alderglen High School Mathematics Department Higher Mathematics Home Eercise Programme R A Burton June 00
Home Eercise The Laws of Indices Rule : Rule 4 : ( ) Rule 7 : n p m p q = = = ( n p ( p+ q) ) m q = m n Rule : Rule 5 : p 0 q = = ( p q) Rule : ( Rule 6 : p ) p q = = p pq. Simplif each of the following 7 4 4 5 (c) 5 4 (d) 4 6 (e) 4 ( ) (f) 4 ( ) (g) Epress with a positive inde (Standard Grade : The Laws of Indices) (7). Simplif ( ) (Standard Grade : The Laws of Indices) (). Epress each of the following as a surd in its simplest form 44 89 (Standard Grade : Simplifing Surds) () 4. Solve each of the following equations. Show all relevant working. An answer onl will receive no marks. = 8 9 = 4 (Standard Grade : Solving Eponential Equations) (5) Time : 5 minutes Total : 6 End of Home Eercise
Home Eercise The Laws of Indices Rule : Rule 4 : ( ) Rule 7 : n p m p q = = = ( n p ( p+ q) ) m q = m n Rule : Rule 5 : p 0 q = = ( p q) Rule : ( Rule 6 : p ) p q = = p pq. Simplif each of the following 4 4 6 5 4 (c) 4 (d) (e) (g) ) (6 (f) 4 5 Epress with a positive inde 6 ( 4 9 (Standard Grade : The Laws of Indices) (7) ). Simplif ( 5) (Standard Grade : The Laws of Indices) (). Epress each of the following as a surd in its simplest form 4 00 8 (c) 5 5 (d) ( 5 )( 5 ) (Standard Grade : Simplifing Surds) (7) 4. Solve each of the following equations. Show all relevant working. An answer onl will receive no marks. 5 = 5 (Standard Grade : Solving Eponential Equations) (7) 4 = Time : 5 minutes Total : End of Home Eercise
Home Eercise. Differentiate each of the following each of the following ( ) 4 f = f ( ) = (c) f ( ) = 5 + 4 5 + (d) f ( ) = 4 5 (Unit : utcome : Differentiation : Using the rules). D(-6,), E(4,5) and F(8,-) are the vertices of triangle DEF. Find the equation of the median from D. (,,,6) K(6,6), L(-,0) and M(0,-) are the vertices of triangle KLM. Find the equation of the altitude from K. (c) P is the point (-,) and Q is (5,5). Find the equation of the perpendicular bisector of PQ. (Unit : utcome : Straight Lines : Altitudes, medians and perpendicular bisectors of a triangle) (,,4) Time : 5 minutes Total : End of Home Eercise
Home Eercise 4. Find the equation of the perpendicular bisector of the line joining A(,-) and B(8,) (Unit : utcome : Straight Lines : Altitudes, medians and perpendicular bisectors of a triangle). P(-4,5), Q(-,-) and R(4,) are the vertices of triangle PQR as shown in the diagram. P(-4,5) (4) Find the equation of PS, the altitude from P Q(-,-) S R(4,) (Unit : utcome : Straight Lines : Altitudes, medians and perpendicular bisectors of a triangle). A triangle AB has vertices A(4,), B(6,) and (-,-) as shown in the diagram. A(4,) () Find the equation of AM, the median from A. M B(6,) (-,-) (Unit : utcome : Straight Lines : Altitudes, medians and perpendicular bisectors of a triangle) 4...over ()
Home Eercise 4 4. B(-,) B M D A(-,-) (7,-) A E Show that triangle AB is right angled at B. The medians AD and BE intersect at M (i) Find the equations of AD and BE. (ii) Hence find the coordinates of M. (Unit : utcome : Straight Lines : Altitudes, medians and perpendicular bisectors of a triangle) (,5,) Time : 5 minutes Total : End of Home Eercise 4
Home Eercise 5. Differentiate each of the following with respect to + 4 + f ( ) = f ( ) = 4 (Unit : utcome : Differentiation : Using the rules). Show that there is onl one tangent to the curve = 4 6 + 5 with a gradient of 8 Find the point of contact on the curve and the equation of the tangent. (5,4). (Unit : utcome : Differentiation : Finding the equation of a tangent to a curve) f ( ) = g( ) = h( ) = (5 ) 4 Find a formula for k() where k()=f [g()] Find a formula for h[k()] (c) What is the connection between the functions h and k. (,) (Unit : utcome : Functions : omposition of functions and inverse functions) 4. The functions f and g are defined on a suitable domain b Find an epression for f [g()] Factorise f [g()] f ( ) = g( ) = + (,,) (Unit : utcome : Functions : omposition of functions) (,) 5. The functions f and g, defined on a suitable domains, are given b f ( ) = 4 g( ) = + Find an epression for h() = g[f()]. Give our answer as a single fraction. State a suitable domain for h. (Unit : utcome : Functions : omposition of functions) (,) Time : 40 minutes Total : 7 End of Home Eercise 5
Home Eercise 6. Find the derivative of each of the following. f ( ) = + 4 f ( ) = +. (Unit : utcome : Differentiation : Using the rules) Find the equation of the tangent to the curve = + at the point where =. (5,6) (Unit : utcome : Differentiation : Finding the equation of a tangent to a curve). Show that the tangents to the curve = at the points where = - and = are parallel. (5) (Unit : utcome : Straight Line : Parallel lines) (Unit : utcome : Differentiation : The equation of a tangent to a curve) 4. For the function defined b f ( ) = 6 + 5 find the intervals on which it is increasing. () (Unit : utcome : Differentiation : Increasing/decreasing functions) 5. Sketch the graph of the curve with equation =. (5) (Unit : utcome : Differentiation : urve sketching) (9) Time : 50 minutes Total : End of Home Eercise 6
Home Eercise 7. Find the derivative of each of the following (4 )(5 ) f ( ) = f ( ) = + (Unit : utcome : Differentiation : Using the rules). A fishing club uses a loch which has been stocked with 000 fish. (5,4) Due to fish being caught b members and ding of natural causes, the fish population is reduced b 8% per month. If members are allowed to fish, without the loch being re-stocked, during which month will the number of fish in the loch fall below 750? The club committee decide that the number of fish in the loch should not be allowed to fall below 00. In order to ensure that this does not happen, it is decided that 80 fish should be added at the end of each month. Will this maintain a level of at least 00 fish at all times? Give a reason for our answer. (Unit : utcome 4 : Recurrence relations : Limit of a sequence). A sequence is defined b the recurrence relation (,5) U n = 0.9U n + U = alculate the value of U. What is the smallest value of n for which U n > 0? (c) Find the limit of this sequence as n. (Unit : utcome 4 : Recurrence relations : The limit of a sequence) 4...over (,,)
Home Eercise 7 4. A zookeeper wants to fence off si individual pens, as shown in the diagram opposite. Each pen is a rectangle measuring metres b metres. metres metres (i) Epress the total length of fencing in terms of and. (ii) Given that the total length of fencing is 60 metres, show that the total area, A m, of the si pens is given b 6 A( ) = 40 Find the value of and which give the maimum area and write down this maimum area. (Unit : utcome : Differentiation : ptimisation) (4,6) Time : 45 minutes Total : End of Home Eercise 7
Home Eercise 8. Trees are spraed weekl with the pesticide, Killpest, whose manufacturers claim it will destro 65% of all pests. Between weekl spraings, it is estimated that 500 new pests invade the trees. A new pesticide, Pestkill comes on to the market. The manufacturers claim that it will destro 85% of eisting pests but it is estimated that 650 new pests per week will invade the trees. Which pesticide will be more effective in the long term? (Unit : utcome 4 : Recurrence relations : Limit of a sequence). Epress f()=( )( + 5) in the form a( + b) + c (7) (Unit : utcome : Functions : ompleting the square). A curve has equation = 4 4 + () Find algebraicall the coordinates of the stationar points. Determine the nature of the stationar points. (5,) (Unit : utcome : Differentiation : urve Sketching) 4. Two sequences are defined b the formulae U n+ = 0.4U n and T n+ = T n + 0.4 The initial values are U = 4 and T = 4. Find the values of U, U, T and T. Eplain wh one of the sequences has a limit as n tends to infinit, and calculate this limit algebraicall. (Unit : utcome 4 : Recurrence relations : Limit of a sequence) 5...over (,)
Home Eercise 8 5. Functions f and g are defined on the set of real numbers b f() = and g() = Find formulae for (i) f [g()] and (ii) g[f()] The function h is defined b h() = f [g()] + g[f()]. Show that h() = and sketch the graph of h. (Unit : utcome : Functions : omposition of functions and sketching the graph of a quadratic) 6. A function f is defined on the set of real numbers b (4,) f ( ) = Find, in its simplest form, an epression for f [f ()]. (Unit : utcome : Functions : omposition of functions) () Time : 50 minutes Total : End of Home Eercise 8
Home Eercise 9. For what values of a does the equation a + 0 + 40 = 0 have equal roots? (Unit : utcome : Quadratic Theor : The discriminant). A sequence is defined b the recurrence relation U n+ = 0.U n + 5 with first term U. () Eplain wh this sequence has a limit as n tends to infinit. Find the eact value of this limit (Unit : utcome 4 : Recurrence relations : Limit of a sequence). Find, algebraicall, the values of for which the function f () = - 6 is increasing. (,) 4. (Unit : utcome : Differentiation : Increasing/decreasing functions) If is an acute angle such that sin ( + 0) is 4 + 0 tan = 4, show that the eact value of (4) 5. (Unit : utcome : Trigonometr : The addition formulae) Given that = d +, find and hence show that d d + = d () (Unit : utcome : Differentiation : Using the rules) 6. Show that the function f() = + 8 can be written in the form f() = a( + b) + c where a, b and c are constants. () Hence, or otherwise, find the coordinates of the turning point of the function. (Unit : utcome : Functions : ompleting the square) 7...over (,)
Home Eercise 9 7. A and B are acute angles such that Find the eact value of tan A = and 4 5 tan B =. sin A cos A (c) sin (A + B) (Unit : utcome : Trigonometr : The double angle formulae) (,,) Time : 5 minutes Total : 4 End of Home Eercise 9
Home Eercise 0. Differentiate ( + ) with respect to. (Unit : utcome : Differentiation : Using the rules). Two sequences are defined b the recurrence relations () U n+ = 0.U n + p U 0 = and V n+ = 0.6V n + q V 0 = If both sequences have the same limit, epress p in terms of q (Unit : utcome 4 : Recurrence relations : Limit of a sequence). f() = + and g() = + k, where k is a constant. (i) Find g[f ()] (ii) Find f [g()] () (i) Show that the equation g[f ()]- f [g()]= 0 simplifies to + 4 k = 0 (ii) Determine the nature of the roots of this equation when k = 6. (iii) Find the value of k for which + 4 k = 0 has equal roots. (Unit : utcome : Functions : omposition of functions) (Unit : utcome : Quadratic Theor : The discriminant) 4. Show that cos cos = sin (,,,,) Hence solve the equation cos cos = sin in the interval 0 60 (Unit : utcome : Trigonometr : Solution of trigonometric equations) (,4) Time : 5 minutes Total : 4 End of Home Eercise 0
Home Eercise. Given that f() = ( ), find f (-). (Unit : utcome : Differentiation : Using the rules, evaluating a derivative). Show that f() = 4 + 5 can be written in the form f() = a( + b) + c () Hence write down the coordinates of the stationar point of = f() and state its nature (Unit : utcome : Functions : ompleting the square) (,). Using triangle PQR, shown opposite, find the eact value of cos. R 7 Q P (Unit : utcome : Trigonometr : The double angle formulae) 4. A curve has equation = - 4 + 4 - () Find the coordinates of the stationar points. Determine the nature of the stationar points (Unit : utcome : Differentiation : urve Sketching) 5. Write the equation cos θ + 8cos θ + 9 = 0 in terms of cos θ and show that, for cos θ, it has equal roots. (5,) Show that there are no real roots for θ. (Unit : utcome : Trigonometr : The double angle formulae) (Unit : utcome : Quadratic Theor : The discriminant) (,) Time : 5 minutes Total : End of Home Eercise
Home Eercise. The point A has coordinates (7, 4). The straight lines with equations + + = 0 and + 5 = 0 intersect at the point B. Find the gradient of AB. Hence show that AB is perpendicular to onl one of these two lines. (Unit : utcome : Straight Line : Gradient formula) (Unit : utcome : Straight Line : Perpendicular lines). f() = 5 - (,5) (i) Show that ( + ) is a factor of f(). (ii) Hence or otherwise factorise f() full. ne of the turning points of the graph of = f() lies on the -ais. Write down the coordinates of this turning point. (Unit : utcome : Polnomials : Snthetic division). Prove that the roots of the equation +p = 0 are real for all values of p. (5,) (Unit : utcome : Quadratics : Discriminant : Real roots) 4. A sequence is defined b the recurrence relation U n+ = ku n +. (4) Write down the condition for this sequence to have a limit. The sequence tends to a limit of 5 as n. Determine the value of k. (Unit : utcome 4 : Recurrence Relations : Limit of a sequence) (,) Time : 5 minutes Total : End of Home Eercise
Home Eercise. The diagram shows the graph of = g() Sketch the graph of = - g() n the same diagram, sketch the graph of = g(). (0,) (b,) (a,-) = g() (Unit : utcome : Functions : Associated graphs). Write 0 + 7 in the form ( + b) + c. Hence show that the function g ( ) = 5 + 7 is alwas increasing. (,) (Unit : utcome : Functions : ompleting the square) (Unit : utcome : Differentiation : Increasing/decreasing functions). The diagram shows the line A with equation = 0. The angle between A and the - ais is a. Find the value of a. a A (,4) The second diagram shows lines A and B. The angle between the two lines is 0. alculate the gradient of the line B correct to one decimal place. B 0 a A (Unit : utcome : Straight line : m = tan θ ) (,)
Home Eercise 4. The point P(,) lies on the curve with equation = 6. Find the value of for which the gradient of the tangent at P is. Hence find the equation of the tangent at P. (Unit : utcome : Differentiation : Equation of a tangent to a graph) (5,) Time : 5 minutes Total : End of Home Eercise
Home Eercise 4. Find the equation of the line which passes through the point (-,) and is perpendicular to the line with equation 4 + = 0. (Unit : utcome : The Straight Line : Perpendicular lines). In the diagram angle DE ˆ = EB ˆ = and DE ˆ = BEˆ A = 90. D = unit and DE = units. B writing DE ˆ A in terms of, find the D eact value of cos( DE ˆ A ). B E A () (Unit : utcome : Trigonometr : Addition formulae). The graph of the cubic function = f() is shown in the diagram. There are turning points at (,) and (,5). (,5) (7) Sketch the graph of = f (). (,) = f() (Unit : utcome : Differentiation : The graph of the derived function) 4. An open cuboid measures internall units b b h units and has an inner surface area of units () Show that the volume, V units, of the cuboid is given b V ( ) = (6 ) Find the eact value of for which the volume is a maimum. h (Unit : utcome : Differentiation : ptimisation) (,5) Time : 5 minutes Total : End of Home Eercise 4
Home Eercise 5. Write f() = + 6 + in the form ( + a) + b. Hence or otherwise sketch the graph of = f(). (Unit : utcome : Functions : ompleting the square) (Unit : utcome : Quadratics : Sketching the graph of a quadratic\). A recurrence relation is defined b U n+ = pu n + q, where - < p < and U 0 =. (,) If U = 5 and U = 6, find the values of p and q. Find the limit of this recurrence relation as n. (Unit : utcome 4 : Recurrence relations : Solving recurrence relations to find m and c, and the limit of a sequence. f() = 6 5 7 + 6 (,) Show that ( ) is a factor of f(). Epress f() in its full factorised form. (Unit : utcome : Polnomials : Snthetic division) (4) 4. Solve the equation cos + 0cos = 0 for 0 π (Unit : utcome : Trigonometr : Solution of trigonometric equations) (4) Time : 5 minutes Total : 6 End of Home Eercise 5
Home Eercise 6. Given that f ( ) = +, find f (4). (Unit : utcome : Differentiation : Using the rules. Evaluating a derivative). Show that the line with equation = + does not intersect with the parabola with equation = + + 4. (5) (Unit : utcome : The quadratic : The discriminant). The diagram shows the graph of the function f. f has a minimum turning point at (0,-) and a point of infleion at (-4,). -4 - (5) Sketch the graph of = f (-). n the same diagram, sketch the graph of = f (-). - (Unit : utcome : Functions : Associated graphs) 4. An open water tank, in the shape of a triangular prism, has a capacit of 08 litres. The tank is to be lined on the inside in order to make it watertight. l (,) The triangular cross-section of the tank is right-angled and isosceles, with equal sides of length cm. The tank has a length of l cm. Show that the surface area to be lined, A cm, is given b Find the value of which minimises this surface area. A( ) = + 4000 (Unit : utcome : Differentiation : ptimisation) (,5) Time : 5 minutes Total : End of Home Eercise 6
Home Eercise 7. Functions f ( ) = and g() = + are defined on suitable domains. 4 Find an epression for h() where h() = f [g()]. Write down a restriction on the domain of h. (Unit : utcome : Functions : omposite functions). A is the point (8,4). The line A is inclined at an angle p radians to the -ais. Find the eact values of: (i) sin (p) (ii) cos (p) p A(8,4) (,) The line B is inclined at an angle p radians to the -ais. B Write down the eact value of the gradient of B p (Unit : utcome : Trigonometr : The Double angle formulae). The incomplete graphs of f() = + and g() = 6 are shown in the diagram. The graphs intersect at A(4,4) and the origin. Find the shaded area enclosed between the curves. = f() A(4,4) = g() (5,) (Unit : utcome : Integration : Area between two curves) (5)
Home Eercise 7 4. Triangle AB has vertices A(-,6), B(-,-) and (5,). A(-,6) Find the equation of the line p, the median from of triangle AB. the equation of the line q, the perpendicular bisector of B. B(-,-) (5,) (c) the coordinates of the point of intersection of the lines p and q. (Unit : utcome : The straight line : Equation of a median and perpendicular bisector) (,4,) Time : 5 minutes Total : End of Home Eercise 7
Home Eercise 8. The point P(,) lies on the circle ( + ) + ( ) =. Find the equation of the tangent at P. (Unit : utcome 4 : The circle : Tangent to a circle). Functions f and g are defined on suitable domains b f() = sin and g() =. (4) Find epressions for: (i) f [g()] (ii) g[f()] Solve f [g()] = g[f()] for 0 60 (Unit : utcome : Functions : omposite functions) (Unit : utcome : Trigonometr : Solution of trigonometric equations). The diagram shows part of the graph of the curve with equation = 7 + 4 + 4 = f() (,5) Find the -coordinate of the maimum turning point. Factorise 7 + 4 + 4 (c) State the coordinates of the point A and hence find the values of for which 7 + 4 + 4 < 0 A (,0) (Unit : utcome : Differentiation : urve sketching) (Unit : utcome : Polnomials : Snthetic division and solution of inequations) (5,,) Time : 5 minutes Total : End of Home Eercise 8
Home Eercise 9. Find the coordinates of the point on the curve = 7 + 0 where the tangent to the curve makes an angle of 45 with the positive direction of the -ais. (Unit : utcome : Differentiation : Tangent to a curve). In triangle AB, show that the eact value of sin (a + b) is 5 a A b B (4) (Unit : utcome : Trigonometr : The addition formulae). A man decides to plant a number of fast-growing trees as a boundar between his propert and the propert of his net door neighbour. He has been warned, however, b the local garden centre that, during an ear, the trees are epected to increase in height b 0.5 metres. In response to this warning he decides to trim 0% off the height of the trees at the start of an ear. 4 If he adopts the 0% pruning polic, to what height will he epect the trees to grow in the long run? His neighbour is concerned that the trees are growing at an alarming rate and wants assurances that the trees will grow no taller than metres. What is the minimum percentage that the trees will need to be trimmed each ear so as to meet this condition? (Unit : utcome 4 : Recurrence relations : The limit of a sequence) 4...over (,)
Home Eercise 9 4. alculate the shaded area enclosed between the parabolas with equations = + 0 and = + 5. = + 0 = + 5 (Unit : utcome : Integration : The area between two curves) (6) Time : 0 minutes Total : 0 End of Home Eercise 9
Home Eercise 0. The graph of the function f intersects the -ais at (-a,0) and (e,0) as shown. There is a point of infleion at (0,b) and a maimum turning point at (c,d). Sketch the graph of the derived function f. (-a,0) (0,b) (c,d) (e,0) = f() (Unit : utcome : Differentiation : The graph of the derived function). Epress f() = 4 + 5 in the form f() = ( a) + b n the same diagram sketch: (i) the graph of = f() (ii) the graph of = 0 f() (c) Find the range of values of for which 0 f() is positive. (Unit : utcome : Functions : ompleting the square and associated graphs). Show that the equation ( k) 5k k = 0 has real roots for all integer values of k. (,4,) (Unit : utcome : The quadratic : The discriminant) (5)
Home Eercise 0 4. The shaded rectangle on this map represents the planned etension to the village hall. It is hoped to provide the largest possible area for the etension. The Vennel 6 m Village Hall 8 m Manse Lane The coordinate diagram represents the right angled triangle of ground behind the hall. The etension has length l metres and breadth b metres, as shown. ne corner of the etension is at the point (a,0) (0,6) l b (a,0) (8,0) (i) 5 Show that l = a. 4 (ii) Epress b in terms of a and hence deduce that the area, A m, of the etension is given b A = a(8 a) 4 Find the value of a which produces the largest area of the etension. (Unit : utcome : Differentiation : ptimisation) (,4) Time : 5 minutes Total : End of Home Eercise 0
Home Eercise. Find the equation of the straight line which is parallel to the line with equation + = 5 and which passes through the point (,-). (Unit : utcome : The Straight Line : Parallel lines). For what value of k does the equation 5 + (k + 6) = 0 have equal roots? () (Unit : utcome : The Quadratic : The discriminant). Given that + is a factor of + + k +, find the value of k. Hence solve the equation + + k + = 0 when k takes this value. () (Unit : utcome : Polnomials : Snthetic division and finding the roots of a cubic equation) 6 4. A curve has equation =, > 0 Find the equation of the tangent at the point where = 4. (,) (Unit : utcome : Differentiation : The equation of a tangent to a curve) (6) Time : 0 minutes Total : 7 End of Home Eercise
Home Eercise. Given that f() = + 8, epress f() in the form ( + a) b. (Unit : utcome : Functions : ompleting the square). Solve the equation sin cos = 0 in the interval 0 80. () The diagram shows parts of two trigonometric graphs, = sin and = cos. Use our solution in to write down the coordinates of the point P. =sin 90 80 P =cos (Unit n : utcome : Trigonometr : Solution of trigonometric equations). n the first da of March, a bank loans a man 500 at a fied rate of interest 0f.5% per month. This interest is added on the last da of each month and is calculated on the amount due on the first da of the month. He agrees to make repaments on the first da of each subsequent month. Each repament is 00 ecept for the smaller final amount which will pa off the loan. (4,) The amount that he owes at the start of each month is taken to be the amount still owing just after the monthl repaments have been made. Let U n and U n+ represent the amounts that he owes at the starts of two successive months. Write down a recurrence relation involving U n+ and U n. Find the date and the amount of the final pament. (Unit : utcome 4 : Recurrence relations : Finding the n th term of a sequence) 4...over (,4)
Home Eercise 4. A firm had a new logo designed. A mathematical representation of the final logo is shown in the coordinate diagram. (4,5) The curve has equation = ( + )( )( ) and the straight line has equation - - 0 4 = 5 5 The point (,0) is the centre of half-turn smmetr. alculate the total shaded area. (-,-5) (Unit : utcome : Integration : The area between two curves) (7) Time : 0 minutes Total : 0 End of Home Eercise
Home Eercise. A compan spends thousand pounds a ear on advertising and this results in a profit of P thousand pounds. A mathematical model, illustrated in the diagram, suggests that P and are related b P P() = 4 for 0 Find the value of which gives the maimum profit. (,0) (Unit : utcome : Differentiation : ptimisation). The diagram shows the graph of two quadratic functions = f() and = g(). = f() = g() (5) Both graphs have a minimum turning point at (,). (,) Sketch the graph of = f () and on the same diagram sketch the graph of = g (). (Unit : utcome : Differentiation : The graph of the derived function). The diagram shows a sketch of a parabola passing through (-,0), (0,p) and (p,0). Show that the equation of the parabola is = p + (p ). For what value of p will the line = + p be a tangent to the curve (0,p) (-,0) (p,0) (Unit : utcome : Polnomials : Finding the equation of a polnomial) 4...over (,)
Home Eercise 4. The circle with centre A has equation + + = 0. The line PT is a tangent to this circle at the point P(5,-) T + A P Show that the equation of this tangent is + =. The circle with centre B has equation + + 0 + + 6 = 0. T + B Q + A P Show that PT is also a tangent to this circle. (c) Q is the point of contact. Find the length of PQ. (Unit : utcome 4 : The ircle : Tangent to a circle and Distance formula) (4,5,) Time : 5 minutes Total : 4 End of Home Eercise
Home Eercise 4. ircle P has equation + 8 0 + 9 = 0. ircle Q has centre (-,-) and radius. (i) Show that the radius of circle P is 4. (ii) Hence show that circles P and Q touch. Find the equation of the tangent to circle Q at the point (-4,). (c) The tangent in intersects circle P in two points. Find the -coordinates of the points of intersection, epressing our answers in the form a ± b. (Unit : utcome 4 : The ircle : Touching circles, the equation of a tangent to a circle an the intersection of a line and a circle.. Find all the values of in the interval 0 π for which tan () = (4,,) (Unit : utcome : Trigonometr : Solution of trigonometric equation). An architectural feature of a building is a wall with an arched window. The curved edge of the window is parabolic. 0.5 m (4) The window is shown in the diagram. The shaded part represents the glass. The top edge of the window is part of the parabola with equation =. Find the area in square metres of the glass in the window. 4 m.5 m (Unit : utcome : Integration : The area between two curves) (8) Time : 5 minutes Total : End of Home Eercise 4