1. ζ Year 9 – End of Year Revision Dr Frost

2. Percentages Be careful: Are you trying to find the new value or the old value? In the first case, you multiply, in the second case, you divide. Percentage change is based on the old value. A jumper is bought in for £30 and marked up by 40%. What is it sold for? Answer: 30 x 1.4 = £42 After one year the value of a care fell by 20% to £9600. What was its original value? Answer: £12000 I put £15,000 into a savings account. It accrues 2.6% interest. What is in my account in one year’s time? Answer: £15390 Lucy made 20% profit on the picture frame she sold at £35. What did she buy it in for? Answer: £29.17 ? ? ? ?

3. Percentages The interest rate for a savings account is 2.5% p.a. with compound interest. The principal is £1500. How much do I have in 10 years time? Answer: £1500 x 1.025 10 = £1920.13 My Bentley depreciates in value 10% each year. It is bought new for £150,000. How much is it worth in 5 years time? Answer: £150,000 x 0.9 5 = £88573.50 ? ?

4. Compound Measures A cat travels at 15km/s. It races around a 50km track. How much time did it take him? Answer = 3.33s The density of a hamster is 1.3kg/m 3. Its volume is 0.03m 3. What is the hamster’s mass? Answer = 0.039kg ? ?

5. Graphs Match the graphs with the equations, and identify what type of equation it is. 1 2 3 4 5 6 7 8 9 10 11 y = -2x 3 + x 2 + 6x y = 4 x y = 2x - 3 y = x 2 + x – 2 y = 5 – 2x 2 y = 2x 3 y = 5 – x y = x 3 – 7x + 6 y = -3x 3 6 Cubic 11 Exponential 9 Straight Line 1 Quadratic 2 Quadratic 8 Reciprocal 5 Cubic 10 Straight Line 3 Cubic 4 Cubic 7 Reciprocal ? ? ? ? ? ? ? ? ? ? ?

6. Graphs y = x 3 – 2x 2 - 5x + 6 x-3-201234 y-240860-4018 When sketching, ensure you sketch a curvy line (i.e. don’t join up your points with lines), or you’ll lose a mark. ????????

7. Changing the Subject Change the subject of the formula to the indicated letter. ? ? ? ? ? ? ? ? ? ? (b)

8. Changing the Subject ? ? ? ? ? ? ? ? ? ?

9. The following require you to factorise at some point. Make a the subject of the formula: n = _3a_ a+1 a = n 2 -Pn P-1 ? ? ??

10. Simultaneous Equations You can either use elimination or substitution. 3x + 2y = 10 5x – 2y = 14 3x + 2y = 10 5x – 2y = 14 3x + 2y = 4 4x + 3y = 7 3x + 2y = 4 4x + 3y = 7 x = 3, y = 0.5x = -2, y = 5 ??

11. Probability Question: Give there’s 5 red balls and 2 blue balls. What’s the probability that after removing two balls from the bag, we have a red ball and a blue ball? R B R B R B 5757 2727 4646 2626 5656 1616 ? ? ? ? ? ? Answer = 10 21 ?

12. Probability What’s the probability that when I roll 10 dice, I see the same number on every die? What’s the probability that when I roll 10 dice, the total of the dice is 10? ? Difficult: What’s the probability that when I roll 3 dice, I see exactly two sixes. ? Total outcomes Matching outcomes ?

13. 6 2 9 3 Probability If I have two dice, one numbered 1, 2, 3 and the other numbered 2, 3, 4, what’s the probability the sum is at least 5? +234 1345 2456 3567 Second Die First Die p(sum ≥ 5) = = ? ?

14. Sequences Determine the formula for the following sequences. 5, 8, 11, 14, 17,...10, 8, 6, 4, 2, 0, -2,... 3, 9, 17, 27, 39,... U n = n 2 + 3n - 1 U n = 3n + 2U n = 12 – 2n 1, 3, 6, 10, 15,... U n = 0.5n(n+1) ?? ??

15. Expanding brackets Expand the following. (x+1)(x-2) = x 2 – x – 2 (x-4)(x-8) = x 2 – 12x + 32 (x+1)(y+1) = xy + x + y + 1 (x 2 +1)(y 2 -1) = x 2 y 2 – x 2 + y 2 – 1 (2x+1)(2x-1) = 4x 2 – 1 (x + 1)(x + y + 1) = x 2 + xy + 2x + y + 1 x(y-x)-y(x-y) = y 2 – x 2 ? ? This is known as the: difference of two squares ? ? ? ? ? ?

16. Factorisation Factorise the following x 2 + 7x + 12 = (x + 4)(x + 3) x 2 + 2x – 3 = (x – 1)(x + 3) x 2 – 10x + 24 = (x – 4)(x – 6) 2x 2 – 5x – 12= (2x + 3)(x – 4) 12x 2 + 5x – 3= (4x + 3)(3x – 1) x 2 – 9 = (x + 3)(x – 3) 4 – y 2 = (2 + y)(2 – y) x 3 – x = x(x + 1)(x – 1) 16x 2 y 2 – 9z 4 = (4xy + 3z 2 )(4xy – 3z 2 ) x 4 + 2x 2 – 143 = (x 2 + 13)(x 2 – 11) ? ? ? ? ? ? ? ? ? ?

17. x-5-4-3-2-10123456 y 4 3 2 1 -2 -3 -4 Object Enlarge the shape by a scale factor of 2 about the point (0,-2) Enlargement Image

18. x-5-4-3-2-10123456 y 4 3 2 1 -2 -3 -4 Object Enlarge the shape by a scale factor of -1 about the point (0,2) Enlargement

19. x-5-4-3-2-10123456 y 4 3 2 1 -2 -3 -4 Object Enlarge the shape by a scale factor of -0.5 about the point (0,2) Enlargement

20. Trigonometry 60 ° x 12 30 ° 4 x x = 13.96 x = 3.46 65 ° x 15 x = 6.99 ?? ?

21. 2 3 θ 1 3 1 1 θ 6 θ 8 a b c d θ Trigonometry θ = 33.69 ° θ = 70.53 ° θ = 45 ° θ = 48.59 ° ? ? ? ?

22. Trigonometry What is the cosine of the angle between the internal diagonal of a cube and the bottom face of the cube? Answer = √ 2 √ 3 ?

23. Solving Equations Solve the following equations for x. x(2x + 1)(x – 2) = 0x = 0 or -0.5 or 2 x 2 = 4x = 2 or -2 x 2 = 3xx = 0 or 3 x 2 + 5x – 6 = 0x = -6 or 1 x 3 = xx = -1, 0 or 1 x 2 + 32 = 12xx = 4 or 8 25x 2 – 4 = 0x = 2/5 or -2/5 ? ? ? ? ? ? ?

24. 2x + 2 x 3x - 2 Determine x Answer: By Pythagoras, x 2 + (2x+2) 2 = (3x-2) 2 Expanding and simplifying, we get 4x 2 – 20x = 0 Solving, x = 5 (we reject the 0 solution). Solving Equations ?

25. Similarity 5 3 10 x x = 16 ?

26. Similarity A square is inscribed in a right-angled triangle with length 4 and height 3. Find the width of the square. 3 4 Length of square = 12 7 ?

27. Loci A B 3km 4km A Spotted Studdert Sheep is known to be within 3km of A and 4km of B. What region could the sheep be in?

28. Loci A B 3km 4km Now the sheep is also known to be of equal distance from A and B. Where can it be?

29. Loci A B 3km 4km Now the sheep is within 3km of A, but at least 4km away from B. Where could it be?

30. Loci I’m equidistance from two lines AB and AC. Where could I be? A B C

31. Loci I’m the indicated distance away from the walls of a building. Where could I be? Circular corners. Straight corners.

32. Loci My sheep is attached to a fixed point A on a square building, of 10m x 10m, by a piece of rope 20m in length. Both the sheep and rope are fire resistant. What region can he reach? 10m 20m A

33. Dimensional Analysis (all variables are lengths) ExpressionLengthAreaVolumeNone of these 2rh πr + 4h (r+h) 2 3b b 3 + rh πr 2 (h + r) bhr_ (b+h) Click your choice.  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ? ? ? ? ? ? ?

34. Ratio My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Sanjotitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What’s the new ratio of black to yellow fish? Answer = 5 : 1 ? Method 1: Suppose a full tank has 12 fish. Then 9 fish are black and 3 yellow. The plague leaves 6 black fish and 2 white. Then if we fill up the rest of the tank with black fish, we have 10 black fish and 2 yellow. This ratio is 5:1.

35. Proportion x16824 y10515 Given that y is proportional to x, find the missing values. ? ?

36. Inverse proportion x16500.25 L52.8340 Given that L is inversely proportional to √ x, fill in the missing values in this table. ? ?

37. Inequalities Solve the following. 2x > x - 6 -x + 1 ≤ 6 x > - 6 x ≥ -5 ? ? 1 ≤ 2x + 3 < 9-1 ≤ x < 3 ? ?

38. Inequalities on a number line. 2 ≤ x < 4x 4 0 1 2 3 4 5 ? -1 0 1 2 3 4 ?

39. Inequalities on a number line. 2 ≤ x < 5 x 4 0 1 2 3 4 5 Draw the range of x on the number line given that both of these inequalities hold. ?

40. -10 -8 -6 -4 -2 2 4 6 8 10 8 6 4 2 -2 -4 -6 -4 < y ≤ - 2

41. -10 -8 -6 -4 -2 2 4 6 8 10 8 6 4 2 -2 -4 -6 y ≤ x + 1 and x ≤ 6 and y > 2

42. Inequalities When all of x, y and z are negative, or one of x, y and z are negative. ?

43. Arcs and Sectors 5 Sector area = 10.91 Arc length = 4.36 Area = 20 Radius = 4.12 2.1cm Sector area = 4.04cm 2 Arc length = 3.85cm ? ? ? ? ? 50 ° 105 ° 135 ° (Hint: Plug values into your formula and rearrange)

44. The shape PQR is a minor sector. The area of a sector is 100cm 2. The length of the arc QR is 20cm. a)Determine the length PQ. Answer: 10cm b)Determine the angle QPR Answer: 114.6 ° P Q R ? ? Arcs and Sectors

45. Volume of a prism 10cm 4cm 6cm 8cm Volume = 400cm 3 ? 1m 5m 3m 5m 6m Volume = 17m 2 x 6 = 102m 3 ?

46. Surface Area 8m 4m 2m Surface Area = 112m 3 ?