ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2004) 1. Diagram below shows the arrangement of the first three of an infinite series of similar triangles. The first triangle has a base of x cm and a height of y cm. the measurements of the base and height of each subsequent triangle are half of the measurements of its previous one 2. Diagram below shows the arrangement of the first three of an infinite series of similar triangles. The first triangle has a base of x cm and a height of y cm. The measurements of the base and height of each subsequent triangle are half of the measurements of its previous one y cm x cm (a) Show that the area of the triangles form a geometric progression and state the common ratio . [3 marks] (b) Given that x = 80 cm and y = 40 cm, 1 2 (i) determine which triangle has an area of 6 cm . 4 (ii) find the sum to infinity of the area, in cm2, of the triangles. [5 marks] [r = 1 1 , n = 5, 2133 ] 4 3 (a) Show that the area of the triangles form a geometric progression and state the common ratio . [3 marks] (b) Given that x = 16 cm and y = 18 cm, (i) determine which triangle has an area of 9 cm2 1 6 (ii) find the sum to infinity of the area, in cm2, of the triangles [5 marks] [r= 1 , n = 5, 192 ] 4 3. The diagram shows the arrangement of the first three circles of an infinite series of similar circles. The first circle has a radius of r cm. The radius of each subsequent circle is half of the radius of its previous one. 2. The diagram shows the arrangement of the first three of an infinite series of similar triangles. The first triangle has a base of b cm and a height of h cm. The measurements of the base and height of each subsequent triangle are two thirds of the measurements of its previous one r cm h cm (a) Show that the areas of the circles from a geometric progression and state the common ratio. (b) Given that the diameter if the first circle is 40 cm, (i) determine which circle has an area of 2 b cm 25 π 256 (a) Show that the area of the triangles form a geometric progression and state the common ratio. (b) Given that b = 27 cm and h = 36 cm, (i) determine which triangle has an area of 18 (ii) find the sum to infinity of the area of the triangles. [r= 4 4 , n = 5, 874 cm2] 9 5 cm , (ii) find the sum to infinity of the areas, in cm2, of the circles, in terms of π . [r = 1 1 ; 7th, 533 π cm2] 4 3 26 cm2 27 1 Progression Paper 2 ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2005) 1. 2. 20 cm The diagram above shows part of an arrangement of bricks of equal size. The number of bricks in the lowest row is 100. For each of the other rows, the number of bricks is 2 less than in the row below. The height of each brick is 6 cm. Ali builds a wall by arranging bricks in this way. The number of bricks in the highest row is 4 , calculate (a) the height, in cm, of the wall. [3marks] (b) the total price of the bricks used if the price of one brick is 40 sen. [3marks] The diagram shows part of the arrangement of a building structure built of bricks of equal size. The number of bricks in the lowest row is 120 and the number of bricks in each subsequent row is less 1 than the previous row. Each brick is 12 cm thick. The highest row consists of 8 bricks. Calcualte (a) the height, in cm, of the whole structure, [3] (b) the total price of bricks used if each brick costs 35 sen. [3] [1356 cm; RM 2 531.20] [n = 49, 294cm; 2 548, RM1 019.20 ] 3. The diagram shows part of an arrangement of 4. The diagram shows a part of an arrangement of bricks of equal size. bricks of equal size. The number of bricks in the lowest row os 73. In each of the other rows, the number of bricks is 2 less than the row below. The height of each brick is 10 cm. 7 cm 8 cm The number of bricks in the lowest row is 80. For each of the other rows, the number of bricks is 2 less than that in the row below. Each brick has a height of 7 cm and a width of 8 cm. Asrul builds a wall by arranging the bricks in this way. The height of the wall is 2.45 m. (a) How many bricks are there in the highest row? (b) Asrul wants to paint the wall. What is the total area of the wall that he needs to paint? 10 cm Sani builds a wall by arranging the bricks in this way. The number of bricks in the highest row is 9.. (a) Find the height of the wall. (b) Calculate the total price of the bricks used if the price of each brick is 80 sen. [12, 90 160 cm2] 2 [330 cm, RM1 082.40] Progression Paper 2 ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2006) 1. Two companies, Delta and Omega, start to sell cars at the same time. (a) Delta sells k cars in the first month and its sales increase constantly by m cars every subsequent month. It sells 240 cars in the 8th month and the total sales for the first 10 months are 1 900 cars. Find the value of k and of m. [5 marks] (b) Omega sells 80 cars in the first month and its sales increase constantly by 22 cars every subsequent month. If both companies sell the same number of cars in the nth month, find the value of n. [2 marks] 2. Nora and Azman start working for two different companies at the same time. (a) Nora is paid RMp in the first year and her annual salary increases constantly by RMq every subsequent year. She is paid RM38 000 in the 9th year and her total salary for the 12 years is RM426 000. Find the values of p and q. [5] (b) Azman is paid RM28 000 in the first year and his annual salary increases constantly by RM1500 every subsequent year. If Nora and his salaries are the same in the nth year, find the value of n. [k = 100, m = 20; n = 11] 3. Two companies, Ali and Hassan, start to sell air conditioners at the same time. (a) Ali sells p air conditioners in the first month and its sales increase constantly by d air conditioners every subsequent month. Its sells 88 air conditioners in the 5th month and the total sales for the first 12 months are 1362. (b) Hassan sells 38 air conditioners in the first month. If both companies sell the same number of air conditioners in the 10th month, find the sales of Ali in the 11th month. [p = 30 000, q = 1000; n = 5] 4. Two shops, Alfa and Beta, start to sell computers at the same time. (a) Alfa sells p computers in the first month and its sales increase constantly by q computers every subsequent month. It sells 184 computers in the 6th month and the total sales for the first 10 months are 1 740 computers. Find the values of p and q. (b) Beta sells 60 computers in the first month and its sales increase constantly by 24 computers everu subsequent month. If both shops sell the same number of computers in the nth month, find the value of n. [p = 20, d = 3; 188] 3 [p = 84, q = 20; n = 7] Progression Paper 2 ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2007) 1. Diagram shows the side elevation of part of stairs built of cement blocks. The thickness of each block is 15 cm. The length of the first block is 985 cm. The length of each subsequent block is 30 cm less than the preceding block as shown in the diagram. 2. 700 cm 20 cm 750 cm 800 cm (a) If the height of the stairs to be built is 3 m, calculate (i) the length of the top most block (ii) the total length of the blocks. [5marks] (b) calculate the maximum height of the stairs. [3marks] [ 415 cm, 14 000 cm; 495 cm] 3. The diagram shows part of the arrangement of a model built of wooden blocks. The height of each bloc is 20 cm. The length of the lowest block is 50 cm less than the previous block and so on. (a) If the height of the model is 2.6 m. calculate (i) the length of the highest block, (ii) the total length of the blocks. [5] (b) Calculate the maximum height the model. [3] [200 cm, 55 m; 320 cm 4. The following diagram shows part of the side elevation of some stairs built of cement blocks. The thickness of each block is 20 cm. The length of the first block is 935 cm. x cm The length of the first block is 1000 cm. The length of each subsequent block is 45 cm less than the preceding block. (a) If the height of the stairs is 250 cm and the length of the highest block is 595 cm, find (i) the thickness, x, in cm, of each block, (ii) the total length, in cm, of the blocks. (b) If the thickness of each block is 30 cm, calculate the maximum height of the stairs. 875 cm 905 cm 935 cm 20 cm The length of each subsequent block is 30 cm less than the preceding block. (a) If the height of the stairs is 5 m, find (i) the length of the topmost block. (ii) the total length of the blocks. (b) Calculate the maximum height of the stairs. [25 cm, 7 975 cm; 690 cm] 4 Progression Paper 2 [215 cm, 14 375 cm; 640 cm] ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2008) 1. Muthu started working for a company on 1 January 2002 with an initial annual salary of RM18,000. Every January, the company increased his salary by 5% of the previous year’s salary. Calculate (a) his annual salary, to the nearest RM, for the year 2007 [3marks] (b) the minimum value of n such that his annual salary in the nth year will exceed RM36,000 [2marks] (c) the total salary, to the nearest RM, paid to him by the company, for the years 2002 to 2007. [2marks] 2. The total sales of a company was RM24 000 in the year of 2005. The company planned to increase its sales at a rate of 4% every subsequent year. Calculate (a) the total sales, to the nearest RM, for the year 2010, [3] (b) the least value of n for the total sales in the nth year to exceed RM40 000, [2] (c) the total sales, to the nearest RM, for years 2005 to 2010. [2] [RM22 973; 16; RM122 434] 3. Alice signed up an insurance policy on 1 February 2003 with an initial annual payment of RM880. Every February, her insurance payment increased by 2.5% of the previous payment. Calculate (a) her annual payment, to the nearest RM, for the year of 2010, (b) the maximum value of n such that her annual payment in the nth year will be less than RM1 150, (c) the total payment, to the nearest RM, for the years 2006 to 2010. [RM29 200; 15; RM159 191] 4. Ramli started working for a company on 1 Januari 2003 with an initial annual salary of RM24 000. Every January, the company increased his salary by 6% of the previous year’s salary. (a) Calculate his annual salary, to the nearest RM, for the year 2009. (b) Find the minimum value of n such that his annual salary in the nth year will exceed RM50 000. (c) Calculate his total salary, to the nearest RM, for the years 2003 to 2009. [RM1 046, 11, RM4 981] 5 [RM34 044, n = 14, RM201 452] Progression Paper 2 ENRICHMENT EXERCISES 3 SPM FORMATTED QUESTIONS (spm 2009) 1. A ball is dropped from a cupboard at a height of p cm above the floor. After the first bounce, the ball reaches a height of p1 cm, where p1 = 0.7p. After the second bounce, the ball reaches a height of p2 cm,where p2 = 0.7p1. The ball continues bouncing in this way untl its stops. Given that p = 120, find (a) the number of bounce when the maximum height of the ball from the floor is less than 45 cm for the first time. (b) total distance travelled by the ball until it stop, in cm. 2. A ball is released from a height of D cm above the ground. After the first bounce, it reaches a height of D1 cm where D1 = 0.9D. After the second bounce, it reaches a height or D2 cm where D2 = 0.9D1 and this is continued until it stops. Given the D = 400, find (a) the number of bounces when the maximum height of the ball from the ground is less that 100 cm for the first time. (b) the total distance, in cm, travelled by the ball until it stops. [4; 680 cm] 3. Ahmad drops a ball from a height of y cm above the floor. After the first bounce, the ball reaches a height of y1 cm, which is 70% of y. After the second bounce, the ball reaches a height of y2 cm, which is 70% of y1. The ball continues bouncing in this way until it stops. Show that y2 = (0.70)2y, and hence, given that y = 210, find (a) the number of bounces when the maximum height of the ball from the floor is less than 40 cm for the first time, (b) the total distance, in cm, travelled by the ball until it comes to rest on the floor. [14; 7600 cm] SPM 2010 1. The diagram shows the arrangement of cylinders having the same radius, r cm. The height of the first cylinder is 2 cm and the height of each subsequent cylinder increases by 3 cm. [Volume of cylinder = π r2h] 8 cm 5 cm (a) Calculate the volume, in cm3, of the 14th cylinder, in terms of π and r. (b) Given the total volume of the first n cyclinders is 301 π r2 cm2, find the value of n. [5; 1190 cm] 6 [41π r2 cm3; 14] Progression Paper 2 ENRICHMENT EXERCISES 3 7 Progression Paper 2