UGANDA ADVANCED CERTIFICATE OF EDUCATION INTERNAL MOCK 2016 PURE MATHEMATICS. 3 hours

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1 P/ PURE MATHEMATICS PAPER JULY 0 HOURS UGANDA ADVANCED CERTIFICATE OF EDUCATION INTERNAL MOCK 0 PURE MATHEMATICS hurs INSTRUCTIONS TO CANDIDATES: Attempt ALL the EIGHT questins in sectin A and any FIVE frm sectin B. All wrking must be clearly shwn. Mathematical tables with list f frmulae and squared paper are prvided. Silent, nn-prgrammable calculatrs shuld be used. State the degree f accuracy at the end f each answer using CAL fr calculatr and TAB fr tables. Clearly indicate the questins yu have attempted in a grid n yur answer scripts. DONOT hand in questin paper. Qn Marks GHS MOCK EXAMINATION 0

2 SECTION A (0 marks). The sum f the first n terms f an A.P is n n. Find the first three terms f the series. ( marks). Prve that the circles x y x y 0 0 and x y y are rthgnal. ( marks). Given that x is a repeated rt f the equatin x px qx 0, find the value f p and q. ( marks) x y x y z 8 at pint Q, find the crdinates f Q. ( marks). A pint P,, parallel t the line z meets the plane. Prve that: ct ct ct sin. ( marks). Slve the differential equatin: yx y, given that y when dy ( marks) 7. Find the vlume generated when the area bunded by the curve y x x and the line y is rtated thrugh 0 abut the line y. ( marks) x. x x x x 8. Evaluate:. ( marks) SECTION B 9. Given that the first three terms in the expansin in ascending pwers f x f 8x are the same as the first three terms in the expansin f ax, find bx the values f aand b. Hence, find an apprximatin t 0. in the frm p. q ( marks) GHS MOCK EXAMINATION 0

3 0. Express 9 7x x in partial fractins and hence find 8 9x 7x 8 ( marks) a) Express i in the frm x iy. ( marks) b) Given that z i a i b i where a and b are real numbers and that arg z and z 7, find the values f a and b. ( marks) a) Given the pints A,, D 9, and B such that D divides AB externally in the rati :, find the crdinates f pint B. ( marks) b) The vectr equatin f tw lines are r and r where t is 0 t a cnstant. If the tw lines intersect find: (i) t and the psitin vectr f the pint f intersectin. (ii) the angle between the tw lines. (7 marks) a) If ABC is a triangle such that A B C 80, prve that sin A sin B sin C cs AcsBcsC ( marks) b) Express 0sin xcsx csx in the frm Rsin x, hence r therwise slve 0sin x csx csx 7 0 in the range 0 x 0. ( marks). A pint P n the curve is given parametrically by x cs and y sec. Find the: (i) equatin f the nrmal t the curve at the pint (ii) Cartesian equatin f the curve. (7 marks) (b) Pint Pap, ap and Q aq, aq lie n the parabla y ax. Find the lcus f the midpint f the chrd PQ fr which pq a. ( marks) GHS MOCK EXAMINATION 0

4 . Sketch the curve the asympttes. y x 8x x x, by finding the turning pints and clearly state ( marks). The rate at which a bdy lses temperature at any instant is prprtinal t the amunt by which the temperature f the bdy at that instant, exceeds the temperature f its surrundings. A cntainer f ht liquid is placed in a rm f temperature 8 C and in minutes the liquid cls frm 8 C t 0 C. Hw lng des it take fr the liquid t cl frm C t 0 C? GHS MOCK EXAMINATION 0

5 . S MARKING GUIDE MOCK TERM 0 n S n a n d, S a d... i ii S a d..., a d 8 d, d, a ALT th The n term f the prgressin is u n Sn Sn Thus n n n n n S, the terms are y x y 0 0 x, x y 0 9 Thus C,, r x y y, x 0 y 0 Thus C 0,, r 0 d 0 r r 0 Therefre, since d r r, then the circles are rthgnal.. x px qx 0, p q 0, p q (i) x px q 0, p q 0, p q (ii) p 8, p 9, q. The equatin f the line thrugh P is r i j k i j k Thus x y z, s x, y, z. 8, Q,, sin cs sin sin cs sin thus cs cs cs sin sin, y, z x s GHS MOCK EXAMINATION 0

6 sin cs cs sin sin sin cs sin ct Separating the variables, y dy y x sin ct ct y Inx Ink In y when x In 8 In Ink, In Ink, k In y Inx thus, y x T get y x 7. The pint f intersectin, x x, x 0 x, x 0, x. V y, x x 0 V, x V x x x, V x x 0 0 V 0, V cubic units 8. Let u x x, du x x 9 x x du x, u 9 u x x x, u Inu 9 9 In 0 GHS MOCK EXAMINATION 0

7 9. 0. a) Or 7 ( 8x) = 8x.. 8x...! = x x... ax ax bx = ( ax) bx b x bx = ( a b) x ( b ab) x... Since the first three terms f the expansin are the same. a b, b ab Hence b = - and a = - Thus; x ( 8x) x Substituting x 0.0, we have ( 0.) = hence (0.) 7 9x 9x 7x 8 x 9 x x 9x A Bx C 9 9 x x x x x x Let 9x A9 x x Bx C x x In 9x x x :9A B 0, x : A B C 9, x : A C Slve t get; A, B, C 9x x x 9x x x 9x x 9x Thus 7x 8 x In x In 9x x Let c c, z z i arg z tan Thus i cs isin i cs isin i 0i 0i GHS MOCK EXAMINATION 0 x 9x x..

8 b) a) b) z 7cs i sin 7i 7 i i a i b i i a b i a b a b (i) a b a b 70 a and b 0 OA OB : (ii) OD fr AB rati is where and x 9 Let B x, y, y 9 9 x, 9 y Thus 9 9 x s x 0 and 9 y s, y The crdinates are B 0,. r and r, if they d intersect then r r t Thus implies that, s t Als, fr, then s, Using t, then t, gives t Psitin vectr fr pint f intersectin is given by r 0 r r i k. ii) Let d i j k and d i j k be the directinal vectrs. i j k. i j k Thus the angle is given by cs. cs,. a) Frm the L.H.S sin A sin B sin C = csa csb sin C = csa csb cs C = cs A BcsA B cs C, 8 GHS MOCK EXAMINATION 0

9 cs( A B) csc = csc csa B cs C = csccs A B csc = csccsa B csa B = cs AcsBcsC b) 0sin xcsx csx = sinx csx Let sinx csx Rsinxcs Rcsxsin Rcs, Rsin, thus tan 7.8. R sinx csx sin(x 7.8 ) as required. 0sin x csx csx 7 0, sin(x 7.8 ) 7 x 7.8.9, 7., x 7., 0. 0 x., 0. 0 Thus, When, x cs and y sec, thus the pint is P,. dy sin and sec tan thus the gradient functin is given by d d dy sec tan when sin cs, dy y Gradient f the nrmal is, s the equatin f the nrmal is x T get 8y x 7 cs x, and frm y, s cs cs y Thus x t get y x xy 7 y ap aq a p q M,, s x, ap aq y ap q y p q, and x a p q pq but pq a a Let the midpint be x y 9 GHS MOCK EXAMINATION 0

10 y Thus x a a is the lcus f the midpint. a. x y xy x 8x, y x y 8x 0 Fr n real values, b ac 0 8 y 0 y, y y y 80 0 y 0, y y 0 y y sign y y y Thus the curve desnt lie in the regin y When y, 9x x 0, x 0, thus, x,, max As y As s y, x 0 x, x 0, thus, x,, min y, x 0, x, x x x,, x, s vertical asympttes: x 0, & x y hrizntal asymptte is y, s when Curve des nt cut any f the axes. y, x, 0 GHS MOCK EXAMINATION 0

11 . dt dt T 8, K T 8 dt When T In dt dt k T 8 dt T kt c 8 thus T When T 8, t 0. A T 8 Ae kt 8 e. When T 0 C, t 0 8 e k, k=.. C 8 e kt, t 9.min When T 0 C, t 9.7min Time t cl is given by min kt. GHS MOCK EXAMINATION 0