Solutionbank Edexcel AS and A Level Modular Mathematics

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1 Page of Exercise A, Question Use the binomial theorem to expand, x <, in ascending powers of x, as far as the term in x, giving each coefficient as a simplified fraction. (6) ( + x ) = + ( + x ) = x ( ) ( ) x ( ) ( ) ( 4 ) +... = x + x x = x 4 x 6 x x 8 x Pearson Education Ltd 009

2 Page of Exercise A, Question The curve C has equation x + y 4x 6yx + = 0 Find the gradient of C at the point (, ). (7) x + y 4x 6yx + = 0 Differentiate with respect to x: dy dy x + 4y 4 6x + 6y = 0 At the point (, ), x = and y =. dy dy = = 0 dy dy = = the gradient of C at (, ) is. Pearson Education Ltd 009

3 Page of Exercise A, Question Use the substitution u = x +, to find an exact value for 0x 0 (9) ( x + ) u = x + du = and x = u 0x ( u ) = ( x + ) = du u u = du u u u = u u du u = u + u Change the limits: x = 0 u = and x = u = 8 Integral = + + = 8 du 8 08 Pearson Education Ltd 009

4 Page of Exercise A, Question 4 (a) Find the values of A and B for which ( x + ) ( x ) A + () x + B x (b) Hence find ( x ). (4) ( x + ) ( x ), giving your answer in the form y = ln f (c) Hence, or otherwise, obtain the solution of dy x + x = 0y, y > 0, x > for which y = at x =, giving your answer in the form y = f ( x ). () (a) + ( x + ) ( x ) A ( x ) + B ( x + ) Substitute x =, then B = B = Substitute x =, then A = A = (b) Integral = + A ( x + ) x + x B ( x ) A ( x ) + B ( x + ) ( x + ) ( x ) = ln x + + ln x + C = ln k (c) Separate the variables to give dy y = x x + 0 ( x + ) ( x )

5 Page of ln y = ln x ln x + + C y = when x = C = ln 7 = ln 49 x y = 49 x + Pearson Education Ltd 009

6 Page of Exercise A, Question A population grows in such a way that the rate of change of the population P at time t in days is proportional to P. (a) Write down a differential equation relating P and t. () (b) Show, by solving this equation or by differentiation, that the general solution of this equation may be written as P = Ak t, where A and k are positive constants. () Initially the population is 8 million and 7 days later it has grown to 8. million. (c) Find the size of the population after a further 8 days. () (a) dp dt dp dt P = m P (b) dp P = m dt ln P = mt + C P = e mt + C = Ae mt where A = e C = Ak t where k = e m (c) When t = 0, P = 8 A = 8 When t = 7, P = = 8k 7 k 7 = 8. 8 When t =, P = 8k = 8 ( k 7 )

7 Page of 8. = 8 8 = 0.8 million (to s.f.) Pearson Education Ltd 009

8 Page of Exercise A, Question 6 Referred to an origin O the points A and B have position vectors i j 7k and 0i + 0j + k respectively. P is a point on the line AB. (a) Find a vector equation for the line passing through A and B. () (b) Find the position vector of point P such that OP is perpendicular to AB. () (c) Find the area of triangle OAB. (4) (d) Find the ratio in which P divides the line AB. () (a) AB = 9i + j + k (or BA = 9i j k) the line may be written 9 0 r = + λ or r = 0 + µ or equivalent λ (b). + λ = λ + + 7λ + 7λ λ = 0 0λ 0 = 0 λ = the point P has position vector (c) OP = and AB = \ = 4 0

9 Page of Area of OAB = base height = = (d) AP = 0 = and PB = = PB = AP i.e. P divides AB in the ratio :. Pearson Education Ltd 009

10 Page of Exercise A, Question 7 The curve C, shown has parametric equations x = cos t, y = t sin t, 0 < t <. (a) Find the gradient of the curve at the point P where t =. (4) (b) Show that the area of the finite region beneath the curve, between the lines x =, x = and the x-axis, shown shaded in the diagram, is given by the integral 9t sin t dt sin t cos t dt. (4) (c) Hence, by integration, find an exact value for this area. (7) 6 (a) x = cos t, y = t sin t dt = sin t and = 4 cos t dy = dy dt 4 cos t sin t

11 Page of When t =, = = 6 dy ( ) (b) The area shown is given by t t y Where t is value of parameter when x = and t is value of parameter when x = i.e. cos t = cos t = t = Also cos t = cos t = 0 t = The area is given by t sin t sin t dt = 9t sin t dt 6 sin t cos t sin t dt Using the double angle formula = 9t sin t dt sin t cos t dt dt dt (c) Area = [ 9t cos t ] + 9 cos t dt [ 4 sin t ] = [ 9t cos t + 9 sin t 4 sin t ] = = + 9 8

12 Page of Pearson Education Ltd 009

Solutionbank Edexcel AS and A Level Modular Mathematics

Solutionbank Edexcel AS and A Level Modular Mathematics Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x