THOMAS WHITHAM SIXTH FORM

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1 THOMAS WHITHAM SIXTH FORM Algebra Foundation & Higher Tier Units & thomaswhitham.pbworks.com

2 Algebra () Collection of like terms. Simplif each of the following epressions a) a a a b) m m m c) d) d d 6d e) n n n f) p 9p p g) t t t h) j j i) 6 k k 8k k j) r r 0r r. Simplif each of the following epressions a) a b a b b) c d c d c) d) e e 6 e) w 8 w f) t s t s g) 6i j i j h) 8p q p 6q i) 0h 6g g h j) u v u v. Simplif a) 6 b) a a 6a c) m n m n d) p q 0p q e) 6v u v u f) c c c c g) 9t s s t h) f e f e i) r 6s r s j) k h k 8h. Simplif a) b) c) d) 6 e) p p 9 p p f) g) h) r r r r i) e 8e e j) h h 6 h h

3 Algebra() Solving simple equations. Solve each of the following equations a) f) 6a 8 b) 9 c) p g) h 6 0 h) q d) 8 i) b e) m j) 9 9 k) d l) t 6 m) k 8 n) 6 o) n p) r 9 q) e r) w 9 s) 8 j 8 t) 6. Solve each of the following equations a) b) p p 9 c) 8m m 8 d) f f e) u u 8 f) 9 g) 8 h) 6 i) r r j) 0e e k) 8 l) 6 m) 8 6 n) 6 o) b 9 b p) 6t t q) m 8 m r) k 8k s) u 6 u t) 6 h h. Solve the following equations a) b) p c) 6 a d) b 6 e) m f) 9 g) h) 0 i) p 8 j) k) 9 l) 6 m) r 9 n) r v 8 o) v p) q) 6 8 b 8 f r) g 6 s) 0 r t) 9 t 8

4 Algebra () Removal of brackets. Remove the brackets for each of the following: a) b) c) 6m d) e) 6 p f) g) 0 h) a b i) j) m n p k) r t l) m) n) a a o) p) m m q) r) b b s) t) aa b c u) pq p v) 6rr s t w) m m n ) e e f ) 8 z) h i j. Remove the brackets and simplif each of the following: a) h) b) i) 6 c) j) d) 6 k) 6 e) 0 l) 8 f) g) m) n) o) p) q) r) s) t)

5 Algebra () Factorisation Factorise each of the following () a 9 () m 6 () 8p () 0 () r 6 (6) a b () f 8g (8) s t (9) 6 z (0) 8i j () m 6n () u 8v w () 0m n p () 9e f g () 6a 0b c (6) () (8) (9) 9 (0) m n mn 0mn

6 Algebra () Factorisation. Factorise each of the following: a. b. 6 8 c. 8 d. 6 e. 6 8 f. g. 8 h. i. j. 0 k. l. m. 6 n. 6 8 o. p. 6 6 q. 0 r.. Factorise each of the following: a) b) c) d) 8 6 e) 6 f) 6 6 g) 9 9 h) 8 i) j). Solve each of the following equations a. 6 0 b. 0 c. 6 0 d. 8 0 e. 0. (a) Factorise (b) Hence, or otherwise solve the equation 0. (a) Factorise (b) Hence, or otherwise solve the equation 0 6. Solve each of the following equations: a) 0 b) c) 0 S. J. Cooper d) e) 8 0 f) 0 g) 6 0 h) 0 i) 9 9 0

7 . (a) Factorise 8 (b) Hence, or otherwise solve the equation (a) Factorise 8 (b) Hence, or otherwise solve the equation (a) Factorise 0 (b) Hence, or otherwise solve the equation Solve each of the following equations: a) b) c) d) e)

8 Algebra (6). Factorise each of the following: a) b) c) d) e) f) a 6b g) h) i) j) 6 k 9 0u v c 08d. Complete the square for each of the following: a) 8 f) b) 6 g) a a c) 6 h) d) 0 i) 6 e) 8 j) 8. Solve each of the following equations b completing the square, leaving our answer as a surd: a) 9 0 f) 0 0 b) 0 0 g) 8 0 c) 0 h) c 0c 0 d) b b 8 0 i) 0 e) e 6e 0 j) 0 0. Using the formula, giving answers to decimal places where appropriate, solve each of the following: a) 0 f) 0 b) 0 g) a a 0 c) 9 0 h) b b 6 0 d) 0 i) r 0r 6 0 e) 6 0 j) u 9u 0

9 Algebra () Substitution into formulae Eercise Answer each of the following questions without the use of a calculator.. If D S T, find the value of D when S and T.. Using the formula A l b, find (i) A when l and b 9 (ii) b when l 6 and A 0. If S n, find the value of S when (i) n (ii) n 6 (iii) n 0. Given P l b, find (i) P when l and b (ii) b when l and P 8. Using the formula v u at find v when (i) u, a 0 and t (ii) u 0, a and t 6 6. Given M D find (i) D when M 0 and V 8 V (ii) V when M 9 and D. When 0, find when (i) (ii) (iii) 8 8. Evaluate n when (i) n (ii) n 8 (iii) n h 9. Find the value of when (i) 8 h (ii) h 0. Find the value of 90 f when (i) f (ii) f 8

10 Eercise Answer each of the following questions with the use of a calculator. u v. Find the value of a when a, given that u. 6, v. 6 and t t. Using the formula E mc, find the value of E given that m and c. Using the formula S u v t, find S when u, v and t.. Using the formula a z find the value of z when 9., a 9 and b. b. If m T find the value of T when 0, m 9. 6 and l 0 l 6. If C mv mu find C when m, v 9 and u. Find the value of T when n n n T and (i) n (ii) n 6 8. Where P find P when (i) and (ii) 8 and 9. Using the formula v u as obtain the value of v when u 0, a. and s 0. Work out the value S at ut when. u, a 9. 8 and t 0

11 Algebra (8) Forming Equations. I think of a number double it and subtract three. If I then have five what was number was I thinking of?. Najid thinks of a number, multiplies it b five and subtracts his answer is. What number did Najid start with?. Susan was told to think of a number. Then she was asked to multipl her number b four and add si. When asked what was her answer she gave. What number did Susan think off?. The perimeter of the rectangle below is 6cm work out the value of a. a a. Given that the perimeter of the triangle below is cm work out the value of If the perimeter of the regular pentagon below has the same perimeter as the rectangle find the value of a. a a. John is ears older than Janet. However in 0 ears time John will be double Janet s age. If John is ears old. Find. 8. Given that the area of the rectangle opposite is 6cm, find.

12 9. 8 Given that the area of the triangle opposite is 8m, find p. p 0. Given that the area of the 0cm find the two possible values for. ( ). Given that the area of the rectangle below is cm find the one possible values for. ( 6). Given that the volume of the cuboid is 0cm, find the one 8 possible values for.

13 Algebra (9) Simultaneous Equations I Eercise Solve each of the following sets of simultaneous equations. Remember to show our working.. b a b a d c d c u t u t. f e f e s r s r q p q p 8. 6 k h k h h g h g b a b a. 9 8 v u v u. 6 6 j i j i. 8 9 q p q p. k k j k 6. 0 e d e d. 8 9 n m n m Eercise. Two numbers have a sum of 8 and a difference of 6. Find these two numbers.. Find two numbers which add to give and have a difference of 8.. Three times one number added to twice another gives. If the difference between the two numbers is find the two numbers.. Alan and Sarah have saved between them. If Alan has more than Sarah how much does each person have?. A bag contains 0 counters red and blue. If there are five more blue counters than red how man red counters are there?

14 6. The cost of si pens and four pencils in the print room is.6, whereas the cost of five pens and two pencils is 9pence. Work out the cost of a pen and a pencil.. For a recent concert Mohammed sold 0 tickets raising 9 for school funds. If the tickets cost per child and per adult, how man tickets did he sell to adults? 8. The cost of three pears and seven apples is.. If the difference between the two prices is pence find the cost of each item. 9. For a recent holida the cost of two adults and three children was quoted for 0. Whereas for two adults and si children the price was 60. What is the price per child and per adult? 0. Angela has 0 spending mone. In a big department store Angela can b four lipsticks and three different nail polishes for 9.. Or alternativel she could bu one lipstick and si nail polishes for 9. What is the cost of each item?

15 Algebra (0) Simultaneous Equations II Eercise Solve each of the following sets of simultaneous equations. Remember to show our working.. 6. r s 8 rs. ab ab9. p q pq. cd cd 8. 6 h k 0 hk 6. tu tu 9. g h 6 gh. e f 0 e f Eercise. (a) Draw the graph of for values of between and (b) On the same set of aes draw the straight line with equation 6 (c) Hence solve the set of simultaneous equations and 6. (a) Draw the graph of for values of between and (b) On the same set of aes draw the straight line with equation (c) Hence solve the set of simultaneous equations and. B drawing suitable graphs solve the set of simultaneous equations given b and

16 Algebra () Simultaneous Equations III Solve each of the following sets of simultaneous equations. Remember to show our working

17 Algebra () Drawing a straight line graph. For each of the following (i) Cop and complete the table (ii) draw the graph for the straight line. (a) 0 (b) 0 (c) 0 (d) 0 (e) 0. Draw the graphs for each of the following: (a) (b) (c) (d) 6 (e) 6 (f) 8 0 (g) 8 (h) (i) 8 (j) (k) (l)

18 . a) Draw on the same set of aes the graphs of and b) Hence solve the simultaneous equations and. a) Draw on the same set of aes the graphs of and b) Hence solve the simultaneous equations and. a) Draw on the same set of aes the graphs of and 6 b) Hence solve the simultaneous equations and 6 6. The graph drawn opposite represents Work out the coordinates of A and B A B 0. The graph drawn below represents 9 Work out the coordinates of P and Q P 0 Q 8. The graph drawn below represents 6 Work out the coordinates of E and F 0 E F

19 Algebra () Gradients and Equations of lines. The graph below represents the graph of 6 a) Find the coordinates of the points A and B A B 0 b) Hence find the gradient of the line.. a) Find the gradient of a line which passes through points,6 and, b) Find the gradient of a line which passes through the points, and,. Write down the gradient and value of the intercept for each of the following graphs. a) f) b) c) 8 d) e) g) h) 6 i) j). Work out the gradient and the value for the intercept of the graph with equation. A straight line passes through the point, and has gradient equal to. a) On a set of aes draw this graph. b) Write down an epression for the gradient of the line 6. A straight line passes the points, and,8 a) Find the gradient of the line b) Find an epression for the equation of the line

20 . A straight line passes the points,9 and, a) Find the gradient of the line b) Find an epression for the equation of the line 8. A straight line passes the points, and 6, a) Find the gradient of the line b) Find an epression for the equation of the line 9. Find an epression for the equation of the straight line drawn below., 0, Find the equation of the line drawn below., 0,

21 Algebra () Drawing quadratics curves. (a) Cop and complete the table below for the graph of for values of from to. 0 (b) Hence draw the graph of.. (a) Cop and complete the table below for the graph of for values of from to. 0 (b) Hence draw the graph of.. (a) Cop and complete the table below for the graph of for values of from to. 0 + (b) Hence draw the graph of.. (a) Cop and complete the table below for the graph of for values of from to. 0 (b) Hence draw the graph of.

22 . (a) Cop and complete the table below for the graph of for values of from to. 0 (b) Hence draw the graph of. 6. (a) Cop and complete the table below for the graph of 6 for from to (b) Draw the graph of 6. (c) Hence state the values at which 0.. (a) Cop and complete the table below for the graph of for from to. 0 + (b) Draw the graph of. 8. (a) Cop and complete the table below for the graph of for from to (b) Draw the graph of. (c) Hence state the coordinates where the curve meets the -ais.

23 Algebra () Quadratic equations. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and. a) Cop and complete the table below for the graph of 6 0 b) Draw the graph of 6 for values of between and

24 6. a) Cop and complete the table below for the graph of 6 0 b) Draw the graph of 6 for values of between and c) Use our graph to determine values of when d) State the minimum value of on the graph of 6. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and c) Use our graph to determine values of when d) State the minimum value of on the graph of 8. a) Cop and complete the table below for the graph of 0 b) Draw the graph of for values of between and c) On the same set of aes draw the graph of d) Using the graph solve the set of simultaneous equation and e) Hence show that the two solutions are the same for the equation 0 9. (a) Draw the graph of for values of between and (d) On the same set of aes draw the straight line with equation 6 (e) Hence solve the set of simultaneous equations and 6 0. (a) Draw the graph of for values of between and (d) On the same set of aes draw the straight line with equation (e) Hence solve the set of simultaneous equations and

25 Algebra (6) Drawing graphs. a) Cop and complete the table below for the graph of 0 b) Hence draw the graph of in the range 0. a) Cop and complete the table below for the graph of b) Hence draw the graph of 0 in the range 0. 0 c) Eplain wh it is not possible to find a value for when 0. a) Cop and complete the table below for the graph of 0 0 b) Hence draw the graph of in the range. a) Cop and complete the table below for the graph of b) Hence draw the graph of in the range a) Cop and complete the table below for the graph of sin b) Hence draw the graph of sin in the range 0 60

26 6. Pair of the following equations with the graphs drawn below a) b) c) d) e) A B C D E. Draw a rough sketch to represent each of the following graphs a) b) c) d) 0 e) f) 6 g)

27 Algebra () Inequalities. Solve each of the following inequalities a) e) b) 0 f) 9 c) 9 g) 8 d) h) i) j). Solve each of the following inequalities: a) e) 8 b) r 8 r f) w 8w 6 c) t t g) f 9 f d) u u h) 0k 9k i) h 9 h j) v 9 6v k) 0z 8z l) 9 9. Solve the following inequalities: a) 8 b) c) 9 d) 9 0 e) 6 f) g) 8 0 h) 6 i) 0 90 j) 0. Solve the following inequalities: a) b) c) d) e)

28 Algebra (8) Inequalities II. Write down the inequalit shaded for each of the following: a) b) 0 c) d) 0 e) f)

29 g) h) 0 i) j) 0 k) l)

30 . For each of the following draw the graph which best each inequalit. (a) (b) (c) (d) (e) (f) (g) 0 (h) 6 (i) (j). Write down an equation which fits each of the following shaded regions a) b) 0 c) d) 0 e) f)

31 g) h) 0 i) j) 0 k) l)

32 . Describe the shaded region in the diagram drawn below.. (a) Write down the equation of the three lines drawn around the shaded region below (b) Write down three inequalities which best describe the shaded region. 6. (a) On one set of aes draw the graphs of (i) (ii) (iii) (b) Shade in the region defined b the set of inequalities,,

33 . (a) On one set of aes draw the graphs of (i) (ii) (iii) (b) Shade in the region defined b the set of inequalities,,

34 Algebra (9) Rearranging formulae Make the subject for each of the following. m c a. a c. t. t d e. w f g k M 8. b a b c d. t 6. a b c Make the subject for each of the following. t r m.. b c f a a t c d. w 8g. w u v. V 8. G T E 9. t 0. A B A 6. g

35 Algebra (0) Inde Notation. Simplif each of the following: a) 8 e) e e e i) a a a a b) r r f) h h h j) c) b b g) n n n d) i 6 i i 6 h). Simplif each of the following: a) a b a b e) p 9 q p q i) e f e f 6 b) m n m n f) j) u u u v c) e f f e g) h 8 k 6 h k 9 d) c d 6 c d h) s t s 6 t 0. Simplif each of the following: a) b) c) d) 9 h t h t m w m w 6 e) f) g) h) 9 8 j j 0 d 6 8 d 6 8 t i) t j) 8. Simplif each of the following: a) 6 b) t c) d) a e) b f) g) c h) p q. Simplif each of the following epressions: a) 8 r r b) 9 8 c) f f d) f 6 6 e) u v f) u v v g) r s r s e f e h) f e f

36 Algebra () Continuing a sequence. Write down the net two terms for each of the following sequences and a rule in words. (a), 9, 6, (b), 6,, 6 (c) 8,, 0, 6 (d), 8,, (e) 6,,, (f), 8,, 6 (g) 9,,, (h),,, (i),,, (j), 9,,. Write down the net two terms for each of the following and give a rule for continuing the sequence. (a) 0,,, (b),,, 9 (c) 99, 88,, 66 (d),,, (e) 6, 9,, (f) 80,, 6, (g) 6,,, 9 (h),, -, - (i),,, 8 (j),, 6, 6. Write down the first five terms for each of the following described sequences. a) Add to the previous term:,. b) Add si to the previous term:,. c) Subtract from the last term: 0, d) Subtract 9 from the previous term: 6, e) Multipl the previous term b :,.. f) Multipl the previous term b :, g) Divide the last term b :, h) Add the net even number each time:,, i) Add a number that increases b each time:,,.

37 . Fill in the missing gaps in the following sequences: (a),,,,,, (b),, 0,,, 8 (c),,,, 0,, (d),,,,, (e) 9,,,, (f),, 8 6, (g),,,,, (h) 8,, 8,, 8, (i), 9,,,, 9 (j),, 6,,, 6. Adding together the previous two terms on the line above generates the PASCAL s triangle. The first four rows of the PASCAL s triangle are shown below 6 (a) Write down the net three rows in this sequence. (b) Form a new PASCAL s triangle starting with, and write down the net five rows. 6. Using the squares in our books draw the net two arrangements for each of the following patterns: a) b)

38 c) d) e) f)

39 Algebra () The nth term. Give the first four terms of the sequence for which a) n U n e) n U n i) U n n b) n U n f) n U n j) n n U n c) U n n g) U n n k) U n n d) n U n h) U n n l) 6n 8 U n. The following collections of dots suggest another sequence of numbers. The sequence is started below. Write down the net seven terms in this sequence.,, 6, 0, etc. a) Write down the values of (i) the nd term (ii) the 0 th term. b) What sorts of numbers belong to this sequence? c) Which term of this sequence has a value of (i)? (ii)?. For each of the following sequences write the net two terms in the sequence and then find a formula for the n th term in the sequence. a), 6, 9,, b),, 8,, c),,, 9, d), 6, 0,, 8 e),, 9,, f),,,, g) 0,, 8,, 6 h), 9, 6,, 0 i), 9,,, j) 0,, 0,, 0 k),,,, l) 8, 0,,, 66 m),,, 9, n),, 0, 9, 8 o),,, 6, p) 9,,,, q),, 9, 6, r) -,, 6,, s), 8,, 6, t),, 8, 6,

40 . Find an epression for the nth term of the sequence 6 8 0,,,, 0 6,... Hence obtain the 00 th term for this sequence.. For each of the following give a formula for the nth term of the sequence a), 9,, 8,.. b) 0,, 8,,,. c),,,, This is the beginning of a sequence of tile patterns. a) Draw the net two patterns b) Write down the first seven terms of the sequence given b the number of tiles in each pattern. c) What sort of numbers does the pattern generate? d) Give the nth term of this sequence.. The diagrams below are made up using matchsticks Diagram Diagram Diagram a. Draw the net pattern in the sequence b. Cop and complete the table below c. Write down an epression for the number of matchsticks (m) in the nth diagram. d. How man matchsticks are their in the 0 th diagram?

41 Algebra () Sequences II. The nth term of the sequence, 0, 8, 8, 0 is given b n an. Find the value of a.. Obtain the nth term for each of the following sequences a) 0,, 6,, 0 b), 9, 6,, 6 c) 0,, 0, 8, 8 d),,, 9,. a) write down the first si terms for the sequence of diagrams below. b) Hence obtain the nth term for this sequence. c) What name is given to this special sequence of numbers?. Obtain the nth term for the sequence below, 6, 6,

42 Algebra () Trial & Improvement. Obtain the value of, correct to one decimal place given that. Helen is using trial and improvement to find a solution to the equation + = 6 The table shows her first two tries. Continue the table to find the solution to the equation, Giving our answer correct to decimal place + Comment 6 too small 6 too big. Find the value of, correct to decimal places for the equation 6. Alan is using trial and improvement to find a solution to the equation The table shows his first trial. = Comment. Too low Continue the table to find a solution to the equation correct to decimal place.

43 . Given that 0, find the value of correct to one decimal places. 6. A rectangle has width p and a length cm longer than its width. Given that its area is equal to 6cm a) Show that the p p 6 0 b) Using trial and improvement find the value of p correct to three decimal places.. Find the value of correct to decimal places such that 8. A semicircle has radius r and perimeter cm. (i) Show that r r (ii) Using trial and improvement find the value of r correct to decimal places. 9. A cuboid has a square base area with side and length cm more than its height. Given the volume of the cuboid is 00cm. 0. Asif and Ben are working out this question. a) Show that 00 0 b) Using trial and improvement find a value for correct to decimal places. A solution of the equation + = 00 lies between and 8. Use trial and improvement to find this solution, correct to one decimal place. Asif s answer is.6 Ben s answer is. Which answer is correct? You must show all our working.

44 Algebra() transformations of curves I. On the same set of aes draw the graphs of (a) (b) (c) (d). on the same set of aes drawn the graphs of (a) (b) (c) (d). The graph drawn is of f, with points A and B labelled. Sketch on separate aes each of the following, paing particular attention to the points A and B. i) f B, ii) f iii) f iv) f 0 A,. The graph drawn is of cos Sketch each of the following on separate sets of aes a) cos 0. b) cos c) cos 90 d) cos

45 Algebra(6) transformations of curves II. On the same set of aes draw the graphs of (a) (b) (c). on the same set of aes drawn the graphs of (a) f and (b) f (c) f (d) f f. The graph drawn is of f, with points A, B and C labelled. Sketch on separate aes each of the following, paing particular attention to the points A and B. i) f ii) f iii) f iv) f B,0 0 C, A,0. The graph drawn is of sin Sketch each of the following on separate sets of aes a) sin b) sin