AQA GCSE Further Maths Topic Areas

Transcription

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 The grade is in Simplifying expressions * Simplify the following expression 6a2 b 3 c 3ab 4 c 3 Factorise the following expression 3a 2 b + 6ab 2 Simplify the expression x 4t 2y 5t + z 2t Solving linear equations * Solve the equation 3(3x 17) = 2(x 1) Solve the equation 1 2 (x + 8) = 2x + 1 (4x 5) 3 Expanding * 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. Expand (x + 5)(2x 3) Multiply (x 3 x 2 + x 2)by (x 2 + 2x 1) Simplify (x 2 1)(x + 1) (x 2 + 1)(x 1) page. 1

2 The grade is in Manipulating surds * Simplifying expressions Simplify the expression 8 Simplify the expression Simplify the expression 3 6 Simplify the expression (4 + 3)(4 3) Simplify the expression 2 3 Simplify the expression by rationalising the denominator Factorising * Factorise xa + xb + ya + yb Factorise x 2 + 6x + 8 Factorise x 2 16 Factorise 4x 2 9y 2 Factorise 2x 2 11x + 15 Rearranging formula * (C) Make r the subject of the equation C = 2πr page. 2

3 The grade is in *Simplifying algebraic fractions Make x the subject of the equation h = (x 2 + y 2 ) Make x the subject of the equation y = x x Simplify a2 a 6 a 2 8a + 15 Simplify 4n2 9 n + 1 2n + 3 n 2 1 Simplify 2 x x 1 *Solving equations involving fractions (C) Solve the following x = x 6 2 Solve the following 2 3x x + 8 = 1 2 Quadratic Identities Function Notation Work out the values of p and q such that x 2 6x + 2 (x p) 2 + q Work out the values of a, b, and c such that 3x 2 + 5x 1 a(x + b) + c f(x) = 2x 1, what is f(5) g(x) = x + 6, solve g(x) = 3 and solve g(2x) = 1 2x page. 3

4 The grade is in Domain and range of a function f(x) = 6 4x and 2 x 3. Work out the range of f(x) *Gradients of a straight line and quadratic functions (C) Write down the domain and range for f(x) Find the equation of the line joining (-1, 4) to (2, 3) For the graph y = x 2 + 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. 4

5 The grade is in Graphs of functions in parts 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 *Coefficients greater than 1 Solve x 2 + 3x 18 = 0. Hence state the coordinates where the function crosses the x axis. Solve 3x 2 + 8x 3 = 0. Hence state the coordinates where the function crosses the x axis. page. 5

6 The grade is in *Completing the Square Solve x 2 8x + 3 = 0. Hence state the coordinates where the function crosses the x axis. *Quadratic formula Solve 2x 2 + x 8 = 0. Hence state the coordinates where the function crosses the x axis. *Simultaneous Equations 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 x 2 2x + 3y 2 = 11 Factor theorem Given that x 3 + 3x 2 x 3 Show that (x+1) is a factor of f(x) Factorise f(x) * Inequalities Linear Solve 2y + 6 < 5y + 12 Solve 5 < 3x 1 17 page. 6

7 The grade is in Quadratic inequalities Solve x 2 2x 3 0 Index Laws * Solve x 3 2 = 8 Write as a single power of x: x 2 x 3 x 8 Sequences * - Linear (C) Work out the nth term of the sequence: 4, 10, 16, 22, 26, Quadratic Work out the nth term of the sequence 4, 13, 26, 43, 64,. Limits of a sequence 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. 7

8 The grade is in Midpoint of a line segment (C) Gradient of a line between 2 points (C) Equation of a straight line * (C) What is the midpoint PQ with the two points P(6, 5) and Q(3, 1)? What is the gradient between the two points P(6, 5) and Q(3, 1)? Points P and Q have coordinates P(3, 1) and Q(5, 7) find the equation of the straight line Perpendicular lines * 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 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. 8

9 The grade is in Pythagoras Theorem * (C) Trigonometry * Trig ratios page. 9

10 The grade is in Circle geometry and circle theorems A (12, 6) and B (14, 4) are two points on a circle, centre C (20, 12). y Not drawn accurately C (20, 12) A (12, 6) M B (14, 4) O x Work out the coordinates of the midpoint M, of AB. Show that the length CM = 7 2 Work out the radius of the circle. page. 10

11 The grade is in Circle theorems * Find the angles n, m and k Find the angles n and m page. 11

12 The grade is in Sine and cosine rule * Find side marked x Find the side marked x B x m A 61⁰ 8m 7m C Special Triangles (30, 60, 90 ) (45, 45, 90 ) 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. Lines and planes in three dimensions 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. page. 12

13 The grade is in Trigonometrical functions for angles of any size Draw the graph y = sinx Draw the graph y = cosx Graphs 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. 13

14 The grade is in Solve the equation 3sinθ = 2 in the interval 0 θ 360 to 1 decimal place Solve the equation 2 sin 2 θ + sinθ 1 = 0 in the interval 0 θ 360 to 1 decimal place Trigonometrical identities Solve the equation 2 cos 2 θ + sinθ 1 = 0 in the interval 0 θ 360 to 1 decimal place Use the identity tanθ = sinθ to solve the equation 2sinθ + cosθ = 0 for 0 θ 360 cosθ Differentiation Differentiate the following: y = x x 3 + 3x y = 3x Work out th gradient of the curve y = x 3 (x 2) at the point (3, 27) page. 14

15 The grade is in For the function y = 5x x 2 on which there is the point P (3, 6). Find: i) The gradient function dy ii) The gradient at point P iii) The equation at the tangent at point P iv) The equation of the normal at point P dx Work out the values of x for which y = x is increasing Work out the values of x for which y = 2x 3 3x 2 72x is decreasing For the curve y = x 3 12x + 3 i) Find dy dy and the values of x for which dx dx ii) State the type of turning points and their x-coordinates iii) Find the corresponding y values Matrices Calculate [ ] [ ] Work out the value of p [ ] p 3 = ( 2 4 ) page. 15

16 The grade is in 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 1 ) followed by a transformation by matrix 0 1 ( ). i) Work out the matrix for the combined transformation ii) Work out the coordinates of the image point of P. page. 16