4016/02 October/November 2010

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1 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 ELEENTARY ATHEATICS aper Suggested Solutions 4016/0 October/November Topics: Trigonometry (Trigonometric Ratios, ythagoras' Theorem, earings) (a) (i) + ythagoras Theorem 7 a + b c 3335 c a m (3 sig. ig.) b (ii) tan 6 c θ b c θ b a a tan θ a b A A tan A A m (3 sig. ig.) (iii) sin 3 7 C sin θ a c 7 C sin (b) sin O 184 m (3 sig. ig.) 7 1 O sin A earing o rom (1 d.p.) is due east o A bearing o A rom A 7 3 C North Topic: Algebra (Solutions to Quadratic Equations, Formulae) (a) (b) 50 8 x 8(50) x ± or 0 t + p q (t + p) 4q 5t + 5p 4q 5t 4q 5p t 4q 5p 5 (c) y a y: cost per copy; x: total no. o copies (i) Sub x 50, y 17, 17 a a (ii) Sub x 100, y When 100 copies are printed, the cost o each copy is $11. (iii) Sub x 300, y Cost per copy Total cost $100 (iv) Sub y 5.0, x copies were printed / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 1 / 7

2 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 3. Topic: Arithmetic (Application o athematics in ractical Situations) (a) (i) Total amount Alan will pay or the computer 1 (199) + 4(40.30) 3 4 monthly instalments eposit $ (ii) Extra cost o computer (as % o cash price) $ $ % $199 extra cost as % o cash price : % 7.79% (3 sig. ig.) (b) Total amount etty will pay $ Interest etty will pay $ $199 $48.19 $48.13 ( d. p.) (c) 115% $759 1% $ % $ $ The trader paid $660 or the camera. hire purchase price cash price 100% cash price Given in ormula sheet (compound interest): Total amount 1 + r 100 n Total interest Total amt. rincipal amt. Selling price Cost price (100%) + roit (15%) 115% Cost price 4. Topic: Coordinate Geometry; Vectors in Two imensions (a) Equation o A: Equation o straight line passing y 4 4 through A(x 1, y 1 ) and (x, y ): ( 5) 3 y y y (x + 5) y y 1 gradient m 3 x x 1 x x 1 y x y 4 0 x Alternative ethod 3 3 y 4 x y 4x + 3 (b) From (a), 3y 4x + 3 (1) Given x + 9y 68, x 68 9y () Sub () into (1), 3y y + 3 3y y + 3 1y 168 y 8 Suby 8 into (),x 68 9(8) Coordinates o (, 8) Equation o straight line with gradient m & y- intercept c: y mx + c y 4 3 x + c Sub ( 5, 4), ( 5)+ c c 3 3 y 4 3 x (c) (i) AO (3 sig.ig.) agnitude o u v : u v u + v / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. / 7

3 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 (ii) (iii) AO OO OA OO AO + OA Coordinates oe(1, 5) (a) From (c)(ii), OO 1 5 (b) From (b), OO 8 O 4 O OO O O OO O (iv) O O Since O O ko AA k A,, C are collinear (straight line), E and are collinear (i.e. they all lie on a straight line) and since O O E is the mid-point o and. 5. Topic: Angles o olygon (a) (i) XC (ii) Since ACEF is a regular polygon and XC & XC are exterior angles, XC XC 4 XC is isosceles CX 180 XC XC (sum o s in ) (b) Given C E a From (a)(ii), XC is a isosceles XC X b X C + CX a + b XE E + X a + b X HenceX XE. (c) From (b), XE is isosceles, XE XE 180 CX 4 EF 180 exterior XE Alternative ethod EF EF XE (15 ) Each exterior o a regular n-sided polygon 360 n NOTE: X is not part o polygon ACEF! XC is an exterior 4 Exterior 4 rom (a) Exterior Each interior o a regular n-sided polygon (n ) 180 n E F / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 3 / 7

GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 6. Topics: Solutions to Quadratic Equations (a) Number o hours.

4 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 6. Topics: Solutions to Quadratic Equations (a) Number o hours. John took 4 (b) Number o hours eter took (c) ( 1) ( 1) (6) x x x x 5 0 (Shown) (d) x x x 1± ( 1) 4()( 5) () 1± or or (3 d.p.) (e) Taking x rom (d), time that John took to complete the race hours 3 hrs min 3 hrs 39 min 33 seconds Time taken istance Speed General solution to a quadratic equation ax +bx+c: x b ± b 4aa a Question simply asks to solve the equation. o NOT reject the negative value o x here! 7. Topic: Trigonometry (a) cos Q R (95)(10) Q R cos 1 ( ) (1 d.p.) (b) In Q, tan x 3 95 x (1d.p.) Angle o depression o Q rom 13.6 (c) Area o RS 500 m (d) (i) 1 1 (R)(RS) sin (170)(RS) sin RS m (3 s..) Cosine rule: c a + b ab cos C Number o panels that needs to be bought 6 (ii) Number o posts required 7 Total cost o the panels and post $ Angle o depression o Q rom Q Q 95 x Area o 1 ab sin C m 10 R x m m S R Rounded up to 6 need to buy 6 panels (o unit length 3 m) to ence up the ull distance o RS. 6 posts or each o the panels + 1 extra post at the end / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 4 / 7

5 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 8. Topics: Trigonometry, ensuration (a) (i) erimeter o sector 44 m Length o major arc RQ + radius (r) 44 m Arc length rθ Note: θ must be in radians and can be relex. rθ + r 44 8θ + (8) 44 θ radians (ii) Obtuse OQ (α) π relex OQ (θ) ( s at a pt.) α π 3.5 radians Area o OQ 1 (O)(OQ) sin α 1 (8) sin(π 3.5) Area o 1 ab sin C m (3 s..) R θ O 8 α 8 Q (b) (i) Volume o the bollard Volume o pyramid + Volume o cuboid 1 (10)(10)(1) + (10)(10)(30) cm 3 V (ii) Let be the midpoint o C. Using ythagoras Theorem in VN, 1 V VN + N Surace area o N 5 pyramid (excl. base) 13 cm Surace area o the bollard [4 Area o VC] + [erimeter o AC AE] ( ) cm Volume o pyramid 1 base area height 3 C Surace area o cuboid (excl. top &base) (iii) Area o major sector 1 r θ Calculator must be in RA mode to perorm this sin operation! 1 (8) (3.5) 11 m Total area o the cross-section o the tunnel Area o major sector + Area o OQ m (3 s..) / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 5 / 7

GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 9. Topic: Graphical Solution o Equations (a) (b) From the graph (i) ass o the baby ater 63 days 5.

6 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version.1 9. Topic: Graphical Solution o Equations (a) (b) From the graph (i) ass o the baby ater 63 days 5.15 kg (ii) ays since birth when the baby s mass was least 18 days (iii) ays since birth when the baby regained its birth mass 31 days (c) (i) From the tangent drawn in the graph, gradient o the curve at (7, 3.10) (3 s..) (ii) This gradient represents the rate o change o the baby s mass at seven days since birth (i.e. t 7). (d) As the graph is non-linear, it is not appropriate to estimate the mass o the baby when it is 1 year old by extending the graph linearly up to t / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 6 / 7

7 GCE O Level October/November 010 Suggested Solutions Elementary athematics (4016/0) version Topics: Statistics, Simple robability (a) (i) a b 60 ( ) 9 c d e (ii) ean Standard deviation (.) (3 sig. ig.) (b) (One pupil read exactly 6 books) (c) (oth had read more than 4 books) rom (a)(ii) no. o pupils who had read exactly 6 books total no. o pupils in group [1 st pupil (chosen rom the 60) had read > 4 books] AN [ nd pupil (chosen rom the remaining 59) had read > 4 books] / missloi@exampaper.com.sg acebook.com/jossstickstuition twitter.com/issloi Copyright 010 ϕ exampaper.com.sg. All rights reserved. 7 / 7

4038/02 October/November 2008

4038/02 October/November 2008 GCE O Level October/November 008 Suggested Solutions Additional Mathematics (408/0) version 1.1 ADDITIONAL MATHEMATICS Paper Suggested Solutions 1. Topic: Exponential Functions (i) Given: V =10000e -pt