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1 9-94 Idividual ::0: Group 97 cm Idividual Evets I Suppose log p to a ifiite umer of terms. Fid p. 4 8 Referece: 0 HI a log p = (sum to ifiity of a G.P. =, a =, r = ) r p = = 9 I Two umers are draw from the set of umers 4,, 6, 7, 8, 9, 0, ad. Fid the proaility that the sum of these two umers is eve. P(sum = eve) = P(odd + odd or eve + eve) = P(odd + odd) + P(eve + eve) 4 4 = = = 7 9 I Give a * a, fid the value of * = = 4 *(*) = 4 = 6; (*)* = 4 = 6 *(*(*)) = 6 ; ((*)*)* = 6 = 8 6 *(*(*)) = ((*)*)* 8 = 8 = 6 *(*(*)). ((*)*)* I4 If loga x = ad a + x = 8, fid a + x. x = a. a + a = 8 a + a 8 = 0 (a + 4)(a ) = 0 a = 4 (rejected, ecause 4 caot e the ase) or a = x = a = 4 a + x = + 4 = 6 I If a : :, : c : ad c : d :, fid a : : c : d. a : = 0 :, : c = : 0, c : d = 0 : 6 a : : c : d = 0 : : 0 : 6 Page
2 I6 A, B, C, D are differet itegers ragig from 0 to 9 ad A B C D 9 D C B A Fid C. Referece: 987 FG9 Cosider the thousads digit. 9A = D A =, D = 9 there is o carry digit i the hudreds digit B = 0 Cosider the tes digit. 9C (mod 0) 9C (mod 0) C = 8 I7 Fid the last digit of the umer 99. =, = 9, = 7, 4 = 8 The patter of uits digit repeat for every multiples of = ( 4 ) 498 (mod 0) The uits digit is. I8 I figure, CD isects BCA, BE // CA, BC 0, CA ad CD 0.. Fid the legth of DE. Let BCD = = ACD (agle isector) BED = (alt. s, AC // EB) BE = BC = 0 (sides opp. equal agles) Let CAD = = EBD (alt. s, AC // EB) ACD ~ BED (equiagular) DE 0 (ratio of sides, ~) 0. DE = 6.8 I9 I figure, XY, YZ 4 ad ZX. Semi-circles are costructed with M, N, O as cetres as show where M, N, O are mid-poits of XY, YZ ad ZX respectively. Fid the sum of the shaded areas. (Referece: 009 FG4.) Sum of the shaded areas = Ssemi-circle XMY + Ssemi-circle YNZ + S XYZ Ssemi-circle XYZ 4 = 4 = 6 I0 I figure, O is the cetre of the circle, OE DE ad AOB = 84. Fid a if ADE = a. DOE = a (ase s isos. ODE) OEB = a (ext. of ODE) A OBE = a (ase s isos. OBE) BOE = 80 4a (s sum of OBE) a + a = 80 (adj. s o st. lie) a = 8 X Y M N O (Figure ) B E 84 O (Figure ) Z D (Figure ) Page
3 G Fid the least value of x so that x + x + x =. Referece: 00 HG9, 004 FG4., 008 HI8, 008 FI., 00 HG6, 0 FGS., 0 FG. Whe x <, x + x + x = x = (rejected) Whe x, x + x + x = 0 = 0 x Whe < x, x + x + x = 6x = x = (rejected) Method By triagle iequality a + a + x + x + x x + x + x = Equality holds whe x, x ad x i.e. x ad i.e x x ad x The miimum value of x =. Whe < x, x + x + x = x = (rejected) The least value of x =. G A solid cue with edges of legth 9 cm is paited completely o the outside. It is the cut ito 7 cogruet little cues with edges cm. Fid the total area of the upaited faces of these cues. Referece: 99 FI. There are 8 cues each paited with sides. Numer of upaited surfaces = /cue There are cues each paited with sides. Numer of upaited surfaces = 4/cue There are 6 cues each paited with side. Numer of upaited surfaces = /cue There is cue which is upaited. Total area of the upaited faces = ( ) cm = 97 cm Method Total surface area of the 7 little cues = 76 cm = 48 cm Total area of paited surface = 69 cm = 486 cm The total area of the upaited faces of these cues = (48 486) cm = 97 cm Page
4 G G4 I a race of 000 m, A fiishes 00 m ahead of B ad 90 m ahead of C. If B ad C cotiue to ru at their previous average speeds, the B will fiish x metres ahead of C. Fid x. Let the speeds of A, B C e a m/s, m/s, c m/s respectively. Suppose A, B, C fiishes the race i t s, t s, t s. at = 000 t + 00 = 000 t = 800 () ct + 90 = 000 ct = 70 () () (): t = c 000 (4) () 0 x = 000 ct c = y (4) = c = 000 y () 0 = 000 = 00 0 Give that the perimeter of a equilateral triagle iscried i a circle is. Fid the area of the circle i terms of. The legth of the triagle = 4 Let the radius of the circumscried circle e r. r cos 0 = 4 4 r = 4 6 The area of the circle = = x log x logx y logx G Give that x > 0 ad y > 0, fid the value of y if log x x. log x logx log y log x By the chage of ase formula, = log log x logx log x log y = log y = 9 G6 There are rectagles i figure. Fid. Let the legth ad the width of the smallest rectagle e a ad respectively. Numer of the smallest rectagles = 7 Numer of rectagles with dimesio a = 4 Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Numer of rectagles with dimesio a = Total umer of rectagles = = 7 (Figure ) Page 4
5 G7 G8 The ase of a triagle is 80 cm ad oe of the ase agles is 60. The sum of the legths of the other two sides is 90 cm. The legth of the shortest side of this triagle is a cm. Fid a. Let B = 60, BC = 80 cm, AB = c cm, AC = cm, the + c = 90. By cosie rule, = (90 c) = 80 + c c80 cos c = c 700 = 00c c = 7, = 90 7 = 7 The shortest side is 7 cm (c = 7). A studet o a vacatio of d days oserved that: (i) it raied 7 times, morig or afteroos; (ii) whe it raied i the afteroo, it was clear i the morig; (iii) there were clear afteroos; (iv) there were 6 clear morigs. What is the value of d? Suppose it raied i the morig for x days, raied i the afteroo i y days ad the umer of clear days (oth i the morig ad the afteroo) e z. x + y = 7 () x + z = () y + z = 6 () () + () + (): (x + y + z) = 8 d = x + y + z = 9 G9 [a] deotes the greatest iteger ot greater tha a. For example, [], [ ], [ ]. If Referece: 00 FI.4 [ x ] x, fid the value of x. [ x] x x = x + + a, where 0 a < a = x 0 x < x < 4 x < 4 9 x + < 4 4 x + is a iteger x + = x = Page
6 Page 6 G0 Give that ) (. Fid the value of a if ) ( 7 a. a = 7 = = 6
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