Aiming for Grade 6-8: Study Programme

Transcription

1 Aiming for Grade 6-8: Study Programme Week A1: Similar Triangles Triangle ABC is similar to triangle PQR. Angle ABC = angle PQR. Angle ACB = angle PRQ. Calculate the length of: i PQ ii AC Week A: Enlargement Shape A is shown in the diagram. Shape A is enlarged to obtain the shape B. a Write down the scale factor of the enlargement. b Complete the drawing of shape B on the diagram.

2 Week B: Trigonometry A: The diagram shows a house and a garage on level ground. A ladder is placed with one end at the bottom of the house wall. The top of the ladder touches the top of the garage wall. The distance between the garage wall and the house is 1.4 m. The angle the ladder makes with the ground is 62º. (a) Calculate the height of the garage wall. Give your answer correct to 3 significant figures. A ladder of length 3.5 m is then placed against the house wall. The bottom of this ladder rests against the bottom of the garage wall. (b) Calculate the angle that this ladder makes with the ground. Give your answer correct to 1 decimal place.

3 B: In the diagram AB = 17.9 m, BD = 8.2 m, angle CBD = 37º and angle BDC = 90º. ADC is a straight line. (a) Calculate the length of DC. Give your answer, in metres, correct to 3 significant figures. (b) Calculate the size of angle DAB. Give your answer correct to 1 decimal place. Week C: Pythagoras Theorem A: Calculate the length of AB. Give your answer correct to 1 decimal place.

4 B: The diagram is part of a map showing the positions of three Nigerian towns. Kaduna is due North of Aba. (a) Calculate the direct distance between Lagos and Kaduna. Give your answer to the nearest kilometre. (b) Calculate the distance between Kaduna and Aba. Give your answer to the nearest kilometre. Week D: Probability 1. A fair coin is tossed and a fair dice is thrown. One possible outcome is (Heads, 4). List all the possible outcomes. 2. A fair dice is to be thrown. (a) Write down the probability of the dice landing on (i) a six (ii) an even number A second dice is to be thrown. The probability that this dice will land on each of the numbers 1 to 6 is given in the table. Number Probability x The dice is to be thrown once. (b) Calculate the value of x. (c) Calculate the probability that the dice will land on a number higher than 3. The dice is thrown 1000 times. (d) Estimate the number of times the dice is likely to land on a six.

5 3. There are 4 red balls, 5 blue balls and 3 green balls in a bag. A ball is to be chose at random and replaced. A second ball is then to be chosen at random. (a) Complete the tree diagram below. (b) Use the tree diagram to calculate the probability that both balls chosen will be (i) red, (ii) the same colour. (c) Calculate the probability that exactly one of the balls chosen will be red. Week E: Circle Geometry 1. Two tangents are drawn from a point T to a circle centre O. They meet the circle at points A and B. Angle AOB is equal to 128º. In this question you MUST give reasons for your answers. Work out the size of the angles i APB, ii BAO, iii ABT.

6 2. The diagram shows a circle centre O. PQ and QR are tangents to the circle at P and Q respectively. S is a point on the circle. Angle PSR = 70º. PS = SR. a. i) Calculate the size of angle PQR. ii) State the reason for your answer. b. i) Calculate the size of angle SPO. ii) Explain why PQRS cannot be a cyclic quadrilateral. Week F: Rearranging Formulae A: Rearrange each of the following to make the letter indicated in (brackets) the subject 1. C = D (D) 2. y = mx + c (c) 5. N = 2 l (l) 6. x = 2a + b 2 (b) 3. v = r 2 h (h) 4. s = ut + 1 / 2 at 2 (a) 7. Make f the subject of the formula u v f B. The velocity of a particle is given by the formula: v 2 = u 2 + 2as 1. Calculate the value of the velocity v when u = -5, a = 2. Rearrange the formula to make u the subject. 2 3 and s = 5.67.

7 Week G: Locus The scale diagram shows the position of a radio mast, M. 1 cm on the diagram represents 20 km. M Signals from the radio mast can be received up to a distance of 100 km. a Shade the region on the scale diagram in which signals from the radio mast can be received. The distance of a helicopter from the radio mast is 70 km correct to the nearest kilometre. b Write down i the maximum distance the helicopter could be from the radio mast, ii the minimum distance the helicopter could be from the radio mast.

8 Week H: Pie Charts & Scatter Graphs Information about oil was recorded each year for 12 years. The scatter graph shows the amount of oil produced (in billions of barrels) and the average price of oil (in per barrel). (a) Draw a line of best fit on the scatter graph. In another year the amount of oil produced was 10.4 billion barrels. (b) Use your line of best fit to estimate the average price of oil per barrel in that year. Week K: The Sine Rule & The Cosine Rule A straight road UW has been constructed to by-pass a village V. The original straight roads UV and VW are 4 km and 5 km in length respectively. V lies on a bearing of 052º from U. W lies on a bearing of 078º from V. The average speed on the route UVW, through the village is 30 kilometres per hour. The average speed on the by-pass route UW is 65 kilometres per hour. Calculate the time saved by using the by-pass route UV. Give your answer to the nearest minute.

9 Week L: Area of a Triangle using Area = 1 / 2 bcsina Triangle ABC is isosceles. AB = AC = 12 cm and Angle ABC is 55º Calculate the area of the triangle. Give your answer to 3 significant figures. Week M: Grouped Data & the Estimate of the Mean Andrew did a survey at the seaside for his science coursework. He measured the lengths of 55 pieces of seaweed. The results of the survey are shown in the table. Length of seaweed Frequency Mid-Point f x x (L cm) f x 0 < L < L < L < L < L < L < L Totals - Andrew needs to calculate an estimate for the mean length of the pieces of seaweed. a. Work out an estimate for the mean length of the piece of seaweed. Give your answer correct to 1 decimal place. b. Write down the class interval that contains the median length of a piece of seaweed.

10 Week N: Direct & Inverse Proportion 1. y is directly proportional to x. If y = 5 when x is 25, find y when x = The Area of a circle is proportional to the square of the radius. If the Area is 113cm 2 when the radius is 6cm find: (a) the Area of a circle with radius 5cm (b) the radius of a circle with Area 29cm 2 Give your answers to 1 decimal place. 3. The Temperature from a factory furnace varies inversely as the square of the Distance from the furnace. The Temperature 2 metres from the furnace is 50ºC. Calculate the Temperature 3.5 metres from the furnace. Give your answer to 2 decimal places Week O: Simultaneous Equations with Quadratics Solve the following equations simultaneously to find the values of both x and y a + 3b = 7 a 3b = 25 2x y = 7 x y = 3 7x + 5y = 66 3x 4y = 16

11 x 2 + y 2 = 5 x + y = 3 xy = 5 x + y = 6 Week P: Plotting Graphs of Trigonometric Functions Here is a graph of y = sin x On the axes above, draw the graphs of: (a) y = sinx + 2 (b) y = sinx 1 (c) y = 3sinx (d) y = cosx

12 Week Q: Transformation of Graphs A: The diagram represents the graph of a function of x Draw and label on the same axes the graphs of i y = f(-x) ii y = f(x + 2) B: a. i Factorise x 2-4x a. ii Solve x 2-4x - 12 = 0. The diagram shows a sketch of the graph of y = x 2-4x The curve cuts the x-axis at the points A and B. b. Write down the coordinates of A and B. c. Sketch on the axes above the graph of y = f(x - 2).

13 Week R: Plans & Elevations 1. Draw the plan view, front elevation and side elevation of a 3D capital letter: (a) T (b) F Week S: Surds & Indices Simplify the following Surds and rationalise the denominator if you have to x x / x / x / 2 Put a tick in the box underneath those numbers that are rational. 2 / / 17 Put a tick in the box underneath the rational numbers

14 Week T: Linear Equations & Quadratics Equations Factorising & Expanding Brackets A: Factorise completely put the brackets back in. 1. 2p st + 8tu - 2tv 15x + 3x 2 B: Expand and simplify the following - get rid of the brackets first. 1. 2a(4 - a) 2. (x + 2)(x + 3) 3. (2c + 3)(c - 4) C: Simplify the following expression 1 3 ( x 2) ( x 1) Solving Quadratics by Factorising Example: x 2 + 5x + 6 = 0 factorised is (x + 2)(x + 3) = 0 such that x = -2 and x = x x + 18 = 0 3. x 2-7x + 10 = 0 2. x 2 + 5x - 6 = 0 4. x 2-8x - 20 = 0

15 Solving Quadratic Equations by Completing the Square Write in the form (x + p) 2 c = 0 and then solve to find x to 2.d.p Example: for x 2 + 8x + 1 = 0 we write, (x + 4) = 0, giving (x + 4) 2 15 = 0 The 4 inside the brackets is from half of 8 and the 16 is from x 2 + 6x + 9 = 0 3. x 2 - x - 3 = 0 2. x 2-6x + 7 = 0 4. x 2 + 3x - 1 = 0 Solving Quadratic Equations using The Formula Solve the following Quadratic Equations using The Formula: x = b b 2 4ac 2a Example: for 3x 2-2x + 5 = 0, a = 3, b = -2 and c = 5 1. x 2 + 6x + 3 = 0 2. x 2 +3x - 1 = x 2 + 8x - 1 = 0

16 Solving Problems using Quadratics 1. In the diagram, each side of the square ABCD is (3 + x) cm. a. Write down an expression in terms of x for the area, in cm 2, of the square ABCD. The actual area of the square ABCD is 10cm 2. b. Show that x 2 + 6x = 1 2. In triangle ABC, AB = 5 cm, AC = x cm, BC = 2x cm and angle BAC = 60. a Show that 3x 2 + 5x - 25 = 0. b Solve the equation 3x 2 + 5x - 25 = 0. Give your answers correct to 3 significant figures. D is the point on AC such that angle ADB = 104. c Calculate the length of BD.

17 Solving Quadratic Equations using Graphs 1. Here is a graph of y = x 2 4x + 3 for 3 x 3. Use the graph to find the solutions to the following equations. Note: there are two solutions to each equation. 1. x 2 4x + 3 = 0 x = and x = 2. x 2 4x + 3 = 2 x = and x = 3. x 2 4x + 2 = 0 x = and x = 4. x 2 4x - 3 = 1 x = and x =

18 2. a. On the grid below, draw the graph of y = 5 + 2x - x 2 for -2 < x < 4. x y b. Use your graph to solve the following equations: (i) 5 + 2x - x 2 = 0 (ii) 5 + 2x - x 2 = 2 c. By drawing a suitable straight line on your graph, find the approximate solutions to x + 4 = 5 + 2x - x 2

19 Mix-Bag I know the square numbers to 15 15? Ok, what is: (a) 14 2 (b) 13 x 13 = (c) Square of 7 = (d) 11 2 = (e) (6 2 ) = (f) 121 = and what about cube numbers? (a) 2 3 = (b) Cube of 4 = (c) = (d) 4 3 = Estimate the answer to the following: (a) ( ) (b) ( ) (c) Equations and Inequalties a 3y + 7 = 28 b 2(3p + 2) = 19 c 3t - 4 = 5t 10 d. On the number line below show the solution to this inequality -7 2x - 3 < 3 e. Solve the inequality 7y > 2y 3 Number Sequences: Write down the next two terms and work out the n th -term for these sequences: i. 2, 4, 6, 8, 10 (g) 2, 4, 8, 16, 32 (b) 1, 3, 5, 7, 9 (h) 1, 3, 6, 10, 15 (c) 1, 1, 2, 3, 5, 8 (d) 5, 7, 9, 11. (i) -8, -2, 4, 10 (e) 1, 4, 9, 16, 25 (j) 2, 8, 18, 32, 50 (f) 1, 8, 27, 64, 125 Dimensions of Formula 1. Draw a circle around each of the expressions which can be used to calculate an area. r( 2 ) 4 2 r h r(r + 4h) rh 4 4r r 3 ( r 2 h) 3r 3 h r 2 ( h r)

20 2. The diagram below represents a solid shape. From the expressions below, choose the one that represents the volume of the solid shape. and 1 are numbers which have no dimensions. 3 a, b and h are lengths. 1 3 (b2 - ab + a 2 ), 1 3 h(b2 + ab + a 2 ), 1 3 h2 (b 2 - a 2 ), 1 3 (a2 + b 2 ), 1 3 h2 (b 2 - ab + a 2 ). Write down the correct expression. Areas & Volumes 1. The diagram shows a cylinder. 2. The diagram shows a triangular prism. The height of the cylinder is 26.3 cm. The diameter of the base of the cylinder is 8.6 cm. Calculate the volume of the cylinder. Give your answer correct to 3 significant figures. BC = 4 cm, CF = 12 cm and angle ABC = 90º. The volume of the triangular prism is 84 cm 3. Work out the length of the side AB of the prism.