1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH MID YEAR EXAMINATION 2022 MATHEMATICS PAPER FOUR DATE: TIME : 13 APRIL 2021 (WEDNESDAY ) 1305 - 1535 (2 Hour 30 Minutes) NAME：______________________ REG NO:_________________ CLASS: : S2HE/S2QIN/S2AI/S2ZI/S2YI/S2PIN 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 130. 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 18 printed pages including cover page. Prepared by ： ………………… (MS. KOO KAR KEI) Checked by： Ong ………………… (MR.ONG EIK HOOI) Koo
2 1. 12000 vehicles drive through a road toll on one day. The ratio cars : trucks : motorcycles = 13 : 8 : 3. (a) (i) Show that 6500 cars drive through the road toll on that day. [1] (ii) Calculate the number of trucks that drive through the road toll on that day. [1] (b) The toll charges in 2014 are shown in the table. Vehicle Charge Cars $2 Trucks $5 Motorcycles $1 Show that the total amount paid in tolls on that day is $34500. [2]
3 (c) This total amount is a decrease of 8% on the total amount paid on the same day in 2013. Calculate the total amount paid on that day in 2013. [3] (d) 2750 of the 6500 car drivers pay their toll using a credit card. Write down, in its simplest terms, the fraction of car drivers who pay using a credit card. [2]
4 2. (a) x is an integer. ξ = {: 1 ≤ x ≤ 10} A = {x:x is a factor of 12} B = {x:x is an odd number} C = {x:x is a prime number} (i) Complete the Venn diagram to show this information. [3] (ii) Use set notation to complete each statement. 6 . . . . . . . . . . . . . A A ∩ B ∩ C = . . . . . . . . . . . . . . A ∩ A' = . . . . . . . . . . . . . [3] (iii) Find n(B). [1] b) (i) Use set notation to complete the statement. {, }. . . . . . . . . . . . . . [1] (ii) Shade ∩ ( ∪ )' [1]
5 3. 120 students take a mathematics examination. (a) The time taken, m minutes, for each student to answer question is shown in this table. Time (m minutes) 0 < m ≤ 1 1 < m ≤ 2 2 < m ≤ 3 3 < m ≤ 4 4 < m ≤ 5 5 < m ≤ 6 Frequency 72 21 9 11 5 2 Calculate an estimate of the mean time taken. [4] (b) (i) Using the table in part (a), complete this cumulative frequency table. Time (m minutes) m ≤ 1 m ≤ 2 m ≤ 3 m ≤ 4 m ≤ 5 m ≤ 6 Frequency 72 120 [2]
6 (ii) Draw a cumulative frequency diagram to show the time taken. [3] (iii) Use your cumulative frequency diagram to find (a) the median, [1] (b) the inter-quartile range, [2] (c) the 35 th percentile. [2]
7 (c) A new frequency table is made from the table shown in part (a). Time (m minutes) 0 < m ≤ 1 1 < m ≤ 3 3 < m ≤ 6 Frequency 72 (i) Complete the table above. [2] (ii) A histogram was drawn and the height of the first block representing the time 0 < m ≤ 1 was 3.6cm. Calculate the heights of the other two blocks. [3]
8 4. The diagram shows the graph of y = f (x) for -2.5≤ x ≤2.
9 (a) Find f(1). [1] (b) Solve f(x) = 3. [1] (c) The equation f(x) = k has only one solution for -2.5≤ x ≤2. Write down the range of values of k for which this is possible. [2] (d) By drawing a suitable straight line, solve the equation f(x) = x – 5. [3] (e) Draw a tangent to the graph of y = f(x) at the point where x = 1. Use your tangent to estimate the gradient of y = f(x) when x = 1. [3]
10 5. OAB is a triangle and ABC and PQC are straight lines. P is the midpoint of OA, Q is the midpoint of PC and OQ : QB = 3 : 1. = 4 and = 8. (a) Find, in terms of a and/or b, in its simplest form. i. [1] ii. [1] iii. [1] (b) By using vectors, find the ratio AB : BC. [3]
11 6. (a) A, B and C are points on horizontal ground. BT is a vertical pole. AT = 60m, AB = 50m, BC = 70m and angle ABC = 130°. (i) Calculate the angle of elevation of T from C. [5] (ii) Calculate the length AC. [4]
12 (iii) Calculate the area of triangle ABC. [2] (b) A cuboid has length 45cm, width 22cm and height 12cm. Calculate the length of the straight line XY. [4]
13 7. David buys potatoes in small sacks, each of mass 4 kg, and large sacks, each of mass 10kg. He buys x small sacks and y large sacks. Today, he buys less than 80kg of potatoes. (a) Show that 2x + 5y < 40. [1] (b) He buys more large sacks than small sacks. He buys no more than 6 large sacks. Write down two inequalities to show this information. [2] (c) On the grid, show the information in part (a) and part (b) by drawing three straight lines and shading the unwanted regions. [5] (d) Find the greatest mass of potatoes the cook can buy today. [2]
14 8. (a) Simplify i. 3 ÷ 3 5 [1] ii. 5xy 8 × 3x 6y−5 [2] iii. (64x 12) 2 3 [2] (b) Factorise 121 2 − 2 [2] (c) Show that 2 2+11 − 1 −4 = 1 2 simplifies to 2x 2 + 3x − 6 = 0. [3]
15 9. (a) One of these 7 cards is chosen at random. Write down the probability that the card. (i) shows the letter A, [1] (ii) shows the letter A or B, [1] (iii) does not show the letter B. [1] (b) Two of the cards are chosen at random, without replacement. Find the probability that (i) both show the letter A, [2] (ii) the two letters are different. [3] (c) Three of the cards are chosen at random, without replacement. Find the probability that the cards do not show the letter C. [2]
16 10. (a) On the grid, draw the image of (i) triangle T after a reflection in the line x = -1, [2] (ii) triangle T after a rotation through 180˚ about (0,0). [2] (b) Describe fully the single transformation that maps (i) triangle T onto triangle U, [2] (ii) triangle T onto triangle V. [3]
17 11. The first four terms of sequences A, B, C and D are shown in the table. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 1 3 2 4 3 5 4 6 B 3 4 5 6 C -1 0 1 2 D -3 0 5 12 (a) Complete the table. [8] (b) Which term in sequence A is equal to 36 37 ? [2] (c) Which term in sequence D is equal to 725? [2]
18 12. Brad travelled from his home in New York to Chamonix. He left his home at 1630 and travelled by taxi to the airport in New York. This journey took 55 minutes and had an average speed of 18km/h. He then travelled by plane to Geneva, departing from New York at 2215. The flight path can be taken as an arc of a circle of radius 6400km with a sector angle of 55.5°. The local time in Geneva is 6 hours ahead of the local time in New York. Brad arrived in Geneva at 1125 the next day. To complete his journey, Brad travelled by bus from Geneva to Chamonix. This journey started at 1300 and took 1 hour 36 minutes. The average speed was 65km/h. The local time in Chamonix is the same as the local time in Geneva. Find the overall average speed of Brad’s journey from his home in New York to Chamonix. Show all your working and give your answer in km/h. [11] -----END OF PAPER----
