1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH MID YEAR EXAM 2023 ADDITIONAL MATHEMATICS DATE: TIME : 13 JUNE 2023 (TUESDAY) 1335 – 1535 ( 2 Hours) NAME：______________________ REG NO:_________________ CLASS: S1YI / S1PIN / S1HE / S1AI 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 and student registration 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. • 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. • For , use either your calculator value or 3.142. INFORMATION • The total mark for this paper is 100. • The number of marks for each question or part question is shown in brackets [ ]. Do Not Turn Over This Page Until You Are Told To Do So This paper consists of 15 printed pages including a cover page. Prepared by ： ………………… (MS. KOO KAR KEI) Checked by： ………………… (MR.ONG EIK HOOI) Koo Ong
2 1. a) It is given that () = 2 + 3 , () = 4 − 1 for ∈ ℝ. i) State the range of f. [1] ii) Solve () = 4. [3] b) A function h is such that ℎ() = 2+1 −4 for ∈ ℝ , ≠ 4. i) Find ℎ −1 () and state its range. [3] iii) Find ℎ 2 (), giving your answer in its simplest form. [3]
3 2. Find the set of values of k for which the line = (4 − 3) does not intersect the curve = 4 2 + 8 − 8. [4]
4 3. a) Show that = 3 2 − 6 + 5 can be written in the form = ( − ) 2 + , where a, b and c are constants to be found. [3] b) Hence, sketch the graph of = |3 2 − 6 + 5| on the axes below and showing its maximum/ minimum point. [3] 0 y x
5 4. a) Solve the equation 163−1 = 8 +2 . [3] b) Given that ( 1 3 − 1 2) 3 − 2 3 1 2 = , find the value of each of the constants p and q. [2]
6 5. The diagram shows triangle ABC with the side AB = (4√3+1) cm. Angle B is a right angle. It is given that the area of this triangle is 47 2 2 . a) Find the length of the side BC in the form (√3 + ), where a and b are integers. [3] b) Hence find the length of the side AC in the form √2 cm, where p is an integer. [2] A B (4√3+1) cm C Not to scale
7 6. It is given that () = 2 3 + 2 + 4 + , where a and b are constants. It is given that 2 + 1 is a factor of () and that when () is divided by − 1 there is a remainder of -12. a) Find the value of a and b. [5] b) Using the value of a and b, write () in the form (2 + 1)(), where () is a quadratic expression. [2] c) Hence find the exact solution of the equation () = 0. [3]
8 7. Variables and are such that when lg is plotted against 2 , a straight-line graph passing through the points (1,0.73) and (4,0.10) is obtained. a) Given that = 2 , find the value of each of the constants A and b. [5] b) Find the value of y when x = 1.5. [2] c) Find the positive value of x when y = 2. [2]
9 8. The diagram below shows a parallelogram OABC such that O is the origin and A is point (2,6). The equations of the straight lines OA, OC and CB are = 3, = 1 2 and = 3 − 15 respectively. The perpendicular line from point A to OC meets OC at point D. a) Find the coordinates of i) point D, [3] ii) point B, [3] 0 y C A(2,6) x B D y =3x y =3x-15 y = 1 2 x
10 iii) point C. [2] b) Find the area of the shape ABCD. [3]
11 9. The diagram shows a circle PQRT, centre O and radius 10 cm. AQB is a tangent to the circle at Q. The straight lines, AO, and BO, intersect the circle at P and R respectively. OPQR is a rhombus. ACB is an arc of a circle, centre O. Calculate a) the angle α, in terms of π, [2] b) the length, in cm, of the arc ACB, [4]
12 c) the area, in 2 , of the shaded region. [4] 10. The diagram shows a sector AOB with centre O. The length of the arc AB is 7.5cm and the perimeter of the sector AOB is 25cm. Find the value of θ, in radian. [3]
13 11. Without using calculator, find the exact value of a) cos 315° [2] b) cot ( 5 3 ) [2] 12. Given cos A = 2 5 and 3 2 ≤ ≤ 2, find the exact value for each of the following. a) tan A [2] b) cosec A [2]
14 13. a) The curve = + sin has and amplitude of 4 and a period of 3 . Given that the curve passes through the point ( 12 , 2), find the value of each of the constants a, b and c. [4] b) Hence, by using your values of a, b, and c, sketch the graph of = | + sin | for 0 ≤ ≤ radians. State clearly any points where your graph meets the coordinates axes. [5] 0 x y
15 14. The table shows the experimental values of two variables x and y which are related by = 2 + where p and q are constants. x 0.8 1 1.3 1.4 1.5 1.7 y 108.75 79 45.38 36.5 26.67 8.19 a) On the provided graph paper on next page, plot xy against 3 and draw a straight-line graph. [5] b) From the graph, estimate the value of i) p and q, [3] ii) x when y = 45 . [2] -End of paper-
