Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find the values of x for which f ( x ) = x x is a decreasing function. f ( x ) = x x f ( x ) = x x Find f ( x ) and put this expression < 0. x x < 0 x ( x ) < 0 f ( x ) is a decreasing function for 0 < x <. Solve the inequality by factorisation, consider the three regions x < 0, 0 < x < and x >, looking for sign changes. dy < 0 for 0 < x < Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Given that A is an acute angle and cos A =, find the exact value of tan A. cos A = Draw a diagram and put in the information for cos A. Use Pythagoras theorem: a + b = c with b = and c =. so a + = a + = 9 a = 5 a = \ 5 so tan A = \ 5 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Evaluate x x. x Remember ax n = axn + n + Change into index form and integrate: x x x = x x = x + + = x = x = x x = [ + ] x Evaluate the integral: substitute x =, then x =, and subtract. ( ) = ( + ) ( ( ) + ) ( ) ( ) = ( 9 + ) ( + ) = 8 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question x Given that y = + x x +, find the values of x when =. dy x y = + x x + d Remember ( ax n ) = anx n dy x = + x x + x = dy Put = and solve the equation. x + x 8 = 0 ( x + ) ( x ) = 0 so x =, x = Factorise x + x 8 = 0 : ( + ) ( ) = 8 ( + ) + ( ) = + so x + x 8 = ( x + ) ( x ) Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 5 Solve, for 0 x<80, the equation cos x = 0., giving your answers to decimal place. cos x = 0. x =.87 cos x is negative, so your need to look in the nd and rd quadrants. Here the angle in the nd quadrant is 80.87 = 5. x =.87,. Read off the solutions from your diagram so x =.,. Find x: divide each value by. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find the area between the curve y = x x, the x-axis and the lines x = and x =. Area = x x Remember ax n = axn + n + x = [ x ] = ( ( ) ) ( Use the limits: Substitute x = and x = into x ( ) ) ( ) = ( ) ( 8 ) = 0 ( 8 ) = 0 ( ) = ( ) and subtract. x Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 7 Given f ( x ) = x x x, (a) find (i) f ( ), (ii) f ( ), (iii) f ( ) (b) interpret your answer to part (a). f ( x ) = x x x f ( x ) = x x f ( x ) = x d Remember f ( x ) = f ( x ), f ( x ) = f ( x ) d (a) (i) f ( ) = ( ) ( ) ( ) = 8 8 8 = 8 Find the value of f ( x ) where x = ; substitute x = into x x x (ii) f ( ) = ( ) ( ) = 8 = 0 Find the value of f ( x ) where x = ; substitute x = into x x (iii) f ( ) = ( ) = = 8 Find the value of f ( x ) when x = ; Substitute x = into x. (b) On the graph of y = f ( x ), the point (, 8 ) is a minimum point. f ( ) = 0 means there is a stationary point at x = f ( ) = 8 > 0 means the stationary point is a minimum. f ( ) = 8 means the graph of y = f ( u ) passes through the point (, 8 ). Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 8 Find all the values of θ in the interval 0 θ<0 for which sin (θ 0 ) = \. sin (θ 0 ) = \ sin (θ 0 ) = \ Divide each side by. θ 0 = 0 Solve the equation: let X =θ 0 \ sin X =, so X = 0 i.e. θ 0 = 0 sin (θ 0 ) is positive so you need to look in the st and nd quadrants. θ 0 = 0, 0 Read off the solutions from your diagram so θ = 0 + 0 Find θ : add 0 to each value. = 90 and θ = 0 + 0 = 50 θ = 90, 50 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 9 The diagram shows the shaded region T which is bounded by the curve y = ( x ) ( x ) and the x-axis. Find the area of the shaded region T. Area = ( x ) ( x ) Expand the brackets so that ( x ) ( x ) = x x x + = x 5x + = x 5x + x = [ + x ] 5x Remember ax n = axn + ( ) = ( 5 ( ) Evaluate the integral. Substitute x =, then x =, into + ( ) ) x 5x + x and subtract. ( ) ( + ( ) ) = 5 ( ) n + The negative value means the area is below the x-axis, as can be seen in the diagram. Area = Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 0 Find the coordinates of the stationary points on the curve with equation y = x x +. dy y = x x + = x Remember dy = 0 at a stationary point. x = 0 x = x = = x = \ = ± When x =, y = ( ) ( ) + = ( ) + = 0 When x = 8 y = ( ) ( ) + = ( ) + + 8 = + + = So the coordinates of the stationary points are (, 0 ) and (, ). Find the coordinates of the stationary points. Substitute x = and x = into the equation for y. Find the coordinates of the stationary points. Substitute x = and x = into the equation for y. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question (a) Given that sin θ = cos θ, find the value of tan θ. (b) Find the value of θ in the interval 0 θ< for which sinθ = cos θ, giving your answer in terms of. (a) sin θ = cos θ sin θ cos θ (b) = tan θ = θ = cos θ cos θ Remember tan θ = sin θ cos θ Divide each side by cos θ. tan θ =, so θ = 5. Remember (radians) = 80 so 5 = (radians). tan θ is positive in the st and rd quadrants. Read off the solutions, in 0 θ<, from your diagram. θ =, 5 + = 5 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question (a) Sketch the graph of y =, x > 0. x (b) Copy and complete the table, giving your values of x to decimal places. (c) Use the trapezium rule, with all the values from your table, to find an estimate for the value of x. (d) Is this an overestimate or an underestimate for the value of x? Give a reason for your answer. (a) Draw the sketch for x > 0. (b) x....8 x 0.8 0.7 0.5 0.55 0.5 (c) Area 0. [ + ( 0.8 + 0.7 + 0.5 + 0.55 ) + 0.5 ] 0.95 0.70. The width of each strip is 0. The values of x are to decimal places, so give the final answer to
Heinemann Solutionbank: Core Maths C Page of (d) This is an overestimate. Due to the shape of the curve, each trapezium will give an area slightly larger than the area under the graph. decimal places. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Show that the stationary point on the curve y = x x + x + is a point of inflexion. dy y = x x + x + = x x + x x + = 0 x x + = 0 ( x ) ( x ) = 0 x = dy Find the stationary point. Put = 0 Simplify. Divide throughout by. Factorise x x + ac =, and ( ) + ( ) = so x x x + = x ( x ) ( x ) = ( x ) ( x ) When x = 0, Find the gradient of the tangent when x = 0 and x =. dy When x = dy = ( 0 ) ( 0 ) + = > 0 = ( ) ( ) + = > 0 Look at the gradient of the tangent on either side of the stationary point. The Stationary point is a point of inflexion. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find all the values of x in the interval 0 x<0 for which tan x =. tan x = tan x = tan x = ± \ Rearrange the equation for tan x. Divide each side by. Take the square root of each side. (i) tan x = ± \ \ = = x = 0 Remember \ \ \ so tan 0 = \ Tan x is positive in the st and rd quadrants. so, x = 0, 0 (ii) tan x = \ x = 0 ( i.e. 0 )
Heinemann Solutionbank: Core Maths C Page of Tan x is negative in the nd and th quadrants. x = 0, 50 Write down all the solutions in 0 x<0 in order of size. so, x = 0, 50, 0, 0 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 5 Evaluate 8 x x. 8 x x Remember ax n = axn + n + = [ x x ] 8 x x = = x ( ) x x = = x ( ) = ( ( 8 ) ( 8 ) ) ( ( ) ( ) ) = ( ( ) ( ) ) ( ( ) ( ) ) = ( ) ( ) ( 8 ) = ( 8 ) = ( \ 8 ) = = = Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question The curve C has equation y = x x + 8x +. (a) Find the coordinates of the turning points of C. (b) Determine the nature of the turning points of C. y = x x + 8x + dy Find the x-coordinate. Solve = 0. (a) dy = x x + 8 x x + 8 = 0 Divide throughout by. x x + = 0 ( x ) ( x ) = 0 Factorize x x + = 0. ac =, ( ) + ( ) = so x x x + = x ( x ) ( x ) = ( x ) ( x ) x =, x = When x =, y = ( ) ( ) + 8 ( ) + When x = = 8 7 Find the y-coordinates. Substitute x = and x = into y = x x + 8x + y = ( ) ( ) + 8 ( ) + = 7 so (, ), (, 7 ). 8 7 Give your answer as coordinates (b) d y = x Remember < 0 is a maximum stationary point, and d y
Heinemann Solutionbank: Core Maths C Page of When x =, d y = ( ) = < 0 (, ) is a maximum. 8 7 When x =, d y = ( ) = > 0 (, 7 ) is a minimum. d y > 0 is a minimum stationary point. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 7 The curve S, for 0 x<0, has equation y = sin x 0. (a) Find the coordinates of the point where S meets the y-axis. (b) Find the coordinates of the points where S meets the x-axis. y = sin ( x 0 ) (a) x = 0 y = sin ( ( 0 ) 0 ) (b) = sin ( 0 ) = so, ( 0, ) y = 0 sin ( x 0 ) = 0 sin ( x 0 ) = 0 x 0 = 0, 80, 0, The curve y = sin ( 0 ) meets the y-axis when x = 0, so substitute x = 0 into y = sin ( x 0 ). x The curve y = sin ( 0 ) meets the x-axis when y = 0, so substitute y = 0 into y = sin ( x 0 ). x (i) x 0 = 0 x = 0 x = 5 Let x 0 = X so sin X = 0 Now, X = 0, 80, 0 Solve for x: X = 0, so x 0 = 0. (ii) X = 80 x, so 0 = 80.
Heinemann Solutionbank: Core Maths C Page of x 0 = 80 x = 0 x = 5 (iii) x 0 = 0 x = 90 x = 585 so ( 5, 0 ), ( 5, 0 ) X = 0 x, so 0 = 0. Solution not in 0 x<0. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 8 Find the area of the finite region bounded by the curve y = ( + x ) ( x ) and the x-axis. Sketch a graph of the curve y = ( + x ) ( x ). Find where the curve meets the x-axis. The curve meets the x-axis when y = 0, so substitute y = 0 into y = ( + x ) ( x ) ( + x ) ( x ) = 0 so x = and x =. Area = ( + x ) ( x ) Find the area under the graph and the x- axis. Integrate y = ( + x ) ( x ) using x = and x = as the limits of the integration. = + x x Expand ( + x ) ( x ). ( + x ) ( x ) = x + x x x = [ x + ] x = ( ( ) + ( ) ) ( ( ) ( ) + ( ) ) ( ) = + x x Remember ax n = axn + n + Evaluate the integral. Substitute x =, then x = into x + and subtract. = 8 ( ) 8 / ( ) = 8 / + / = 0 5 / x = 0 5 / x Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 9 The diagram shows part of the curve with equation y = x ( x ). The curve meets the x-axis at the points O and A. The point B is the maximum point of the curve. (a) Find the coordinates of A. dy (b) Show that = x ( x ). (c) Find the coordinates of B. y = x ( x ) (a) x ( x ) = 0 The curve meets the x-axis when y = 0, so substitute y = 0 into y = x ( x ). (i) x = 0 x = 0 (ii) x = 0 x = so A (, 0 ). d Remember ( ax n ) = anx n (b) y = x ( x ) = x x Expand the brackets.
Heinemann Solutionbank: Core Maths C Page of x = x x x = x x = x + = x dy = x x Differentiate. d ) ( x = x d ) ( x = x = <semantics> </semantics> x = x = x ( x ) as required Factorise. Divide each term by x so that x x = x x x = <semantics> </semantics> x = x ( ) = x + = x = x (c) x ( x ) = 0 x = 0 x = When x =, y = ( ) ( ( ) ) = = so B (, ). Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 0 (a) Show that the equation cos x = 5sin x may be written as sin x 5sin x + = 0. (b) Hence solve, for 0 θ<0, the equation cos x = 5sin x. (a) cos x = 5sin x ( sin x ) = 5sin x sin x = 5sin x sin x 5sin x + = 0 (as required) Remember cos x + sin x = so cos x = sin x. (b) Let sin x = y y 5y + = 0 ( y ) ( y ) = 0 so y =, y = Factorise y 5y + = 0 ac =, ( ) + ( ) = 5 so y 5y + = y y y + = y ( y ) ( y ) = ( y ) ( y ) (i) sin x = x = 0, 50 Solve for x. Substitute (i) y = and (ii) y = into sin x = y. sinx is positive in the st and nd quadrants. Read off the solutions in 0 x<0.
Heinemann Solutionbank: Core Maths C Page of (ii) sin x = (Impossible) No solutions exist as sinx. so x = 0, 50 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Use the trapezium rule with 5 equal strips to find an estimate for 0 x\ ( + x ). 0 x\ ( + x ) 0. [ 0 + ( 0.9 +. ] + 0.7 + 0.759 +.07 ) 0. or 0.5 Divide the interval into 5 equal strips. Use h = b a n. Here b =, a = 0 and 0 n = 5. So that h = = 0. The trapezium rule gives an approximation to the area of the graph. Here we work to an accuracy of decimal places. Remember A h [ y 0 + ( y + y + ) + y n ] The values of x\ + x are to decimal places, so give your final answer to decimal places. 5 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question A sector of a circle, radius r cm, has a perimeter of 0 cm. (a) Show that the area of the sector is given by A = 0r r. (b) Find the maximum value for the area of the sector. (a) Remember: The length of an arc of a circle is l = rθ. The area of a sector of a circle is A = r θ. Draw a diagram. Let θ be the center angle and l be the arc length. r + l = 0 l = rθ so r + rθ = 0 rθ = 0 r 0 θ = r Area = r θ = r ( ) 0 r = r r 0 r = 0r r (as required) The perimeter of the sector is r + r + l = r + l so r + l = 0 Expand the brackets and simplify. r r 0 r = 0 = 0r = 0r = 0r = r = r r r (b) Find the value of r for the area to have a maximum. Solve da dr = 0.
Heinemann Solutionbank: Core Maths C Page of da dr = 0 r 0 r = 0 r = 0 r = 5 when r = 5 Area = 0 ( 5 ) 5 Find the maximum area. Substitute r = 5 into A = 0r r = 50 5 = 5 cm Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Show that, for all values of x: (a) cos x ( tan x + ) = (b) sin x cos x = ( sin x cos x ) ( sin x + cos x ) (a) cos x ( tan x + ) sinx Remember tan x = so tan sinx sinx x = = (b) = cos sin x ( x + ) cos x = cos sin x x + cos x cos x sin x cos x cos x Expand the brackets and simplify. cos x sin x cos x = <semantics> cos x sin x cos x = sin x </semantics> = sin x + cos x Remember sin x + cos x = = (as required) sin x cos x = ( sin x cos x ) ( sin x + cos x ) cos x Remember a b = ( a b ) ( a + b ). Here a = sin x and b = cos x = ( sin x cos x ) Remember sin x + cos x = = sin x cos x Use a b = ( a b ) ( a + b ) again. Here = ( sin x cos x ) ( sin x + cos x ) a = sin x and b = cos x. (as required) cos x Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question The diagram shows the shaded region R which is bounded by the curves y = x ( x ) and y = 5 ( x ). The curves intersect at the points A and B. (a) Find the coordinates of the points A and B. (b) Find the area of the shaded region R. y = x ( x ), y = 5 ( x ) x ( x ) = 5 ( x ) Solve the equations y = x ( x ) and y = 5 ( x ) simultaneously. Eliminate y so that x ( x ) = 5 ( x ). x x = 5 ( x x + ) Expand the brackets and simplify. x x = 5x 0x + 0 Rearrange the equation into the form ax + bx + c = 0 9x x + 0 = 0 Factorise 9x x + 0 = 0 ( x 0 ) ( x ) = 0 0 x =, x = ac = 80, ( ) + ( 0 ) = 9x x 0x + 0 = x ( x ) 0 ( x ) = ( x ) ( x 0 ). (i) 0 When x =, 0 y = ( ) ( ) 0 = = 80 0 Find the coordinator of A and B. Substitute (i) x = 0 and (ii) x =, into y = x ( x ). (ii) When x = Find the coordinator of A and B. Substitute (i) x =
Heinemann Solutionbank: Core Maths C Page of y = ( ) ( ) = = 80 0 0 and (ii) x =, into y = x ( x ). so A (, ), B (, ) 80 0 80 (b) Area = 0 00 ) ) ) ) ) 0 x ( x ) 5 ( x ) = x x 5 ( x x + ) 0 Remember Area = a b ( y y ). Here y = x ( x ), y = 5 ( x ), a = 0 b =. Expand the brackets and simplify. 0 = x x 5x + 0x 0 Remember ax n = axn + = x 9x 0 0 = [ 8x x 0x ] 0 = ( 8 ( ) ( ) 0 ( 0 ( 8 ( ) ( ) 0 ( 00 = ( 8 ( ) ( ) 9 0 000 7 ( 8 ( ) ( ) ) 9 8 7 n + and Evaluate the integral. Substitute x =, then x = = ( ) ( ) 9 9 ( ) 0, into 8x x 0x and subtract. 9 9 = + 9 9 0 = 8 9 = 8 9 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 5 The volume of a solid cylinder, radius r cm, is 8. (a) Show that the surface area of the cylinder is given by S = + r. 5 r (b) Find the minimum value for the surface area of the cylinder. Draw a diagram. Let h be the height of cylinder. (a) Surface area, S = rh + r (volume =)8 =r h h = 8 r = 8 r 8 so S = r + r r Find expressions for the surface area and volume of the cylinder in terms of, r and h. Eliminate h between the expressions S = rh + r and 8 =r h. Rearrange 8 =r h for h so that r h = 8 h = 8 r 5 = + r (as required) r Substitute h = expression. = 8 r 8 r into S = rh + r and simplify the (b) ds dr = r 5 Find the volume of r for which S = + r has a r r stationary value. Solve 5 r ds dr = 0. Differentiate + r with respect to r, so that 5
Heinemann Solutionbank: Core Maths C Page of r 5 r = 0 r = 5 r r = r = cm When r =, 5 S = + ( ) ( ) = + d dr d 5 ( ) r d = 5r dr = 5r = 5r = 5 r ( r ) = r dr = r = r Find the value of S when r =. Substitute r = into S = 5 r + r. = 9 cm Give the exact answer. Leave your answer in terms of. Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question The diagram shows part of the curve y = sin ( ax b ), where a and b are constants and b <. Given that the coordinates of A and B are (, 0 ) and (, 0 ) respectively, (a) write down the coordinates of C, (b) find the value of a and the value of b. 5 (a) AB = BC 5 AB = = AB is half the period, so AB = BC Find the coordinates of C. Work out the length of AB, AB = OB OA. Work with exact values. Leave your answer in terms of. = 5 so, OC = + 5 = + OC = OB + BC and AB = BC. So, OC = OB + AB. = 9 = so, C (, 0 ) (b) (i) sin ( a ( ) b ) = 0 so a ( ) b = 0 sin ( 0 ) = 0 and sin () = 0. So, at A, x = and a (
Heinemann Solutionbank: Core Maths C Page of 5 ) b = 0 and at B, x = and a ( ) b =. 5 (ii) 5 sin ( a ( ) b ) = 0 5 so a ( ) b = Solving Simultaneously 5 a ( ) b = a ( ) b = 0 a ( ) = Solve the equations a ( ) b = 0and a ( ) b = simultaneously. Subtract the equations. 5 a = ( ) = = When a = Find b. Substitute a = into a ( ) b = 0. ( ) ( ) b = 0 b = check sub a =, b = into a ( 5 ) b ( ) ( ) = 5 so a = and b =. 5 = (as required) Check answer by substituting a = and b = into a ( 5 ) b. ( ) ( ) = = 5 5 5 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 7 Find the area of the finite region bounded by the curve with equation y = x ( x ) and the line y = 0 x. y = x ( x ), y = 0 x x ( x ) = 0 x x x = 0 x Find the x-coordinates of the points where the line y = 0 x meets the curve y = x ( x ). Solve the equations simultaneously, eliminate y. x 7x + 0 = 0 Factorise x 7x + 0. ( 5 ) ( ) = 0 and ( 5 ) + ( x 5 ) ( x ) = 0 ( ) = 7 so x 7x + 0 = ( x 5 ) ( x ). so x = and x = 5. Finite area = 5 x ( x ) ( 0 x ) = 5 x x 0 + x = 5 7x x 0 Use area = a b ( y y ). Have y = x ( x ), y = 0 x, a = and b = 5. Remember ax n = axn + n + 7x = [ 0x ] 5 x 7 ( 5 ) = ( ( 5 ) 0 ( 5 ) ) 0x and subtract. ( 0 ( ) ) 7 ( ) ( ) Evaluate the integral. Substitute x = 5, then x =, into x 7x = ( ) ( 8 ) ( 8 ) = + 8 = = Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 8 A piece of wire of length 80 cm is cut into two pieces. Each piece is bent to form the perimeter of a rectangle which is three times as long as it is wide. Find the lengths of the two pieces of wire if the sum of the areas of the rectangles is to be a maximum. Draw a diagram. Let the width of each rectangle be x and y respectively. Total length of wire = 80 so 80 = 8x + 8y Write down an equation in terms of x and y for the total length of the wire. x + y = 0 Divide throughout by 8. Total Area = A A = x + y Write down an equation in terms of x and y for the total area enclosed by the two pieces of wire. A = x + ( 0 x ) Solve the equations x + y = 0 and A = x + y simultaneously. Eliminate y. Rearrange = x + ( 00 0x + x ) x + y = 0, so that y = 0 x, and substitute into = x + 00 0x + x A = x + y da = x 0x + 00 = x 0 x 0 = 0 x = x = 5 cm The length of each piece of wire is ( 8x = ) 0 cm. Find the value of x for which A is a maximum. Solve da = 0. Total length is 80 cm, so 0 + 0 = 80 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 9 The diagram shows the shaded region C which is bounded by the circle y = \ ( x ) and the coordinate axes. (a) Use the trapezium rule with 0 strips to find an estimate, to decimal places, for the area of the shaded region C. The actual area of C is. (b) Calculate the percentage error in your estimate for the area of C. Remember A h [ y 0 + ( y + y + ) + y n ] Area + 0.59 ) + 0 ] 0. [ + ( 0.9950 + 0.9798 + Divide the interval into 0 equal stufs. Use h = b a n 0. Here a = 0, b = and n = 0, so that = = 0. 0 0 The trapezium rule gives an approximation to the area under the graph. Here we round to decimal places. 0.77 or 0.77 The values of \ x are rounded to decimal place. Give your final another to decimal places. (b) =
Heinemann Solutionbank: Core Maths C Page of 0.77 % error = 00 ( ) =. % Use percentage error = True value 00 Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question 0 The area of the shaded region A in the diagram is 9 cm. Find the value of the constant a. 0 a ( x a ) = 9 Write down an equation in terms of a for the area of region A. 0 a x ax + a = 9 x [ ax + a x ] a 0 = 9 Expand ( x a ) so that ( x a ) ( x a ) = x ax ax + a = x ax + a Remember ax n =. Here ax n + n + x = x ( a ( a ) + a ( a ) ) ( a ) ( a ( 0 ) + a ( 0 ) ) = 9 ( 0 ) a ( a + a ) 0 = 9 ax a = ax = ax = a x Evaluate the integral. Substitute x = a, then x = 0, into x ax + a x, and subtract. a = 9 a = 7 a = Pearson Education Ltd 008
Heinemann Solutionbank: Core Maths C Page of