CFE National 5 - Pack 1

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1 CFE National - Pack Unit : Epressions & Formulae (EF) N A T I O N A L WORKSHEETS Worksheets covering all the unit topics Answers

2 INDEX (EF) EXPRESSIONS and FORMULAE. Working with surds Simplification Rationalising denominators. Simplifying epressions Multiplication and division using positive and negative indices including fractions Calculations using Scientific Notation. Rounding to a given number of significant figures. Working with algebraic epressions with brackets a(b c) d(e f) (a b)(c d) a(b c) (a b)(c² d e) a, b, c, d and e are integers. Factorising an algebraic epression Common factor Difference of two squares ² y² ; p² y² Common factor with difference of two squares Trinomials with unitary ² coefficient Trinomials with non-unitary ² coefficient

3 . Completing the square in a quadratic epression with unitary ² coefficient. Reducing an algebraic epression to its simplest form a/b where a and b are of the form ( p)n or ( p)( q). Applying the four operations to algebraic fractions a/b c/d where a, b, c and d are simple constants or variables * can be add, subtract, multiply or divide. Determining the gradient of a straight line given two points m = y y /. Working with the length of arc and area of a sector of a circle. Working with the volume of a solid sphere, cone, pyramid

4 . WORKING with SURDS. Epress each of the following in its simplest form: (a) 8 (c) (h) 7 00 (j) 7 (k) 9 (l) 8 (n) 98 (o) 90 (p) 8 (q) 8 (r) 80 (s) (t) 0 (u) 0 (v) (w) () 7. Simplify: (a) 8 (c) (h) (j) (k) (l) 0. Epress each of the following in its simplest form: (a) 7 7 (c) 8 (h) (j) (k) (l) 7. Epress each of the following in its simplest form: (a) 7 8 (c) (h) (j) (k) 7 08 (l) (n) 8 (o) 7 0 (p) 98 (q) 80 0 (r) (s) 8 (t) 7 (u) 8 8

5 . Simplify: (a) (c) a a c c k k (h) 8 (j) (k) (l) y 8 (n) (o) 0 (p) (q) a b (r) 0 (s) p q (t) k (u) 0 (v) (w) 0 () (y) 0 (z) 8. (a) (c) 7 8 (h) 7. Simplify: (a) 8 7 (c) (h) (j) (k) 88 8 (l) (n) (o) 98 7 (p) 0 0

6 8. Epand and simplify: (a) ( ) ( ) (c) ( ) ( ) ( ) ( 8 ) ( 8) (h) ( ) ( 8) (j) 8( ) (k) ( ) (l) ( 00 0) ( ) (n) ( 8 ) (o) ( ) (p) ( ) 9. Epand and simplify where possible: (a) ( )( ) ( )( ) (c) ( )( ) ( )( ) ( )( ) ( )( ) ( )( - ) (h) ( 8 )( 8 ) ( )( ) (j) ( ) (k) ( ) (l) ( ) ( 7 ) (n) ( ) (o) ( )( ) (p) ( 7 ) (q) ( ) (r) ( )( ) 0. Epress each of the following with a rational denominator and simplify where possible: (a) (c) 0 (h) 0 (j) (k) (l) 0 (n) 7

7 . Epress each of the following with a rational denominator and simplify where possible: (a) (c) (h) 0. Epress each of the following in its simplest form with a rational denominator. (a) (c) (h) 8 (j) (k) 7 (l) 8 (n) (o) (p) (q) 8 (r) 0 0 (s) (l). Epress each of the following with a rational denominator and simplify where possible: (a) (c) (h) 0 (j) (k) (l)

8 . Rationalise the denominator, in each fraction, using the appropriate conjugate surd. (a) (c) (h) (j) (k) 7 (l) (n) 0 (o) (p) 9

9 SURDS PROBLEMS. A right angled triangle has sides a, b and c as shown. For each case below calculate the length of the third side, epressing a c your answer as a surd in its simplest form. b (a) Find a if b = and c =. Find c if a = and b =. (c) Find c if a = 8 and b = Find b if a = 8 and c =.. Given that = and y =, simplify: (a) y y (c) y ( y)( y). Given that p = and q =, simplify: (a) p q pq (c) p q. A rectangle has sides measuring ( ) cm and ( ) cm. Calculate the eact value of (a) its area the length of a diagonal.. A curve has as its equation y =. (a) If the point P(, k) lies on this curve find the eact value of k. Find the eact length of OP where O is the origin.

10 . In ABC, AB = AC = cm and BC = 8cm. Epress the length of the altitude from A to BC as a surd in its simplest form. [The line AM in the diagram] A cm cm 7. An equilateral triangle has each of its sides measuring a metres. (a) Find the eact length of an altitude of the triangle in terms of a. Hence find the eact area of the triangle in terms of a. [Draw a diagram to help you with this question] B M 8 cm C 8. The eact area of a rectangle is ( ) square centimetres. Given that the breadth of the rectangle is cm, show that the length is equal to ( ) cm. 9. (a challenge) Given that tan 7 o = o, show that tan 7 =.

11 . INDICES. Write each of the following in its simplest inde form. (a) (c) (h) 8 (j) c c 9 (k) a a (l) y y b 0 b 0 (n) p p 9 (o) d d (p) q q 9 (q) t t 7 (r) f f (s) k k (t) z 0 z 0 (u) 0 0 (v) y 9 y (w) a a () b b 0. Write each of the following in its simplest inde form. (a) 8 (c) (h) 7 (j) a 9 a (k) y 0 y 0 (l) b b p p (n) c 7 c 7 (o) q 8 q (p) d d 9 (q) 8 a (r) a (s) m 7 s (t) 7 m s 0 d (u) d 00 y (v) 0 y (w) t 00 0 w () 0 t w. Write each of the following in its simplest inde form. (a) ( ) (8 ) (c) (0 ) ( ) ( ) ( 7 ) ( ) (h) ( ) ( ) (j) (y 8 ) (k) (a ) 7 (l) (m ) (b ) (n) (p ) (o) (k ) 0 (p) (z ) 0

12 . Write the following without brackets. (a) (7a) (c) () (y) (ab) (y) 7 (wz) (h) (st) (pq ) (j) ( y) (k) (a b ) (l) (a ) (0 ) (n) (c ) (o) (ab ) (p) (m k). Simplify these epressions. (a) a a (c) p 7 p 0b 0b y (y ) (q ) q (c ) 8c (h) 7z (z ) k (k k ) (j) m (m m ) (k) ( ) (l) a (a a ) ( m (n) m ) c c (o) c 7 (q ) q (p) 7 q (q) (y ) 9 y (a b ) (r) (ab) ( p ) (s) p 8 p (ab ) (t) a b ab. Write down the value of (a) 0 0 (c) 00 0 () 0 0 ½ 0 a 0 (h) k 0 (mn) 0 (j) (ab ) 0 (k) (0 ) 0 (l) (y z ) 0 7. Rewrite the following with positive indices. (a) (c) a (h) p 7 (j) y 0 (k) b (l) 0 q (n) w (o) a 0 (p) 8 c (q) t (r) y

13 8. Rewrite the following with negative indices. (a) 9 (c) 7 0 (h) a p (j) 0 y (k) q (l) 8 c 9. Simplify the following epressions. (a) m m 7 (c) p 8 p a a (y ) (c ) (q ) (h) (w ) b b (j) 9 (k) k k (l) 8d d ( ) (n) p (p p 8 ) (o) a (a a ) (p) ½ m (m 0m v v ) (q) v 7 h h (r) h c 9c (s) c (t) 8 0. Find the value of (a) 8 (c) (h) 8 (j) (k) (l) (n) (o) 9 (p) 7 (q) (r) 000 (s) (t) 8 (u) 8 (v) ( 8) (w) () 00 (y) ( ) (z) ( ) 8

14 . Simplify the following epressions, giving your answers with positive indices. (a) ( ) ( p ) (c) ( ( a ) 8 ( y ) 9 q ) 0 ( k ) (g ) (h) ( m ) (c 9 ) (j) ( h ) (k) ( z ) (l) ( b ) (n) y y (o) d d (p) s s (q) (r) (u) (v) (s) (t) (w) 8 () 9 7. Write the following in surd form. (a) y (c) a y b c (h) a c (j) z (k) m (l) k 7 p (n) (o) w (p) d. Write the following in inde form. (a) a (c) y z c p (h) m a (j) z (k) (l) a b (n) m (o) y (p) c

15 . Simplify each of the following by... changing root signs to fractional powers; (ii) moving 's onto the numerators; (iii) epanding brackets where necessary. (a) ( ) ( ) (c) ( ) ( ) ( ) ( ) (h) (j) (k) (l) ( ) INDICES EXAM QUESTIONS. (a) Simplify 7a b a b If a = and b =, find the value of the epression in part (a).. Given that y =, find y when = 8.. Simplify ( ) m. (a) Simplify m Evaluate p 8 p. Epress p in its simplest form.. Simplify, writing your answer with a positive inde: a a 7. Simplify the fraction, giving your answer in positive inde form: 9 a a 8. Simplify a. 9. (a) Remove the brackets and simplify: p ( p ). Hence, or otherwise, find the value of p ( p ) when p =.

16 . CALCULATIONS USING SCIENTIFIC NOTATION. Rewrite these sentences with the numbers written out in full (a) (c) The speed of light is 0 8 metres per second. The diameter of the earth is 8 0 kilometres. A Building Society has. 0 9 in its funds. The radius of the orbit of an electron is 0 8 mm. A space probe reached a speed of 9 0 m.p.h. The earth weighs 0 tonnes. A film of oil is mm thick.. Use your calculator to answer the following, giving your answers in Standard Form. (a) ( 0 ) ( 0 ) ( 0 7 ) ( 0 8 ) (c) ( 8 0 ) ( 0 ) (9 0 ) ( 0 ) ( 0 ) ( 9 0 ) ( 0 ) ( 0 7 ) ( 0 ) ( 8 0 ) (h) ( ) ( 0 9 ) (8 7 0 ) (7 0 0 ) (j) ( 0 0 ) ( ) (k) ( 0 ) ( 0 8 ) (l) ( 0 ) ( 0 ) ( 8 0 ) ( 0 7 ) (n) ( 0 0 ) ( 0 8 ) (o) ( 0 8 ) ( ) (p) (8 0 ) ( 0 7 ) (q) ( 0 ) ( 0 ) (r) (8 8 0 ) ( 0 ) (s) (9 7 0 ) ( 0 0 ) (t) ( 8 0 ) ( 0 ) 8 0 (u) (v) (w) 0 0

17 . Answer each of the following questions leaving your answers in standard form. (a) Light travels at 8 0 miles per second. How far will it travel in an hour? The radius of the earth is 0 metres. What is its circumference (in km)? (c) If a heart beats 70 times a minute, how many times will it beat in a lifetime of 80 years?[take all years to have days] 00 grams of water contains 000 drops. How many drops would there be in a tank containing tonne of water? In gram of carbon there are there in kg of pure carbon? 0 atoms. How many carbon atoms are. Answer each of the following questions leaving your answers in standard form (a) (c) The weight of a droplet of water is grams. Calculate the weight of droplets. A space probe can travel at a speed of 0 miles per day. What distance will it travel in a week? A biscuit factory produces 7 0 teacakes every day. How many teacakes were produced in the month of February 008? The speed of light is approimately 99 million metres per second. How far can light travel in a minute? Last year 8 0 copies of a DVD were sold on its first day of release. If the cost of one DVD was, how much money was collected on that first day? In a reality TV show there were calls made to vote for the contestants. If each call cost p calculate how much the calls cost in total. Give your answer in pounds. There are of April? 8 0 seconds in one day. How many seconds are there in the month (h) Organisers of the London Marathon provide enough water to give each runner 7 litres during the race. If runners take part, how many litres of water are provided? The echange rate in Turkey is = Turkish Lira. Stephen is going on an Adriatic cruise and changes 700 into Turkish Lira. How much will he get in Lira?

18 SCIENTIFIC NOTATION EXAM QUESTIONS. The distance between the earth and mars is on average approimately 0 8 miles. A spaceship has been designed to travel between the earth and mars at an average speed of miles per hour. How many days will the spaceship take to reach mars? Give your answer correct to the nearest day.. Uranium is a radioactive isotope which has a half-life of 9 0 years. This means that only half of the original mass will be radioactive after 9 0 years. How long will it take for the radioactivity of a piece of Uranium to reduce to one eighth of its original level? Give your answer in scientific notation.. The population of Scotland in June 00 was.0 0 people. The population of China in June 00was approimately 0 times larger than that of Scotland. Calculate, correct to three significant figures, the population of China in 00, epressing your answer in standard form.. The Blackbird is a two-seater high speed jet. In December 9 it broke a world speed record by travelling at 0 0 metres per second. Calculate, correct to three significant figures, the distance travelled if the jet were to maintain this speed for one hour. Epress your answer in scientific notation.

19 . SIGNIFICANT FIGURES. Round to significant figure : (a). (c) (h) 8. 7 (j) 98 (k) 800 (l) (n) (o) 98 (p) (q) 0 (r) 9 (s) 0 8 (t) 0. Round to significant figures : (a) (c) (h) 8 7 (j) 8 7 (k) 97 (l) (n) 0 00 (o) (p) 8 (q) (r) 9 (s) 9 0 (t) 0 7. Round to significant figures : (a) 9 (c) (h) (j) 8 (k) (l) 7 8 (n) (o) 7 (p) 0 0 (q) 79 (r) 08 (s) 007 (t) Round 88 correct to (a) sig. figs sig. figs (c) sig. figs sig. fig. Round correct to (a) sig. figs sig. figs (c) sig. figs sig. fig

20 . Calculate and give your answer correct to significant figures (a) (c) (h) (0 08 ) (j) (0 08 ) (k) ( 0) (l) 7 7. Calculate and give your answer correct to significant figures (a) (c) ( 0 ) (h) ( 9) (j) ( 9 8) (k) 0 9 (l) 0 8. The speed of light is approimately 8 0 times faster than the speed of sound in air. If the speed of sound in air is 7 metres per second, calculate the speed of light. Give your answer in scientific notation correct to significant figures.

21 . ALGEBRAIC EXPRESSIONS with BRACKETS. Multiply out the brackets: (a) ( ) (y 7) (c) 8 (a ) ( t) ( 9) y ( y) b (b ) (h) p ( p) a (b c) (j) ( y) (k) p (q r) (l) a (a ). Epand the brackets: (a) (a ) 7 (y ) (c) ( ) 9 (c 7) a (a ) ( 8) 0y ( y) (h) t (t ) ( 9) (j) y (7 y) (k) b (b 8) (l) ( ). Epand and simplify: (a) (a ) a ( ) (c) 8(b ) 9 (h ) 7 ( ) (c ) 8 (t ) 0t (h) p(p q) pq 7( c) 0 (j) ( ) (k) 7a (a ) (l) ( 7) (y ) (n) 9b (b ) (o) 8 ( 7) (p) ( ) (q) c ( c) (r) 7 (a ). Multiply out the brackets: (a) ( )( ) (y )(y ) (c) (a )(a ) (b )(b ) ( 9)( ) (s )(s 8) (y 7)(y ) (h) (b )(b ) (c )(c 7) (j) (a 8)(a ) (k) (y )(y ) (l) ( 9)( 8) (p )(p 7) (n) (c )(c ) (o) (t 7)(t 9) (p) ( )( 9) (q) (y )(y ) (r) (a )(a 9)

22 . Multiply out the brackets: (a) ( )( ) (c )(c ) (c) (y )(y 7) (b )(b 8) ( )( ) (s 8)(s ) (y )(y 9) (h) (a )(a ) (t )(t ) (j) ( )( ) (k) (b )(b ) (l) (c 0)(c ) (a )(a 9) (n) (y 8)(y 7) (o) ( )( ) (p) (s )(s 7) (q) (d )(d ) (r) (b 0)(b ). Multiply out the brackets: (a) ( )( ) (a )(a 7) (c) (t )(t ) (y 8)(y ) (c )(c 7) ( )( ) (b )(b 9) (h) (p 0)(p ) (y 8)(y 7) (j) (z )(z ) (k) ( )( ) (l) (a )(a ) (c )(c ) (n) (p 7)(p ) (o) (b 0)(b ) 7. Multiply out the brackets: (a) ( ) (w ) (c) (a ) (c 8) (y ) (a ) (b ) (h) (s 7) (b 9) (j) ( 0) (k) (c ) (l) (y ) ( ) (n) (y ) (o) ( ) (p) (b ) 8. Multiply out the brackets: (a) (a b)(c d) ( )( y) (c) (a )(b ) (p q)(r s) ( a)(7 b) (c )(d 8) 9. Multiply out the brackets: (a) ( ) ( ) (c) ( 8) ( ) ( 8 ) ( 7)

23 0. Multiply out the brackets and simplify: (a) ( )( ) ( )( ) (c) ( )( ) ( )( ) ( 8)( ) ( )( 7 ) ( )( 7) (h) ( 0)( 9) ( 9)( 7) (j) ( 7)( 9 ) (k) ( )( ) (l) ( )( ) ( )( 8 ) (n) ( )( 7) (o) ( 0)( ) (p) ( 9)( ) (q) ( )( ) (r) ( 7)( 8 ). Multiply out the brackets and simplify: (a) ( )( ) ( 7)( ) (c) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (h) ( )( 7) ( 9)( ) (j) ( )( 8 ) (k) ( 8)( 7) (l) ( )( 9 ) ( )( ) (n) ( 0)( 8) (o) ( )( 7 ) (p) ( )( 7 ). Multiply out the brackets and simplify: (a) ( )( 9) ( )( ) (c) ( )( 7) ( 7)( 9 ) ( )( 8) ( )(7 ) ( )( ) (h) ( )( ) ( )( 7) (j) ( )( )

24 . Epand and simplify each of the following epressions: (a) ( ) ( ) ( )( ) ( ) (c) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) (h) ( ) ( ) ( ) ( ) ( )( ) (j) ( ) ( ) (k) ( ) ( ) (l) ( ) ( )

25 . FACTORISING an ALGEBRAIC EXPRESSION. Factorise by first finding a common factor: (a) y c d (c) s t y 9a 9b 8b 8c p q (h) 7g 7h m n (j) 9e 9f (k) j k (l) v w. Factorise by finding the common factor: (a) d 9 (c) s 9a b 8 y 0 (h) 0 c (j) 8m (k) 0 a y. Factorise by finding the common factor: (a) y 8 (c) 8a 0c 9s b 0 (h) m 0 (j) 8 y (k) b 0 (l) 8d 0. Factorise by finding the common factor: (a) a b 0 y (c) 8m n 0c d a 9 8s t y (h) a 7b c 0d (j) 9b y (k) 8 y a 8b. Factorise by finding the common factor (a) a ay y a (c) pqr pst ay bac pq p y y a ab (h) ab bc n n (j) y y (k) abc abd (l) fgh efg

26 . Factorise by finding the highest common factor: (a) a a y 9y (c) a ab pq pq y 9z b b a 7ah (h) abc 0abd s 9s (j) yz (k) 0b c bcd (l) πr πrh 7. Factorise by finding the highest common factor: (a) ap aq ar a b c (c) e f g p pq p ab bc 9bd ½ ah ½ bh ½ ch 8y (h) ac ad 0a p 0pq 0ps 8. Factorise the following epressions, which contain a difference of squares: (a) a b y (c) p q s t a p 9 (h) c b (j) y (k) m (l) a 9 d (n) q (o) 9 w (p) 9. Factorise the following epressions, which contain a difference of squares: (a) a b y (c) p q c d 8 g w y a (h) g 8h 9 y (j) 9c d (k) p 9q (l) b 00c a (n) d (o) 9k (p) Factorise the following epressions which contain a common factor and a difference of two squares: (a) a b p (c) d y 0 b 00 q 7 (h) 8a b ab a (j) y (k) abc ab (l) 8p 0q 88 (n) ak a (o) 0s (p) ½ y 0

27 . Factorise the following quadratic epressions: (a) a a (c) y y b 8b a 9a (h) w 0w 9 d 7d 0 (j) 0 (k) p 9p 0 (l) c 0c s s (n) 8 (o) y 0y. Factorise the following quadratic epressions: (a) a 8a 9 8 (c) c 9c 8 y y b b c 0c (h) 7 y n (j) p p (k) a a (l) b b (n) q q 0 (o) a 7y. Factorise the following quadratic epressions: (a) b b 0 7 (c) y y a a 0 q q d d (h) c 9c p p (j) y 7y 8 (k) a a (l) b b (n) s s (o) d d. Factorise the following quadratic epressions: (a) 7 a a (c) c 8c p p 9 y y d d q 9q (h) b 8b (j) a a (k) 0 7 (l) 9c c y y (n) b b (o) 8

28 . Factorise the following quadratic epressions: (a) 7 a a (c) p 7p b 7b 7 y y 7c 9c (h) m 9m a 0a (j) 8y y (k) p 7p (l) a a (n) c c (o) b b. Factorise the following quadratic epressions: (a) a a (c) p p c 7c y y w 0w 8 m m (h) q q b 7b 0 (j) t t (k) z z (l) d d 7s 7s (n) (o) v v (p) v 0v 7 (q) l l (r) m m 7 (s) n 9v 8 (t) b 0b (u) 9c 8c 8 (v) q q (w) a a () 8b b (y) m 8m (z) n n 8 7. Fully factorise these epressions: (a) p p 0 (c) 9 0 a a a y y c 7c (h) b 8b 9q q 8 (j) 0s s (k) 8m 0m (l) 8a a t t (n) 90d 0d 80 (o) 00

29 . COMPLETING THE SQUARE. Write the following in the form and write down the minimum value of each one. (a) (c) (h). Write the following in the form and write down the minimum value of each one. (a) (c) (h) (j) (k) (l). Write the following in the form and write down the maimum value of each one. (a) (c)

30 . REDUCING an ALGEBRAIC FRACTION to SIMPLEST FORM. Epress these fractions in their simplest form: (a) 8 (c) 0 7 0a 9b 8 (h) y c c a (j) 8a p (k) p (l) ab bc a a (n) 0 v t (o) y 9vt (p) 0ab a b 0 p q (q) pq 8 y (r) y (s) mn mn (t) 8def 0e f (u) ab c a c k m (v) 8km efg (w) 0e fg y (). Simplify by first finding the common factor: (a) a b y (c) a a ab y y y y y ab b 9b b b 0b (h) p 0q s a ab ac (j) 9 9y (k) st rs st (l) c 0ac bc p 8 p 8 p (n) 8c ac d ad (o) 8n n n (p) y 0 y

31 . Simplify the following by first factorising the numerator and/or denominator: (a) b b 9 8 (c) a a y y 9 c c a a p p (h) 9 9 q q b a b a (j) y y (k) 8 m m (l) 8 8 d d (n) p p p (o) a a (p) a a a (q) 9 9 b p b (r) c c c (s) (t) 8 y y y y (u) p p p p (v) c c c c (w) 8 9 () a a a a (y) b b b b

32 . APPLYING the FOUR OPERATIONS to ALGEBRAIC FRACTIONS. Epress each sum as a fraction in its simplest form: (a) (c) 0 8 (h) 9 (j) (k) (l) 8 7 (n) (o) (p) Epress each difference as a fraction in its simplest form: (a) (c) (h) 7 7 (j) (k) (l) (n) (o) (p) Epress each product as a fraction in its simplest form: (a) (c) (h) 7 8 (j) (k) (l) (n) (o) (p)

33 . Epress as a single fraction: (a) (c) (h) (j) (k) (l) (n) (o) (p) Epress each sum as a fraction in its simplest form: (a) (q) a a b b p p (c) 0 8 y y (h) 9 m m a a 8 (j) (k) (l) y y p p a b y (n) (o) (p) m n p q c d y 9 7 (r) (s) (t) a b a b m n p q (u) a a (v) (w) b b () 8 m m. Epress each difference as a fraction in its simplest form: (a) a a b b p p (c) 0 8 8y y 7 (h) 9 m m a a 8 (j) (k) (l) 8 y y p p a b y

34 . (continued) (q) (n) (o) (p) m n p q c d y 7 (r) (s) (t) a b a b m n p q (u) a a (v) 7 (w) b b () 7 p p 7. Epress each product as a fraction in its simplest form (a) y y a b p q (c) 7 8 c c 0 (h) a a y p p m (j) (k) (l) m m b c m 7 y (n) 9 y a 7a (o) p p (p) t s s t pq 7ab c m (q) (r) (s) pq c a mn n yz z ab a cd a (t) (u) (v) 9 y c bc 7a cd 0y y st (w) () y 8s t (y) pq a a p 8. Epress as a single fraction: (a) a a y ab a (c) p p c c 0 t t

35 8. (continued) 9 (h) k m y y bc c (j) (k) y 0y q 9q (l) z z p 0 p 8ab 9b (n) c ac 0m 8mn 0a (o) n 9 y ay 9. Simplify the following: (a) a a (c) d d a a a b a b u v u v (h) 7 (j) (k) (l) ALGEBRAIC FRACTIONS EXAM QUESTIONS. Write as a single fraction in its simplest form :, 0.. Simplify this fraction 9. Simplify fully the fraction e e e. Simplify. Write as a single fraction in its simplest form: a a. Epress as a single fraction in its simplest form:.

36 . DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS. (a) Calculate the gradient of each line in the diagram opposite. (ii) (iii) (iv) (v) Copy and complete each statement below: The gradient of any horizontal line is. The gradient of any vertical line is. A line sloping upwards from left to right has a gradient. A line sloping upwards from right to left has a gradient.. Find the gradients of the lines shown in each of the diagrams below: a b c g h i j e k d f l

37 . Find the gradients of the lines below: a y b c d y O O e f. Calculate the gradient of the line joining each pair of points below: (a) (, ) and (, ) (, ) and (, ) (c) (, 0) and (, ) (, ) and (8, ) (, 9) and (, ) (7, ) and (, ) (, ) and (, ) (h) (, ) and (, ) (, ) and (, ) (j) (, ) and (, ) (k) (, ) and (, ) (l) (, ) and (0, ). Calculate the gradient of the line joining each pair of points below: (a) A(, ) and B(8, 8) C(, ) and D(, ) (c) E(, 9) and F(8, ) G(0, ) and H(, ) I(, ) and J(7, 9) K(, 0) and L(, ) M(, ) and N(, ) (h) P(, ) and Q(, 0) R(, ) and S(8, ) (j) T(, ) and U(7, ) (k) V(, ) and W(, ) (l) X(, 7) and Y(, ) J(, 8) and K(, ) (n) S(, ) and T(, 8) (o) D(, ) and E(0, ) (p) F(, 9) and G(, )

38 . Prove that the following sets of points are collinear: (a) A(,), B(, ) and C(, ) P(, ), Q(, ) and R(7, 0) (c) E(, ), F(, ) and G(7, ) K(, ), L(, ) and M(9½, 0) 7. Given that each set of points are collinear, find the value of k in each case: (a) P(, ), Q(, ) and R(8, k) A(, ), B(, k) and C(, ) (c) E(, ), F(k, ) and G(8, 7) S(k, ), T(9, ) and U(, ) 8. The points E and F have coordinates (, ) and (, a) respectively. Given that the gradient of the line EF is, find the value of a. 9. If the points (, ), (, 0) and (, k) are collinear, find k. 0. Given that the points (, ), (, ) and (, a) are collinear, find the value of a.. The line which passes through (, ) and (, ) is parallel to the line through (, 7) and (k, ). Find the value of k.. The line which passes through (, ) and (, 9) is parallel to the line through (, k) and (, ). Find the value of k.

39 . WORKING with the LENGTH of an ARC of a CIRCLE. Calculate the length of the arc in each diagram below, giving your answer correct to d.p. A (a) D (c) E O 90 o 8cm B C mm 0 o o O O m F. Calculate the perimeter of each sector in Question. Giving your answers correct to d.p.. Find the length of the minor arc AB in each of the following circles, giving your answers correct to d.p. (a) A (c) O 90 o O O cm 9 cm 90 o cm 0 o B A B A B O 7 cm 0 o A 7cm B (h) h. A A B 0 o cm O O 80 o cm A O 8 cm O 0 o B 7 o 0 cm B A B. Calculate the length of the major arc in the circles shown in Question, giving your answers correct to d.p.

40 . WORKING with the AREA of a SECTOR of a CIRCLE. Calculate the area of the sector in each diagram below, giving your answer correct to significant figures. A E (a) (c) D O 90 o 8cm B C mm 0 o o O O m F. Calculate the area of minor sector OAB in the circles shown below, giving your answers correct to significant figures. (a) (c) A O 90 o O O cm 9 cm 90 o cm 0 o B A B A B (h) A O 7 cm 0 o A 7cm B A 0 o B cm O O 80 o cm A O 8 cm O 0 o B 7 o 0 cm B A B. Calculate the area of the major sector for the circles in Question, giving your answers correct to significant figures.. The length of minor arc CD is 7 cm. Calculate the area of the circle. O 0 o C D

41 . WORKING with the ARCS and SECTORS of a CIRCLE EXAM QUESTIONS Give your answers correct to significant figures unless otherwise stated.. Calculate the area of the sector shown in the diagram, given that it has radius 8cm. O o o. A table is in the shape of a sector of a circle with radius m. m The angle at the centre is 0 o as shown in the diagram. 0 O o Calculate the perimeter of the table. o. The door into a restaurant kitchen swings backwards and forwards through 0 o. 0 o 90cm The width of the door is 90cm. Calculate the area swept out by the door as it swings back and forth.

42 . The YUMMY ICE CREAM Co uses this logo. It is made up from an isosceles triangle and a sector of a circle as shown in the diagram. The equal sides of the triangle are cm The radius of the sector is cm. 00 o Calculate the perimeter of the logo. cm. A sensor on a security system covers a horizontal area in the shape of a sector of a circle of radius m. 0º The sensor detects movement in an area with an angle of 0º. Calculate the area covered by the sensor.. A biscuit is in the shape of a sector of a circle with triangular part removed as shown in the diagram. The radius of the circle, PQ, is 7 cm and PS = cm. R Q Angle QPR = 80 o. Calculate the area of the biscuit. P S

43 7. Two congruent circles overlap to form the symmetrical shape shown below. Each circle has a diameter of cm and have centres at B and D. A D B C Calculate the area of the shape. O 8. A sector of a circle with radius cm is shown opposite. o Angle AOB o = If the eact area of the sector is π square centimetres, calculate the size of the angle marked. A cm B 9. A hand fan is made of wooden slats with material on the outer edge. 9cm cm 0 (a) Calculate the area of material needed for the hand fan. Calculate the perimeter of the shaded area in the diagram above.

44 0. O 7 o The area of the shaded sector is ˑ0 cm. Calculate the area of the circle. P Q. The area sector OPQ is 78. cm. Calculate the size of angle o. of the circle. O 0 o P Q. A school baseball field is in the shape of a sector of a circle as shown. Given that O is the centre of the circle, calculate: (a) the perimeter of the playing field; O 80 o the area of the playing field. 80m. In the diagram opposite, O is the centre of two concentric circles with radii cm and 0cm as shown. Angle o AOB = 0. A Calculate: (a) The perimeter of the shaded shape. 0cm 0 o B The shaded area. O cm

45 . A Japanese paper fan is fully opened when angle o PQR = 0 as shown. (a) Using the dimensions shown in diagram, calculate diagram 9cm cm the approimate area of paper material in the fan. P 0 o Q R diagram Decorative silk bands are placed along the edges as shown in diagram, calculate the approimate total length of this silk edging strip.. A grandfather clock has a pendulum which travels along an arc of a circle, centre O. The arm length of the pendulum is 0cm. O The pendulum swings from position OA to OB. The length of the arc AB is cm. Calculate the size of angle AOB to the nearest degree. A B. The shape opposite is the sector of a circle, centre P, radius 0m. The area of the sector is square metres. Q Find the length of the arc QR. 0m P 0m R

46 7. A metal strip has been moulded into an arc of a circle of radius centimetres which subtends an angle of 8 o at the centre of the circle as shown in the diagram below. cm 0cm 8 o o Metal strip The same strip of metal has now been remoulded to form an arc of a circle of radius 0 centimetres as shown. Calculate the size of, the angle now subtended by the metal strip. 8. Draw a diagram to help you answer these questions. (a) A circle, centre O, has an arc PQ of length 0cm. If the diameter of the circle is 80cm, calculate the size of angle POQ correct to d.p. A circle, centre O, has a sector EOF with an area of 0cm. If the radius of the circle is 8cm, calculate the size of angle EOF correct to d.p. (c) An arc AB on a circle, centre O, has a length of mm. If angle AOB = 7 o, calculate the radius of this circle. A sector of a circle has an area of cm. If the angle at the centre is 0 o, calculate the diameter of the circle correct to -decimal places.

47 . WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID. Calculate the volume of each sphere described below, rounding your answer to decimal place. r (a) (c) r = cm r = m r = 9mm r = cm. Find the volume of a sphere for the following values of r and d. (give your answers correct to significant figures) (a) r = 0cm d = 8cm r r = cm r = 80mm (c) d = m (h) d = cm r = 00mm r = m d = cm (j) d = 8cm. A sphere has a diameter of 8cm. Calculate its volume giving your answer correct to significant figures.. Find the volume of a cone for the following values of r and h. (give your answers correct to significant figures) (a) r = cm h = cm r = 7cm h = cm (c) r = cm h = cm r = cm h = 7cm h r

48 . Find the volume of a cone for the following values of d and h. (give your answers correct to significant figures) (a) d = cm h = 0cm d = cm h = 7cm (c) d = cm h = cm d = 8ˑ8cm h = 0cm. Calculate the volume of each cone described below, rounding your answers to decimal place. h r (a) (c) r = cm and h = cm r = 8mm and h = mm r = cm and h = cm r = m and h = m 7. A cone has a base diameter of 8cm and a height of cm. Calculate the volume of this cone. 8. A cone has a base diameter of 0cm and a slant height of cm. Calculate the volume of the cone. cm cm 9. A cone has a base radius of 9cm and a slant height of cm. Calculate the volume of the cone. 0. A pyramid has a square base of side cm and a vertical height of 7cm. Calculate the volume of the pyramid correct to significant figures.. A pyramid has a rectangular base measuring mm by mm and a vertical height of 0mm. Calculate the volume of the pyramid.

49 WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID EXAM QUESTIONS. The Stockholm Globe Arena is the largest hemispherical building in the world. The radius of the building is 0 m. Calculate the volume of the building in cubic metres, giving your answer in scientific notation correct to significant figures.. A metal bottle stopper is made up from a cone topped with a sphere. The sphere has diameter cm. The cone has radius 0 9cm. Radius = 0 9cm The overall length of the stopper is cm. Calculate the volume of metal required to make the stopper. Give your answer correct to significant figures. cm. The volume of this sphere is cm. Calculate the diameter, d cm. dcm. Non Calculator! Calculate the volume of this sphere which has radius m. [Take π = ] m

50 . Sherbet in a sweet shop is stored in a cylindrical container like the one shown in diagram. 0cm cm Diagram The volume of the cylinder, correct to the nearest 000cm, is cm. The sherbet is sold in conical containers with diameter cm as shown in diagram. cm 0 of these cones can be filled from the contents of the cylinder. d cm Calculate the depth, d cm, of a sherbet cone. Diagram. Non Calculator! The diagram shows a cone with radius 0 centimetres and height 0 centimetres. 0 cm Taking π =, calculate the volume of the cone. 0 cm 7. A children s wobbly toy is made from a cone, cm high, on top of a hemispherical base of diameter 0 cm. cm The toy has to be filled with liquid foam. Calculate the volume of foam which will be required. 0 cm

51 8. The lamp cover in a street lamp is in the shape of a cone with the bottom cut off. The height of the cone is 0cm and its radius is cm. The height of the lamp is 0cm and the base of the lamp has a radius of 8cm Calculate the volume of the lamp cover. [Answer to significant figures.] 9. A glass candle holder is in the shape of a cuboid with a cone removed. The cuboid measures cm by cm by cm. The cone has a diameter of cm and a height of cm. cm Calculate the volume of glass in the candle holder. cm cm 0. For the Christmas market a confectioner has created a chocolate Santa. It consists of a solid hemisphere topped by a solid cone. Both have diameter cm and the height of the cone is cm as shown in the diagram. cm cm Calculate the volume of chocolate required to make one chocolate Santa, giving your answer correct to significant figures.

52 . The diameter of an ordinary snooker ball is cm. Calculate the volume of a snooker ball giving your answer correct to significant figures.. A dessert is in the shape of a truncated cone [a cone with a slice taken from the top]. The radius of the base is cm and is cm at the top. The other dimensions are shown in the diagram. 7cm cm Calculate the volume of the dessert.. A young child was given a slab of moulding clay. It was a cuboid and measured cm by 8cm by cm. (a) Calculate the volume of the cuboid rounding your answer to significant figures. The clay was made into identical spheres. Using your answer from part (a), calculate the radius of one of the spheres.

53 National Epressions and Formulae ANSWERS. WORKING WITH SURDS. (a) (c) (h) 0 (j) (k) (l) (n) 7 (o) 0 (p) (q) 7 (r) (s) (t) 0 (u) (v) (w) 7 () 7. (a) 0 (c) (h) 0 (j) 9 (k) 7 (l) 8. (a) 8 7 (c) 9 (h) 7 0 (j) (k) (l). (a) (c) 8 (h) 7 0 (j) (k) 0 (l) 7 8 (n) (o) (p) 8 (q) (r) (s) 7 (t) 8 (u). (a) (c) a c k (h) (j) (k) (l) y (n) (o) 0 (p) 8 (q) ab (r) 0 (s) pq (t) k (u) (v) (w) () (y) (z). (a) 0 (c) 0 8 (h) 7. (a) (h) (c) (j) (k) (l) 0 (n) (o) (p)

54 8. (a) (c) (h) 8 (j) 8 (k) (l) 0 (n) (o) (p) 9. (a) (c) 0 (h) (j) (k) (l) 0 (n) 7 0 (o) (p) 8 7 (q) 8 (r) 0. (a) (c) (h) 0 (j) (k) (n) 7 (l). (a) 0 0 (h) (c) 7. (a) 0 (c) (h) 0 (j) (k) (l) (n) (o) (p) (q) (r) (s) (l) 0. (a) 0 (c) 0 (h) 0

55 (j) (k) (l). (a) ( ) ( ) (c) ( ) ( ) ( ) (h) ( ) ( 7 ) (j) ( ) (k) 7 (l) ( ) (n) ( 0 ) (o) ( ) (p) (9 79 ) SURDS PROBLEMS. (a) (c). (a) 0 (c). (a) 8 (c). (a) cm cm. (a) (a) a a 8. Proof 9. Proof. INDICES. (a) (c) (h) 8 (j) c (k) a (l) y 0 b 0 (n) p 0 (o) d (p) q 0 (q) t 0 (r) f 7 (s) k (t) z 00 (u) 80 (v) y 0 (w) a 90 () b. (a) (c) (h) (j) a (k) y 0 (l) b p (n) (o) q (p) d (q) (r) a (s) m (t) (u) 8 d (v) 90 y (w) 99 t () 0 w

56 . (a) 8 8 (c) (h) 8 (j) y 0 (k) a (l) m b 8 (n) p (o) k 00 (p). (a) b a (c) 8 y a b 7 y 7 w z (h) s t p q (j) 8 y (k) a 0 b (l) a (n) c 0 (o) 7a b (p) m k. (a) 0a 8 9 (c) p b y 7 80q 0 8c 7 (h) 8z k k 7 (j) m 7 m 8 (k) 7 (l) 0a 7 a 8 (n) m (o) 0c (p) q (q) y (r) 0 8 a (s) b p (t) 8 a b. (a) (c) (h) (j) (k) (l) 7. (a) (c) a (h) 7 p (j) 0 y (k) b (l) 0 q (n) w (o) a (p) 8 0c (q) t (r) y 8. (a) 9 (c) 7 0 (h) a p (j) 0 y (k) q (l) 8 c 9. (a) m (c) p a 8 y c q (h) w 8 0b (j) 7 (k) k (l) d - (n) p p (o) a 9a (p) m m (q) 0 v (r) h (s) c (t) 0

57 0. (a) (c) 9 0 (h) 7 (j) 8 (k) (l) (n) (o) (p) 9 (q) (r) 00 (s) (t) (u) (v) (w) () 000 (y) (z). (a) y p (c) a q k m g (h) 8 c (j) h (k) z (l) b (n) y (o) d (p) s (q) (r) (s) 0 (t) (u) (v) (w) (). (a) (c) a y b c (h) a c (j) z (k) m (l) k p (n) (o) w (p) d 7. (a) a (c) y z c p (h) m a (j) z (k) (l) a b (n) m (o) y (p) c 9. (a) (c) (h) (j) (k) (l)

58 INDICES EXAM QUESTIONS. (a) 7a b... (a) m 8. p 9. a a 0 9. (a) p p 0. CALCULATIONS USING SCIENTIFIC NOTATION. (a) The speed of light is metres per second. (c) The diameter of the earth is 80 kilometres. A Building Society has in its funds. The radius of the orbit of an electron is mm. A space probe reached a speed of m.p.h. The earth weighs tonnes. A film of oil is mm thick.. (a) (c) (h) 0 0 (j) (k) 0 (l) (n) 9 0 (o) 0 7 (p) 0 (q) (r) 0 (s) (t) (u) 0 (v) 9 0 (w) 9 0. (a) (c) (a) grams. 0 7 (c) (h) SCIENTIFIC NOTATION EXAM QUESTIONS. days. 0 0 years

59 . SIGNIFICANT FIGURES. (a) 0 (c) (h) (j) 000 (k) 8000 (l) 000 (n) 00 (o) (p) 0000 (q) (r) 90 (s) 0 9 (t) 00. (a) (c) (h) (j) 9 (k) 00 (l) (n) 0 00 (o) (p) 000 (q) (r) 000 (s) 9 (t) 0 7. (a) 9 (c) (h) (j) 00 (k) (l) 8 (n) 0 (o) 70 (p) 0 0 (q) 8 (r) 090 (s) 0 (t) (a) (c) (a) (c) (a) 0 0 (c) (h) (j) 7 (k) 00 (l) 7. (a) 78 (c) (h) 0 9 (j) (k) 07 (l)

60 . ALGEBRAIC EXPRESSIONS with BRACKETS. (a) y (c) 8a 8 8 t ² 9 y y² b² b (h) p p² ab ac (j) ² y (k) pq pr (l) a² a. (a) 8a 0 y 8 (c) c a² a ² 0 0y 0y² (h) t² 8t ² 7 (j) y 0y² (k) b² b (l) ² 0. (a) a 7 (c) 8b 7 8h 9 c t (h) p² pq c (j) (k) a 9 (l) 9 y (n) b (o) (p) 0 (q) 7c (r) 0a. (a) y 7 y 0 (c) a 0a b 7b s s y y 8 (h) b b 9 c c (j) a a (k) y y 8 (l) 7 7 p 9 p 8 (n) c c 0 (o) t t (p) (q) y 7y 0 (r) a 0a 9. (a) c c 8 (c) y 0y b b s s 0 y y 8 (h) a 8a t 9t 8 (j) 0 (k) b 8b (l) c c 0 a a 7 (n) y y (o) (p) s s 8 (q) d d (r) b b 0. (a) a a (c) t t 0 y y c c

61 b 7b 8 (h) p 8p 0 y y (j) z z (k) (l) a a 0 c 9 (n) p p 7 (o) b b 0 7. (a) ² 9 w w (c) a 0a c² c y 8y a ² a b² b (h) s² s 9 b 8b 8 (j) 0 00 (k) c c (l) y y 9 (n) y² 0y (o) 9² (p) b² 0b 8. (a) ac bc ad bd y y (c) ab b a 0 pr qr ps qs 7 7a b ab cd d 8c 8 9. (a) 9 (c) (a) (c) (h) 9 90 (j) 7 (k) (l) 7 7 (n) (o) 0 (p) 9 (q) 9 (r). (a) (c) 8 (h) (j) 0 (k) 7 (l) (n) 80 (o) 0 (p) 8

62 . (a) 9 8 (c) (h) (j) 9. (a) 7 8 (c) (h) 8 (j) 0 (k) 9 (l). FACTORISING an ALGEBRAIC EXPRESSION. (a) ( y) (c d) (c) (s t) ( y) 9(a b) 8(b c) (p q) (h) 7(g h) (m n) (j) 9(e f) (k) (j k) (l) (v w). (a) ( ) (d ) (c) (s ) ( ) ( a) (b ) (y ) (h) ( c) ( ) (j) (m ) (k) ( a) 7(y ). (a) ( ) (y ) (c) 8( a) (c ) (s ) (b 7) ( ) (h) (m ) ( ) (j) ( y) (k) (b ) (l) (d ). (a) (a b) ( y) (c) (m n) (c d) (a ) (s t) ( y) (h) 7(a b) (c d) (j) (b y) (k) ( y) (a b). (a) a ( y) (y a ) (c) p(qr st) a(y bc) p(q ) y(y ) a(a b) (h) b(a c) n(n ) (j) y ( y) (k) ab(c d) (l) fg(h e). (a) a( ) y( y) (c) 8a( b) pq(q ) (y z) b(b ) a(a 9h) (h) ab(c d) s (s ) (j) (7 yz) (k) bc(b d) (l) πr(r h)

63 7. (a) a(p q r) (a b c) (c) (e f g) p(p q ) b(a c d) ½ h(a b c) ( 8y ) (h) a(c d a) p(p q s) 8. (a) (a b)(a b) ( y)( y) (c) (p q)(p q) (s t)(s t) (a )(a ) ( )( ) (p 9)(p 9) (h) (c )(c ) (b )(b ) (j) (y )(y ) (k) (m )(m ) (l) (a )(a ) ( d)( d) (n) ( q)( q) (o) (7 w)(7 w) (p) ( 8)( 8) 9. (a) (a b)(a b) ( y)( y) (c) (p 8q)(p 8q) (c d)(c d) (9 g)(9 g) (w y)(w y) (a )(a ) (h) (g 9h)(g 9h) (7 y)(7 y) (j) (c d)(c d) (k) (p q)(p q) (l) (b 0c)(b 0c) ( a)( a) (n) (d )(d ) (o) ( 7k)( 7k) (p) ( 0 )( 0 ) 0. (a) (a b)(a b) (p )(p ) (c) ( )( ) (d )(d ) (y )(y ) (b )(b ) (q )(q ) (h) 8(a b)(a b) a(b 8)(b 8) (j) (y )(y ) (k) ab(c )(c ) (l) (p q)(p q) ( )( ) (n) a(k )(k ) (o) (s (s ) (p) ½( y 0)(y 0). (a) ( )( ) (a )(a ) (c) (y )(y ) ( 7)(a ) ( )( ) (b )(b ) (a 7)(a ) (h) (w )(a 9) (d )(d ) (j) ( 7)( ) (k) (p )(p ) (l) (c )(c ) (s )(s ) (n) ( 7)( ) (o) (y )(y ). (a) (a )(a ) ( )( 8) (c) (a )(a ) (y )(y ) (b )(b ) ( )( ) (c )(c 8) (h) ( )( ) (y )(y 8) (j) (p 8)(p ) (k) (a 9)(a ) (l) ( )( ) (b )(b ) (n) (q 0)(q ) (o) (a )(a )

64 . (a) (b )(b ) ( 7)( ) (c) (y )(y ) (a )(a ) (q )(q ) ( )( 0) (d 7)(d ) (h) (c )(c ) (p )(p 8) (j) (y )(y 8) (k) (a )(a ) (l) ( )( 9) (b )(b ) (n) (s )(s ) (o) (d 8)(d ). (a) ( )( ) (a )(a ) (c) (c )(c ) (p 9)(p ) (y )(y ) (d )(d ) (q )(q ) (h) (b )(b ) ( )( ) (j) (a )(a ) (k) ( )( ) (l) (c )(c ) (y )(y ) (n) (b )(b ) (o) ( )( ). (a) ( )( ) (a )(a ) (c) (p )(p ) (b )(b ) ( )( ) (y )(y ) (7c )(c ) (h) (m )(m ) (8a )(a ) (j) (y )(y ) (k) (p )(p ) (l) ( )( ) (a )(a ) (n) (c )(c ) (o) (b )(b 9). (a) ( )( ) (a )(a ) (c) (p )(p ) (c )(c ) (y )(y ) (w )(w ) (m )(m ) (h) (q )(q ) (b )(b ) (j) (t )(t ) (k) (z )(z ) (l) (d )(d ) (7s )(s ) (n) ( )( ) (o) (v )(9v ) (p) ( v 7)( v ) (q) ( l )( l ) (r) ( m 7)(m ) (s) ( n 7)( n ) (t) ( b )(b ) (u) ( c )(c ) (v) ( q )( q ) (w) ( a )(a ) () ( b )(b ) (y) ( m )(m ) (z) ( n 7)( n ) 7. (a) ( )( ) ( p )( p ) (c) 9 ( )( ) ( )( ) a ( )( ) ( y )( y ) (c )( c ) (h) (b )(b ) (q )( q ) (j) (s )( s ) (k) (m )( m ) (l) (a )( a ) (t 7)( t ) (n) 0(d )(d ) (o) (0 )(0 )

65 . COMPLETING THE SQUARE. (a) ( ) [] ( ) [] (c) ( ) [ ] ( ) 0 [0 ] ( ) 9 [9] ( ) [] ( ) [ ] (h) ( ) 0 [0 ]. (a) ( ) [] ( ) 7 [7] (c) ( ) 7 [7] ( ) [] ( ) [] ( 8) 7 [7] ( ) [] (h) ( ) [] ( ) 8 [8] (j) ( 7) [] (k) ( ) [] (l) ( 0) 0 [0]. (a) ( ) [] ( ) [] (c) ( ) [] ( ) [] ( ) [ ] 7 ( ) [7 ]. REDUCING an ALGEBRAIC FRACTION to SIMPLEST FORM. (a) (h) y (c) 8 c (j) a (k) a p (l) b a c a (n) y (o) v t (p) b a (q) p q (r) 7 (s) n (t) df e (u) b (v) a k 7m (w) eg 7 y (). (a) a b ( y) (c) a b y y y a b b b b (h) 7 p q s b c (j) ( y) (k) t r t (l) a b 7 p (n) a (o) n (p)

66 . (a) b 9 (c) a y c 7 a 8 p (h) q a b (j) y (k) m (l) d (n) (s) (t) p y y (o) (u) a p p (p) (v) a a c c 7 (q) (w) b b (r) () c c a a (y) b b. APPLYING the FOUR OPERATIONS to ALGEBRAIC FRACTIONS. (a) 7 0 (h) (c) 7 8 (j) 9 0 (k) 7 9 (l) 7 (n) 0 (o) (p) 9. (a) (c) 7 (h) 0 0 (j) (k) 7 (l) 9 0 (n) 7 0 (o) (p). (a) 7 7 (h) 0 (c) 7 8 (j) 9 (k) 0 0 (l) 7 (n) (o) 0 7 (p). (a) 9 (h) 7 (c) (j) (k) 8 (l) 8 7 (n) 8 (o) 9 (p)

67 . (a) a b (c) 7 8 p 7y 9 m (h) n 7m mn 9 a (n) y q p pq (j) (o) (k) p 9 d 7c cd b a ab (l) (p) 9 y y y y (q) 8 b a ab (r) b 9a ab (s) n m mn (t) 7 q p pq a (u) a (v) b (w) b m () m. (a) a b 0 (c) 8 p y 9 m (h) 9 a y (j) 7 p (k) b a ab (l) y y 7n m mn (n) q p pq (o) 9d 7c cd (p) 9y y (q) 0b 9a ab (r) b a ab (s) n m mn (t) 7q p pq a (u) a 7 (v) b 9 (w) b 8p () p 7. (a) 8 0 y y 8 (h) p (c) ab 8 pq m (j) bc (k) c 0 a (l) 0 8 y (n) 7a 0 (o) p (p) s t (q) 8q (r) 7b c 9a 8m (s) n z (t) y a b (u) p (v) a d (w) 8 y () s t 8q a (y) p 8. (a) y (c) b p c m k (h) y b (j) 8 (k) 8 (l) 8pq a 7 (n) 8n (o) a y 9

68 9. (a) 7 a b (j) ( )( ) 7 a 0 u 7v 8 ( )( ) (c) (k) d ( )( ( )( ) ) (h) (l) a ( )( ) 7 ( )( ) ALGEBRAIC FRACTIONS EXAM QUESTIONS. 9 8 ( ). ( ). e e. ( )( ).. ( ). DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS. (a) (ii) (iii) (iv) (v) 0; undefined; positive; negative. (a) (c) (h) (j) (k) (l) 8. (a) (c). (a) (c) (h) (j) (k) (l). (a) (c) (h) 7 (j) (k) 7 (l) 9 (n) 7 (o) (p). (a) both gradients both gradients (c) both gradients both gradients 7. (a) k = k = (c) k = k = 8. a = 9 9. k = 0. a = 0. k =. k = 9

69 . WORKING with the LENGTH of an ARC of a CIRCLE. (a) cm mm (c) m. (a) 8 cm mm (c) m. (a) 7 8cm 7cm (c) 8 8cm 7cm 9cm 7cm 0 9cm (h) cm. (a) cm cm (c) 7 7cm 0 cm 7 7cm 8 cm 9 cm (h) 0 cm. WORKING with the AREA of a SECTOR of a CIRCLE. (a) 0 cm² 9mm² (c) m². (a) 9 cm² 7 07cm² (c) 8 8cm² 8cm² 88cm² 00cm² 8 7cm² (h) 8cm². (a) 8 9cm² cm² (c) 70cm² cm² 7 8cm² cm² 7cm² (h) cm². 8cm². WORKING with the ARCS and SECTORS of a CIRCLE EXAM QUESTIONS. cm². 8m. 7770cm². 7cm. m². cm² 7. 0 cm² 8. 0 o 9. (a) 7cm² cm 0. cm². 90 o. (a) 7m 70m². (a) 8 cm cm². (a) 0cm² 77cm. 0 o. cm 7. 0 o 8. (a) 7 o 89 o (c) mm 9 8cm. WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID. (a) 90 cm³ m³ (c) 0 mm³ 0cm³. (a) 90cm³ 00cm³ (c) 9m³ 00000mm³ 97cm³ 00cm³ 0000mm³ (h) 8700cm³ 80m³ (j) 7900cm³. 8cm³

70 . (a) cm³ 80cm³ (c) 07cm³ 00cm³. (a) 9cm³ 70cm³ (c) 800cm³ 08cm³. (a) cm³ 80 8mm³ (c) 7 cm³ cm³ cm³ 8. cm³ 9. 00cm³ 0. 7cm³. 0mm³ WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID EXAM QUESTIONS m³. 0cm³. 0cm. 0m³. cm. 0cm³ 7. 9cm³ cm³ 9. 8 cm³ cm³. 7 7cm³. 9 cm³. (a) 0cm³ cm