1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH MID-YEAR EXAMINATION 2022 ADDITIONAL MATHEMATICS PAPER 1 DATE: 21 APRIL 2022 (THURSDAY) TIME: 1035 - 1235 (2 hours) NAME：______________________ REG NO: _________________ CLASS： S2QIN, S2HE, S2ZI, S2AI You must answer on the question paper. No additional materials are needed. INSTRUCTIONS Answer all questions. Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. Write your name, centre number and candidate number in the boxes at the top of the page. Write your answer to each question in the space provided. Do not use an erasable pen or correction fluid. You should use a calculator where appropriate. You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION The total mark for this paper is 80. The number of marks for each question or part question is shown in brackets [ ]. This paper consists of 13 pages, including this cover page. Prepared by ： ………..…………….. (Ms Leow Sook Wan) Checked by： ……………………….. (Mr Ong Eik Hooi) Do Not Turn This Page Until You Are Told To Do So
2 Mathematical Formulae 1. ALGEBRA 2. TRIGONOMETRY
3 1. Given that (6 3 2 4 5) 4 2 1 2−1 = , find the values of the constants a, p and q. [3] 2. Express √ 1−2 42−4 in the form where k cos is a constant to be found. [4]
4 3. It is given that lg 3 = 10 and lg( 2 ) = , (i) Find, in terms of , expressions for lg and lg . [5] (ii) Find the value of . [1]
5 4. A curve has equation = (3 2 + 15) 2 3. Find the equation of the normal to the curve at the point where = 2 . [5]
6 5. Variables x and y are such that, when 2 is plotted against 2 , a straight line graph is obtained. This line has a gradient of 5 and passes through the point (16, 81). (i) Express 2 in terms of 2 . [3] (ii) Find the value of x when y = 6. [3]
7 6. (i) Given that (3 + ) 5 + (3 − ) 5 = + 2 + 4 , find the value of , of and of . [4] (ii) Hence, using the substitution = 2 , solve, for x, the equation (3 + ) 5 + (3 − ) 5 = 1086 [4]
8 7. (i) Show that (4−√) 2 √ can be written in the form − 1 2 + + 1 2, where p, q and r are integers to be found. [3] (ii) A curve is such that = (4−√) 2 √ for x > 0. Given that the curve passes through the point (9, 30), find the equation of the curve. [5]
9 8. The figure shows a right-angled triangle ABC, where the point A has coordinates (−4, 2), the angle B is 90° and the point C lies on the x-axis. The point M (1, 3) is the midpoint of AB. Find the area of the triangle ABC. [7]
10 9. (i) Given that y = x sin 4x, find . [3] (ii) Hence find ∫ cos 4 and evaluate ∫ cos 4 8 0 . [6]
11 10. (i) Solve 2 2 = 5 + 5, for 0° < < 360°. [5] (ii) Solve √2 sin( 2 + 3 ) = 1, for 0 < < 4 radians. [5]
12 11. Vector a and b are such that a = ( 3 + m 5 − 2 ) and b = ( 4 − 2n 10 + 3 ) . (i) Given that 3a + b = ( 1 + n −5 ), find the value of m and of n. [3] (ii) Show that the magnitude of b is √5, where k is an integer to be found. [2] (iii) Find the unit vector in the direction of b. [1]
13 12. A particle moves in a straight line so that, t second after leaving a fixed point O, its velocity −1 is given by = 3 2 + 4. (i) Find the initial velocity of the particle. [1] (ii) Find the initial acceleration of the particle. [3] (iii) Find the distance travelled by the particle in the third second. [4] -End of paper-