Forming & Solving Equations (Cambridge (CIE) IGCSE Maths: Extended): Exam Questions

The perimeter of the rectangle is 80 cm.
The area of the rectangle is $A$cm².

i) Show that  $x^2–40x+A=0$.

[3]

ii) When $A=300$, solve the equation $x^2–40x+A=0$ by factorising.  

$x$ = .................. or $x$ = .................  [3]

iii) When $A=200$, solve the equation $x^2–40x+A=0$ using the quadratic formula.
Show all your working and give your answers correct to 2 decimal places.  

$x$= .................. or $x$ = .................  [4]

The area of triangle DEF is 70 m².

i) Show that  $4x^2+11x-300=0$.

[4]

ii) Use the quadratic formula to solve $4x^2+11x-300=0$.
Show all your working and give your answers correct to 2 decimal places.

$x$ = .......................... or $x$ = .......................... [4]

iii) Find the length of DE.

DE = ............................................... m [1]

A cone with height 14.8 cm has volume 275 cm³.  
Calculate the radius of the cone.
[The volume, $V$, of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\mathrm{\pi }{r}^{2}h$.]  

............................................... cm

Paulo and Jim each buy sacks of rice but from different shops. Paulo pays $72 for sacks costing $$m$ each.
Jim pays $72 for sacks costing $$(m+0.9)$ each.

i) Find an expression, in terms of $m$, for the number of sacks Paulo buys.  

[1]

ii) Find an expression, in terms of $m$, for the number of sacks Jim buys.  

[1]

Paulo buys 4 more sacks than Jim.  
Write down an equation, in terms of $m$, and show that it simplifies to  $10m^2+9m-162=0$.

i) Solve  $10m^2+9m-162=0$.  

$m$ = .................... or $m$ = .................... [3]

ii) Find the number of sacks of rice that Paulo buys. 

[1]

The diagram shows a right-angled triangle ABC.

The area of this triangle is 30 cm².
Show that  $2x^2+3x-20=0$.

Use factorisation to solve the equation  $2x^2+3x-20=0$.   

$x$ = ...................... or $x$ = ...................... 

Calculate BC.  

BC = .......................................... cm 

In this question, all measurements are in metres.

The diagram shows a right-angled triangle.

Show that $5x^2-12x-27=0.$

Solve $5x^2-12x-27=0.$
Show all your working and give your answers correct to 2 decimal places.

$x=$......................... or $x=$......................... 

Calculate the perimeter of the triangle.

............................................ m 

Calculate the smallest angle of the triangle.

At a football match, the price of an adult ticket is $$x$ and the price of a child ticket is $$(x-2.50)$.
There are 18 500 adults and 2400 children attending the football match.
The total amount paid for the tickets is $320 040.
Find the price of an adult ticket.

$ ................................................ 

Solve.

$\frac{3y}{2y-1}=\frac{3}{4}$

$y$ = ..............................................

Here is a rectangle.

The length of the rectangle is 7 cm longer than the width of the rectangle.

4 of these rectangles are used to make this 8-sided shape.

The perimeter of the 8-sided shape is 70 cm.
Work out the area of the 8-sided shape.

$ABCD$ is a rectangle.
$EFGH$is a trapezium.

All measurements are in centimetres.
The perimeters of these two shapes are the same.

Work out the area of the rectangle.

3 teas and 2 coffees have a total cost of £7.80 5
teas and 4 coffees have a total cost of £14.20

Work out the cost of one tea and the cost of one coffee.

A cinema sells adult tickets and child tickets.

   The total cost of 3 adult tickets and 1 child ticket is £30   
 The total cost of 1 adult ticket and  3 child tickets is £22

   Work out the cost of an adult ticket and the cost of a child ticket.

The diagram shows a trapezium.

All the measurements are in centimetres.

The area of the trapezium is 351 cm².

Show that  $2x^2+x—351=0$

Work out the value of x.

Alan invests $200 at a rate of $r\%$ per year compound interest. After 2 years the value of his investment is $206.46.

i) Show that     $r^2+200r-323=0$.

[3]

ii) Solve the equation   $r^2+200r-323=0$  to find the rate of interest.
Show all your working and give your answer correct to 2 decimal places.

$r=$................................................ [3]

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $\$x$ and every large cake costs $\$(2x+1)$.

Petra spends $20 on small coffee cakes and $10 on large coffee cakes.
The total number of cakes is 45.

Write an equation in terms of $x$.
Solve this equation to find the cost of a small coffee cake.
Show all your working.   

$ ................................................ 

The total perimeter of a semicircle is 19.02 cm.  
Calculate the radius of the semicircle.  

................................................ cm

Paddy and Anna each invest $2000 for 5 years.
Paddy earns simple interest at a rate of 1.25% per year.
Anna earns compound interest at a rate of $r$% per year.
At the end of 5 years, Paddy’s investment is worth the same as Anna’s investment.  

Calculate the value of $r$.  

$r=$................................................ 

All the lengths in this question are in centimetres.

The diagram shows a shape ABCDEF made from two rectangles. The total area of the shape is 342 cm².

Show that $x^2+x-72=0$.

Solve  $x^2+x-72=0$ by factorisation.  

$x=$ .................... or  $x=$.................... 

Work out the perimeter of the shape ABCDEF.  

 ............................................ cm 

Calculate angle DBC.  

Angle DBC = ................................................ 

A, B and C lie on the circle, centre O.
Angle AOC = $(3x+22)°$ and angle ABC = $5x°$.

Find the value of $x$.

$x$ = ................................................... 

In a shop, the price of a monthly magazine is $$m$ and the price of a weekly magazine is $$(m-0.75)$.
One day, the shop receives
•  $168 from selling monthly magazines
•  $207 from selling weekly magazines.
The total number of these magazines sold during this day is 100.

i) Show that  $50m^2-225m+63=0$.

[3]

ii) Find the price of a monthly magazine. Show all your working.

$ ............................................... [3]

Here are two right-angled triangles.

Given that

      $\tan e=\tan f$

find the value of $x$.

You must show all your working.

Here is a right-angled triangle.

All measurements are in centimetres.
The area of the triangle is 2.5 cm² .

Find the perimeter of the triangle.
Give your answer correct to 3 significant figures.
You must show all of your working.

There are only $r$ red counters and $g$ green counters in a bag.

A counter is taken at random from the bag.
The probability that the counter is green is $\frac{3}{7}$.

The counter is put back in the bag.

2 more red counters and 3 more green counters are put in the bag.
A counter is taken at random from the bag.

The probability that the counter is green is $\frac{6}{13}$
Find the number of red counters and the number of green counters that were in the bag originally.