H2 Mathematics 2017 Promotion Exam Paper 1 Question (VJC Answer all questions [100 marks]. By using an algebraic approach, solve. 3 x.

Transcription

1 H Mathematics 07 Promotion Eam Paper Question (VJC Answer all questions [00 marks]. By using an algebraic approach, solve 0. [] Verify that = 5 i is a root of the equation = 0. Hence, without the use of a calculator, find the other roots. [] (a) the eact value of cos 0 e ( ) ( ) d. [] (b) + d. [] the epansion of in ascending powers of, up to and including the term in. [] By substituting = into the epansion in part, find an approimation for in the form a, where a and b are integers to be determined. [] b 5 A curve C has parametric equations 5 t t y t t = +, = +. The tangent to C at the point P where t = p is denoted by l. It is given that l is parallel to the y-ais. the equation of l. [] the least value of. [] (iii) Sketch C, indicating the coordinates of the points where C crosses the -ais. []

2 6 The diagram below shows the cross section of a cooling tower of height 5 m. The radii of its circular base and its circular top are 50 m and 7.5 m respectively. The narrowest part of the tower occurs at a height of 80 m above the base of the tower. 7.5 m 5 m 80 m Take the origin to be at the center of the base, and m to be unit on both aes. The walls of the tower is part of the curve C with equation a ( y k ) =. b the values of a, b and k. [] For the rest of the question, use 80, 900 and 600 as the values of k, a and b respectively. Sketch C, indicating the equations of any asymptotes and coordinates of the points where C crosses the aes, where appropriate. [] (iii) Given that r is a positive constant and C intersects the curve with equation ( 80) 50 m + y = r at eactly two distinct points, state the value of r. [] 7 Given that y = tan, π < < π, show that d y dy = y. [] d d By further differentiation of this result, find the Maclaurin series for y up to and including the term in. [] tan d. [] (iii) the Maclaurin series for ln sec up to and including the term in. []

3 8 (a) m h m r m m m m Water is poured at a rate of 9 m per minute into an open container in the form of a frustum of a right circular cone as shown in the diagram above. The container has bottom radius m, top radius m and height m. After t minutes, the radius of the water surface is r m and the depth of water is h m, where 0 h <. h Show that r = +. [], in eact form, the rate of increase of the depth of water when h =. [] [The volume of a frustum of a right circular cone of bottom radius r, top radius r and height h is given by V = π ( r + r + r r ) h.] (b) The vertical viewing angle is the angle between two lines from the eye level of the viewer; one from the eye level to the top of the screen, and the other from the eye level to the bottom of the screen. Research has shown that viewer comfort will be compromised if this angle eceeds 5 π radians. The diagram below shows a movie screen which is m high with its lowest point mounted at m above the eye level of a viewer. The viewer positions himself at a variable horizontal distance m from the vertical wall, resulting in a vertical viewing angle θ. Determine, by differentiation, if the viewer comfort will be compromised based on what the research has found. [6] Movie m Wall Eye level of viewer θ m m

4 9 Show that = r r r r r r r ( + ) ( + )( + ) ( + )( + ). [] Epress N in the form r= r r + r + ( )( ) B A + + ( N + )( N + ) ( N + )( N + ) C, where the eact values of A, B, C are to be determined. [] By writing as k + r ( r + )( r + ) where k is a constant to be r r r determined, show that N r + 6r + 8r + < N + for all N. [] r r + r + 96 r= ( )( ) (iii) Using your result in part, find N. [] r= r + r + r + ( )( )( 6) 0 The diagram below shows the graph of y = f ( ). The graph cuts the - and y-aes at the points (,0 ) and ( 0, ) respectively. It has a turning point at ( 5, 5) the asymptotes are =, y = 0 and y =.. The equations of On separate diagrams, sketch the graph of y ( ) y ( ) (iii) y f ( ) (iv) = f +, [] = f, [] =, [] y = f, given that 0 f =. [] ( ) ( )

5 The functions f and g are defined by + f : a,,, g : a e,. Sketch the graph of y = f ( ), stating the equations of the asymptotes and the coordinates of the points of intersection with the aes. Hence, show that f eists. [] f ( ). Hence or otherwise, find f ( ). [] (iii) the range of gf. (You do not need to show that gf eists.) [] (a) A particular bond is issued at $00 per unit with a 5% annual coupon rate. The bondholder will receive a fied amount of $(00 5%) as coupons for bond that he holds, at the end of every year until the bond matures after 0 years. At the start of 06, Mr Reech purchased 0 units of this particular bond. At the start of each subsequent year, he purchases another 0 units of the same bond. Assuming that Mr Reech receives the coupons as cash at the end of each year, show that the total amount of cash he receives over a period of 0 years is $0,500. Mr Puwer, on the other hand, puts his money in a savings account that pays him k% compound interest at the end of every year. At the start of 06, Mr Puwer deposits $800 in the savings account. At the start of each subsequent year, he deposits $800 into the same account. Mr Puwer does not withdraw any money from this account. Show that the total amount of money Mr Puwer has in his n 800( 00 + k ) 00 + k savings account at the end of n years is $. [] k 00 (iii) Hence, find the range of values of k such that at the end of 0 years, Mr Reech receives more cash in coupons than what Mr Puwer receives in total interest paid. [] (b) Mr Soo has $00,000 in his savings account, and the prevailing interest rate is % per annum. Mr Soo does not deposit any more money in his account. Interest paid at the end of every year is withdrawn by him from the account, but the annual interest rate is halved every year. the minimum number of years it takes for the total interest withdrawn by Mr Soo to be at most $0 less than the theoretical maimum total interest payable by the bank. [] [] End Of Paper 5