Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level

Transcription

1 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex.html Page of /0/0 Solutionbank C Exercise, Question The sector AOB is removed from a circle of radius 5 cm. The AOB is. radians and OA = OB. (a) Find the perimeter of the sector AOB. () (b) Find the area of sector AOB. () (a) Arc length = rθ = 5. = 7 cm Perimeter = 0 + Arc = 7 cm (b) Sector area = r θ = 5. = 7.5 cm Pearson Education Ltd 008

2 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex.html Page of /0/0 Solutionbank C Exercise, Question Given that log x = p: (a) Find log ( 8x ) in terms of p. () (b) Given also that p = 5, find the value of x. () (a) log x = p log ( 8x ) = log 8 + log x = + log x = + p (b) log x = 5 x = 5 x = Pearson Education Ltd 008

3 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex.html Page of /0/0 Solutionbank C Exercise, Question (a) Find the value of the constant a so that ( x ) is a factor of x ax 6. () (b) Using this value of a, factorise x ax 6 completely. () (a) Let f ( x ) = x ax 6 If ( x ) is a factor then f ( ) = 0 i.e. 0 = 7 a 6 So a = a = 7 (b) x 7x 6 has ( x ) as a factor, so x 7x 6 = ( x ) ( x + x + ) = ( x ) ( x + ) ( x + ) Pearson Education Ltd 008

4 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex.html Page of /0/0 Solutionbank C Exercise, Question (a) Find the coefficient of x and the coefficient of x in the binomial expansion of ( + x ) 5. () The coefficient of x and the coefficient of x in the binomial expansion of ( + kx ) 5 are equal. (b) Find the value of the constant k. () (a) ( + x ) 5 = x + x + Coefficient of x = 6 = 65 6 = 80 Coefficient of x = 8 = 55 8 = 60 (b) 80k = 60k So k = i.e. k = Pearson Education Ltd 008

5 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 5.html Page of /0/0 Solutionbank C Exercise, Question 5 (a) Prove that: cos θ sin θ + sin θ sin θ, 0 <θ<80. () sin θ (b) Hence, or otherwise, solve the following equation for 0 <θ<80 : cos θ sin θ + sin θ = Give your answers to the nearest degree. () (a) LHS = cos θ sin θ + sin θ = (using sin θ + cos θ ) = (factorising) = (cancelling [ + sin θ ] ) = RHS sin θ sin θ + sin θ ( sin θ ) ( + sin θ ) sin θ ( + sin θ ) sin θ sin θ (b) = cos θ sin θ + sin θ = sin θ = sin θ (can multiply by sin θ 0<θ<80) sin θ = sin θ = sin θ sin θ So θ = 9.7, 60.5 = 9, 6 (to nearest degree) Pearson Education Ltd 008

6 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 6.html Page of /0/0 Solutionbank C Exercise, Question 6 (a) Show that the centre of the circle with equation x + y = 6x + 8y is (, ) and find the radius of the circle. (5) (b) Find the exact length of the tangents from the point (0, 0) to the circle. () (a) x 6x + y 8y = 0 ( x ) + ( y ) = i.e. ( x ) + ( y ) = 5 Centre (, ), radius 5 (b) Distance from (, ) to (0, 0) = \ 7 + = \ 65 Length of tangent = \ \ 65 5 = \ 0 = \ 0 Pearson Education Ltd 008

7 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 7.html Page of /0/0 Solutionbank C Exercise, Question 7 A father promises his daughter an eternal gift on her birthday. On day she receives 75 and each following day she receives of the amount given to her the day before. The father promises that this will go on for ever. (a) Show that after days the daughter will have received 5. () (b) Find how much money the father should set aside to ensure that he can cover the cost of the gift. () After k days the total amount of money that the daughter will have received exceeds 00. (c) Find the smallest value of k. (5) (a) Day = 75, day = 50, total = 5 (b) a = 75, r = geometric series a S = = r 75 Amount required = 5 (c) S k = a ( r k ) r 75 [ ( ) k ] Require > 00 i.e. 5 k > 00 k > > k 9 Take logs: log > k log Since log is negative, when we divide by this the inequality will change around.

8 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 7.html Page of /0/0 So k > log ( ) 9 log ( ) i.e. k > 5.9 So need k = 6 Pearson Education Ltd 008

9 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 8.html Page of /0/0 Solutionbank C Exercise, Question 8 Given I = + x dx : (a) Use the trapezium rule with the table below to estimate I to significant figures. () (b) Find the exact value of I. () x x.5.5 y (c) Calculate, to significant figure, the percentage error incurred by using the trapezium rule as in part (a) to estimate I. () (a) h = I = 6.7 = (b) I = x + x dx x = + = + x = + + = 6 x x (c) Percentage error = 00 = 0.79 % = 0. % 6 Pearson Education Ltd 008

10 Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex 9.html Page of /0/0 Solutionbank C Exercise, Question 9 The curve C has equation y = 6x 7x +. 7 dy (a) Find. () dx d y (b) Find. () dx (c) Use your answers to parts (a) and (b) to find the coordinates of the stationary points on C and determine their nature. (9) 7 (a) y = 6x 7x + dy dx dy dx 7 = 6 x x = x x d y 56 dx (b) = x dy (c) = 0 x x = 0 x x = 0 dx So x = 0 or x = 0 d y dx = < 0 (0, ) is a maximum d y 56 dx x = = > 0 (, ) is a minimum Pearson Education Ltd 008