Eton Education Centre JC 1 (2010) Consolidation quiz on Normal distribution By Wee WS (wenshih.wordpress.com) [ For SAJC group of students ]

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1 JC (00) Cosolidatio quiz o Normal distributio By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Sped miutes o this questio. Q [ TJC 0/JC ] Mr Fruiti is the ower of a fruit stall sellig a variety of fruits, oe of which is watermelo. The weight W, i kg, of a watermelo follows a ormal distributio with mea µ kg ad stadard deviatio σ kg. P W <. 5 = P W > 4. 5 = 0. 05, state the value of µ ad show that σ = 0. 69, (i) If ( ) ( ) (ii) correct to 3 sigificat values. [3] Fid the probability that out of 5 radomly chose watermelos, two weigh betwee.5 kg ad 3.5 kg each ad the other three weigh at least 3.5 kg each. [3] Mr Fruiti also sells rock melo. The weight of a rock melo follows a ormal distributio with mea.6 kg ad stadard deviatio 0.3 kg. (ii) Fid the least value of k for which the probability that the weight of a radomly chose rock melo exceeds k kg is at most 0.7. [] (iii) What is the probability that twice the weight of a radomly chose rock melo differs from the weight of a radomly chose watermelo by less tha 300g? [4]

2 JC (00) H Mathematics Cosolidatio quiz o Differetiatio ad itegratio techiques By Wee WS (weshih.wordpress.com) [ For YJC group of studets ] Aswer all questios i 7 miutes. Total marks: 5. If y = l( + si x), show that d y dx d 3 y dy d y = ad that + = 0. [3] + si x dx 3 dx dx Fid the derivative of si x + x 4x, expressig your aswer i the form k 4x, where k is a costat to be determied. State the rage of values of x for which the derivative is udefied. [4] /4 Hece fid the exact value of 4x dx. [] 0 3 Fid (a) (b) 5 ta θ dθ, [3] d x. x(6 x) [3]

3 JC (00) H Mathematics Cosolidatio quiz o Applicatios of differetiatio By Wee WS (weshih.wordpress.com) [ For AJC group of studets ] Aswer all questios i 7 miutes. Total marks: 5. Fid, without the use of a graphig calculator, the equatio of the ormal to the curve y x + 3e y = 3 at the origi. [4] The equatio of a circle is give by x + y =. The poit A lies o the circle ad the poit Q lies o the x-axis at (, 0). The agle AOQ is deoted by θ, where O is the origi. Give that A moves alog the circumferece of the circle, i the aticlockwise directio, at the rate of 0. radias per secod, fid the rate of chage of 7 the y-coordiate of A whe x =. [4] 5 3 A right circular coe has base radius r, height h ad curved surface area 9π. Show 4 that the volume V of the coe is V = π r 8 r. [] 3 If r varies, determie the exact value of r at which the volume of the coe is maximum. [5] [Curved surface area of a coe = π rl, where l is the legth of the slat edge ad volume of a coe = 3 π r h.]

4 JC (00) H Mathematics Cosolidatio quiz o AP/GP By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Aswer all questios i 9 miutes. Total marks: 6. (a) The cost of Soia s ew car was $P. She accepted a iterest-free loa of $P, which she agreed to repay by mothly istalmets. The first istalmet was $00. The istalmets were icreased by $50 per moth so the secod ad third istalmets were $50 ad $300 respectively. Give that the loa was repaid i k istalmets, ad that the fial istalmet was $350, (i) fid the value of k ad P. [3] The value of Soia s car at the ed of the first year was $7000. After the first year, the value of the car depreciated, each moth, by % of its value at the start of that moth. (ii) Calculate, to the earest dollar, the value of Soia s car at the ed of the third year. [] (iii) Give that the value of the car depreciated to less tha 50% of the cost of the ew car by the th moth. Fid the value of. [3] (b) Give that the terms of the sequece a, a, a 3,, a are i arithmetic progressio a r ad br =, for r =,, 3,,, show that the sequece b, b, b 3,, b is 3 geometric. Give that b = 7 ad the commo differece of the arithmetic progressio is, fid a expressio for a r. [4]

5 The atural umbers,, 3, are arraged i rows i the followig way:, 3 4, 5, 6, 7 8, 9, 0,,, 3, 4, 5 The umber of items i the ext row is always twice that of the umber of items i the precedig row. (i) Write dow the umber of items i the -th row. [] (ii) What is the last iteger i the -th row? [] (iii) Determie the sum of all itegers i the -th row. []

6 JC (00) H Mathematics Cosolidatio quiz o Sigma otatio, method of differeces, mathematical iductio By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Aswer all questios i 7 miutes. Total marks: 5. r Fid ( 4r + ) i terms of. [3] r= 3 ( ) e e e Show that if u =, the u u =. []!! 3 N e( e ) e ( e 3) e ( e 4) e ( e N ) Hece fid [3]! 3! 4! N! r + 3 Prove by iductio, that for all positive itegers, ( r + ) = ( ) Hece fid ( r + ) Evaluate r i terms of. r= + r= r= + ( r + ) ( r + ) r r. r= as. [8]

7 JC H Mathematics [ SAJC group ] Cosolidatio quiz o fuctios (by Wee WS) Sped 9 miutes o this questio. (iv) (v) Without fidig f, sketch the graphs of y = f ( x) ad y f f ( x) their relatioship with the graph of y f ( x) Explai, with a reaso, whether f = to show =. [3] g is a fuctio. []

8 JC H Mathematics Cosolidatio quiz o trasformatios (by Wee WS) Sped 7 miutes o this questio. (a) (v) y f ( x) =. [3] (b) State the sequece of trasformatios which maps the graph of e x + x y = e x to the graph of y =. []

9 JC H Mathematics Cosolidatio quiz o method of limits, method of differeces & recurreces (by Wee WS) Sped 7 miutes o these questios. Fid 4x + lim. [] x 3 x + A sequece of real umbers satisfy the recurrece relatio x = + x ( x + )e for =,, 3, (i) If x k as, fid the value of k correct to 3 decimal places. [3] (ii) State a possible value of x such that the sequece will ot coverge to k. [] (iii) Usig a graphical method, show that if x > k, the x < + x. [] 3 (i) Show that if u r = r + 4r u r u r 4r [] r r (ii) Usig the result i (i) or otherwise, fid ( ). r= 4r [4] r r (iii) Hece, write dow the value of ( ). 4 r [] r=

10 JC H Mathematics: Cosolidatio quiz o AP/GP for YJC group (by Wee WS) Sped 5 miutes o these questios. A circle is cut ito sectors whose areas form a arithmetic progressio. The area of the largest sector is 5 times the area of the smallest sector. Fid the agle betwee the straight edges of the smallest sector, givig your aswer i radias exactly. [4] A coverget geometric series, T, T, T3,.. has first term, such that T5 = ( T4 T ). T3 3 Fid the possible value of the commo ratio. Hece fid the possible value of the sum to ifiity. [3] Give also ks + = T, where Z ad k R. Fid a simplified expressio for k. [] 3 Two caoeists from two differet colleges are udergoig two types of traiig programme i preparatio for the A-divisio champioships. Caoeist A: Caoes 5000 m o day, ad o each successive day, the distace covered is icreased by of the previous day. Caoeist B: Caoes 5000 m o day, ad o each successive day, the distace covered is icreased by d m. (i) (ii) Fid the least value of N for which the total distace covered by caoeist A up to N days exceeds 00 km. [3] Caoeist B aspires to cover a distace of 6 km o day 6. Fid exactly, the least value of d. []

CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01

CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01 CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ