MTHEMTIS (Three hours ad quarter) swer Questio from Sectio ad 4 questios from Sectio. ll workig, icludig rough work, should e doe o the same sheet as, ad adjacet to, the rest of the aswer. The iteded marks for questios or parts of questios are give i rackets []. Mathematical formulae are give at the ed of this questio paper. The use of calculator (f-8/ f-00) is allowed. Sectio swer LL questios. irectios: Read the followig questios carefully. For each questio there are four alteratives,, ad. hoose the correct alterative ad write it i your aswer sheet. Questio (5=0 Marks) (i) The epaded form of 7 ( i) is 4 - + 0 + 4-9 + 6. +4+9+6+ 5+ 6+49. 4 + + 0 + + 4 + 9 + 6. + - 0 - + 4 5 + 6 7. (ii) The adjoit of the matri 5 is 5. 5. 5. 5. HSE/0/008 This ooklet cotais 0 pages. Page of 0 opyright reserved
+ (iii) The restrictios o the epressio 6 7 are 7,. 7,.. 7. (iv) If ga ( ) = has a vertical asymptote at a = 0, what is the vertical asymptote a of f( a) =? ( a ) - 0 4 (v) ody moves such that its distace metres, after t secods is give y = t 4t + 5t 6. Its acceleratio after 4 secods would e 4 m/sec. 64 m/sec. 7 m/sec. 7 m/sec. (vi) 7si 7 The value of lim is 0. 7.. 7. (vii) Which oe of the followig is the derivative of e? e 6 e 6 e HSE/0/008 Page of 0 opyright reserved
(viii) The itegral value of -. 0... π si d is π (i) The legth of the arc i the diagram alogside is.75 cm. 5.5 cm. cm. 44 cm. 45 7 cm () + i Which oe of the followig is the modulus of? 4i 5 5 5 5 (i) Which oe of the followig is the greatest surd? 6 7 HSE/0/008 Page of 0 opyright reserved
(ii) The regressio co-efficiet of y o of the regressio lie y = 7 is.... (iii) If f() = 7, f '() = 5, g() = ad g'() = 7, the value of ( f o g)'() is -09. -.. 49. (iv) The polar form of + 4i is 4c is 6.9. 5c is 6.9. 4c is 5.. 5c is 5.. (v) Which oe of the followig is the coic represeted y the equatio 6 +9y = 44? hyperola paraola ellipse circle HSE/0/008 Page 4 of 0 opyright reserved
Questio Sectio (70 Marks) swer ay 4 questios. ll questios i this sectio have equal marks. Uless otherwise stated, you may roud aswers to decimal places. + log a) Evaluate d. [] 5 ) If y = v v, where v =. etermie dy [] d = Questio a) car depreciates y 0% per year. If a car is purchased for NU., 50,000. Whe will its value e half of the origial price? [] ) Fid the derivative of y = e y usig the first priciple method. [] Questio 4 a) etermie cos5d. [] ) Epress Questio 5 4+ 9 5 i the form a + i. [] a) etermie the cetre ad radius of the equatio of the circle + y + 6 4y+ 9= 0. [] ) etermie the equatio of the ellipse whose focus is (-, ), eccetricity ad directri is y+ = 0. [] Questio 6 a) etermie the equatio of the plae which passes through the poits P(,, ), Q(, -, 4) ad R(-,, -4). [] π ) etermie the equatio of the taget lie to y = cosθ whereθ =. [] HSE/0/008 Page 5 of 0 opyright reserved
Questio 7 Prove that =, N y mathematical iductio method. [5] ii ( + ) + Questio 8 The sides of a equilateral triagle decrease at the rate of 0cm/sec. etermie the rate of decrease of the area of the triagle, whe the area is 00 cm. [5] Questio 9 Fid all the values of ( ) 4 Questio 0 + i y usig e Moivre s Theorem. [5] For the hyperola, with equatio 4 y + 6+ y = 0, determie: Questio (i) the cetre; (ii) the vertices; (iii) the eccetricity; (iv) the legths of cojugate ad trasversal ais. [5] Solve the followig system of liear equatios usig matri method. -4 + y 9z =, + 4y + z = 5, -y + z = 8. [5] Questio rectagular uildig is to e uilt o a 40m y 70 m rectagular plot i such a way that there is a path metres wide surroudig the uildig. The uildig ca occupy up to 70% of the plots area. What is the rage of possile itegral values for the width of the path? [5] Questio a) If - is the root of the fuctio y = 8 a+, fid the value of a. [] ) etermie the square root of 7 + 8 7. [] HSE/0/008 Page 6 of 0 opyright reserved
Questio 4 a) car attery loses % of its charge every day. Write a epoetial equatio i asic form ad i ase e form to model this situatio, takig as the remaiig charge of the attery. [] ) revolvig watch tower torch is situated 850m from a plae surface. It turs oe revolutio per miute. How fast does it sweep alog the surface at a poit 550m from the earest poit? [] Questio 5 Fid the area ouded y y = +, = 0, ad = usig the limit of sums. [5] Questio 6 a) etermie the volume of the solid geerated y revolvig the curve y = aout the - ais from = to =. [] a) etermie the chages i the cost of livig ide of the livig figures i Paro as give elow. [] Items Food Ret lothig Fuel Others Percetage 5 0 5 0 0 epediture Price i 50 60 80 50 00 000 Price i 00 70 80 00 50 50 Questio 7 For the fuctio y = 9+, fid the: (i) itercepts; (ii) critical values; (iii) iflectio poits; (iv) maimum ad miimum poits. (v) sketch the graph. [5] HSE/0/008 Page 7 of 0 opyright reserved
Questio 8 etermie the co-efficiet of correlatio of the data ad iterpret the result. [5] X 4 5 6 7 8 9 0 Y 4 7 0 6 9 5 8 4 HSE/0/008 Page 8 of 0 opyright reserved
MTHEMTIS FORMULE FOR HSE Fuctios ad Equatios () ( a± ) = a + ± a () () ( a± ) = a ± a + a ± a = a+ a ( )( ) a ± = ( a± )( a m a+ ) (5) ± 4 = a (6) vt () = h'() t Sequece ad series () () () (5) ac ( + ) ( + )( + ) i = 6 ( + ) i = t = ar S a( r ) a( r ) = =, r r where r> (6) t = a + ( )d (7) S = [ a+ ( ) d] ifferetiatio () f ( + h) f( ) f '( ) = lim h 0 h () y =, y = () y = cf(), y = cf ( ) y = f ( ) ± g( ), y = f ( ) ± g ( ) (5) F() = f()g(), F () = f()g () + f ()g(). f ( ) (6) F( ) =, g ( ) gf ( ) ( ) f( g ) ( ) F'( ) = g ( ) [ ] (7) ( f o g) ( ) = f g( ) ( g ) (8) dy dy du = d du d oordiate Geometry () ( y y) = m( ) () d = ( a) + ( y ) Trigoometry () si( ± ) = si cos ± cos si () cos( ± ) = cos cos m si si () ta ± ta ta( ± ) = m ta ta si θ + cos θ = Logarithmic Epoetials () y = y ( ) 0 + r () k y = y0e () = P( + r) Itegratio () d f ( gd ) ( ) = f( ) gd f( ) gdd ( ) d () () a V f ( d ) = lim f( ) = π a y d a = yd i = Measuremet () π oe: V = r h () oe: S = π rl + π r () 4π Sphere: V = r Sphere: S = 4π r (5) ylider: S = πr + πrh (6) ylider: V = πr h (7) ircle: =πr (8) ircle: =πr (9) Triagle: =, =, 4 = ss ( a)( s )( s c) (0) Rectagle: =lw, () Rectagle: P = l + w () Square: = s, i Page 9 of 0
MTHEMTIS FORMULE FOR HSE () Square: P = 4S Rectagular Prism: V = lwh omple umers () r = a + () taθ = a θ = ta a () If z = rcisθ the z = r cisθ 60 z = r θ cis( + k ) f or k = 0,,,,... Secod egree Relatios () X y Ellipse: + = a () X y Hyperola: = a () c e = a Geometry () = ( ) + ( y y ) + ( z z ) () l + m ly + my lz + mz ( yz,, ) =,, l + m l + m l + m () For a+ y+ cz= 0 ad a+ y+ cz= 0, y z = = c c ca c a a a θ l = π r 0 60 (5) θ = r 0 π 60 Matrices () ( ) i + j ij = Mij () = = I () Iverse of = = adj det ata & Proaility f () = () () (5) (6) (7) Media = l + ( m c) f = = l l ( ) i + + d + d m + = m+ P 00 P0 I = PW i I = 00 PW 0 + (8) ov ( X, Y) = ( X X)( Y Y) (9) (0) () () () (5) r = r ( )( y y) ( )( y y) ( ) ( y y) = YX cov = y ( X, Y) y = r ( XY) ( ) cov, Y Y = X X = ( ) r X X y y y y r = r S τ = ( ) 6 d r = ( ) y Page 0 of 0