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1 Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without using clculus nd show tht this point is minimum point. Find the minimum vlue of Hence, show tht the curve f ( β ) touches the -is if, where nd re constnts to e determined in terms of k. f in terms of k. α lies entirel ove the -is if < k <, k or k, γ cuts the -is in two distinct points if k < or k >. ii. Prove tht the stright line m f in two rel nd distinct points for intersects the curve ll rel nd finite vlues of m if nd onl if k <. () Let g 7 6. Using Reminder theorem repetedl show tht ( ) is fctor of Epress g in the form ( ) ( c) Deduce tht g for ll rel vlues of. g., where, nd c re constnts to e determined. nswer () f k k ( k ) k k ( k ) ( k k ) This is of the form f ( ), where k nd k k. When ( k,) Let k, f k k Is the turning point of f, where k k. k Δ nd k Δ, where Δ > f ( ) ( k Δ k) ( k ) f ( ) ( k Δ k) ( k ) k ( Δ ) ( Δ ) k f ( ) > f ( k) f ( ) > f ( k) k is the turning point nd the minimum point of f. - - Done B : Chndim Peiris (B.Sc - Specil)

2 Minimum vlue of f f ( k) k k If k k > then, the curve f lies entirel ove the -is. k k > k k < Sign of ( k )( k ) ( k )( k ) < < k < If < < k then, the curve f k then, the curve f If k k k k k k k k or k If k or If k < k then, the curve f k then, the curve f k k k < k > ( k )( k ) > k < or k > If k < or > k then, the curve f lies entirel ove the -is. touches the -is. touches the -is. cuts the -is in two distinct points. cuts the -is in two distinct points. Sign of ( k )( k ) Let m nd k k intersect ech other. m k k ( k m) ( k ) Here the discriminnt Δ ( k m) ( k ) If the line m intersects the curve f discriminnt ( Δ ) should e positive. Since ( k m) > for ll vlues of m, ( k ) ( k ) < ( k ) < k < in two rel nd distinct points for ll Rthen the should e less thn ero. Similrl, if k < then, the ove discriminnt will e positive. The stright line m finite vlues of m if nd onl if k < intersects the curve f in two rel nd distinct points for ll rel nd - - Done B : Chndim Peiris (B.Sc - Specil)

3 () g 7 6 g g (ccording to the reminder theorem) is fctor of g ( )( ) Let h h ( ) ( ) is fctor of h (ccording to the reminder theorem) g ( ) ( ) ( ) is fctor of g h ( )( ) g ( ) ( ) g of the form ( ) ( c) Here Let f. The discriminnt of for ll But R, >, Equlit holds when, where g for ll R, Equlit holds when, nd c. f (S Δ ) is. i.e. Δ < - - Done B : Chndim Peiris (B.Sc - Specil)

4 . B for ll R. () Find constnts nd B such tht Hence, determine () r Show tht n u r f for r Ζ, such tht u r f () r f ( r ), whereu r. r r ( n ) r r. Show tht the series r ur is convergent nd find the vlue of r () Sketch, in the sme figure, the grphs of nd. Hence, find the set of vlues of for which 6. B considering the grph of k, for n k R, in the sme figure find for wht vlue of l the eqution 6 l hs onl one rel solution. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur u. r nswer B When, When B ( ) 8 B () B B u r r ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) - - Done B : Chndim Peiris (B.Sc - Specil)

5 ( r ) ( r ) u r f () r f ( r ), where f () r. r Let S () r f () () () u r u f f u f f f r u f f f n f ( n ) f ( n ) f ( n) ( n) f u n u n u n n r n f n u u r f () f ( n ) ( ) ( n ) ( ) r r n S n n u lim S n ( ) r r n n lim n lim n ( n ) S This is finite vlue. Since S is finite vlue, the series ur is convergent. r The vlue of u r r - - Done B : Chndim Peiris (B.Sc - Specil)

6 () ; when ; when < ; when ; when < 8 6 Finding the vlue of ; Finding the vlue of ; The required rnge of is, When k, k ecomes. Then there is onl one non-trivil position where nd intersect ech other. 6 6 Here l. 6 6 When l, the eqution 6 l hs onl one non trivil solution Done B : Chndim Peiris (B.Sc - Specil)

7 - 7 - Done B : Chndim Peiris (B.Sc - Specil). () Let e mtri. Show tht O I, where I is the identit mtri nd O is the ero mtri. Hence, find. Let 6 B e mtri. Show tht B B. Hence or otherwise find non-ero mtri C such tht O BC. () Let e comple numer. Prove tht nd Re. Hence, show tht for n two comple numers nd. Deduce tht. If < i then, show tht < <. Shde the region R consisting the set of points in the rgnd digrm which represent the comple numer for which i nd rg. nswer () I ) ( O I (); O I O I I 6 B 6 B

8 6 B B Let C c d c d BC 6 c d 6c 6d Since BC O, c nd 6c c d nd 6d d t Let t, where t is prmeter. c k Let k, where k is prmeter. d C () Let i, where, R i () ( i)( i) i () ( Q i ) From () nd (); Since >, > Re, when. When, Re Re, Equlit holds when. Let i nd i, where,,, R. i Wnt: ( )( ) [( ) i( )]( [ ) i( )] ( ) ( ) ( ) Done B : Chndim Peiris (B.Sc - Specil)

9 - 9 - Done B : Chndim Peiris (B.Sc - Specil) Since,. i.e. Let i i i < < i This is circle whose centre, nd rdius. ccording to the ove rgnd digrm, O is the minimum distnce from O to the nd OB is the mimum distnce from O to the. Min nd M < < Where rg i R O,,, B Re Im O, Re Im, B R

10 . () B considering onl first derivtive find the minimum nd mimum vlue of Sketch the grph of. 7 Hence, find for wht vlues of k, the eqution k 7k, where k is rel, hs (i) two coincident rel roots, (ii) three coincident rel roots, (iii) two distinct rel roots, (iv) no rel roots.. 7 () Consider rectngle BCD with B nd BC ( < ). Let P e movle point on CD. The length of PB L, where DP. P is L. Show tht Find the minimum length of L nd the position of P on CD corresponding to this minimum length. lso, find the mimum length of L. nswer () Let 7 Differentiting with respect to. d ( 7) d 7 d d ( 8 ) ( 7) ( ) ( 7) ( )( ) ( 7) ( )( )( ) ( 7) For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur d Criticl points of re given d, nd re the criticl points of. - - Done B : Chndim Peiris (B.Sc - Specil)

11 When,. When, When, < < < < < > ( ) - ( ) - d d (, ) is inflection point. - -, is reltive minimum point., is reltive mimum point. lim nd lim. - - Done B : Chndim Peiris (B.Sc - Specil)

12 Therefore the following figure illustrtes the grph of 7... Let k. i.e. k 7 Let f ( k ) k 7 k 7k. (i) (ii) (iii) (iv) When k or k, f hs two coincident rel roots. When k, re two coincident rel roots of f nd when k, coincident rel roots of f. When k, f hs three coincident rel roots. Here re three coincident rel roots. When < k < or < k <, f hs two distinct rel roots. When k > or k <, f hs no rel roots. re two - - Done B : Chndim Peiris (B.Sc - Specil)

13 () D ( ) P C B PB ( ) P Let L P PB L Differentiting L with respect to dl d dl d Criticl points of ( ) ( ) ( ) ( ) dl L re given d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( )( ) ( ) Since nd,. - - Done B : Chndim Peiris (B.Sc - Specil)

14 - - Done B : Chndim Peiris (B.Sc - Specil) d dl d L d d L d > d L d L hs reltive minimum point when, ccording to the nd derivtive test. L Min f Min f When, L hs mimum vlue. L M f M f

15 . () Show tht ( sin cos ) d () Using integrtion prts, or otherwise find tn d. 8. (c) Using prtil frctions find ( ) nswer () Let I ( sin cos )d sin cos I sin d d. ( sin cos )( sin cos sin cos ) ( sin cos )( sin cos ) sin cos sin cos d sin cos cos cos d cos sin sin d [ cos ] [ sin ] sin cos d cos ( sin )d [ cos cos] [ sin sin ] sin d ( sin ) cos d( cos ) sin cos [ ] sin sin cos cos [ ] 6 I I [ ] [ ] () Let I tn d I tn d d tn d tn d d I d - - Done B : Chndim Peiris (B.Sc - Specil)

16 tn tn I tn tn C, where C is constnt C D (c) ( ) ( ) ( ) ( )( ) B( ) ( C D)( ) B When When When 8 B B D B C D B D () C D () When 6 B 9C 9D 6 9C 9D C D () C D () () (); C D D C () From () nd (); ( D) C D D C D D C () () ; C C 8 From () ; D D 6 From () ; ( ) ( ) d d d ( ) ( ) ( ) d ln ln( ) tn C, where C is constnt Done B : Chndim Peiris (B.Sc - Specil)

17 6. () Find the equtions of the isectors of the ngle etween two non prllel stright lines l c nd l c. Show tht the isector of the cute ngle etween two stright lines given nd is the isector of the otuse ngle etween two stright lines given 7 8 nd 8. () Show tht, for ll vlues of g nd f the circle g f r isects the circumference of the circle r., touching the stright line nd isecting the circumference of the circle. Find the equtions of these two circles. Show tht two circles cn e drwn through the point (,) nswer () l c d d (, ) l l c d d c c l is isector of the ngle etween two stright lines l ndl., is n point on the linel. l is isector of the ngle etween two stright lines l ndl, d d. Let Let ( ) Since c c c c ± B considering nd - of the right hnd side of the ove eqution, we hve two equtions of the stright lines. These re the equtions of the isectors of the ngles etween two non prllel stright lines l c nd l c. One of these equtions is the isector of the cute ngle etween l nd l nd other one is the eqution of the isector of the otuse ngle Done B : Chndim Peiris (B.Sc - Specil)

18 l θ θ l The equtions of the isectors of the ngle etween l nd l re given, ± ± B considering ; B considering -; () () If the eqution () is the eqution of the isector of the cute ngle etween l ndl then, θ <. i.e. tnθ should e less thn one. m, m m m tnθ m m < Here tn θ < is the eqution of the isector of the cute ngle etween l 7 8 φ φ l 8 Done B : Chndim Peiris (B.Sc - Specil)

19 The equtions of the isectors of the ngle etween l ndl re given, ± ± ( 8 ) B considering ; B considering -; () () Let us ssume tht the eqution () is the eqution of the isector of the cute ngle etween l nd l. m, m 8 m m tnφ m m 8 8 > Here tnφ > φ > Therefore is not the eqution of the isector of the cute ngle etween l nd l. i.e. the eqution of the isector of the otuse ngle etween l nd l. is the eqution of the isector of the cute ngle etween nd nd lso it is the eqution of the isector of the otuse ngle etween 7 8 nd 8. () g f r () r () () ; g f g f g (*) f This is of the form m Done B : Chndim Peiris (B.Sc - Specil)

20 (*) is the eqution of the line joining the points of intersection of () nd (). For ll vlues of g nd f, stright line (*) isects the circumference of the eqution circle. For ll vlues of g nd f, the circle g f r isects the circumference of the circle r. (Q(*) is the eqution of dimeter of the circle r ) Let S g f r, where g, f R, is the eqution of the required circle. Centre C ( g, f ) Since the circle S psses the point (,), g f r g f r () Let R is the rdius of the circle S. R g f r Since the circle S touches the line, f R f R ( f ) ( f ) g f r () Since the circle S isects the circumference of the circle, r. Sustituting r in (); g f g f () Sustituting f g f From () nd (); f f r in (); () f f f f f f f 8 f This is qudrtic eqution of f. There re two circles cn e drwn ccording to the ove given conditions. f 8 f f f f or f When f, g When f, g Their equtions: 7. () For tringle BC, prove in the usul nottion, tht - - Done B : Chndim Peiris (B.Sc - Specil)

21 c sin sin B sin C B C Deduce tht ( c) cos cosec () Show tht, for n rel vlue ofθ, the epression tnθ tnθ cnnot tke n vlue etween -7 nd. (c) Epress cos θ 8cosθ sinθ 9sin θ in the form of the cos( θ α ), where nd re constnts nd α is n ngle independent ofθ. Hence or otherwise find the generl solution of the eqution. 8 ( cos sin ) ( cos sin ) 9. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur nswer () Sine rule c sin sin B c sin C B C Proof: Cse : For n cute ngled tringle Drw the line D perpendiculr from to the side BC. c Then BD is right ngled tringle. B D sin B B D B sin B c sin B () C D Similrl, in the right ngle tringle CD, D sin C C D C sin C sin C () c From () nd (); csin B sin C sin B sin C - - Done B : Chndim Peiris (B.Sc - Specil)

22 c In the sme w, it cn e proved tht drwing perpendiculr line from B to the side BC. sin sin C c For n cute ngled tringle, sin sin B sin C Cse : For right ngled tringle Let B is the right ngle of the tringle BC. Drw the line BD perpendiculr from B to the side C. D c Then BD is right ngled tringle. BD sin B BD B sin c sin () B C Similrl, in the right ngled tringle BCD, BD sin C BC BD BC sin C sin C () c From () nd (); sin C csin sin sin C B Since BC is right ngled tringle, sin C B C sin C sin C () C But sin B sin9 We cn write B c sin B (6) c From () nd (6); sin C csin B sin B sin C c For right ngled tringle, sin sin B sin C Cse : For n otuse ngled tringle c B Let the ngle C of the tringle BC is n otuse ngle. Drw the line BD perpendiculr from B to produced C. BCD ˆ C D In the right ngled tringle BCD, BD sin ( C ) BC BD BC sin C sin C (7) Similrl, in the right ngled tringle BD, BD sin B BD B sin c sin (8) c From (7) nd (8); sin C csin sin sin C In the sme w, it cn e proved tht drwing perpendiculr line from C to the side B. sin B sin C - - Done B : Chndim Peiris (B.Sc - Specil)

23 For n otuse ngled tringle, sin sin B ccording to the ove three cses, for n tringle c sin C sin c. sin B sin C c Let k, where k is constnt. sin sin B sin C k sin, k sin B, c k sinc Let us consider ( c) k sin ( c) k( sin B sin C). sin sin B sin C sin B C B C cos sin sin cos Q B C B C cos sin sin cos Q cos sin B C sin sin cos B C sin B C cos cosec ( c) B C B C cos cosec c () Let tnθ tnθ tnθ tn tnθ tnθ tn ( tnθ ) tnθ tnθ - - Done B : Chndim Peiris (B.Sc - Specil)

24 ( tnθ ) tnθ ( tnθ ) tnθ tnθ tnθ tn θ tnθ ( ) tnθ ( ) tn θ This is qudrtic eqution of tn θ. Since tn θ R, the discriminnt of the ove eqution should e. ( ) ( ) 8 Sign curve of ( 7)( ) 6 7 ( 7)( ) 7 [, ) (, 7] The vlue of cnnot tke n vlue etween -7 nd. i.e. The epression tnθ tnθ cnnot tke n vlue etween -7 nd. (c) Let E cos θ 8cosθ sinθ 9sin θ E cos θ sin θ 8cosθ sinθ sin θ 9sin θ cosθ ( Q cos θ sin θ sin θ cos θ ) 7 9sin θ cosθ 9 7 sin θ cos θ E 7 cos θ sin θ If cosα then sin α E 7 ( cosθ cosα sin θ sinα ) E 7 cos θ α, where 7, ndsin α, 8 cos sin cos sin This is of the form cos( θ α ) 9 cos α. ( cos sin cos ) ( cos sin sin cos sin ) 9 ( cos ) 6sin cos ( sin cos ) sin cos 8sin 9 8 sin 8 sin cos 6sin cos 8sin sin 9 cos 6sin cos 8sin 9 ( cos 8sin cos 9sin ) 9 ( 7 cos( α )) 9, wheresin α, cos α. ( α ) 9 ( α ) cos cos cos ( α ) cos - - Done B : Chndim Peiris (B.Sc - Specil)

25 α n ± n ± α α n ±, n Ζ. Where 6 sin α, cos α. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur - - Done B : Chndim Peiris (B.Sc - Specil)

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