determine the net percentage change.

Let the number be $x$. Increasing by 25% gives $x + 0.25x = 1.25x$. Decreasing the result by 20% gives $1.25x - 0.20(1.25x) = 1.25x - 0.25x = x$.

Q6 (4 marks)

If 60 is 75% of a number, find the number.

Let the number be $x$. Then $0.75x = 60$.

Section 2: Algebra (20 marks)

Q7 (3 marks)

Solve for $x$: $3x - 7 = 2x + 5$.

Q8 (3 marks)

Factorise $x^2 - 9$.

This is a difference of squares: $a^2 - b^2 = (a - b)(a + b)$.

Q9 (3 marks)

Expand and simplify $(2x + 3)(x - 4)$.

Use the distributive property (FOIL method).

Q10 (3 marks)

Solve the quadratic equation $x^2 - 5x + 6 = 0$.

Factorise the quadratic equation or use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$.

Q11 (4 marks)

Solve the simultaneous equations:

$x + y = 10$
$x - y = 2$

Solve for $x$ and $y$.

Q12 (4 marks)

If $a = b + 3$ and $2a - 4b = 6$, find $a$ and $b$.

Substitute the first equation into the second equation to solve for $b$.

Section 3: Functions (20 marks)

Q13 (3 marks)

Given $f(x) = 2x + 1$, evaluate $f(5)$.

Substitute $x = 5$ into the function.

Q14 (3 marks)

For $g(x) = x^2 - 4$, compute $g(3)$.

Substitute $x = 3$ into the function.

Q15 (3 marks)

Find $x$ for which $f(x) = 0$, where $f(x) = 2x + 1$.

Solve $2x + 1 = 0$ for $x$.

Q16 (3 marks)

Write the function $h(x) = 3x - 2$ in function notation and state its slope and y-intercept.

• Slope: The coefficient of $x$.

• y-intercept: The value of $h(x)$ when $x = 0$.

Q17 (4 marks)

If $f(x) = x^2$ and $g(x) = x + 3$, find $(f \circ g)(2)$.

$(f \circ g)(x) = f(g(x))$. First, find $g(2)$, and then find $f(g(2))$.

Section 4: Coordinate Geometry (20 marks)

Q19 (3 marks, Calculator Allowed)

Find the gradient of the line through $(1, 2)$ and $(4, 8)$.

Gradient $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

Q20 (3 marks, Calculator Allowed)

Find the equation of the line through $(1, 2)$ with a gradient of 3.

Use the point-slope form: $y - y<em>1 = m(x - x</em>1)$, where $m$ is the gradient and $(x<em>1, y</em>1)$ is the point.

Q21 (3 marks)

Find the midpoint of the segment joining $(3, 4)$ and $(7, 10)$.

Midpoint $= (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$.

Q22 (3 marks)

Calculate the distance between (2, -1) and (-3, 4).

Distance $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$.

Q23 (4 marks)

The line $y = 2x - 1$ intersects the x-axis at A. Find the coordinates of A.

The x-axis is where $y = 0$. Set $y = 0$ and solve for $x$.

Q24 (4 marks)

Given that the line $y = mx + c$ passes through $(2, 3)$ and $(4, 7)$, determine $m$ and $c$.

Substitute the points into the equation to form two simultaneous equations and solve for $m$ and $c$. Solve using substitution or elimination. $m$ is the gradient and $c$ is the y-intercept.