Solutions to O Level Add Math paper

Transcription

1 Solutions to O Level Add Math paper 4. Bab food is heated in a microwave to a temperature of C. It subsequentl cools in such a wa that its temperature, T C, t minutes after removal from the microwave, is given b kt T Ae, where A and k are constants. (i) Eplain wh A = 6. [] When t = the temperature of the food is 6 C. (ii) Find the value of k correct to significant figures. [] A bab should onl be given this food when the temperature of the food is less than 4 C. (iii) Determine, with working, whether it is safe to give the food 4 minutes after removal from the microwave. [] [Analsis] Understand the modelling of cooling with eponential function. Solution : (i) When t, T, Ae A 6 (ii) When t, T 6, 6 6e 4 k e 6 ln k 4 4 k ln k. k (iii) For T 4, kt 6e 4 kt e B KL Ang, Jan Page

2 Solutions to O Level Add Math paper 4 kt ln ln t k ln t ln 4 ln t. ( s.f) Therefore, when t 4, T 4. It is safe to feed the bab. In Summar: Modelling question similar to this will continue to be in the future paper. B KL Ang, Jan Page

3 . Given that f 6 (i) Solutions to O Level Add Math paper 4, find the remainder when (ii) show that f is divided b, [] is a factor of f and hence solve the equation f. [4] [Analsis] Appling Remainder and Factor Theorems. Solution : (i) Given that f 6, f 6 6 (ii) f 6 6, therefore Let 6 k When, 6 k 6 k k 7 is a factor of Therefore, or or f. In Summar: A ver tpical question on R/F theorems. B KL Ang, Jan Page

4 Solutions to O Level Add Math paper 4. The area of a quadrilateral is 4 cm. (i) In the case where the quadrilateral is a rectangle with width cm, find, without using a calculator, the length of the rectangle in the form a b cm. [4] (ii) In the case where the quadrilateral is a square with side ccm, find, without using a calculator, the value of the constant c. [] [Analsis] Simplifing surd. Solution : (i) (ii) c 4 4c c 4 4c c c 4 In Summar: Surd will still be in the future paper. B KL Ang, Jan Page 4

5 Solutions to O Level Add Math paper 4 4. The quadratic equation 4 has roots α and β. (i) Find the value of α + β. [] (ii) Find the quadratic equation whose roots are α and β. [] [Analsis] SOR: ; POR:. Solution : (i) Given that 4, (ii) SOR: POR:. The quadratic equation is or 64 In Summar: As epected for the new cubic identit. B KL Ang, Jan Page

6 . (a) Epress log log 4 Solutions to O Level Add Math paper 4 as a quadratic equation in and eplain wh there are no real solutions. [] (b) Given that log, epress as a power of. [4] log [Analsis] Log rules, discriminant. base changing. Solution: (a) Given that log log 4 log log 4 log log log 4 log log log 4 log log 4, Taking discriminant, Hence, has no real root. log log (b) Given that log log log log log log log log log, log,,,, B KL Ang, Jan Page 6

7 Solutions to O Level Add Math paper 4 log In Summar: The last part in (ii) is a little non-routine. B KL Ang, Jan Page 7

8 Solutions to O Level Add Math paper 4 6. C E A B F D The diagram shows a triangle ABC whose vertices lie on the circumference of a circle. The triangle DEF is formed b tangents drawn to the circle at the points A, B and C. (i) Prove that angle DEF = angle ABC. [4] (ii) Make a similar deduction about angle DFE. [] (iii) Prove that angle BAC = + angle EDF. [] [Analsis] Properties of tangents to a circle are test in this question. Solution: (i) Hence, Therefore, Therefore, Therefore, (ii) Similarl, ABC ACE (Alternate Segment Theorem) EC EA (Tangents from an eternal point) AEC is an isosceles triangle. CAE ACE ABC ABC AEC ABC DEF ABC DEF DFE ACB B KL Ang, Jan Page

9 Solutions to O Level Add Math paper 4 (iii) BAC ABC ACB (Sum of angles in a triangle) DEF DFE BAC BAC 6 DEF DFE BAC 6 EDF (Sum of angles in a triangle) BAC EDF In Summar: Geometrical Proof is a weak link common to man students. Kids, do more practice, it is not at all that difficult. B KL Ang, Jan Page 9

10 Solutions to O Level Add Math paper 4 7. A curve has the equation 4. The point ( p, q ) is the stationar point on the curve. (i) Determine the value of p and of q. [4] (ii) Determine whether is increasing or decreasing (a) for values of less than p, [] (b) for values of greater than p. [] (iii) What do the results of part (ii) impl about the stationar point? [] d (iv) What is the value of at the stationar point? d [] [Analsis] Question on differentiation with graph. Solution: (i) Given that 4, d 4 d d 4 d d Let, d 4 Therefore, p, q 4 (ii) (a) When, d 4 d, is increasing. (b) When, B KL Ang, Jan Page

11 d 4 d Solutions to O Level Add Math paper 4, is decreasing. (iii) The turning point is a maimum point. (iv) d d When, d d In Summar: Part (ii) ma have been challenging for some students. B KL Ang, Jan Page

12 Solutions to O Level Add Math paper 4. A particle travelling in a straight line passes through a fied point O with a speed of m/s. The acceleration, a m/s t, of the particle, t s after passing through O, is given b a e.. The particle comes to instantaneous rest at the point P. (i) Show that the particle reaches P when t ln. [6] (ii) Calculate the distance OP. [4] (iii) Show that the particle is again at O at some instant during the fiftieth second after first passing through O. [] [Analsis] Kinematics with integration. Solution: t (i) Given that a e., The velocit, v e e. t dt. t v C where C is an integration constant.. When t, v. C e. C e. t v. When v,.t e..t. e ln.t ln.t t ln B KL Ang, Jan Page

13 Solutions to O Level Add Math paper 4 e. t (ii) Since v,. Let the distance, s t e. dt..t e s t D where D is an integration constant.. When t, s e D. D.t e s t. When t ln, s OP.ln e OP ln. OP ln. OP ln 47. m ( s.f.) (iii) When t,. e s. s e s.674 ( s.f.) Since, when t ln, s 47. > and when t, s. 674 <, the particle must have return to point O, at least once. In Summar: Part (iii) ma have given problem to some students. B KL Ang, Jan Page

14 Solutions to O Level Add Math paper 4 9. (i) Solve the equation cosa sin A for A 6. [4] (ii) On the same aes sketch, for, the graphs of cos6 and sin. [6] (iii) Eplain how the solutions of the equation in part (i) could be used to find the -coordinates of the points of intersection of the graphs of part (ii). [] [Analsis] Trigo equation and its related graph. Solution: (i) Given that cosa sin A for A 6 cos A sin A sin A sin A sin A sin A 6sin A sin A 6sin A sin A sin A sin A sin A or sin A sin A A sin Principal angles, A 9. A A 99. or A 4. A or A B KL Ang, Jan Page 4

15 Solutions to O Level Add Math paper 4 (ii) sin O 6 9 cos6 (iii) Given that cos6 and sin, the intersections sin cos6 cos6 sin cos6 sin Let A, 6, A 6 cosa sin A From (i), the solutions of the equation are A 99., A 4., A, A Therefore, 99., 4.,, 66.,.,, B KL Ang, Jan Page

16 Solutions to O Level Add Math paper 4. A circle, C, has equation = 6. (i) Find the radius and the coordinates of the centre of C. [] A second circle, C, has a diameter AB. The point A has coordinates (, ) and the equation of the tangent to C at B is 4. (ii) Find the equation of the diameter AB and hence the coordinates of B. [4] (iii) Find the radius and the coordinates of the centre of C. [] (iv) Eplain wh the point (4, 6) lies within onl one of the circles C and C. [] Solution: (i) Given that = 6, Centre,, radius 7 units A second circle, C, has a diameter AB. The point A has coordinates (, ) and the equation of the tangent to C at B is 4. (ii) Find the equation of the diameter AB and hence the coordinates of B. [4] 4 4 Therefore, gradient of AB, 4 equation of the diameter AB, B KL Ang, Jan Page 6

17 Solutions to O Level Add Math paper 4 Let B, (iii) radius of C centre of C, C, (iv) The distance between point (4, 6) and C , The point (4, 6) is inside circle C. The distance between point (4, 6) and C 4 6 4, The point (4, 6) is outside circle C. B KL Ang, Jan Page 7

18 Solutions to O Level Add Math paper 4 B KL Ang, Jan Page. (a) Show that d d. [] (b) The diagram shows the line = and part of the curve. The curve intersects the -ais at the point A. The line through A with gradient intersects the curve again at the point B. (i) Verif that the -coordinate of B is. [] (ii) Determine the area of the shaded region bounded b the curve, the line, the -ais and the line AB. [4] Solution: (a) Let u. d d u A O B

19 Solutions to O Level Add Math paper 4 B KL Ang, Jan Page 9 (b) (i) When, where, A Equation of the line AB, Let or 64 4, B (ii) area of the shaded region d d 9

20 Solutions to O Level Add Math paper 4 B KL Ang, Jan Page square units ( s.f.) In Summar: A simple routine question on integration. Part (a) is to facilitate the integration.