Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS

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1 Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS

2 USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form ( + p)n or ( + p)( + q) EF. Appling the four opertions to lgeric frctions / c/d where,, c nd d re simple constnts or vriles * cn e dd, sutrct, multipl or divide REL. Working with qudrtic equtions Fctorising Grphicll Qudrtic formul Discriminnt

3 EF. REDUCING n ALGEBRAIC FRACTION to SIMPLEST FORM. Epress these frctions in their simplest form: c c 8 p p c (n) 0 v t (o) vt 0 0 p q (q) pq 8 (r) (s) mn mn (t) 8def 0e f (u) c c k m (v) 8km efg (w) 0e fg (). Simplif first finding the common fctor: 0 p 0q s c st rs st c 0c c p 8 p 8 p (n) 8c c d d (o) 8n n n 0

4 . Simplif the following first fctorising the numertor nd/or denomintor: 8 c c p p q q 8 m m 8 8 d d (n) p p p (o) (q) p (r) c c c (s) (t) 8 (u) p p p p (v) c c c c (w) 8 () () 0 0

5 EF. APPLYING the FOUR OPERATIONS to ALGEBRAIC FRACTIONS. Epress ech sum s frction in its simplest form: (n) 8 (o) 8. Epress ech difference s frction in its simplest form: 8 (n) (o) 8 8. Epress ech product s frction in its simplest form: (n) (o) 8 8

6 . Epress s single frction: (n) 0 8 (o) Epress ech sum s frction in its simplest form: (q) p p 0 8 m m 8 p p (n) (o) m n p q c d (r) (s) (t) m n p q (u) (v) (w) () 8 m m. Epress ech difference s frction in its simplest form: p p m m 8 8 p p

7 . (continued) (q) (n) (o) m n p q c d (r) (s) (t) m n p q (u) (v) (w) () p p. Epress ech product s frction in its simplest form p q 8 c c 0 p p m m m c m (n) (o) p p t s s t pq c m (q) (r) (s) pq c mn n z z cd (t) (u) (v) c c cd 0 st (w) () 8s t () pq p

8 8. Epress s single frction: p p c c 0 t t k m c c 0 q q z z p 0 p 8 (n) c c 0m 8mn 0 (o) n. Simplif the following: d d u v u v ALGEBRAIC FRACTIONS EXAM QUESTIONS. Write s single frction in its simplest form :, 0.. Simplif this frction. Simplif full the frction e e e. Simplif 8

9 . Write s single frction in its simplest form:. Epress s single frction in its simplest form:. REL. WORKING with QUADRATIC EQUATIONS DRAWING GRAPHS. Use the sketches elow to solve the qudrtic equtions. = = 0 = = = 0 = Cop nd complete the tles. (ii) Mke sketch of the grph. (iii) Write down the roots of the qudrtic eqution = 0. = ² = ² + 0 0

10 = ² + + = 8 ² 0 0. For ech eqution, drw suitle sketch nd find the roots. = 0 + = 0 = = = 0 + = = = = 0 + = 0 + = 0 = 0 = 0 (n) + = 0 (o) = 0 FACTORISING. Solve these qudrtic equtions, which re lred in fctorised form. ( ) = 0 ( + ) = 0 ( ) = 0 ( ) = 0 ( + ) = 0 m(m ) = 0 ( )( ) = 0 ( )( ) = 0 (c )(c ) = 0 (w + )(w + ) = 0 (s + )(s + ) = 0 (z + )(z + 8) = 0 ( + )( ) = 0 (n) (t + )(t ) = 0 (o) ( + )( ) = 0 ( )( + ) = 0 (q) (p )(p + ) = 0 (r) (c )(c + ) = 0 (s) (d )(d ) = 0 (t) ( + )( + ) = 0 (u) (s + )(s ) = 0. Solve these qudrtic equtions fctorising first. + = 0 c c = = 0 p p = 0 z + z = 0 n + n = 0 t + t = 0 0 = 0 8 = 0 = 0 + = 0 + = 0 = 0 (n) = 0 (o) m m = 0 w w = 0 (q) c c = 0 (r) 0 = 0 0

11 . Solve these qudrtic equtions fctorising first. = 0 = 0 = 0 = 0 z = 0 k = 0 = 0 p = 0 m 00 = 0 t = 0 8 = 0 s = 0 8 = 0 (n) c 80 = 0 (o) = 0. Solve these qudrtic equtions fctorising first. + + = = = 0 m + m + = 0 c + c + 8 = 0 z + z + = 0 ² = = = 0 w w + = ² = 0 k 0k + = 0 8 ² = 0 (n) + m m² = 0 (o) t t 0 = 0 + = 0 (q) c c = 0 (r) p p² = 0. Solve these qudrtic equtions fctorising first. + + = 0 p + p + = 0 t + 0t + = 0 k + k + = = 0 ² = 0 w w² = 0 d d + = 0 + = 0 m m + 8 = 0 + c c² = 0 ² = 0 = (n) q + q = (o) t(t ) =0 m + m = (q) v = v + (r) s = + s USING QUADRATIC FORMULA. Solve these equtions using the qudrtic formul. + + = = 0 c + 8c + = 0 p + p + = = 0 d + d + = 0 + = 0 + = 0 p p + = 0 + = 0 + = 0 + = 0 = 0 (n) = 0 (o) p p = 0 c + c = 0 (q) = 0 (r) w + 0w 8 = 0

12 . Solve these equtions using the qudrtic formul, giving our nswers correct to deciml plces. + + = = 0 p + p + = 0 c + c + = = = 0 z z + = 0 q q + = 0 w w + = 0 d 0d + 8 = 0 + = 0 m m + = = 0 (n) k + k = 0 (o) c + c = 0. Solve these equtions using the qudrtic formul, giving our nswers correct to deciml plces = = 0 p + p + = 0 + c c = = = 0 8z z + = 0 + q q = 0 w w + = 0 d 0d + = 0 + = 0 + 8m m = = 0 (n) k k = 0 (o) 0c + c = 0 8 t t = 0 (q) + = 0 (r) z + z = 0. Solve these equtions using the qudrtic formul, giving our nswers correct to significnt figures. + + = 0 c + c + = 0 m + 8m + = = 0 p + p + = = 0 + = 0 z z + = 0 q q + = = 0 c 8c + 8 = 0 w w + = 0 k + k 0 = 0 (n) d + d = 0 (o) s + 8s = 0 + = 0 (q) + = 0 (r) c + c = 0 (s) = (t) + = (u) p p = (v) m = m (w) = 8 () c = + c

13 MORE QUADRATICS. Sketch the grphs of the following qudrtic functions mrking ll relevnt points. Then. for ech function nswer the following questions (ii) (iii) (iv) Stte the roots (or zeros) of the function; write down the eqution of the is of smmetr; stte the coordintes nd nture of the turning point; give the coordintes of the -intercept point; f ( ) g ( ) 8 h ( ) f ( ) g( ) h( ) 8 f ( ) 8 g( ) h ( ) 0 f ( ) g ( ) h( ) f ( ) 0 (n) g( ) (o) h ( ). Find the coordintes of the points mrked with letters in the digrms elow. L A o B E o F I J o D C G H K D M o N R o S o A B P Q U C T

14 PROBLEMS INVOLVING QUADRATIC EQUATIONS. The digrm opposite shows right-ngled tringle with sides mesuring, nd centimetres. Form n eqution nd solve it to find. Hence clculte the perimeter of the tringle.. The sides of right ngled tringle re n, n + n nd n centimetres long. Find n solving n eqution, nd hence clculte the re of this tringle.. Repet question. for right ngled tringle with short sides mesuring n nd n nd the hpotenuse mesuring n millimetres.. A rectngulr sheet of glss hs n re of 00 cm nd perimeter of 0 cm. Tking the length of the glss sheet s, write down n epression for the width of the sheet in terms of. Form n eqution in for the re of the sheet nd solve it to find. Hence stte the dimensions of the glss sheet.. A frmer hs 0 metres of cler plstic fencing. He uses ll the fencing to crete rectngulr holding pen. Tking s the length of the pen, write down n epression for the width of the pen in terms of. Given tht the re of the pen is 000 squre metres, form n eqution nd solve it to find, the length of the pen.

15 WORKING with QUADRATICS EXAM QUESTIONS. For the qudrtic function = ( ¾ ) + ½, write down: the turning point its nture the eqution of the is of smmetr. A qudrtic function is defined the formul f () = ( + ). Write down the turning point of the grph of the function.. Solve qudrtic function 0 + = 0 using n pproprite formul. Give our nswers correct to deciml plce.. The grph shown is function of the form = ( )( c). Estlish the eqution of the function. Stte the coordintes nd nture of the turning point. 0 Write the eqution of the function in the form = p( q) + r.. Solve the qudrtic eqution + 0 = 0 using n pproprite formul. Give our nswers correct to deciml plce.. For the qudrtic function = ( + ½) ¾, write down the turning point nd its nture.

16 . Solve the qudrtic eqution 0 = 0 giving the roots correct to significnt figures. 8. The eqution of the prol is of the form = ( + p) + q. (, ) Write down the eqution of the prol. 0. Solve the qudrtic eqution + 8 = 0 Give our nswers correct to deciml plce. 0. The grph shows qudrtic function of the form = ( + ) +. Stte: The coordintes of the turning point nd its nture. The eqution of the is of smmetr. The eqution of the function. 0 (, ). The digrm shows the grph of Find the - coordintes of the points where the grph crosses the - is giving our nswers correct to deciml plce. = + o

17 . For the qudrtic function = (¾ ) +, write down the turning point nd its nture. the eqution of the is of smmetr. Solve the eqution 8 = 0 giving the roots correct to significnt figures.. The grph shown hs the eqution of the form = ( )( c). Find the eqution of the prol. Find the coordintes of where the grph cuts the -is. Stte the coordintes nd nture of the turning point. Stte the eqution of the is of smmetr of the prol. 0. A qudrtic grph hs eqution ( ). Wht re the coordintes nd nture of the turning point of the grph? Which of the following is the eqution of its is of smmetr? A = B = ` C = D =. Solve the qudrtic eqution 0 Give our nswers correct to deciml plce.

18 . There is n rch uilt over the new Wemle footll stdium in London. It cn e represented on suitle es the prol with eqution ( ). A 0 B Write down the coordintes of A, the mimum turning point of the prol. Wht is the eqution of the is of smmetr of the prol? The prol cuts the - is t the origin nd the point B. Find the coordintes of B. 8. The grph shown is function of the form = ( ). Write down the turning point of the grph nd stte its nture. Wht is the eqution of the is of smmetr of the grph? Write the co-ordintes of the points where the grph cuts the -is. 0. A prol hs eqution ( ) Wht re the coordintes nd nture of its turning point? Write down the eqution of the is of smmetr of the prol, 8

19 0. The digrm shows prt of the grph of the prol which hs minimum turning point t (, ). 0 B A(, ). Write down the eqution of the prol in the form ( ) The prol cuts the - is t the origin nd the point B. Wht is the length of OB?. The prol in the digrm hs minimum turning point (, ) nd crosses the - is t points A nd B. Write its eqution in the form ( ). Stte the eqution of the is of smmetr of the prol. Find the coordintes of the point where the prol cuts the - is. A B B is the point (, 0). Wht re the coordintes of A?. Show tht the re of this L shped room is given the function A( ). Given tht the re is m, clculte the vlue of. m m m m

20 DISCRIMINANT. Find the discriminnt for ech of these qudrtic equtions w w² = = 0 + = 0 = (n) 0 (o) (q) 0 (r) 8 0. Use the discriminnts from Q to stte the nture of the roots of ech of the qudrtic equtions.. Here re some grphs of qudrtic functions. Wht cn ou s out the discriminnt for ech one?

21 Find the vlue of so tht these qudrtic equtions hve equl roots ( ) Find the vlue of k so tht these qudrtic equtions hve equl roots. 8 0 k k 8 0 k 0 k k k 0

22 DISCRIMINANT EXAM QUESTIONS. The following words cn e used to descrie the roots of qudrtic. I Rel II Equl III Distinct IV Non-rel V Rtionl VI Irrtionl Which of the ove words cn e used to descrie the roots of the eqution 0?. Find the vlue of the discriminnt for the qudrtic eqution 0 Use the discriminnt to stte the nture of the roots in prt.. For wht vlues of p does the eqution p 0 hve equl roots?. The roots of qudrtic eqution cn e descried s: I Rel II Equl III Distinct IV Non-rel V Rtionl VI Irrtionl Which of the ove cn e used to descrie the roots of the eqution 0?. For the qudrtic eqution 0, find the vlue of the discriminnt. Use the words from question to descrie the nture of the roots of the eqution. Revise for USAP

23 Answers EF. REDUCING n ALGEBRAIC FRACTION to SIMPLEST FORM. 8 c p c (n) (o) v t (q) p q (r) (s) n (t) df e (u) (v) k m (w) eg (). ( ) p q s c ( ) t r t p (n) (o) n. c 8 p q m d (n) (s) (t) p (o) (u) p p (v) c c (q) (w) (r) () c c ()

24 EF. APPLYING the FOUR OPERATIONS to ALGEBRAIC FRACTIONS (n) 0 (o) (n) 0 (o) (n) (o) (n) 8 (o). 8 p m n m mn (n) q p pq (o) p d c cd (q) 8 (r) (s) n m mn (t) q p pq (u) (v) (w) () m m. 0 8 p m

25 p n m mn (n) q p pq (o) d c cd (q) 0 (r) (s) n m mn (t) q p pq (u) (v) (w) 8p () p p 8 pq m c c (n) 8m z (s) (t) n 0 (o) p (u) p s t (v) (q) d 8q (w) (r) c 8 () s t 8q () p 8. p c m k pq (n) 8n (o). ( )( ) u v 0 8 ( )( ) d ( )( ) ( )( ) 0 ( )( ) ( )( ) ALGEBRAIC FRACTIONS EXAM QUESTIONS

26 . 8 ( ). ( ). e e. ( )( ).. ( ) REL. WORKING with QUADRATIC EQUATIONS DRAWING GRAPHS. = 0 or = or = or = or 8 = 0 or = or. 8,, 0,, 0,, 8; roots 0 nd, 0,,,, 0 ; roots nd 8,, 0,, 0, 8; roots nd 0,, 8,, 8,, 0; roots nd. Grphs drwn with roots: 0 nd 0 nd 0 nd nd (twice) (twice) nd nd nd nd nd nd nd (n) nd` (o) nd FACTORISING. 0 nd 0 nd 0 nd 0 nd 0 nd 0 nd nd nd nd nd nd nd 8 nd (n) nd (o) nd nd (q) nd (r) nd (s) nd ½ (t) nd (u) nd

27 . 0 nd 0 nd 0 nd 8 0 nd 0 nd 0 nd 0 nd 0 nd 0 nd 0 nd 0 or 0 nd (n) 0 nd (o) 0 nd 0 nd (q) 0 nd (r) 0 nd 0 or. nd nd nd nd nd 8 nd 8 nd nd 0 nd 0 nd nd nd nd (n) nd (o) nd. nd nd nd nd nd nd nd (twice) nd nd nd nd nd (n) nd (o) nd 0 nd (q) nd (r) nd. nd nd nd nd nd nd nd nd (q) nd nd nd nd nd (n) nd (r) nd (o) nd nd nd USING QUADRATIC FORMULA. nd nd nd nd nd nd nd nd nd nd nd nd (n) nd (o) nd nd nd

28 (q) nd (r) or. 8 nd 0 nd 8 0 nd 0 nd 0 nd 0 8 nd nd 0 nd 0 nd 0 nd 0 88 nd 0 8 nd 0 0 nd 8 (n) nd (o) nd. nd 0 nd 0 nd 8 0 nd 0 nd 0 nd 0 0 nd 0 8 nd 0 8 nd 0 nd 0 nd 0 0 nd 0 nd 8 (n) 0 nd 0 (o) 0 nd 0 nd 0 8 (q) 0 nd 0 0 (r) 8 nd 8. 0 nd nd 0 8 nd nd 0 nd 0 nd 0 8 nd 0 nd nd 0 0 nd nd nd nd 8 (n) nd (o) nd 8 nd 8 (q) nd (r) 8 nd 8 (s) 0 nd 0 8 (t) 8 nd 08 (u) 0 nd 8 (v) 0 nd 0 (w) nd () nd MORE QUADRATICS 8

29 . nd ; = ; (, ) Minimum; (0, ) nd ; = ; (, ) Minimum; (0, 8) nd ; = ; (, ) Minimum; (0, ) nd 0; = ; (, ) Minimum; (0, 0) 0 nd ; = ; (, ) Minimum; (0, 0) 0 nd 8; = ; (, ) Mimum; (0, 0) nd ; = ; (, ) Mimum; (0, 8) nd ; = ; (, ) Mimum; (0, ) nd ; = ; (, ) Minimum; (0, ) nd ; = ; (, ) Minimum; (0, ) nd ; = ; (, ) Minimum; (0, ) 0 nd ; = ; (, ) Mimum; (0, 0) nd ; = ; (, ) Mimum; (0, 0) (n) nd ; = 0; (0, ) Mimum; (0, ) (o) nd ; = 0; (0, ) Minimum; (0, ). A(, 0); B(8, 0); C(, ); D(0, ) E(, 0); F(, 0); G(, ); H(0, ) I(, 0); J(, 0); K(, ); L(0, ) M(, 0); N(, 0); O(, ); P(0, ) R(, 0); S(, 0); T(, ); U(0, ) A(, 0); B(, 0); C(, ); D(0, ) PROBLEMS INVOLVING QUADRATIC EQUATIONS

30 . ( ) ; = 0 cm. n ; cm². n 0; mm². 80 ( 80 ) 00 ; ( 0 ) 000; 80m WORKING with QUADRATICS EXAM QUESTIONS. (¾, ½ ) minimum = ¾. (, ). 0 nd. = ( + )( ) (, ); minimum = ( ).. 0 nd. ( ½, ¾) ; minimum. 8 nd 8. = ( ) nd 0. (, ) minimum = = ( + ). 0 nd. (¾, ) minimum = ¾. nd. = ( + )( ) (0, 8) (, ); min =. (, ); minimum B. 8 nd 0. (, ) = (, 0) 8. (, ); minimum = (, 0) nd (, 0). (, ) mimum = 0. ( ) units. ( ). = (0, ) A(, 0). proof = DISCRIMINANT 0

31 (n) (o) 0 (q) (r) 0. rel, rtionl, distinct equl rel, rtionl, distinct rel, rtionl, distinct rel, rtionl, distinct equl rel, rtionl, distinct non rel rel, rtionl, distinct rel, rtionl, distinct rel, irrtionl, distinct non rel rel, irrtionl, distinct (n) non rel (o) Rel, rtionl, distinct equl (q) non rel (r) equl. c > 0 c < 0 c = 0 c > 0 c = 0 c < 0 c > 0 c > 0 c > 0 c < 0 c < 0 c > or 0. EXAM QUESTIONS 8., rel, irrtionl nd distinct. rel, irrtionl nd distinct.. Non rel., rel, irrtionl nd distinct

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse