CAMBRIDGE IGCSE MATHS (0580)

Question 1:
A parallelogram has base \( (2x - 1) \) metres and height \( (4x - 7) \) metres. The area of the parallelogram is \( 1m^2 \).

(a) (i) Show that \( 4x^2 - 9x + 3 = 0 \).
Answer (a)(i) ............................................... [3]

(ii) Solve the equation \( 4x^2 - 9x + 3 = 0 \). Show all your working and give your answers correct to 2 decimal places.
Answer (a)(ii) x = ........ or x = ........ [4]

(iii) Calculate the height of the parallelogram.
Answer (a)(iii) ........ m [1]

(b) (i) Factorise \( x^2 - 16 \).
Answer (b)(i) ............................................... [1]

(ii) Solve the equation: \[ \frac{(2x+3)}{(x-4)} + \frac{(x+40)}{(x^2 - 16)} = 2 \]
Answer (b)(ii) x = ........ [4]

Question 2:

(a) Simplify:
(i) \( (2x^2 y^3)^3 \)
Answer (a)(i) ............................................... [2]

(ii) \( \left( \frac{27}{x^6} \right)^{-\frac{1}{3}} \)
Answer (a)(ii) ............................................... [3]

(b) Multiply out and simplify:
\( (3x - 2y)(2x + 5y) \)
Answer (b) ............................................... [3]

(c) Make \( h \) the subject of:
(i) \( V = \pi r^3 + 2\pi r^2 h \)
Answer (c)(i) \( h = \) ............................................... [2]

(ii) \( V = \sqrt{3h} \)
Answer (c)(ii) \( h = \) ............................................... [2]

(d) Write as a single fraction in its simplest form: \[ \frac{x}{2} + \frac{5x}{3} - \frac{7x}{4} \]
Answer (d) ............................................... [2]

Question 3:

(a) Write as a single fraction:
(i) \( \frac{5}{4} - \frac{2x}{5} \)
Answer (a)(i) ............................................... [2]

(ii) \( \frac{4}{x+3} + \frac{2x - 1}{3} \)
Answer (a)(ii) ............................................... [3]

(b) Solve the simultaneous equations:
\[ 9x - 2y = 12 \]
\[ 3x + 4y = -10 \]
Answer (b) \( x = \) ...............................................
\(\quad y = \) ............................................... [3]

(c) Simplify: \[ \frac{7x + 21}{2x^2 + 9x + 9} \]
Answer (c) ............................................... [4]

Question 4:

(a) (i) Solve:
\[ 2(3x - 7) = 13 \]
Answer (a)(i) \( x = \) ............................................... [3]

(ii) Solve by factorising:
\[ x^2 - 7x + 6 = 0 \]
Answer (a)(ii) \( x = \) ................. or \( x = \) ................. [3]

(iii) Solve:
\[ \frac{3x - 2}{5} + \frac{x + 2}{10} = 4 \]
Answer (a)(iii) \( x = \) ............................................... [4]

Question 5:

(a) (i) Show that the equation:
\[ \frac{7}{x+4} + \frac{2x - 3}{2} = 1 \] can be simplified to:
\[ 2x^2 + 3x - 6 = 0 \]
Answer (a)(i) ............................................... [3]

(ii) Solve the equation:
\[ 2x^2 + 3x - 6 = 0 \]
Show all your working and give your answers correct to **2 decimal places**.
Answer (a)(ii) \( x = \) ........................... or \( x = \) ........................... [4]

(b) The total surface area of a **cone** with radius \( x \) and **slant height** \( 3x \) is equal to the area of a **circle** with radius \( r \).
Show that:
\[ r = 2x \]
**[The curved surface area, \( A \), of a cone with radius \( r \) and slant height \( l \) is given by:**
\[ A = \pi r l \]
Answer (b) ............................................... [4]

Question 6:

(a) Make \( x \) the subject of the formula:
\[ A - x = \frac{xr}{t} \]
Answer (a) \( x = \) ................................................. [4]

(b) Find the value of **a** and **b** when:
\[ x^2 - 16x + a = (x + b)^2 \]
Answer (b) \( a = \) ................................................
\( b = \) ................................................. [3]

(c) Write as a **single fraction** in its **simplest form**:
\[ \frac{6}{x-4} - \frac{5}{3x -2} \]
Answer (c) ................................................. [3]

Question 7:

(a) Expand and simplify:
\[ 3x(x - 2) - 2x(3x - 5) \]
Answer (a) ................................................ [3]

(b) Factorise the following completely:
(i) \[ 6w + 3wy - 4x - 2xy \]
Answer (b)(i) ................................................ [2]

(ii) \[ 4x^2 - 25y^2 \]
Answer (b)(ii) ................................................ [2]

(c) Simplify:
\[ \left( \frac{16}{9x^4} \right)^{-\frac{3}{2}} \]
Answer (c) ................................................ [2]

Question 8:

(a) Solve the inequality:
\[ 5x - 3 > 9 \]
Answer: ................................................... [2]

(b) Factorise completely:
(i) \[ xy - 18 + 3y - 6x \]
Answer: ................................................... [2]

(ii) \[ 8x^2 - 72y^2 \]
Answer: ................................................... [3]

(c) Make \( r \) the subject of the formula:
\[ p + 5 = \frac{1 - 2r}{r} \]
Answer: \( r = \) .................................................. [4]

Question 9:

(a) Expand the brackets and simplify:
(i) \[ 4(2x + 5) - 5(3x - 7) \]
Answer: ................................................. [2]

(ii) \[ (x - 7)^2 \]
Answer: ................................................. [2]

(b) Solve:
(i) \[ \frac{2x}{3} + 5 = -7 \]
Answer: \( x = \) ................................................ [3]

(ii) \[ 4x + 9 = 3(2x - 7) \]
Answer: \( x = \) ................................................ [3]

(iii) \[ 3x^2 - 1 = 74 \]
Answer: \( x = \) ..................... or \( x = \) ..................... [3]

Question 10:

(a) In this part, all lengths are in centimetres.

Given a rectangle with sides \(2x + 6\) and \(2x +1\), and a square with side \(3x -1\):

(i) Find the value of \(x\) when the perimeter of the rectangle is equal to the perimeter of the square.

Answer: \( x = \)

(ii) Find the value of \(x\) when the area of the rectangle is equal to the area of the square.

Show all your working.

Answer: \( x = \)

(b) (i) Factorise:

\[ x^2 + 4x - 5 \]

Answer:

(ii) Solve the equation:

\[ \frac{5}{x} - \frac{8}{x+1} = 1 \]

Show all your working.

Answer: \( x = \) or \( x = \)

Question 11:

(a) Solve the simultaneous equations.
You must show all your working.

\[ 2x + 3y = 11 \]

\[ 3x - 5y = -50 \]

Answer: \( x = \) , \( y = \)

(b) Given:

\[ x^2 - 12x + a = (x + b)^2 \]

Find the values of \( a \) and \( b \).

Answer: \( a = \) , \( b = \)

(c) Write as a single fraction in its simplest form:

\[ \frac{x}{2x - 5} + \frac{3x + 2}{x - 1} \]

Answer:

Question 12:

(a) Factorise:
(i) \[ 2mn + m^2 - 6n - 3m \]
Answer: ................................................... [2]

(ii) \[ 4y^2 - 81 \]
Answer: ................................................... [1]

(iii) \[ t^2 - 6t + 8 \]
Answer: ................................................... [2]

(b) Rearrange the formula to make \( x \) the subject:
\[ k = \frac{2m - x}{x} \]
Answer: \( x = \) .................................................. [4]

(c) Solve the simultaneous equations:
\[ \frac{1}{2}x - 3y = 9 \]
\[ 5x + y = 28 \]
Answer:
\( x = \) ..................................................
\( y = \) .................................................. [3]

(d)\( \frac{3}{m+4} - \frac{4}{m} = 6 \)
(i) Show that the equation can be written as:
\[ 6m^2 + 25m + 16 = 0 \]
Answer: ................................................... [3]

(ii) Solve the equation:
\[ 6m^2 + 25m + 16 = 0 \]
Give answers correct to 2 decimal places.
Answer:
\( m = \) ..................... or \( m = \) ..................... [4]

Question 13:

(a) Given the equation: \[ s = ut + \frac{1}{2} a t^2 \]
(i) Find \( s \) when \( t = 26.5 \), \( u = 104.3 \), and \( a = -2.2 \).
Give your answer in standard form, correct to 4 significant figures.
Answer: ................................................. [4]

(ii) Rearrange the formula to express \( a \) in terms of \( u \), \( t \), and \( s \).
Answer: \( a = \) ................................................ [3]

Question 14:

(a) Expand and simplify:
\[ (x + 7)(x - 3) \]
Answer: ................................................. [2]

(b) Factorise completely:
(i) \[ 15p^2 q^2 - 25q^3 \]
Answer: ................................................. [2]

(ii) \[ 4fg + 6gh + 10fk + 15hk \]
Answer: ................................................. [2]

(iii) \[ 81k^2 - m^2 \]
Answer: ................................................. [2]

(c) Solve the equation:
\[ 3(x - 4)+\frac{x + 2}{5} = 6 \]
Answer: \( x = \) ................................................. [4]

Question 15:

(a) Solve:
\[ 5x - 17 = 7x + 3 \]
Answer: \( x = \) ................................................. [2]

(b) Find the integer values of \( n \) that satisfy this inequality:
\[ -7 < 4n \leq 8 \]
Answer: ................................................. [3]

(c) Simplify:
(i) \[ a^3 \times a^6 \]
Answer: ................................................. [1]

(ii) \[ (5xy^2)^3 \]
Answer: ................................................. [2]

(iii) \[ \left(\frac{27x^{12}}{64y^3}\right)^{\frac{-1}{3}} \]
Answer: ................................................. [3]