CAIE IGCSE Mathematics Extended E2.5 Equations

[Maximum number: 7]

(a)

Antonio travels 100 km at an average speed of $x \mathrm{~km} / \mathrm{h}$ .

He then travels a further 150 km at an average speed of (x+10) km / h.
The time taken for the whole journey is 4 hours 20 minutes.

[ 7 ]

(i)

Show that $13x^2-620x-3000=0$ .

[ 4 ]

$\frac{100}{x}+\frac{150}{x+10}=4\frac13$ oe or $150=\left(\frac{13}{3}-\frac{100}{x}\right)(x+10)$ $\frac{100(x+10)+150x}{x(x+10)}$ or better
$300x+3000+450x=13x^2+130x$ oe or better
Allow correct multiples of $13x^2-620x-3000=0$

(ii)

Solve $13x^2-620x-3000=0$ to find the speed Antonio travels for the first 100 km of the journey.
You must show all your working and give your answer correct to 1 decimal place.

[ 3 ]

$\frac{620\pm\sqrt{(-620)^2-4(13)(-3000)}}{2(13)}$
or equivalent quadratic formula
M1 for $(-620)^2-4\times13\times(-3000)$ oe
x=52.1 to 52.2 final answer
Reject negative speed

[Maximum number: 7]

(a)

In a shop the cost of a fiction book is $ x and the cost of a reference book is $(x+2). The cost of 11 fiction books is the same as the cost of 10 reference books.

Find the value of x.

x=

[ 2 ]

(b)

In another shop, the cost of a fiction book is $ y and the cost of a reference book is $(y+2). Maria spends $ 95 on fiction books and $ 147 on reference books.
She buys a total of 12 books.

[ 5 ]

(i)

Show that 6 y^2-109 y-95=0.

[ 4 ]

$\frac{95}{y}+\frac{147}{y+2}=12$
M2
M1 for $\frac{95}{y}$ or $\frac{147}{y+2}$ 95(y+2)+147y=12y(y+2) oe
M1 for clearing their equation with algebraic fractions in y and y+2
leading to $6y^2-109y-95=0$
A1 with all brackets shown expanded and no errors or omissions

(ii)

Find the value of y.

y=

[ 1 ]

[Maximum number: 3]

18 Solve.
$\frac{3y}{2y-1}=\frac34$ y=

$-\frac12$ oe

B2 for 12y=6y-3 or better.
M1 for $4\times3y=3(2y-1)$ .

[Maximum number: 4]

Antonio travels 100 km at an average speed of $x \mathrm{~km} / \mathrm{h}$ .

He then travels a further 150 km at an average speed of (x+10) km / h.
The time taken for the whole journey is 4 hours 20 minutes.

Show that $13x^2-620x-3000=0$ .

$\frac{100}{x}+\frac{150}{x+10}=4\frac13$ oe or $150=\left(\frac{13}{3}-\frac{100}{x}\right)(x+10)$ $\frac{100(x+10)+150x}{x(x+10)}$ or better
$300x+3000+450x=13x^2+130x$ oe or better
Allow correct multiples of $13x^2-620x-3000=0$

[Maximum number: 2]

In a shop the cost of a fiction book is $ x and the cost of a reference book is $(x+2). The cost of 11 fiction books is the same as the cost of 10 reference books.

Find the value of x.

x=

[Maximum number: 2]

Solve the simultaneous equations.

x+2y=13x+5y=22x=y=

[Maximum number: 4]

The line y=x+1 intersects the curve $y=x^{2}+x-3$ at two points.
Find the coordinates of the two points.

Figure for Question 7 — CAIE IGCSE Mathematics Extended

[4]
22 x is inversely proportional to the square root of w. When w=16, x=3.

Find x in terms of w.

x=

(2, 3) and (-2, -1)

4

B3 for x=2 and x=-2

or B2 for $x^{2}-4[=0]$ or better

or for (2,3) or (-2,-1)

or M1 for $x+1=x^{2}+x-3$ oe

22

$\frac{12}{\sqrt{w}}$ oe final answer

2

M1 for $x=\frac{k}{\sqrt{w}}$ oe

[Maximum number: 3]

Antonio travels 100 km at an average speed of $x \mathrm{~km} / \mathrm{h}$ .

He then travels a further 150 km at an average speed of (x+10) km / h.
The time taken for the whole journey is 4 hours 20 minutes.

Solve $13x^2-620x-3000=0$ to find the speed Antonio travels for the first 100 km of the journey.
You must show all your working and give your answer correct to 1 decimal place.

$\frac{620\pm\sqrt{(-620)^2-4(13)(-3000)}}{2(13)}$
or equivalent quadratic formula
M1 for $(-620)^2-4\times13\times(-3000)$ oe
x=52.1 to 52.2 final answer
Reject negative speed

[Maximum number: 4]

Rearrange the formula to make m the subject.
$R=\frac{2(m-k)}{m}$ m=

$m=\frac{2k}{2-R}$
or $m=\frac{-2k}{R-2}$
final answer
M1 for clearing fractions
M1 for expanding brackets (or $\div2$ )
M1 for collecting terms in m on one side and terms not in m on the other
M1 for dividing by a bracket, maximum 3 if final answer incorrect

CAIE IGCSE Mathematics Extended E2.5 Equations Topic Practice

Covered syllabus points

CAIE IGCSE Mathematics Extended E2.5 Equations Topic Practice

Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9