1. GCSE Past Paper Questions & Solutions
Dr J Frost
Last modified: 18th April 2014

2. Index More Topics >> Click to visit section.
Straight Line Equations
Algebra
Factorising, Simplifying & Solving
Non-Right Angled Triangles
Congruent Triangle Proofs
Algebraic Proofs & Algebraic Geometry
Right-Angled Triangles
Number
Includes bounds, direct/indirect proportion, standard form, %s, etc.
Probability
Circle Theorems
Volumes and Surface Area
Functions and Graph Transformations
More Topics >>
Mark Scheme Notes:
M1 Method mark. cao Correct Answer Only oe Or equivalent
A1 Accuracy mark. C ‘Communication Mark’ (used for *-ed questions)
B1 ‘Independent mark’. Sort of a ‘miscellaneous’ mark, often used when an explanation is required.

3. Index Click to visit section. Vectors Loci & Constructions
Angles & Bearings
Mark Scheme Notes:
M1 Method mark. cao Correct Answer Only oe Or equivalent
A1 Accuracy mark. C ‘Communication Mark’ (used for *-ed questions)
B1 ‘Independent mark’. Sort of a ‘miscellaneous’ mark, often used when an explanation is required.

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Straight Lines
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Gradient = -9/4 ?
2y = -x + 1, so y = -0.5x + 0.5
So gradient is -0.5 ?
y = 3x - 4 ? ?
y = -3x + 16 ?
y = 4x + 3
y = 4x – 5 ?

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Straight Lines
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y = 4x - 11 ?
y = (1/3)x + 2 ?
y = -(1/2)x + c (where c can be anything) ?
y = -(1/5)x – 1 ?

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Straight Lines
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4. Finding where a line intercepts the 𝒙 or 𝒚 axis.
At what point does 𝑦=2𝑥+3 intercept:
The 𝑦-axis: (𝟎,𝟑)
The 𝑥-axis: 𝟎=𝟐𝒙+𝟑 𝒔𝒐 𝒙=− 𝟑 𝟐 → − 𝟑 𝟐 ,𝟎
At what point does 𝑦=3𝑥−2 intercept:
The 𝑦-axis: 𝟎,−𝟐
The 𝑥-axis: 𝟐 𝟑 ,𝟎 ? ? ? ?
5. Sketching a line with a given equation.
Sketch the line 𝑦=2𝑥−4
between 𝑥=−1 and 𝑥=3 6 4 2 -2 -4 -6
Tips:
Be careful about the axis.
You only need to plot two points. E.g. use 𝑥=−1 and 𝑥=3, work out the 𝑦 values for these, then join the two points with a line.
Reveal

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-9/5 ?

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Reveal ? 5
1.5

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Reveal

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2/3 ?
Gradient of 2y = 10 – 3x is -3/2
2/3 × -3/2 = -1
therefore perpendicular ?

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1.5 ?
Reveal
-3/2 ?

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-0.6, 5.5 ?
-1.4, 6.4
x = 0.2, y = -3.8
x = 5.8, y = 1.8 ?

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Straight Lines
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y = -(1/2)x – 1
-1/2 ?

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Straight Lines
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[June 2010 NonCalc]

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Algebra
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Algebra
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x10 ?
m12 ? ?

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Algebra
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Algebra
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Algebra
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12.5 ?
4m2 – 1 ?

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Algebra
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Algebra
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9e5f6 ? 3 ?

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Algebra
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Algebra
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Algebra
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3(2 + 3x) ?
(y + 4)(y – 4) ?
(2p – 5)(p + 2) ?

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Algebra
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Algebra
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4 , 5 ?

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Algebra
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Algebra
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Algebra
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-3 < x ≤ 4 ?
t > 7/2 ?

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Algebra
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2x(x – 2y) ?
(p – 4)(p – 2) ?
x + 2 ?
6a5b2 ?

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Algebra
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Algebra
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Algebra
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x + 4
2x – 3 ?
7x – 2
(x+2)(x-2) ?

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Algebra
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(x + p)(x + q) ?
(m + 2)(m – 2) ?
x + 10
(x – 4)(x + 3) ?

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Algebra
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x2 + 3 = 7x
x2 – 7x + 3 = 0
x = (7 ±37) / 2 ?

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Algebra
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-2, -1, 0, 1, 2, 3, 4 ?
x > 5/2 ?

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Algebra
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16n12

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Algebra
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(2x – 1)(x – 4) ? ?
(2x – 1)(x – 4) = (2x – 1)2
x – 4 = 2x – 1
x = -3

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Algebra
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-160 ? ?

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Algebra
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4(3n + 1) ?
3(n + 4) ?
2n + 1 ?

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Algebra
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Algebra
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Algebra
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4x-1 or 4/x ?

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Algebra
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8x2 + 6xy – 20y2 ?
x + 10 ?
x – 5
x + 2 ? 3
-11 ? ?

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Algebra
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Non-Right Angled Triangles
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Non-Right Angled Triangles
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Non-Right Angled Triangles
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60. Non-Right Angled Triangles?

61. Non-Right Angled Triangles? ?

62. Non-Right Angled Triangles? ?

63. Non-Right Angled Triangles? ?

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Congruent Triangles
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65. Congruent Triangles ? ?

66. Congruent Triangles ?

67. Congruent Triangles ?

68. Congruent Triangles ? ?

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Algebraic Proofs and Geometry
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21. ?

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Algebraic Proofs and Geometry
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5x2

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Right Angled Triangles
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Right Angled Triangles
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80.1

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Right Angled Triangles
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11.5
47.2

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3.52

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Right Angled Triangles
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33.7
9.44

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Right Angled Triangles
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13.86

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Right Angled Triangles
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Right Angled Triangles
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48.2

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Right Angled Triangles
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Right Angled Triangles
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Number
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9.56 x 107 ? ?

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Number
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Number
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Remember, you choose the greatest degree of accuracy such that the two bounds are the same.

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Number
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3.76 x 10-4 ?
0.5 x 109
= 5 x 108

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Number
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16 x 10-5
= 1.6 x 10-4

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Number
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2.7 x 1014 ?
2.4 x 1016 ?
6.4 x 108 ?

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Number
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Number
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8.25 x 107 ?
1.456 x 10-15 ?

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Number
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Probability
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Probability
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Probability
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Probability
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Probability
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222
380

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Probability
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110

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Probability
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Probability
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Circle Theorems
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Angle DAB = 180 – 103 = 77 (opposite angles of cyclic quadrilateral)
Angle DBA = 39 (Alternate Segment Theorem)
Angle ADB = 180 – 77 – 39 = 64

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Circle Theorems
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116 ?
Angle OCB = (180 – 116)/2 = 32
Angle OCA = 74 – 32 = 42

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Circle Theorems
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Angle BOA = 152
Angle APB = 360 – 152 – 90 – 90
= 28

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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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236 ?

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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Volumes and Surface Area
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Yes
Yes ?
8 x 1003 =

166. On to questions >>>
Functions and Graph Transformations
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General Tips:
When asked to sketch a transformed graph, e.g. f(x + 3), pick key points on the original graph to transform first (e.g. ones that go exactly through grid points, or y-intercepts, etc.) then join up with a line. This will ensure you draw it accurately.
Remember that changes inside the function brackets affect the x-axis and do the opposite of what you expect.
Learn the shape of y = sin(x), y = cos(x) and y = tan(x). In particular, learn that coordinates for which the graphs cross the x-axis, and the maximum/minimum points.
On to questions >>>

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Functions and Graph Transformations
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Reveal
To get the curve perfectly mirrored, mirror points that go through squares first, i.e. (-4, 4), (-3, 1), (-1, 1), (0.4), then join up with a line.

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Functions and Graph Transformations
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y = f(x – 6)

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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Functions and Graph Transformations
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Functions and Graph Transformations
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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Reveal

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Functions and Graph Transformations
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Reveal ?
x = -1.6, y = 2.6
x = 2.6, -1.6

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Functions and Graph Transformations
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f(x – 5) ?

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Reveal

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Vectors
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Vectors
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Vectors
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Vectors
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Vectors
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Vectors
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Vectors
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Vectors
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Loci & Constructions
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#1: Use compass the get some fixed distance across the lines AC and AB.
Reveal
#2: Find perpendicular bisector of these two points.

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Loci & Constructions
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Reveal

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Loci & Constructions
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Reveal Step #1
#1: Use two arcs with radius the width of the line to form an equilateral triangle (only one side needed).
This gives you an angle of 60.
Reveal Step #2
#2: Find the angle bisector of these two lines in the usual way, in order to find the angle half of 60.

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Loci & Constructions
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Loci & Constructions
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Reveal
Angle bisector of angle DAB
5cm
3cm

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Angles & Bearings
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40 ?
Angle PQT = 70 (angles on straight line add to 180). Angle PTQ = 70 (base angles of isosceles triangle are equal)
Angle TPQ = 40 (angles in triangle add to 180) ?

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Angles & Bearings
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360 ÷ 5 = 72

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Angles & Bearings
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112 ? ?

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Angles & Bearings
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55 ? ?
Corresponding angles.

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Angles & Bearings
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360 ÷ 30 = 12 ?

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Angles & Bearings
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Angles & Bearings
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360 – 90 – 120 = 150 ?

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Angles & Bearings
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42 ?

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Angles & Bearings
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Angle PBA = 180 – (x + 50) – (2x – 10) = 140 – 3x
y = 180 – (140 – 3x) = 3x + 40 ? ?
3x + 40 = 145
x = 35 ?
= 85

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Angles & Bearings
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3x – 15 = 2x + 24
x = 39

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Angles & Bearings
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Interior angle of hexagon = 180 – (360 / 6) = 120
Interior angle of octagon = 180 – (360 / 8) = 135
x = 360 – 120 – 135 = 105

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Angles & Bearings
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Angles & Bearings
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Angles & Bearings
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150 ?

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Angles & Bearings
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Angles & Bearings
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Angles & Bearings
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Angles & Bearings
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