( ) ( ) ( ) ( ) (b) (a) sin. (c) sin sin 0. 2 π = + (d) k l k l (e) if x = 3 is a solution of the equation x 5x+ 12=

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Adelia George
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1 Eesio Mahemaics Soluios HSC Quesio Oe (a) d 6 si 4 6 si si (b) (c) 7 4 ( si ).si +. ( si ) si + 56 (d) k + l ky + ly P is, k l k l , , ( 5,9) if is a soluio of he equaio 5+ Therefore + is a facor of 5+ d d (e) By he facor heorem, + is a facor of he polyomial For : (f) Le u + u du d d 5 u udu 5 u u d u u d d si +.sicos si ( + ) 4+ ( ) + These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

2 Eesio Mahemaics Soluios HSC Quesio Two (a) + ( a) f (c) f lim h h ( + ) ( a h ) a+ h + a+ ha a lim h h 6 lim h h 6ah+ h + h lim h h lim a+ f a h f h h a + ah+ h + a+ h a a (b) (i) There are 9! possible permuaios of 9 uique leers Bu he wo As are idisiguishable. Therefore he umber of arragemes as required 9! 844 (ii) The 5 uique cosoas ca be arraged i 5! ways, 4! he 4 vowels wih a repeaed A i ways. 4!5! Therefore oal arragemes 44 (d) 9 r 9 9r Cr ( ) r 9 r r r ( ) ( ) r 9 r r 9 r 8r Cr Therefore for cosa erm, he power o which is raised 8 r r C6 The erm is: 84 C e (i) d l( + e ) + C + e (ii) cos d + cos6 d si ( ) ( ) These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

3 Eesio Mahemaics Soluios HSC Quesio Three (a) f si+ cos f cossi ( ) ( ) f Newo's mehod:,. + f si. + cos.. Approimae roo. cos. si..6 ( sigifica figures) (b) (i) AOB APB (I circle C, he agle a he cere is wice he agle a he circumferece subeded by he same arc ¼AB) AOB θ (ii) TAB AOB (I C, he agle bewee he age TA ad he chord BA equals he agle i he alerae segme AOB ) TAB θ (iii) TAB BPA+ APB (eerior agle of a riagle heorem) θ θ + APB APB θ VBAP is isosceles(base agles APB, ABP are equal) Therefore PA BA (equal sides of similar riage, VBAP) These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

4 Eesio Mahemaics Soluios HSC (c) (i) si θ+θ siθ cosθ + cosθsiθ si 4si si 4si θ siθ siθ si θ + cos θ.siθ cosθ si si + si cos θ θ θ θ si si + si si θ θ θ θ si si + si si siθ 4si θ (ii) siθ siθ θ θ θ 4si θ θ θ θ For siθ, θ,, θ For siθ : θ siθ ± 5 7 θ,,, θ Therefore for siθ si θ, θ 5 7 θ,,,,,, These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma 4

5 Eesio Mahemaics Soluios HSC Quesio Four (a) LHS has he same sig as y +,. [Draw graph] For V >, V ms - Therefore for < - (b) o Sice a45, he raio bewee he magiudes of he y ad compoes of velociy Bu due o he egaive direcio of moio of he paricle verically, hey are i fac opposie: & y& & V y& V Bu we also kow ha 4m a his poi. 4 V 4 V V V 4 These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma 5

6 Eesio Mahemaics Soluios HSC (c) Le v & v 4 + C dv d dv dv d v dv d v && v d d d d dv d d d v 4 d v C + A, v C C v d v d For v < : d 6 d d d 6 6 cos + C 6 A, : d C cos cos + 6 6cos + 6si + 6 These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma 6

7 Eesio Mahemaics Soluios HSC Quesio Five (a) f (i) () cos (ii) f (iii) ( ) f cos ( ) ( ) cos f A f d (b) cos. si 6u LHS q+ p q p r r r r q p q ( p) r r r r r q r r r p ( ) p r r r r r q + p r r + r+ r r For eve r, ( ) ad such a erm q p [ ] r r+ For odd r, ( ) ad such a erm q p [ + ] q p r r r r r+ Therefore he overall sum, LHS q p + ( ) r r q p + q p +... RHS r r r r These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Therefore, if is odd, he las erm is q p p If is eve, he las erm cacels o ad so r becomes he fial erm i he epasio: q qp p + Joel Nohma 7

8 Eesio Mahemaics Soluios HSC (c) (i) Probabiliy of rollig r6s is: r 5 Pr r 6 6 r (ii) 5 Le p, q 6 6 The probabiliy a odd umber of 6s are rolled is he probabiliy ha si is rolled or sies are rolled or 5 sies are rolled ad so o... P P + P + P +... odd 5 ( p) ( q) + ( p) ( q) +... {( q+ p) ( q p) } from par (b) as required These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma 8

9 Eesio Mahemaics Soluios HSC Quesio si (a) For : which is divisible by 9. Therefore he proposiio is rue for Assume he proposiio rue for k, k Ie, assume 9, k + k + + k + N N We eed o prove he proposiio rue for k + + ( k + ) + ( k + ) Ie, prove ha k+ is divisible by 9 k+ + k + + k + k+ + k + + k + k + k + k + k+ + k + + 9k + 7k + 7 N + k + k by assumpio ( N k k ) which is divisible by 9 Therefore he proposiio is rue for k + if i is rue for k Bu i is also rue for. Therefore by mahemaical iducio i is rue for all, ie,,,... (b) (i) dy dy d d d d a a Therefore he gradie of he ormal a P is Hece he equaio of he ormal is: ( ) y a a y a a + y a+ a These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma 9

10 Eesio Mahemaics Soluios HSC (ii) ( aq aq ) Le Q be, The gradie of he ormal mpr For PR QR he age a Q mus be P o he ormal a P dy Ie, a Q, d q a a Q is herefore, (iii) The equaio of PR is y a a () + + The equaio of is QR + qy aq + aq Subsiuig q : y a a y a a + () Addig () ad () o elimiae we ge: y a a + y + + a + a y + a y a a + + () Sub () io (): a + a a + + a a + a a a a These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

11 Eesio Mahemaics Soluios HSC (iv) a + a + + y a a a aya y + a a These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

12 Eesio Mahemaics Soluios HSC Quesio Seve (a) (i) dv d v d d d v d v C + v + C For, v, C v (ii) + ( ) v Takig v > a d d d d l + C A, : l+ C C e For v iiially, e holds rue Bu uder his moio, he paricle always has a posiive velociy e + These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Joel Nohma

13 Eesio Mahemaics Soluios HSC (b) (i) By he cosie rule: +. cos AP AO PO AOPO + + AO PO AOPO. AO PO AOPO. o Now AO OTco45 h Ad PO OTcoα hcoα Therefore: AP h + h co α h co α () (ii) +. cosθ AP AT PT AT PT AT + PT AP cos θ () AT. PT AT AO + TO h + h h AT h () PT PO + TO PT h co α + h co h α + Bu cos si α + α co α + cosec α PT hcosec α (4) These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. Subbig (), (), (4) io (): cosθ cosθ ( α ) ( α α) h + h + h + h h co co co h + h coα h cosecα + coα cosecα cosα + siα siα + ( siα cosα). hhcosecα Joel Nohma

14 Eesio Mahemaics Soluios HSC (iii) siα + cosα R cos α φ Feel free o derive his - I wo'! R + φ a a 5 siα + cosα cos α a 8 5 θ 8 ( α ) cos cos a 5 θ si a 5 8 cos ( α a ) 8 Saioary pois where θ ( α ) si a α a for α 5 8 If α a, θ cos ( α ) If α a., θ If α a +., θ Therefore here is a local miimum a a,cos 5 8 These soluios are copyrigh, Joel Nohma, ad may be freely disribued, bu o sold. + - As α, θ icreases owards cos - As α, θ icreases owards cos Joel Nohma 4

12 th Mathematics Objective Test Solutions

12 th Mathematics Objective Test Solutions Maemaics Objecive Tes Soluios Differeiaio & H.O.D A oes idividual is saisfied wi imself as muc as oer are saisfied wi im. Name: Roll. No. Bac [Moda/Tuesda] Maimum Time: 90 Miues [Eac rig aswer carries