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E2.5 Quadratic Equations
Question Paper
Level
Subject
Exam Board
Level
Topic
Sub-Topic
Booklet
IGCSE
Maths (0580)
Cambridge International Examinations (CIE)
Extended
E2. Algebra and Graphs
E2.5 Quadratic Equations
Question Paper
108 minutes
Time Allowed:
Score:
/ 90
Percentage:
/100
Grade Boundaries:
A*
A
B
C
D
E
U
>85%
75%
60%
45%
35%
25%
<25%
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1.
(a)
x cm
NOT TO
SCALE
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, by factorising, the equation
x 2 - 40x + A = 0.
x = ..................... or x = ....................[3]
(iii)
When A = 200, solve, by using the quadratic formula, the equation x 2 - 40x + A = 0.
Show all your working and give your answers correct to 2 decimal places.
x = ..................... or x = ....................[4]
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(b) A car completes a 200 km journey with an average speed of x km/h.
The car completes the return journey of 200 km with an average speed of (x + 10) km/h.
(i)
Show that the difference between the time taken for each of the two journeys is
2000
hours.
x (x + 10)
[3]
(ii)
Find the difference between the time taken for each of the two journeys when x = 80.
Give your answer in minutes and seconds.
.................... min .................... s [3]
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2.
y = x2 + 7x – 5 can be written in the form y = (x + a)2 + b.
Find the value of a and the value of b.
a = .................................................
b = ................................................. [3]
3
(a) Solve the inequality.
5x – 3 > 9
................................................... [2]
(b) Factorise completely.
(i)
xy – 18 + 3y – 6x
................................................... [2]
(ii)
8x 2 - 72y 2
................................................... [3]
(c) Make r the subject of the formula.
p+5 =
1 - 2r
r
r = .................................................. [4]
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4
Alfonso runs 10 km at an average speed of x km/h.
The next day he runs 12 km at an average speed of (x – 1) km/h.
The time taken for the 10 km run is 30 minutes less than the time taken for the 12 km run.
(a) (i) Write down an equation in x and show that it simplifies to x2 – 5x – 20 = 0.
[4]
(ii)
Use the quadratic formula to solve the equation x2 – 5x – 20 = 0.
Show your working and give your answers correct to 2 decimal places.
x = ....................... or x = ....................... [4]
(iii)
Find the time that Alfonso takes to complete the 12 km run.
Give your answer in hours and minutes correct to the nearest minute.
................ hours ................ minutes [2]
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(b) A cheetah runs for 60 seconds.
The diagram shows the speed-time graph.
Speed
(m/s)
NOT TO
SCALE
25
0
10
55
60
Time (seconds)
(i)
Work out the acceleration of the cheetah during the first 10 seconds.
........................................... m/s2 [1]
(ii)
Calculate the distance travelled by the cheetah.
............................................... m [3]
5
Solve (x – 7)(x + 4) = 0.
x= ................................. or x= .................................[1]
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6.
Solve the equation 3x2 - 11x + 4 = 0.
Show all your working and give your answers correct to 2 decimal places.
x=............................ or x= ............................[4]
7
Expand and simplify.
x (2x + 3) + 5(x – 7)
Answer ................................................ [2]
__________________________________________________________________________________________
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8
f(x) = x2 + 4x − 6
(a) f(x) can be written in the form (x + m)2 + n.
Find the value of m and the value of n.
Answer(a) m = ................................................
n = ................................................ [2]
(b) Use your answer to part (a) to find the positive solution to x2 + 4x – 6 = 0.
Answer(b) x = ................................................ [2]
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9
Factorise completely.
9x2 – 6x
Answer ................................................ [2]
10
Factorise
2x2 – 5x – 3.
Answer ................................................ [2]
11
Solve the equation.
2x2 + x – 2 = 0
Show your working and give your answers correct to 2 decimal places.
Answer x = ......................... or x = ......................... [4]
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12
(a) Jamil, Kiera and Luther collect badges.
Jamil has x badges.
Kiera has 12 badges more than Jamil.
Luther has 3 times as many badges as Kiera.
Altogether they have 123 badges.
Form an equation and solve it to find the value of x.
Answer(a) x = ................................................. [3]
(b) Find the integer values of t which satisfy the inequalities.
4t + 7 < 39  7t + 2
Answer(b) ................................................. [3]
(c) Solve the following equations.
(i)
21 - x
=4
x+3
Answer(c)(i) x = ................................................. [3]
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(ii) 3x2 + 7x – 5 = 0
Show all your working and give your answers correct to 2 decimal places.
Answer(c)(ii) x = ......................... or x = ......................... [4]
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13
(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) 4x2 – 25y2
Answer(b)(ii) ................................................ [2]
(c) Simplify.
3
16 - 2
c 4m
9x
Answer(c) ................................................ [2]
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(d) n is an integer.
(i) Explain why 2n – 1 is an odd number.
Answer(d)(i) ................................................................................................................................
..................................................................................................................................................... [1]
(ii) Write down, in terms of n, the next odd number after 2n – 1.
Answer(d)(ii) ................................................ [1]
(iii) Show that the difference between the squares of two consecutive odd numbers is a multiple of 8.
Answer(d)(iii)
[3]