2 4 5 2 · pdf filethis document covers all the specific areas of the aqa gcse further maths...

page. 1

AQA GCSE Further Maths Topic Areas

This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need

practice, and stating your level of confidence

Area – a star shows a topic is both higher and

further The further grade is in

brackets

Examples Confidence (1 – beginner,

5 – star)

Simplifying expressions * (B) (B) (A)

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 6𝑎2𝑏3𝑐

3𝑎𝑏4𝑐3

Factorise the following expression 3𝑎2𝑏 + 6𝑎𝑏2

Simplify the expression 𝑥

4𝑡−

2𝑦

5𝑡+

𝑧

2𝑡

Solving linear equations * (B) (A) (A)

𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3(3𝑥 − 17) = 2(𝑥 − 1)

𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛1

2(𝑥 + 8) = 2𝑥 +

1

3(4𝑥 − 5)

The length, l metres, of a field is 80m greater than its width. The perimeter is 600m

(i) Write the information in the form of an equation for l. (ii) Solve the equation and so find the area of the field.

Expanding brackets * (B) (A) (A)

𝐸𝑥𝑝𝑎𝑛𝑑 (𝑥 + 5)(2𝑥 − 3) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (𝑥3 − 𝑥2 + 𝑥 − 2)𝑏𝑦 (𝑥2 + 2𝑥 − 1) 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 (𝑥2 − 1)(𝑥 + 1) − (𝑥2 + 1)(𝑥 − 1)

page. 2



brackets


5 – star)

Manipulating surds * Simplifying expressions (A) (A*)

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 √8

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 √32 − √18

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 √3 × √6

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 (4 + √3)(4 − √3)

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛2

√3

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑏𝑦 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑖𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟3√2

4 − √5

Factorising * (A) (B) (A) (A) (A)

𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒 𝑥𝑎 + 𝑥𝑏 + 𝑦𝑎 + 𝑦𝑏 𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒 𝑥2 + 6𝑥 + 8 𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒 𝑥2 − 16 𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒 4𝑥2 − 9𝑦2 𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒 2𝑥2 − 11𝑥 + 15

Rearranging formula * (C)

𝑀𝑎𝑘𝑒 𝑟 𝑡ℎ𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝐶 = 2𝜋𝑟

page. 3



brackets


5 – star)

(B) (B)

𝑀𝑎𝑘𝑒 𝑥 𝑡ℎ𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ℎ = √(𝑥2 + 𝑦2)

𝑀𝑎𝑘𝑒 𝑥 𝑡ℎ𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑦 =𝑥 + 2

1 + 3𝑥

*Simplifying algebraic fractions (A) (A) (A)

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦𝑎2 − 𝑎 − 6

𝑎2 − 8𝑎 + 15

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 4𝑛2 − 9

𝑛 + 1÷

2𝑛 + 3

𝑛2 − 1

𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦2

𝑥 + 1+

5

𝑥 − 1

*Solving equations involving fractions (C) (B)

𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔𝑥 + 2

6=

𝑥 − 6

2

𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 2

3𝑥 − 1+

1

𝑥 + 8=

1

2

Quadratic Identities (A*) (A*)

Work out the values of p and q such that 𝑥2 − 6𝑥 + 2 ≡ (𝑥 − 𝑝)2 + 𝑞 𝑊𝑜𝑟𝑘 𝑜𝑢𝑡 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 3𝑥2 + 5𝑥 − 1 ≡ 𝑎(𝑥 + 𝑏) + 𝑐

Function Notation 𝑓(𝑥) = 2𝑥 − 1, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑓(5)

𝑔(𝑥) =𝑥 + 6

2𝑥, 𝑠𝑜𝑙𝑣𝑒 𝑔(𝑥) = 3 𝑎𝑛𝑑 𝑠𝑜𝑙𝑣𝑒 𝑔(2𝑥) = 1

page. 4



brackets


5 – star)

Domain and range of a function

𝑓(𝑥) = 6 − 4𝑥 𝑎𝑛𝑑 − 2 ≤ 𝑥 ≤ 3. 𝑊𝑜𝑟𝑘 𝑜𝑢𝑡 𝑡ℎ𝑒 𝑟𝑎𝑛𝑔𝑒 𝑜𝑓 𝑓(𝑥)

Write down the domain and range for f(x)

*Gradients of a straight line and quadratic functions (C) (B)

Find the equation of the line joining (-1, 4) to (2, 3) For the graph 𝑦 = 𝑥2 + 6𝑥 + 11, state:

(i) The vertex (minimum or maximum point) (ii) The equation of the line of symmetry (iii) The co-ordinates of the point where the graph intersects the y-axis

page. 5



brackets


5 – star)

Graphs of functions in parts (A) (A)

Draw the graph of y=g(x) where: 𝑔(𝑥) = 𝑥 + 3; −3 ≤ 𝑥 < 0 𝑔(𝑥) = 3 − 𝑥; 0 ≤ 𝑥 ≤ 3

Here is the graph of y=f(x). (i) Define f(x), stating clearly the domain for each part (ii) State the range of f(x) and (iii) Solve f(x)=5

*Factorising quadratic equations - Two brackets (B) *Coefficients greater than 1

𝑆𝑜𝑙𝑣𝑒 𝑥2 + 3𝑥 − 18 = 0. 𝐻𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑡𝑒 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠. 𝑆𝑜𝑙𝑣𝑒 3𝑥2 + 8𝑥 − 3 = 0. 𝐻𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑡𝑒 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠.

page. 6



brackets


5 – star)

*Completing the Square (A*) *Quadratic formula (A)

𝑆𝑜𝑙𝑣𝑒 𝑥2 − 8𝑥 + 3 = 0. 𝐻𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑡𝑒 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠. 𝑆𝑜𝑙𝑣𝑒 2𝑥2 + 𝑥 − 8 = 0. 𝐻𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑡𝑒 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠.

*Simultaneous Equations (B) (A)

Vince has £2.20 to spend on fruit for a picnic and can buy either five apples and four pears or two apples and six pairs. Write this information as a pair of simultaneous equations Solve your equations to find the cost of each type of fruit Solve 𝑥 + 2𝑦 = −3

𝑥2 − 2𝑥 + 3𝑦2 = 11

Factor theorem (A*) Given that 𝑥3 + 3𝑥2 − 𝑥 − 3 Show that (x+1) is a factor of f(x) Factorise f(x)

* Inequalities – Linear (B)

Solve 2y + 6 < 5y + 12 𝑆𝑜𝑙𝑣𝑒 5 < 3𝑥 − 1 ≤ 17

page. 7



brackets


5 – star)

Quadratic inequalities (A)

𝑆𝑜𝑙𝑣𝑒 𝑥2 − 2𝑥 − 3 ≥ 0

Index Laws * (A) (B)

𝑆𝑜𝑙𝑣𝑒 𝑥32 = 8

Write as a single power of x: 𝑥2 × 𝑥3 ÷ 𝑥8

Sequences * - Linear (C) Quadratic (A*)

Work out the nth term of the sequence: 4, 10, 16, 22, 26, … Work out the nth term of the sequence 4, 13, 26, 43, 64, ….

Limits of a sequence (A*) The nth term of a sequence is 2𝑛−1

3𝑛+2. Prove that the limiting sequence as 𝑛 → ∞ 𝑖𝑠

2

3

* Distance between 2 points (C)

What is the distance between the following two points (6, 5) and (3, 1)?

page. 8



brackets


5 – star)

Midpoint of a line segment (C)

What is the midpoint PQ with the two points P(6, 5) and Q(3, 1)?

Gradient of a line between 2 points (C)

What is the gradient between the two points P(6, 5) and Q(3, 1)?

Equation of a straight line * (C)

Points P and Q have coordinates P(3, 1) and Q(5, 7) find the equation of the straight line

Perpendicular lines * (A) Points P and Q have coordinates P(3, 1) and Q(5, 7) find the equation of the perpendicular bisector of this straight line

Equation of a circle (A) (A) (A)

Write the equation of the circle with centre (1, -2) and radius 3 The line segment AB is the diameter of a circle. A is the point (1, -4) and B is the point (5, 2). Work out the equation of the circle.

The circle (𝑥 − 2)2 + (𝑦 + 3)2 = 50 intersects the line y = x – 5 at the points P and Q. Work out the coordinates of P and Q.

page. 9



brackets


5 – star)

Pythagoras Theorem * (C)

Trigonometry * (B) Trig ratios

page. 10



brackets


5 – star)

Circle geometry and circle theorems (A) A (12, 6) and B (14, 4) are two points on a circle, centre C (20, 12).

Work out the coordinates of the midpoint M, of AB.

Show that the length CM = 27

Work out the radius of the circle.

Not drawn

accurately

O

x

y

C (20, 12)

A (12, 6)

B (14, 4)

M

page. 11



brackets


5 – star)

Circle theorems * (B)

Find the angles n, m and k

Find the angles n and m

page. 12



brackets


5 – star)

Sine and cosine rule * (B) Find side marked x

Find the side marked x

Special Triangles (30◦, 60◦, 90◦) (45◦, 45◦, 90◦) (A*) (A*)

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4√2 cm and

one of the angles is 45°.

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 cm and

one of the angles is 30°.

Lines and planes in three dimensions (A)

A

C

B

x m

8m

61⁰

7m

page. 13



brackets


5 – star)

Trigonometrical functions for angles of any size (A*) Graphs

Draw the graph y = sinx

Draw the graph y = cosx

Area of a triangle * (C) Find the area of the triangle ABC

Solutions of trigonometrical equations

Solve the equation 𝑐𝑜𝑠𝜃 = 0.5 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 0° ≤ 𝜃 ≤ 360° 𝑡𝑜 1 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑝𝑙𝑎𝑐𝑒

page. 14



brackets


5 – star)

(A*) (A*)

Solve the equation 3𝑠𝑖𝑛𝜃 = 2 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 0° ≤ 𝜃 ≤ 360° 𝑡𝑜 1 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑝𝑙𝑎𝑐𝑒

Solve the equation 2 sin2 𝜃 + 𝑠𝑖𝑛𝜃 − 1 = 0 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 0° ≤ 𝜃 ≤ 360° 𝑡𝑜 1 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑝𝑙𝑎𝑐𝑒

Trigonometrical identities (A*) (A*)

Solve the equation 2 cos2 𝜃 + 𝑠𝑖𝑛𝜃 − 1 = 0 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 0° ≤ 𝜃 ≤ 360° 𝑡𝑜 1 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑝𝑙𝑎𝑐𝑒

Use the identity 𝑡𝑎𝑛𝜃 =𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃to solve the equation 2𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃 = 0 𝑓𝑜𝑟 0° ≤ 𝜃 ≤ 360°

Differentiation (A*) (A*)

Differentiate the following:

𝑦 = 𝑥5 + 12𝑥3 + 3𝑥

𝑦 = 3𝑥5 + 2

Work out th gradient of the curve 𝑦 = 𝑥3(𝑥 − 2) at the point (3, 27)

page. 15



brackets


5 – star)

(A*) (A*) (A*) (A*)

For the function 𝑦 = 5𝑥 − 𝑥2 on which there is the point P (3, 6). Find:

i) The gradient function 𝑑𝑦

𝑑𝑥

ii) The gradient at point P

iii) The equation at the tangent at point P

iv) The equation of the normal at point P

Work out the values of x for which 𝑦 = 𝑥2 + 4 is increasing

Work out the values of x for which 𝑦 = 2𝑥3 − 3𝑥2 − 72𝑥 is decreasing

For the curve 𝑦 = 𝑥3 − 12𝑥 + 3

i) Find 𝑑𝑦

𝑑𝑥and the values of x for which

𝑑𝑦

𝑑𝑥= 0

ii) State the type of turning points and their x-coordinates

iii) Find the corresponding y values

Matrices (A*) (A*)

Calculate [0 −1

−2 3] [

2 4−3 2

]

Work out the value of p [4 21 −1

] ⌊𝑝3

⌋ = (2

−4)

page. 16



brackets


5 – star)

(A*) (A*)

Work out the matrix that represents the following transformations

i) Rotation through 270◦ (anti-clockwise) about the origin

ii) Enlargement, centre the origin, scale factor 3

Point P (3, -2) is transformed by the matrix (1 −10 1

) followed by a further transformation by matrix

(0 21 0

).

i) Work out the matrix for the combined transformation

ii) Work out the coordinates of the image point of P.

2 4 5 2 · pdf filethis document covers all the specific areas of the aqa gcse further maths...

Documents