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GCSE Exam-Style Questions Algebra 8-9: A Comprehensive Guide with Answers, Exercises of Mathematics

Cornwall CollegeMathematics

A collection of gcse exam-style questions focused on algebra, specifically targeting students aiming for grades 8 and 9. It covers a range of topics, including completing the square, finding inverses of functions, solving trigonometric equations, and proving algebraic properties. Each question is accompanied by a detailed solution, offering a valuable resource for students to practice and understand key concepts in algebra.

Typology: Exercises

2024/2025

Available from 01/11/2025

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yacine-ouali 🇬🇧

91 documents

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1 of 5
x + 2
GCSE Exam-Style Questions Algebra 8-9
1.
Write the following expressions in the form a(x + b)
2
+ c
a.
2x2 + 6x + 10
b.
i. 3x
2
+ 3x
–
4
ii. Hence, state the minimum value of 3x2 + 3x – 4
2.
If g(x) = x
+ 1
, work out the value of g
-1
(2).
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download GCSE Exam-Style Questions Algebra 8-9: A Comprehensive Guide with Answers and more Exercises Mathematics in PDF only on Docsity!

1 of 5

x + 2

GCSE Exam-Style Questions Algebra 8-

1. Write the following expressions in the form a ( x + b )^2 + c

a. 2 x^2 + 6 x + 10

b. i. 3 x^2 + 3 x – 4

ii. Hence, state the minimum value of 3 x^2 + 3 x – 4

2. If g ( x ) = x^ + 1^ , work out the value of g -1(2).

GCSE Exam-Style Questions Algebra 8 - 9 2 of 5

y

2

1

(^0) x

  • 1
  • 2

2 x − 1

3. A function f ( x ) is given by f ( x ) = 3 x + 2

Explain why x ≠ 0.

4. The graph y = sin x for 0° ≤ x ≤ 360° is shown below.

y

Write down the equations of the graphs shown: a.

2

1

(^0) 360° x

  • 1
  • 2

GCSE Exam-Style Questions Algebra 8 - 9 4 of 5

  1. Write down the equation of a circle with a centre at the origin and a radius of 4.

9. A circle with equation x^2 + y^2 = 5 has a tangent at the point (1, 2). Find the equation of the

tangent to the circle at this point.

  1. Bella cycled to her friend’s house, stayed there for a while and then cycled to the park. After a short time, she returned home. distance (km)

10 9 8 7 6 5 4 3 2 1

10:00 11:00 12:00 13:00 time

a. Did Bella travel faster before or after she stopped at her friend’s house? Explain your answer.

GCSE Exam-Style Questions Algebra 8 - 9 5 of 5

b. Work out Bella’s average speed for the total time she was cycling.

11. The velocity-time graph shows the velocity (metres per second) of a car after t seconds.

speed (m/s) 30

20

10

1 2 3 4 5 6 7 8 9 10 time (seconds)

a. Work out an estimate for the acceleration of the car at t = 3 seconds.

b. Estimate the total distance travelled by the car in the first 4 seconds.

GCSE Exam-Style Questions Algebra 8 - 9 Answers 2 of 5

2 x − 1

1

3. A function f ( x ) is given by f ( x ) = 3 x + 2

Explain why x ≠ 0.

Remember that whenever you have a fraction, the denominator cannot be zero. Here,

if x = 0.5 then 2 × 0.5 – 1 = 0

We cannot divide by zero so the function cannot be defined when x = 0.

4. The graph y = sin x for 0° ≤ x ≤ 360° is shown below.

y

Write down the equations of the graphs shown: a.

The graph has been translated by the vector ( 0 ). This is the equation y = sin x + 1

2

1

(^0) 360° x

  • 1
  • 2

y

2

1

(^0) x

  • 1
  • 2

GCSE Exam-Style Questions Algebra 8 - 9 Answers 3 of 5

0

3

3

b. y

The graph has been translated by the vector ( 90 ). This is the equation y = sin( x − 90)

5. Solve the equation 3sin x = 2 for 0° ≤ x ≤ 360°

Divide by 3:

sin x = 2

Now, find sin- x = sin-1( 2 ) = 41.810…

Remember that the other values for the function sin x can be found by subtracting this

from 180°. 180 − 41.810... = 138.189…

Correct to 1 decimal place, x = 41.8° and x = 138.2°

  1. Prove, algebraically, that the sum of three consecutive integers is a multiple of 3.

Let the consecutive integers be n , n + 1 and n + 2

Their sum is n + n + 1 + n + 2 = 3 n + 3

We can now factorise this to get 3( n + 1) which is, by definition, a multiple of 3.

  1. Prove, algebraically, that the difference between two consecutive cube numbers has a remainder of 1 when divided by 3.

Let the cube numbers be n^3 and ( n + 1)^3. Their difference is ( n + 1)^3 − n^3

Expand ( n + 1)^3 by writing it as ( n + 1)( n + 1)( n + 1) and expand the brackets:

( n + 1)( n + 1)( n + 1) = ( n^2 + 2 n + 1)( n + 1)

= n^3 + 3 n^2 + 3 n + 1

Find the difference between the cube numbers:

( n + 1)^3 − n^3 = n^3 + 3 n^2 + 3 n + 1 − n^3

= 3 n^2 + 3 n + 1

2

1

(^0) 360° x

  • 1
  • 2

GCSE Exam-Style Questions Algebra 8 - 9 Answers 5 of 5

time

5 − 1

2

a. Did Bella travel faster before or after she stopped at her friend’s house? Explain your answer. Before as the line is steeper. b. Work out Bella’s average speed for the total time she was cycling. speed = distance distance = 5 + 5 + 10 = 20km

time spent cycling = 0.75 + 1 + 0.75 = 2.5 hours Speed = 20 = 8km/hour

11. The velocity-time graph shows the velocity (metres per second) of a car after t seconds.

velocity (m/s) 30

20

10

1 2 3 4 5 6 7 8 9 10 time (seconds)

a. Work out an estimate for the acceleration of the car at t = 3 seconds.

Draw a tangent to the curve and pick two points this line passes through. (1, 10) and (5, 24) might be two such points (though yours will be different depending on your tangent). The acceleration is then given by the gradient. acceleration = 24 −^10 = 3.5m/s^2

b. Estimate the total distance travelled by the car in the first 4 seconds. The distance is the area under the curve. To find this, we draw a line from (0, 0) to (4, 20) and estimate using the formula for area of a triangle. distance = 1 × 4 × 20 = 40 metres