your answer in scientific notation. 3.0

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1 Section A Foundation Questions (%) Evaluate (4.8 ) (0.3 ) ( ) without using a calculator and express your answer in scientific notation. (4.8 ) (0.3 7 ) ( ) ( )[ 0.3 (3)( ) ( 7) ] 3 1 ( a b ). Simplify and express your answers with positive indices. 1 4 ( a b ) 3 ( a b ) 1 ( a b ) a a 1 ( 4) b 1 a b 6 4 b a a 8 4 b b 3. Solve the inequality of 40x > 73 8 x 9 + x x <. 3 Hence, find the smallest integer satisfies the inequality. x 9 + x x < 3 ( x) 3(9 + x) < 1x 0 0x 7 3x < 1x x > Hence, the smallest integer is. (4 marks) 4. Solve the following pair of simultaneous equations. x 3y 4. 3x y x 3y 4KKKKKKKKKK (1) 3x y KKKKKKKKKK () (1) () F3 MY Mathematics Paper I Section A & B _solution Page 1 /8

2 4x 9x 8 1 x 7 x Put 7 7 x into (1), 7 ( ) 3y y 4 + y 6. A camera which costs $ 000 is marked at a profit of 40%. (a) Find the marked price of the camera. (b) If the camera is sold at a discount of %, what is the profit percentage? a) The marked price $ 000(1+ 40%) $000(1.4) $800 b) The selling price $8 000(1 %) $000(0.9) $0 $0 $000 Profit % 0% $ 000 $0 0% $ 000 6% 6. Let A, B, C, D, E and F be interior angles of an 6 sided polygon and A : B : C : D : E : F : 3: 1 : 3 : :. Find the median angle of this 6 sided polygon. Sum of interior angles (6 ) Median angle ( ( ( (70 ) + (70 )) ( sum of polygon) 3 ) + ( )) 7. The figure shows the right triangular prism ABCDEF. Let M be the mid-point of AB F3 MY Mathematics Paper I Section A & B _solution Page /8

3 a) Name the projection of EB. b) Name the projection of EM. c) Name the angle between the line EB and plane ABCD. d) Name the angle between the line EM and plane ABCD. e) Name the angle between the plane EMD and the plane EDA. a) the projection of EB is DB 0.A b) the projection of EM is DM 0.A c) The required angle is EBD 0.A d) The required angle is EMD 0.A e) The required angle is MDA or ADM 8. Consider a pair of dual polyhedra, A and B. If polyhedra A and B have 0 and vertices respectively, find the numbers of faces and edges of polyhedron B. Let V, F and E be the number of vertices, faces and edges of the polyhedron B respectively Then V By duality, number of vertices of polyhedron A number of faces of polyhedron B So F 0 By Euler s formula V E + F So E Thus, the numbers of faces and edges of polyhedron B are 0 and 8 respectively. Section B Short Questions (3%) 9. a) Factorize x 3x. b) Using (a), solve the equation 3 1 x + x. c) Using (a), Solve 0 x < x. (6 marks) a) x 3x ( x )( x + ) b) x + x 3 x x 0 x 3x 0 ( x )( x + ) F3 MY Mathematics Paper I Section A & B _solution Page 3 /8

4 c) Therefore, x or x 0 x + x < x > 3x x < < x < x > x 3x > 0 ( x )( x + ) > 0 1 explanation Therefore, x < or x > For x 0, (0) 3(0) < 0. Hence x 0 is not a solution of given inequality. (a) For the set of data,, 11, 1, 13, 16, find the mean and the mode. (b) Four unknown data are combined with the six data in (a) to form a set of ten data. (i) Find the least and the greatest possible values of the median of the combined set of ten data. (ii) If the mean of the four unknown data is 11, find the mean of the combined set of ten data. ( marks) a) The mean 1 6 The mode b)(i) If 4 unknown data are less than, then the median If 4 unknown data are greater than 16, then the median 14. Hence the least and greatest possible values of the median are and 14. respectively. b)(ii) (11)(4) + 1(6) The mean of the combined set of ten data Four polls have been held to determine who is the most popular singer between Kelly and Sammi. Due to the longer polling time of the third and the forth polls, their weighting factors are higher. The weighted mean of the number of votes in all four polls determines the winner. The table below shows the respective numbers of votes for Kelly and Sammi in the polls and the weighting factors of each poll. Poll Number of votes Weighting factor Kelly Sammi First Second Third Fourth 7 x (a) If Kelly and Sammi got the same weighted mean of the number of votes in all four polls, find the value of x. (b) If the weighting factor of the fourth poll is changed to 3 while the value of x in (a) F3 MY Mathematics Paper I Section A & B _solution Page 4 /8

5 remains unchanged, who is the most popular singer? (4 marks) a) b) x x x Weighted mean of the number of votes for Kelly Weighted mean of the number of votes for Sammi Kelly is the most popular singer A 0.A 1. The salaries tax payable by Mr. Chan is $1 300 for this year. If his allowance is $90000, find his monthly income. The table below shows the salaries tax rate for this year: (4 marks) Net chargeable income Tax rate On the first $ % On the next $ % On the next $ % Remainder 17% Let $X be his net chargeable income. Then (X 000)(17%) (40000(%) (7%) (1%)) (X 000)(17%) ( ) (X 000)(17%) ( ) (X 000)(17%) 0 (X 000) X 000 Hence his total income $( ) $ His monthly income $ $ In the figure, AEB, AFC and AGD are straight lines. A (a) Prove that FG // CD. (b) Hence, find CAD. ( marks) E F 6 4 cm G 4 cm B C D F3 MY Mathematics Paper I Section A & B _solution Page /8

6 a) EF // BC (given) AE EB (given) AF FC (intercept theorem) As AF FC (proved) AG GC 4 cm (given) FG // CD (mid-point theorem) b) ADC AGF (corr. s, FG // CD) 6 In ACD, CAD ( sum of ) CAD In ABC, BDC is a straight line. AD is the altitude of ABC on BC. E is a point on AB such that CE is the angle bisector of ACB. DAC 8 and ABC 74. (a) Find AEC. (b) Show that the circumcentre of ABC is on CE. a) In ACD, b) AB CE (proved) AEC BCE 90 ( marks) ACD ( sum of ) 3 ACD ECB (angle bisector) 3 16 AEC (ext. of ) 90 CE CE (common side) ACE BCE (angle bisector) ACE BCE (A.S.A.) AE BE (corr. sides, s) So, CE is the perpendicular bisector of AB Hence, the circumcentre of ABC is on CE. A E B 8 74 D C F3 MY Mathematics Paper I Section A & B _solution Page 6 /8

7 1. In the figure, ABCD is a rhombus. E is a point outside the rhombus such that CB CE and BCE 40. If DAB 0, find C (a) CED, (b) BDE. (6 marks) D B E a) BCD BAD 0 Q DCE BCD + BCE Prop of rhombus 60 A b) CD CB CB CE CD CE CED CDE CED + CDE + DCE 180 CED 180 DCE 180 DCE CED Q CD CB CDB CBD Q CDB + CBD + BCD 180 CDB 180 BCD 180 BCD CDB Prop of rhombus Given Base s isos sum of Base s isos sum of From (a), BCD 0 BDE CDB CDE BDE From (a), CDE F3 MY Mathematics Paper I Section A & B _solution Page 7 /8

8 - End of Section A, B F3 MY Mathematics Paper I Section A & B _solution Page 8 /8

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale. For all questions, means none of the above answers is correct. Diagrams are NOT drawn to scale.. In the diagram, given m = 57, m = (x+ ), m = (4x 5). Find the degree measure of the smallest angle. 5. The