Background knowledge

Size: px
Start display at page:

Download "Background knowledge"

Transcription

1 Bckground knowledge Contents: A B C D E F G H I J K L Surds nd rdicls Scientific nottion (stndrd form) Numer systems nd set nottion Algeric simplifiction Liner equtions nd inequlities Modulus or solute vlue Product expnsion Fctoristion Formul rerrngement Adding nd sutrcting lgeric frctions Congruence nd similrity Coordinte geometry

2 BACKGROUND KNOWLEDGE This chpter contins mteril tht is normlly covered prior to this course. It is ssumed knowledge. Not ll preliminries re covered within the chpter. However, other necessry work is revised within the chpters. A SURDS AND RADICALS Rel numers like p, p 3, p 5, p 6, etc., re clled surds or rdicls. Surds re present in solutions to some qudrtic equtions. p 4 is rdicl ut is not surd s it simplifies to. Surds re irrtionl rel numers. Definition: p is the non-negtive numer such tht p p =. Properties: ² ² ² ² p p is never negtive, so > 0. p is meningful only for > 0. p p p = for > 0 nd > 0. r p = p for > 0 nd >0. EXERCISE A Exmple 1 Write s single surd: p p 3 p 18 p 6 p p 3 = p 3 = p 6 = p p 18 6 q 18 6 = p 3 or p 18 p 6 = p 6 p 3 p 6 = p 3 1 Write s single surd or rtionl numer: p p p 3 5 ( 3) c p p d 3 p p e 3 p 7 p 7 f p 1 p g p 1 p 6 h p 18 p 3 Exmple Simplify: 3 p 3+5 p 3 p 5 p Compre with x 5x = 3x 3 p 3+5 p 3 p 5 p =(3+5) p 3 = ( 5) p =8 p 3 = 3 p

3 BACKGROUND KNOWLEDGE 3 Simplify the following mentlly: p +3 p p 3 p c 5 p 5 3 p 5 d 5 p 5+3 p 5 e 3 p 5 5 p 5 f 7 p 3+ p 3 g 9 p 6 1 p p p p 6 h + + Exmple 3 Write p 18 in the form p where nd re integers nd is s lrge s possile. p 18 = p 9 f9 is the lrgest perfect squre fctor of 18g = p 9 p =3 p 3 Write the following in the form p where nd re integers nd is s lrge s possile: p p p p 8 1 c 0 d 3 p p p p e 7 f 45 g 48 h 54 p p p p i 50 j 80 k 96 l 108 Exmple 4 Simplify: p 75 5 p 7 p 75 5 p 7 = p p 9 3 = 5 p p 3 =10 p 3 15 p 3 = 5 p 3 4 Simplify: 4 p 3 p 1 3 p + p 50 c 3 p 6+ p 4 d p 7 + p p p p p p 1 e 75 1 f Exmple 5 Write 9 p 3 without rdicl in the denomintor. 9 p3 = 9 p 3 p 3 p 3 = 9p 3 3 =3 p 3

4 4 BACKGROUND KNOWLEDGE 5 Write without rdicl in the denomintor: 1 p 6 p3 c 7 p d 10 p 5 e 10 p f 18 p 6 g 1 p 3 h 5 p7 i 14 p 7 j p 3 p B SCIENTIFIC NOTATION (STANDARD FORM) Scientific nottion (or stndrd form) involves writing ny given numer s numer etween 1 nd 10, multiplied y power of 10, i.e., 10 k where 1 6 <10 nd k Z. EXERCISE B Exmple 6 Write in stndrd form: : = 3: fshift deciml point 4 plces to the =3: left nd g 0: = 8: fshift deciml point 4 plces to the =8: right nd g 1 Express the following in scientific nottion: c :59 d 0:59 e 0: f 40:7 g 4070 h 0:0407 i j k 0: Express the following in scientific nottion: The distnce from the Erth to the Sun is m. c d e Bcteri re single cell orgnisms, some of which hve dimeter of 0:0003 mm. A speck of dust is smller thn 0:001 mm. The core temperture of the Sun is 15 million degrees Celsius. A single red lood cell lives for out four months nd during this time it will circulte round the ody times. Exmple 7 Write s n ordinry numer: 3: 10 5: : 10 =3:0 100 = 30 5: = : =0:

5 3 Write s n ordinry deciml numer: BACKGROUND KNOWLEDGE c : d 7: e 3: f 8: g 4: h Write s n ordinry deciml numer: c : d 7: e 3: f 8: g 4: h Write s n ordinry deciml numer: The wvelength of light is m. The estimted world popultion for the yer 000 ws 6: c The dimeter of our glxy, the Milky Wy, is light yers. d The smllest viruses re mm in size. 6 Find, correct to deciml plces: (3: ) (4: ) (6:4 10 ) c 3: d (9: ) (7: 10 6 ) e 1 3: f (1: 10 3 ) 3 7 If missile trvels t 5400 km h 1 how fr will it trvel in: 1 dy 1 week c yers? Give your nswers in scientific nottion correct to deciml plces, nd ssume tht 1 yer ¼ 365:5 dys. 8 Light trvels t speed of metres per second. How fr will light trvel in: 1 minute 1 dy c 1 yer? Give your nswers with deciml prt correct to deciml plces, nd ssume tht 1 yer ¼ 365:5 dys. C NUMBER SYSTEMS AND SET NOTATION NUMBER SYSTEMS We use: ² R to represent the set of ll rel numers. These include ll of the numers on the numer line. negtives ² N to represent the set of ll nturl numers. N = f0, 1,, 3, 4, 5,...g ² Z to represent the set of ll integers. Z = f0, 1,, 3, 4,...g ² Z + is the set of ll positive integers. Z + = f1,, 3, 4,...g ² Q to represent the set of ll rtionl numers which re ny numers of the form p where p nd q re integers, q q 6= 0. 0 positives

6 6 BACKGROUND KNOWLEDGE SET NOTATION fx j 3 <x<g reds the set of ll vlues tht x cn e such tht x lies etween 3 nd. the set of ll such tht Unless stted otherwise, we ssume tht x is rel. 3 <x< cn lso e written s x ] 3, [. EXERCISE C 1 Write verl sttements for the mening of: fx j x>5, x R g fx j x 6 3, x Z g c fy j 0 <y<6g d fx j 6 x 6 4, x Z g e ft j 1 <t<5g f fn j n< or n > 6g Exmple 8 Write in set nottion: included not included fx j 1 6 x 6 4, x N g or fx j 1 6 x 6 4, x Z g fx j 3 6 x<4, x Z g Write in set nottion: c d e f Sketch the following numer sets: fx j 4 6 x<10, x N g fx j 4 <x6 5, x Z g c fx j 5 <x6 4, x R g d fx j x> 4, x Z g e fx j x 6 8, x R g D ALGEBRAIC SIMPLIFICATION To nswer the following questions, you will need to rememer: ² the distriutive lw ( + c) = + c ² power nottion =, 3 =, 4 =, nd so on. EXERCISE D 1 Simplify if possile: 3x +7x 10 3x +7x x c x +3x +5y d 8 6x x e 7 +5 f 3x +7x 3

7 Remove the rckets nd then simplify: 3(x + 5) + 4(5 + 4x) 6 (3x 5) BACKGROUND KNOWLEDGE 7 c 5( 3) 6( ) d 3x(x 7x +3) (1 x 5x ) 3 Simplify: x(3x) E c p 16x 4 d ( ) When deling with inequlities: ² multiplying or dividing oth sides y negtive reverses the inequlity sign. ² do not multiply or divide oth sides y the unknown or term involving the unknown. EXERCISE E 1 Solve for x: x +5=5 3x 7 > 11 c 5x +16=0 x d 3 7=10 e 6x +11< 4x 9 f 3x =8 5 1 g 1 x > 19 h x +1= 3 x i 3 3x 4 = 1 (x 1) Solve simultneously for x nd y: ½ x +y =9 x y =3 F d LINEAR EQUATIONS AND INEQUALITIES ½ 5x 4y =7 3x +y =9 e ½ x +5y =8 x y = ½ x +y =5 x +4y =1 c f ½ 7x +y = 4 3x +4y =14 8 x >< + y 3 =5 >: x 3 + y 4 =1 The modulus or solute vlue of rel numer is its size, ignoring its sign. For exmple: the modulus (or solute vlue) of 7 is 7, nd the modulus (or solute vlue) of 7 is lso 7. GEOMETRIC DEFINITION The modulus of rel numer is its distnce from zero on the numer line. Becuse the modulus is distnce, it cnnot e negtive. We denote the modulus of x s MODULUS OR ABSOLUTE VALUE jxj. jxj is the distnce of x from 0 on the rel numer line.

8 8 BACKGROUND KNOWLEDGE x If x>0 If x<0 0 x x x 0 ALGEBRAIC DEFINITION jx j cn e considered s the distnce of x from : jxj = ½ x if x > 0 x if x<0 or jxj = p x MODULUS EQUATIONS It is cler tht jxj = hs two solutions, x = nd x =, since jj = nd j j =. EXERCISE F 1 Find the vlue of: If jxj = where > 0, then x =. 5 ( 11) j5j j 11j c j5 ( 11)j d ( ) + 11( ) e j 6j j 8j f j 6 ( 8)j If =, =3, nd c = 4 find the vlue of: jj jj c jjjj d jj e j j f jj jj g j + j h jj + jj i jj j k c jcj l jj 3 Solve for x: jxj =3 jxj = 5 c jxj =0 d jx 1j =3 e j3 xj =4 f jx +5j = 1 g j3x j =1 h j3 xj =3 i j 5xj =1 G PRODUCT EXPANSION y =(x 1)(x +3) cn e expnded into the generl form y = x + x + c. Likewise, y =(x 3) +7 cn lso e expnded into this form. Following is list of expnsion rules you cn use: ² ( + )(c + d) =c + d + c + d ² ( + )( ) = fdifference of two squresg ² ( + ) = + + fperfect squresg

9 BACKGROUND KNOWLEDGE 9 EXERCISE G Exmple 9 Expnd nd simplify: (x + 1)(x +3) (3x )(x +3) (x + 1)(x +3) (3x )(x +3) =x +6x + x +3 =3x +9x x 6 =x +7x +3 =3x +7x 6 1 Expnd nd simplify using ( + )(c + d) =c + d + c + d: (x + 3)(x +1) (3x + 4)(x +) c (5x )(x +1) d (x + )(3x 5) e (7 x)( + 3x) f (1 3x)(5 + x) g (3x + 4)(5x 3) h (1 3x)( 5x) i (7 x)(3 x) j (5 x)(3 x) k (x + 1)(x +) l (x 1)(x +3) Exmple 10 Expnd using the rule ( + )( ) = : (5x )(5x +) (7 + x)(7 x) (5x )(5x +) (7 + x)(7 x) =(5x) =7 (x) =5x 4 = 49 4x Expnd using the rule ( + )( ) = : (x + 6)(x 6) (x + 8)(x 8) c (x 1)(x +1) d (3x )(3x +) e (4x + 5)(4x 5) f (5x 3)(5x +3) g (3 x)(3 + x) h (7 x)(7 + x) i (7 + x)(7 x) j (x + p )(x p ) k (x + p 5)(x p 5) l (x p 3)(x + p 3) Exmple 11 Expnd using the perfect squre expnsion rule: (x +) (3x 1) (x +) (3x 1) = x +(x)() + =(3x) + (3x)( 1) + ( 1) = x +4x +4 =9x 6x +1 3 Expnd nd simplify using the perfect squre expnsion rule: (x +5) (x +7) c (x ) d (x 6) e (3 + x) f (5 + x) g (11 x) h (10 x) i (x +7) j (3x +) k (5 x) l (7 3x)

10 10 BACKGROUND KNOWLEDGE 4 Expnd the following into the generl form y = x + x + c: y =(x + )(x +3) y =3(x 1) +4 c y = (x + 1)(x 7) d y = (x +) 11 e y =4(x 1)(x 5) f y = 1 (x +4) 6 g y = 5(x 1)(x 6) h y = 1 (x +) 6 i y = 5 (x 4) Exmple 1 Expnd nd simplify: 1 (x +3) (3 + x) ( + x)(3 x) The use of rckets is essentil! 1 (x +3) =1 [x +6x +9] =1 x 1x 18 = x 1x 17 (3 + x) ( + x)(3 x) =6+x [6 x +3x x ] =6+x 6+x 3x + x = x + x 5 Expnd nd simplify: 1+(x +3) +3(x )(x +3) c 3 (3 x) d 5 (x + 5)(x 4) e 1 + (4 x) f x 3x (x + )(x ) g (x +) (x + 1)(x 4) h (x +3) +3(x +1) i x +3x (x 4) j (3x ) (x +1) H FACTORISATION Algeric fctoristion is the reverse process of expnsion. For exmple, (x + 1)(x 3) is expnded to x 5x 3, wheres x 5x 3 is fctorised to (x + 1)(x 3). Notice tht x 5x 3=(x + 1)(x 3) hs een fctorised into two liner fctors. Flow chrt for fctorising: Expression Difference of two squres =( + )( ) Tke out ny common fctors Recognise type Perfect squre + + =( + ) Sum nd Product type x + x + c, 6= 0 ² find c ² find the fctors of c which dd to ² if these fctors re p nd q, replce x ypx + qx ² complete the fctoristion Sum nd Product type x + x + c x + x + c =(x + p)(x + q) where p + q = nd pq = c

11 BACKGROUND KNOWLEDGE 11 Exmple 13 Fully fctorise: 3x 1x 4x 1 c x 1x +36 3x 1x f3x is common fctorg =3x(x 4) 4x 1 =(x) 1 =(x + 1)(x 1) fdifference of two squresg Rememer tht ll fctoristions cn e checked y expnsion! c x 1x +36 = x +(x)( 6) + ( 6) fperfect squre formg =(x 6) EXERCISE H 1 Fully fctorise: 3x +9x x +7x c 4x 10x d 6x 15x e 9x 5 f 16x 1 g x 8 h 3x 9 i 4x 0 j x 8x +16 k x 10x +5 l x 8x +8 m 16x +40x +5 n 9x +1x +4 o x x + 11 Exmple 14 Fully fctorise: 3x +1x +9 x +3x +10 3x +1x +9 f3 is common fctorg =3(x +4x +3) fsum =4, product =3g =3(x + 1)(x +3) x +3x +10 = [x 3x 10] fremoving 1 s common fctor to mke the coefficient of x e 1g = (x 5)(x +) fsum = 3, product = 10g Fully fctorise: x +9x +8 x +7x +1 c x 7x 18 d x +4x 1 e x 9x +18 f x + x 6 g x + x + h 3x 4x +99 i x 4x j x +6x 0 k x 10x 48 l x +14x 1 m 3x +6x 3 n x x 1 o 5x +10x +40

12 1 BACKGROUND KNOWLEDGE FACTORISATION BY SPLITTING THE x-term Using the distriutive lw to expnd we see tht: (x + 3)(4x +5) =8x +10x +1x +15 =8x +x +15 We will now reverse the process to fctorise the qudrtic expression 8x +x +15: 8x +x +15 Step 1: Split the middle term =8x +10x +1x +15 Step : Group in pirs =(8x +10x) + (1x + 15) Step 3: Fctorise ech pir seprtely =x(4x + 5) + 3(4x +5) Step 4: Fctorise fully =(4x + 5)(x + 3) The trick in fctorising these types of qudrtic expressions is in Step 1 where the middle term needs to e split into two so tht the rest of the fctoristion proceeds smoothly. Rules for splitting the x-term: The following procedure is recommended for fctorising x + x + c : ² Find c: ² Find the fctors of c which dd to : ² If these fctors re p nd q, replce x y px + qx: ² Complete the fctoristion. Exmple 15 Fully fctorise: x x 10 6x 5x +14 x x 10 hs c = 10 = 0: The fctors of 0 which dd to 1 re 5 nd +4: ) x x 10 = x 5x +4x 10 = x(x 5) + (x 5) =(x 5)(x +) 6x 5x +14 hs c =6 14 = 84: The fctors of 84 which dd to 5 re 1 nd 4: ) 6x 5x +14 =6x 1x 4x +14 =3x(x 7) (x 7) =(x 7)(3x )

13 BACKGROUND KNOWLEDGE 13 3 Fully fctorise: x +5x 1 3x 5x c 7x 9x + d 6x x e 4x 4x 3 f 10x x 3 g x 11x 6 h 3x 5x 8 i 8x +x 3 j 10x 9x 9 k 3x +3x 8 l 6x +7x + m 4x x +6 n 1x 16x 3 o 6x 9x +4 p 1x 10 9x q 8x 6x 7 r 1x +13x +3 s 1x +0x +3 t 15x x +8 u 14x 11x 15 Exmple 16 Fully fctorise: 3(x +)+(x 1)(x +) (x +) 3(x +)+(x 1)(x +) (x +) =(x + )[3 + (x 1) (x + )] fs (x +) is common fctorg =(x + )[3 + x x ] =(x + )(x 1) 4 Fully fctorise: 3(x +4)+(x + 4)(x 1) 8( x) 3(x + 1)( x) c 6(x +) +9(x +) d 4(x +5)+8(x +5) e (x + )(x + 3) (x + 3)( x) f (x + 3) + (x + 3) x(x + 3) g 5(x ) 3( x)(x +7) h 3(1 x)+(x + 1)(x 1) Exmple 17 Fully fctorise using the difference of two squres : (x +) 9 (1 x) (x +1) (x +) 9 (1 x) (x +1) =(x +) 3 = [(1 x) (x + 1)][(1 x)+(x + 1)] =[(x + ) + 3][(x +) 3] = [1 x x 1][1 x +x +1] =(x + 5)(x 1) = 3x(x +) 5 Fully fctorise: (x +3) 16 4 (1 x) c (x +4) (x ) d 16 4(x +) e (x +3) (x 1) f (x + h) x g 3x 3(x +) h 5x 0( x) i 1x 7(3 + x)

14 14 BACKGROUND KNOWLEDGE INVESTIGATION 1 ANOTHER FACTORISATION TECHNIQUE Wht to do: (x + p)(x + q) h pq i 1 By expnding, show tht = x +[p + q]x + : If x (x + p)(x + q) + x + c =, show tht p + q = nd pq = c. 3 Using on 8x +x +15, we hve 8x (8x + p)(8x + q) +x +15= 8 ) p =1, q =10, or vice vers ) 8x +x +15= (8x + 1)(8x + 10) 8 4(x + 3)(4x +5) = 8 =(x + 3)(4x +5) where ½ p + q = pq =8 15 = 10 Use the method shown to fctorise: i 3x +14x +8 ii 1x +17x +6 iii 15x +14x 8 Check your nswers to y expnsion. I FORMULA REARRANGEMENT In the formul D = xt + p we sy tht D is the suject. This is ecuse D is expressed in terms of the other vriles x, t nd p. We cn rerrnge the formul to mke one of the other vriles the suject. We do this using the usul rules for solving equtions. Whtever we do to one side of the eqution we must lso do to the other side. EXERCISE I Exmple 18 Mke x the suject of D = xt + p. If D = xt + p ) xt + p = D ) xt + p p = D p fsutrct p from oth sidesg ) xt = D p xt ) t = D p t ) x = D p t fdivide oth sides y tg

15 BACKGROUND KNOWLEDGE 15 1 Mke x the suject of: + x = x = c x + = d d c + x = t e 5x +y =0 f x +3y =1 g 7x +3y = d h x + y = c i y = mx + c Exmple 19 Mke z the suject of c = m z. c = m z c z = m z z ) cz = m ) cz c = m c fmultiply oth sides y zg fdivide oth sides y cg ) z = m c Mke z the suject of: z = c z = d c 3 d = z d z = z 3 Mke: the suject of F = m r the suject of C =¼r c d the suject of V = ldh d K the suject of A = K : Exmple 0 Mke t the suject of s = 1 gt where t>0. 1 gt = s frewrite with t on LHSg ) 1 gt = s fmultiply oth sides y g ) gt =s ) gt g = s g ) t = s g r s ) t = g fdivide oth sides y gg fs t>0g

16 16 BACKGROUND KNOWLEDGE 4 Mke: r the suject of A = ¼r if r>0 x the suject of N = x5 c r the suject of V = 4 3 ¼r3 d x the suject of D = n x 3 : 5 Mke: p the suject of d = l the suject of T = 1 n 5p l c the suject of c = p r l d l the suject of T =¼ g e the suject of P =( + ) f h the suject of A = ¼r +¼rh g r the suject of I = E R + r 6 Mke the suject of the formul k = d. Find the vlue for when k = 11, d =4, =. h q the suject of A = B p q : 7 The formul for determining the volume of sphere of rdius r is V = 4 3 ¼r3. Mke r the suject of the formul. Find the rdius of sphere which hs volume of 40 cm 3. 8 The distnce (in cm) trvelled y n oject ccelerting from sttionry position is given y the formul S = 1 t where is the ccelertion in cm s nd t is the time in seconds. Mke t the suject of the formul. Consider t>0 only. Find the time tken for n oject ccelerting t 8 cm s to trvel 10 m. 9 The reltionship etween the oject nd imge distnces (in cm) for concve mirror 1 cn e written s f = 1 u + 1 where f is the focl length, u is the oject distnce v nd v is the imge distnce. Mke v the suject of the formul. Given focl length of 8 cm, find the imge distnce for the following oject distnces: i 50 cm ii 30 cm. 10 According to Einstein s theory of reltivity, the mss of prticle is given y the m 0 formul m = r ³, where m v 0 is the mss of the prticle t rest, 1 c v is the speed of the prticle, nd c is the speed of light. c Mke v the suject of the formul given v>0. Find the speed necessry to increse the mss of prticle to three times its rest mss, i.e., m =3m 0. Give the vlue for v s frction of c. A cyclotron incresed the mss of n electron to 30m 0 : With wht velocity must the electron hve een trvelling, given c = ms 1?

17 J BACKGROUND KNOWLEDGE 17 ADDING AND SUBTRACTING ALGEBRAIC FRACTIONS To dd or sutrct lgeric frctions, we comine them into single frction with the lest common denomintor (LCD). x 1 For exmple, 3 EXERCISE J Exmple 1 x +3 hs LCD of 6, so we write ech frction with denomintor 6. Write s single frction: + 3 x x 1 3 x x ³ x = + x 3 x x 1 x +3 3 = µ x 1 3 µ x = x +3 x = = = (x 1) 3(x +3) 6 x 3x 9 6 x Write s single frction: 3+ x 5 d 3 x 4 e 1+ 3 x +x 3 + x 4 5 c 3+ x x +5 f x Exmple 3x +1 Write x s single frction. = = 3x +1 x µ 3x +1 x (3x +1) (x ) x 3x +1 x +4 = x = x +5 x µ x x fthe LCD is (x )g

18 18 BACKGROUND KNOWLEDGE Write s single frction: 1+ 3 x + d + 3 x 4 x 1 x e 3 x x +1 c 3 x 1 f x 3 Write s single frction: 3x x +5 + x 5 x K c 5x x 4 + 3x x +4 CONGRUENCE d 1 x 1 x 3 x +1 x 3 x +4 x +1 CONGRUENCE AND SIMILARITY Two tringles re congruent if they re identicl in every respect prt from position. The tringles hve the sme shpe nd size. There re four cceptle tests for the congruence of two tringles. Two tringles re congruent if one of the following is true: ² corresponding sides re equl in length (SSS) ² two sides nd the included ngle re equl (SAS) ² two ngles nd pir of corresponding sides re equl (AAcorS) ² for right ngled tringles, the hypotenuse nd one other pir of sides re equl (RHS). Exmple 3 A Explin why ABC nd DBC re congruent: B C A B D C D s ABC nd DBC re congruent (SAS) s: ² AC = DC ² ACB = DCB, nd ² [BC] is common to oth.

19 BACKGROUND KNOWLEDGE 19 If congruence cn e proven then ll corresponding lengths, ngles nd res must e equl. EXERCISE K.1 1 Tringle ABC is isosceles with AC = BC. [BC] nd [AC] re produced to E nd D respectively so tht CE = CD. E C D Prove tht AE = BD. A B Point P is equidistnt from oth [AB] nd [AC]. Use congruence to show tht P lies on the isector of B AC. A P B C 3 Two concentric circles re drwn. At P on the inner circle, tngent is drwn which meets the other circle t A nd B. Use tringle congruence to prove tht P is the midpoint of [AB]. A O P B SIMILARITY Two tringles re similr if one is n enlrgement of the other. Similr tringles re equingulr, nd hve corresponding sides in the sme rtio. Exmple 4 Estlish tht pir of tringles is similr nd find x if BD =0cm: 1 cm B A x cm C E ( x + ) cm D 1 cm B A E x cm C 0 cm ( x + ) cm D ² - x + x smll lrge The tringles re equingulr nd hence similr. ) x + = x fsides in the sme rtiog 0 1 ) 1(x +)=0x ) 1x +4=0x ) 4 = 8x ) x =3

20 0 BACKGROUND KNOWLEDGE EXERCISE K. 1 In ech of the following, estlish tht pir of tringles is similr, nd hence find x: A c U P cm B C 5cm X x cm 3cm 6cm x cm Q S 7cm R x cm cm D E 6cm T V 5cm Y 6cm Z d D e X f A 4cm E B 3cm 5cm C x cm Y cm U 5cm x cm Z V S P 8cm x cm 10 cm 50 Q 50 9cm R T A fther nd son re stnding side-y-side. The fther is 1:8 m tll nd csts shdow 3: m long, while his son s shdow is :4 m long. How tll is the son? L COORDINATE GEOMETRY THE NUMBER PLANE The position or loction of ny point in the numer plne cn e specified in terms of n ordered pir of numers (x, y), where x is the horizontl step from fixed point O, nd y is the verticl step from O. The point O is clled the origin. Once O hs een specified, we drw two perpendiculr xes through it. The x-xis is horizontl nd the y-xis is verticl. The numer plne is lso known s either: ² the -dimensionl plne, or ² the Crtesin plne, nmed fter René Descrtes. (, ) is clled n ordered pir, where nd re the coordintes of the point. is clled the x-coordinte. is clled the y-coordinte. y-xis DEMO P(, ) x-xis

21 BACKGROUND KNOWLEDGE 1 THE DISTANCE FORMULA If A(x 1, y 1 ) nd B(x, y ) re two points in plne, then the distnce etween these points is given y AB = p (x x 1 ) +(y y 1 ). Exmple 5 Find the distnce etween A(, 1) nd B(3, 4). A(, 1) B(3, 4) " " " " x 1 y 1 x y AB = p (3 ) +(4 1) = p 5 +3 = p = p 34 units THE MIDPOINT FORMULA If M is hlfwy etween points A nd B then M is the midpoint of [AB]. A M B If A(x 1, y 1 ) nd B(x, y ) re two points then µ x1 + x the midpoint M of [AB] hs coordintes, y 1 + y : Exmple 6 Find the coordintes of the midpoint of [AB] for A( 1, 3) nd B(4, 7). The x-coordinte of the midpoint = 1+4 The y-coordinte of the midpoint = 3+7 ) the midpoint of [AB] is (1 1, 5). =5 = 3 =11 THE GRADIENT OR SLOPE OF A LINE When looking t line segments drwn on set of xes, it is cler tht different line segments re inclined to the horizontl t different ngles. Some pper to e steeper thn others. The grdient or slope of line is mesure of its steepness. If A is (x 1, y 1 ) nd B is (x, y ) then the grdient of [AB] is y y 1 x x 1 :

22 BACKGROUND KNOWLEDGE Exmple 7 Find the grdient of the line through (3, ) nd (6, 4). (3, ) (6, 4) " " " " x 1 y 1 x y grdient = y y 1 = 4 x x = Note: ² horizontl lines hve grdient of 0 (zero) ² verticl lines hve n undefined grdient ² forwrd sloping lines hve positive grdients ² prllel lines hve equl grdients ² ckwrd sloping lines hve negtive grdients ² the grdients of perpendiculr lines re negtive reciprocls, i.e., if the grdients re m 1 nd m then m = 1 m 1 or m 1 m = 1. This is true except when the lines re prllel to the xes. EQUATIONS OF LINES The eqution of line sttes the connection etween the x nd y vlues for every point on the line, nd only for points on the line Equtions of lines hve vrious forms: ² All verticl lines hve equtions of the form x = where is constnt. ² All horizontl lines hve equtions of the form y = c where c is constnt. ² If stright line hs grdient m nd psses through (, ) then it hs eqution y = m or y = m(x ) fpoint-grdient formg x which cn e rerrnged into y = mx + c fgrdient-intercept formg or Ax + By = C fgenerl formg Exmple 8 Find, in grdient-intercept form, the eqution of the line through ( 1, 3) with slope of 5. To find the eqution of line we need to know its grdient nd point on it. The eqution of the line is y 3 x 1 =5 i.e., y 3 x +1 =5 ) y 3=5(x +1) ) y =5x +8

23 BACKGROUND KNOWLEDGE 3 Exmple 9 Find, in generl form, the eqution of the line through (1, 5) nd (5, ). The slope = = So, the eqution is y x 5 = 3 4 y + ) x 5 = 3 4 ) 4y +8=3x 15 ) 3x 4y =3 AXES INTERCEPTS Axes intercepts re the x- nd y-vlues where grph cuts the coordinte xes. The x-intercept is found y letting y =0. The y-intercept is found y letting x =0. y y-intercept x-intercept x Exmple 30 For the line with eqution x 3y =1, find the xes intercepts. When x =0, 3y =1 ) y = 4 So, the y-intercept is 4 nd the x-intercept is 6. When y =0, x =1 ) x =6 DOES A POINT LIE ON A LINE? A point lies on line if its coordintes stisfy the eqution of the line. Exmple 31 Does (3, ) lie on the line with eqution 5x y =0? Sustituting (3, ) into 5x y =0 gives 5(3) ( ) = 0 i.e., 19 = 0 which is flse ) (3, ) does not lie on the line.

24 4 BACKGROUND KNOWLEDGE WHERE GRAPHS MEET Exmple 3 Use grphicl methods to find where the lines x + y =6 nd x y =6 meet. For x + y =6: when x =0, y =6 when y =0, x =6 For x y =6: when x =0, y =6, ) y = 6 when y =0, x =6, ) x =3 The grphs meet t (4, ). Check: 4+=6 X nd 4 =6 X x 0 6 y 6 0 x 0 3 y 6 0 y!-@=6 (4, ) x!+@=6 There re three possile situtions which my occur. These re: Cse 1: Cse : Cse 3: The lines meet in The lines re prllel nd The lines re coincident. single point of never meet. There is There re infinitely mny intersection. no point of intersection. points of intersection. INVESTIGATION FINDING WHERE LINES MEET USING TECHNOLOGY Grphing pckges nd grphics clcultors cn e used to plot grphs nd hence find their point(s) of intersection. This cn e useful if the solutions re not integer vlues. One downside of using grphics clcultors is tht most require the equtions to e entered in the form y = f(x). So, if the eqution of stright line is given in generl form, it must e rerrnged into grdient-intercept form. Suppose we wish to use technology to find the point of intersection of 4x +3y =10 nd x y = 3: GRAPHING PACKAGE If you re using the grphing pckge, click on the icon to open the pckge nd enter the two equtions.

25 BACKGROUND KNOWLEDGE 5 If you re using grphics clcultor, follow the following steps: Step 1: We rerrnge ech eqution into the form y = mx + c: 4x +3y =10 ) 3y = 4x +10 ) y = 4 3 x x y = 3 ) y = x 3 ) y = x + 3 Step : Enter the functions Y 1 = 4X=3+10=3 nd Y = X=+3=. Step 3: Drw the grphs of the functions on the TI sme set of xes. You my hve to chnge the viewing window. C Step 4: Use the uilt in functions to clculte the point of intersection. Thus, the point of intersection is (1, ). Wht to do: 1 Use technology to find the point of intersection of: y = x +4 5x 3y =0 x +y =8 y =7 x d x + y =7 3x y =1 e y =3x 1 3x y =6 c x y =5 x +3y =4 f y = x 3 + x +3y =6 Comment on the use of technology to find the point(s) of intersection in 1end 1f. EXERCISE L 1 Use the distnce formul to find the distnce etween the following pirs of points: A(1, 3) nd B(4, 5) O(0, 0) nd C(3, 5) c P(5, ) nd Q(1, 4) d S(0, 3) nd T( 1, 0). Find the midpoint of [AB] for: A(3, 6) nd B(1, 0) A(5, ) nd B( 1, 4) c A(7, 0) nd B(0, 3) d A(5, ) nd B( 1, 3). 3 By finding y-step nd n x-step, determine the slope of ech of the following lines: c d e f

26 6 BACKGROUND KNOWLEDGE 4 Find the grdient (slope) of the line pssing through: (, 3) nd (4, 7) (3, ) nd (5, 8) c ( 1, ) nd ( 1, 5) d (4, 3) nd ( 1, 3) e (0, 0) nd ( 1, 4) f (3, 1) nd ( 1, ). 5 Clssify the following pirs of lines s prllel, perpendiculr or neither. Give resons for your nswers. c d e f 6 Stte the slope of the line which is perpendiculr to the line with grdient (slope): c 4 d 1 3 e 5 f 0 7 Find, in grdient-intercept form, the eqution of the line through: (4, 1) with slope (1, ) with slope c (5, 0) with slope 3 d ( 1, 7) with slope 3 e (1, 5) with slope 4 f (, 7) with slope 1: 8 Find, in generl form, the eqution of the line through: (, 1) with slope 3 (1, 4) with slope 3 c (4, 0) with slope 1 3 d (0, 6) with slope 4 e ( 1, 3) with slope 3 f (4, ) with slope 4 9 : 9 Find the equtions of the lines through: (0, 1) nd (3, ) (1, 4) nd (0, 1) c (, 1) nd ( 1, 4) d (0, ) nd (5, ) e (3, ) nd ( 1, 0) f ( 1, 1) nd (, 3) 10 Find the equtions of the lines through: (3, ) nd (5, ) (6, 7) nd (6, 11) c ( 3, 1) nd ( 3, 3) 11 Copy nd complete: Eqution of line Grdient x-intercept y-intercept x 3y =6 4x +5y =0 c y = x +5 d x =8 e y =5 f x + y =11 g 4x + y =8 h x 3y =1 If line hs eqution y mx c then the grdient of the line is m.

27 1 Does (3, 4) lie on the line with eqution 3x y =1? Does (, 5) lie on the line with eqution 5x +3y = 5? c Does (6, 1 ) lie on the line 3x 8y =? 13 Use grphicl methods to find where the following lines meet: BACKGROUND KNOWLEDGE 7 x +y =8 y = 3x 3 c 3x + y = 3 y =x 6 3x y = 1 x 3y = 4 d x 3y =8 e x +3y =10 f 5x +3y =10 3x +y = 1 x +6y = 11 10x +6y = 0 Exmple 33 A stright rod is to pss through the points A(5, 3) nd B(1, 8). c Find where this rod meets the rod given y: i x =3 ii y =4 If we wish to refer to the points on the rod (AB) etween A nd B, how cn we indicte this? Does C(3, 0) lie on the rod? B(1' 8) y 4 x 3 A(5' 3) First we must find the eqution of the line representing the rod. Its grdient is m = = 5 4 y 3 So, its eqution is x 5 = 5 4 ) 4(y 3) = 5(x 5) ) 4y 1 = 5x +5 ) 5x +4y =37 i When x =3, 5(3) + 4y =37 ) y =37 ) 4y = ) y =5 1 ) meet t (3, 5 1 ). We restrict the possile x-vlues to 1 6 x 6 5. c ii When y =4, 5x + 4(4) = 37 ) 5x +16=37 ) 5x =1 ) x =4: ) they meet t (4:, 4). If C(3, 0) lies on the line, its coordintes must stisfy the line s eqution. Now LHS = 5(3) + 4( 0) = =35 6= 37 ) C does not lie on the rod.

28 8 BACKGROUND KNOWLEDGE 14 Find the eqution of the: horizontl line through (3, 4) verticl line with x-intercept 5 c verticl line through ( 1, 3) d horizontl line with y-intercept e x-xis f y-xis. 15 Find the eqution of the line which is: through A( 1, 4) nd with grdient 3 4 through P(, 5) nd Q(7, 0) c prllel to the line with eqution y =3x nd psses through (0, 0) d prllel to the line with eqution x +3y =8 nd psses through ( 1, 7) e perpendiculr to the line with eqution y = x + 5 nd psses through (3, 1) f perpendiculr to the line with eqution 3x y =11 nd psses through (, 5). 16 A is the town hll on Scott Street nd D is Post Office on Kech Avenue. Digonl Rod intersects Scott Street t B nd Kech Avenue t C. c Find the eqution of Kech Avenue. Find the eqution of Pecock Street. Find the eqution of Digonl Rod. (Be creful!) A(3,17) d Plunkit Street lies on the mp reference line x =8. Where does Plunkit Street intersect Kech Avenue? Pecock St Scott St Digonl Rd B(7,0) Kech Av D(13,1) E C(5,11) Exmple 34 Find the eqution of the tngent to the circle with centre (, 3) t the point ( 1, 5). C(, 3) The grdient of [CP] is 3 5 ( 1) = 3 = 3 ) the grdient of the tngent t P is 3. Since the tngent psses through ( 1, 5), P( 1, 5) its eqution is y 5 x 1 = 3 ) (y 5) = 3(x +1) ) y 10 = 3x +3 ) 3x y = 13 The tngent is perpendiculr to the rdius t the point of contct. 17 Find the eqution of the tngent to the circle with centre: (0, ) t ( 1, 5) (3, 1) t( 1, 1) c (, ) t(5, ).

29 BACKGROUND KNOWLEDGE 9 Exmple 35 Mining towns re situted t B(1, 6) nd A(5, ). Where should the rilwy siding S e locted so tht ore trucks from either A or B would trvel equl distnces to rilwy line with eqution x =11? B(1, 6) A(5, ) S x 11 rilwy line 18 A(5, 5) nd B(7, 10) re houses nd y =8 is gs pipeline. Where should the one outlet from the pipeline e plced so tht it is the sme distnce from oth houses? 19 (CD) is wter pipeline. A nd B re two towns. A pumping sttion is to e locted on the pipeline to pump wter to A nd B. Ech town is to py for their own service pipes nd they insist on equlity of costs. Where should C e locted to ensure equlity of costs? Wht is the totl length of service pipe required? c Suppose S hs the coordintes (11, ). Now BS = AS ) p (11 1) +( 6) = p (11 5) +( ) ) 10 +( 6) =6 +( ) fsquring oth sidesg ) = ) 1 +4 =4 100 ) 8 = 96 ) =1 So, the rilwy siding should e locted t (11, 1). Exmple 36 A(, 3) B A If the towns gree to py equl mounts, would it e cheper to instll the service pipeline from D to B to A? C D B(5, 4) y Scle: 1 unit 1 km y P N Q(, 4) x x Scle: ech grid unit is 1 km. A tunnel through the mountins connects town Q(, 4) to the port t P. P is on grid reference x =6nd the distnce etween the town nd the port is 5 km. Assuming the digrm is resonly ccurte, wht is the horizontl grid reference of the port?

30 30 BACKGROUND KNOWLEDGE Suppose P is t (6, ). Now PQ =5 ) p (6 ) +( 4) =5 ) p 16 + ( 4) =5 ) 16 + ( 4) =5 ) ( 4) =9 ) 4= 3 ) =4 3=7or 1 But from the digrm, P is further north thn Q ) >4 So, P is t (6, 7) nd the horizontl grid reference is y =7. 0 y 8 Clifton Json s girlfriend lives in house on Clifton Highwy Highwy which hs eqution y = 8. The distnce s the crow flies from Json s house to his girlfriend s house is 11:73 km. If Json lives t (4, 1), wht re the coordintes of his girlfriend s house? J(4, 1) Scle: 1 unit 1 km. 1 A circle hs centre (, ) nd rdius r units. P(x, y) moves on the circle. Show tht (x ) +(y ) = r. Find the eqution of the circle with: i centre (4, 3) nd rdius 5 units ii centre ( 1, 5) nd rdius units iii centre (0, 0) nd rdius 10 units iv ends of dimeter ( 1, 5) nd (3, 1). Find the centre nd rdius of the circle: (x 1) +(y 3) =4 x +(y +) =16 c x + y =7 3 Consider the circle with eqution (x ) +(y +3) =0: Stte the circle s centre nd rdius. Show tht (4, 1) lies on the circle. c Find the eqution of the tngent to the circle t the point (4, 1). 4 Recll tht the perpendiculr isector of chord of circle psses through the centre of the circle. Find the centre of circle pssing through points P(5, 7), Q(7, 1) nd R( 1, 5) y finding the perpendiculr isectors of [PQ] nd [QR] nd solving them simultneously.

31 BACKGROUND KNOWLEDGE (Answers) 31 EXERCISE A p p p p c 4 d 1 e 4 f 6 g h 6 5 p p c p 5 d 8 p 5 e p 5 f 9 p 3 g 3 p 6 h 3 p 3 p p 3 c p 5 d 4 p e 3 p 3 f 3 p 5 g 4 p 3 h 3 p 6 i 5 p j 4 p 5 k 4 p 6 l 6 p 3 4 p 3 8 p c 5 p 6 d 10 p 3 e 3 p 3 f p p 5 p 7 p 3 c d p 5 e 5 p f 3 p 6 g 4 p 3 h 5 p 7 7 i p 7 j EXERCISE B 1 :59 10 : c : d : e : f 4: g 4: h 4:07 10 i 4: j 4: k 4: : m mm c mm d 1: o C e c 100 d e f 86 g h :004 0:05 c 0:001 d 0: e 0: f 0:86 g 0: h 0: : m c light yers d 0: mm 6 1: : c 5: d 1:36 10 e : f 1: : km 9: km c 9: km 8 1: m : m c 9: m EXERCISE C 1 The set of ll rel x such tht x is greter thn 5. The set of ll integers x such tht x is less thn or equl to 3. c The set of ll y such tht y lies etween 0 nd 6. d The set of ll integers x such tht x is greter thn or equl to, ut less thn or equl to 4. x is, 3 or 4. e The set of ll t such tht t lies etween 1 nd 5. f The set of ll n such tht n is less thn or greter thn or equl to 6 fx j x>g fx j 1 <x6 5g c fx j x 6 or x > 3g d fx j 16 x 6 3, x Z g e fx j 0 6 x 6 5, x Z g f fx j x<0g 3 c d p x x x x e x EXERCISE D 1 10x 10 9x c 5x +5y d 8 8x e 1 f cnnot e simplified x x c 4 3 d 3x 3 16x +11x x 3 c 4x d EXERCISE E 1 x =10 x>6 c x = 4 d x =51 e x< 10 5 f x =14 g x 6 9 h x =18 i x = 3 x =5, y = x = 3, y = 8 3 c x =, y =5 d x = 45 11, y = e no solution f x =66, y = 84 EXERCISE F c 16 d 18 e f 3 c 6 d 6 e 5 f 1 g 1 h 5 i 4 j 4 k l 3 x = 3 no solution c x =0 d x =4 or e x = 1 or 7 f no solution g x =1 or 1 3 h x =0 or 3 i x = or 14 5 EXERCISE G 1 x +5x +3 3x +10x +8 c 10x + x d 3x +x 10 e 6x +17x+14 f 6x 13x+5 g 15x +11x 1 h 15x 11x+ i x 17x+1 j 4x 16x +15 k x 3x l 4x x +6 x 36 x 64 c 4x 1 d 9x 4 e 16x 5 f 5x 9 g 9 x h 49 x i 49 4x j x k x 5 l 4x 3 3 x +10x +5 x +14x +49 c x 4x +4 d x 1x +36 e x +6x +9 f x +10x +5 g x x +11 h x 0x+100 i 4x +8x+49 j 9x +1x +4 k 4x 0x +5 l 9x 4x y =x +10x +1 y =3x 6x +7 c y = x +6x +7 d y = x 4x 15 e y =4x 4x +0 f y = 1 x 4x 14 g y = 5x +35x 30 h y = 1 x +x 4 i y = 5 x +0x 40 5 x +1x +19 3x +3x 16 c x +6x 6 d x x +5 e x 16x +33 f 3x +4 g 7x +8 h 7x +18x +1 i x +19x 3 j 7x 16x + EXERCISE H 1 3x(x+3) x(x+7) c x(x 5) d 3x(x 5) e (3x 5)(3x+5) f (4x+1)(4x 1) g (x )(x+) h 3(x+ p 3)(x p 3) i 4(x+ p 5)(x p 5) j (x 4) k (x 5) l (x ) m (4x +5) n (3x +) o (x 11) (x + 8)(x +1) (x + 4)(x +3) c (x 9)(x +) d (x + 7)(x 3) e (x 6)(x 3) f (x + 3)(x ) g (x )(x +1) h 3(x 11)(x 3) i (x +1) j (x+5)(x ) k (x 8)(x+3) l (x 6)(x 1) m 3(x 1) n (x +1) o 5(x 4)(x +) 3 (x 3)(x+4) (3x+1)(x ) c (7x )(x 1) d (3x )(x+1) e (x 3)(x+1) f (5x 3)(x+1) g (x+1)(x 6) h (3x+7)(x 4) i (4x+3)(x 1) j (5x+3)(x 3) k (3x 1)(x+8) l (3x+)(x+1) m (x + 3)(x 1) n (6x + 1)(x 3) o 3(x + 7)(x ) p (3x )(3x 5) q (4x 9)(x +3) r (4x + 3)(3x +1)

32 3 BACKGROUND KNOWLEDGE (Answers) s (6x + 1)(x +3) t (5x 4)(3x ) u (7x + 5)(x 3) 4 (x+4)(x+1) ( x)(5 3x) c 3(x+)(x+7) d 4(x + 5)(x + 11) e x(x +3) f 5(x +3) g (x )(3x + 6) h (x 1)(x 1) 5 (x + 7)(x 1) (x + 1)(3 x) c 1(x +1) d 4x(x +4) e (3x + )(x +4) f h(x + h) g 1(x+1) h 5(3x 4)(x 4) i 3(x+9)(5x+9) EXERCISE I 1 x = x = 0 y e x = f x = 5 h x = c y z = c i z = d c x = y c m 1 3y c z = d 3 x = d g d x = d 3y 7 x = t c 3 = F m r = C ¼ c d = V lh d K = A r r A 4 r = x = 5p q 3V n N c r = 3 d x = 3 ¼ 4¼ D 5 = d n l =5T c = p + c d l = gt 4¼ e = P f h = A ¼r g r = E ¼r I R h q = p B A r 6 = d 3V 1:9 7 r = 3 :1 cm k 4¼ r S 8 t = 15:81 sec 9 v = uf i 9:5 cm ii 10:9 cm u f µ 10 v = sc 1 m 0 = c p m m m 0 m v = p 8 3 c EXERCISE J x x +3 5 x e 8x 4x +17 f 15 1 e 3 c x +5 x + x +3 x +1 f 11 x x 4 x +3 1 x 7x 6x 5 (x 5)(x ) 8x +6x +8 x 16 c : ms 1 EXERCISE K.1 1 Hint: Consider s AEC, BDC d c c x x x 1 d d 1 (x )(x 3) 3x +3x +13 (x 3)(x +1) 14 x 4 5x + x +1 Hint: Let M e on [AB] so tht [PM]? [AB]. Let N e on [AC] so tht [PN]? [AC]. Join [PM], [PN] nd consider the two tringles formed. EXERCISE K. 1 x =:4 x =:8 c x =3 3 d x = e x =7 f x =7: 1:35 m tll EXERCISE L 1 p 13 units p 34 units c p 0 units d p 10 units (, 3) (, 1) c (3 1, 1 1 ) d (, 1 ) 3 c 1 d 0 e 4 f undefined c undefined d 0 e 4 f 5 prllel, slopes 5 prllel, slopes 1 7 c perpendiculr, slopes 1, d neither, slopes 4, 1 3 e neither, slopes 7, 1 5 f perpendiculr, slopes, c d 3 e 5 f undefined 7 y =x 7 y = x +4 c y =3x 15 d y = 3x +4 e y = 4x +9 f y = x x y =4 3x +y =11 c x 3y =4 d 4x + y =6 e 3x y =0 f 4x +9y = 9 x 3y = 3 5x y =1 c x y =3 d 4x 5y =10 e x y = 1 f x +3y = 5 10 y = x =6 c x = 3 11 Eqution of line Slope x-int. y-int. x 3y = x +5y = c y = x d x =8 undef. 8 no y-int. e y =5 0 no x-int. 5 f x + y = g 4x + y =8 4 8 h x 3y = yes no c yes 13 (4, ) (, 3) c ( 3, 6) d (4, 0) e prllel lines do not meet f coincident lines 14 y = 4 x =5 c x = 1 d y = e y =0 f x =0 15 3x 4y = 19 x y =7 c y =3x d x +3y =19 e x y =5 f x +3y =13 16 x 8y = 83 8x + y =41 c 9x y =3 for 5 6 x 6 7 d (8, ) 17 x 3y = 16 x y = 3 c x =5 18 (4 3 4, 8) 19 ( 1, 7) 3 8:03 km c yes (6:16 km) 0 (13:41, 8) or( 5:41, 8) 1 Hint: Use the distnce formul to find the distnce from the centre of the circle to point P. i (x 4) +(y 3) =5 ii (x +1) +(y 5) =4 iii x + y = 100 iv (x 1) +(y 3) =8 centre (1, 3), rdius units centre (0, ), rdius 4 units c centre (0, 0), rdius p 7 3 centre (, 3), rdius p units 0 units Hint: Sustitute (4, 1) into eqution of circle. c x +y =6 4 (3, 3) 1 4

Background knowledge

Background knowledge Bckground knowledge Contents: B C D E F G H I J K L M N Surds nd rdicls Scientific nottion (stndrd form) Numer systems nd set nottion lgeric simplifiction Liner equtions nd inequlities Modulus or solute

More information

Directions for completing the IB Math SL Presumed Knowledge unit

Directions for completing the IB Math SL Presumed Knowledge unit Directions for completing the IB Mth SL Presumed Knowledge unit 2018-2019 Plese complete the Bckground Knowledge unit nd do not wit to strt completing it! Go through ech section nd mke sure you cn do ech

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Stage 11 Prompt Sheet

Stage 11 Prompt Sheet Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper

More information

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12. Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

More information

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd

More information

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

Answers for Lesson 3-1, pp Exercises

Answers for Lesson 3-1, pp Exercises Answers for Lesson -, pp. Eercises * ) PQ * ) PS * ) PS * ) PS * ) SR * ) QR * ) QR * ) QR. nd with trnsversl ; lt. int. '. nd with trnsversl ; lt. int. '. nd with trnsversl ; sme-side int. '. nd with

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

Chapter 1: Fundamentals

Chapter 1: Fundamentals Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

More information

TO: Next Year s AP Calculus Students

TO: Next Year s AP Calculus Students TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

More information

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

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (

More information

Math Sequences and Series RETest Worksheet. Short Answer

Math Sequences and Series RETest Worksheet. Short Answer Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning

More information

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically. Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

Number systems: the Real Number System

Number systems: the Real Number System Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset

More information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

Preparation for A Level Wadebridge School

Preparation for A Level Wadebridge School Preprtion for A Level Mths @ Wdebridge School Bridging the gp between GCSE nd A Level Nme: CONTENTS Chpter Removing brckets pge Chpter Liner equtions Chpter Simultneous equtions 6 Chpter Fctorising 7 Chpter

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+ .1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Precalculus Spring 2017

Precalculus Spring 2017 Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information

Summary Information and Formulae MTH109 College Algebra

Summary Information and Formulae MTH109 College Algebra Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

More information

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

More information

CHAPTER 9. Rational Numbers, Real Numbers, and Algebra

CHAPTER 9. Rational Numbers, Real Numbers, and Algebra CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers

More information

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

Math 154B Elementary Algebra-2 nd Half Spring 2015

Math 154B Elementary Algebra-2 nd Half Spring 2015 Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Trigonometric Functions

Trigonometric Functions Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

More information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 12 Mathematics Extension 2 HSC Trial Examination 2014 Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

More information

Minnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017

Minnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017 Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.

More information

0.1 THE REAL NUMBER LINE AND ORDER

0.1 THE REAL NUMBER LINE AND ORDER 6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression. SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

Lesson 2.4 Exercises, pages

Lesson 2.4 Exercises, pages Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0 - ( ) () 0 c) -7 + d) (7) ( ) 7 - + 8 () ( 8). Expnd nd simplify. ) b) - 7 - + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing

More information

PARABOLA EXERCISE 3(B)

PARABOLA EXERCISE 3(B) PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution

More information

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

More information

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

More information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

Scientific notation is a way of expressing really big numbers or really small numbers.

Scientific notation is a way of expressing really big numbers or really small numbers. Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific

More information

Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Maths. Self Check Booklet. visit   for a wealth of free online maths resources at all levels from S1 to S6 Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

More information

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).

More information