Model Paper WITH ANSWERS. Higher Maths

Transcription

1 Model Paper WITH ANSWERS Higher Maths

2 This model paper is free to download and use for revision purposes. The paper, which may include a limited number of previously published SQA questions, has been specially commissioned by Hodder Gibson, and has been written by eperienced senior teachers and eaminers. This is not SQA material but has been devised to provide further practice for SQA National Qualification eaminations in 06 and beyond. Model Question Papers and Answers Hodder Gibson. All rights reserved. Hodder Gibson would like to thank SQA for use of any past eam questions that may have been used in model papers, whether amended or in original form.

3 H National Qualifications MODEL PAPER Mathematics Paper (Non-Calculator) Duration hour and 0 minutes Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Write your answers clearly in the answer booklet provided. In the answer booklet you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the eamination room you must give your answer booklet to the Invigilator; if you do not you may lose all the marks for this paper. 06 Hodder & Stoughton

4 FORMULAE LIST Circle: The equation + y + g + fy + c = 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( a) + (y b) = r represents a circle centre (a, b) and radius r. Scalar Product: a.b = a b cos q, where q is the angle between a and b a b or a.b = a b + a b + a b where a = a b and b=. a b Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A = sin A cos A cos A = cos A sin A = cos A = sin A ± sin A sin B Table of standard derivatives: f() f () sin a cos a a cos a a sin a Table of standard integrals: f() f ()d sin a a cos a + C cos a a sin a + C 06 Hodder & Stoughton Page two

5 Attempt ALL questions Total marks 60 MARKS. Triangle ABC has vertices A(,), B(, ) and C(7, ). y A(, ) D O C(7, ) B(, ) E (a) Find the equation of the median BD. (b) Find the equation of the altitude AE. (c) Find the coordinates of the point of intersection of BD and AE.. Relative to a suitable coordinate system, A and B are the points (,, ) and (,, ) respectively. A, B and C are collinear points and C is positioned such that BC = AB. C B A Find the coordinates of C. 06 Hodder & Stoughton Page three

6 . The diagram shows two right-angled triangles with angles c and d marked as shown. MARKS c d (a) Find the eact value of sin (c + d). (b) (i) Find the eact value of sin c. (ii) Show that cos d has the same eact value.. The diagram shows a sketch of part of the graph of a trigonometric function whose equation is of the form y = a sin b + c. y y + a sin(b) + c O π Determine the values of a, b and c. 06 Hodder & Stoughton Page four

7 . The graph shown has equation y = The total shaded area is bounded by the curve, the -ais, the y-ais and the line =. y MARKS O S (, 0) (, 0) (a) Calculate the shaded area labelled S. (b) Hence find the total shaded area. 6. Show that the line with the equation y = + does not intersect with the parabola with equation y = Find d 0 ( + ). 8. Find algebraically the values of for which the function f() = 6 is increasing. 9. Evaluate log + log 0 log. 0. Solve sin = 0 for 0 π.. The circles with the equations ( ) + (y ) = and + y k 8y k = 0 have the same centre. Determine the radius of the larger circle. 06 Hodder & Stoughton Page five

8 . The diagram shows a sketch of function y = f(). y MARKS (, 8) (, 8) y = f() O (a) Copy the diagram and on to it sketch the graph of y = f(). (b) Copy the diagram again and, this time, sketch the graph of y = f(). [END OF MODEL PAPER] 06 Hodder & Stoughton Page si

9 H National Qualifications MODEL PAPER Mathematics Paper Duration hour and 0 minutes Total marks 70 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Write your answers clearly in the answer booklet provided. In the answer booklet you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the eamination room you must give your answer booklet to the Invigilator; if you do not you may lose all the marks for this paper. 06 Hodder & Stoughton

10 FORMULAE LIST Circle: The equation + y + g + fy + c = 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( a) + (y b) = r represents a circle centre (a, b) and radius r. Scalar Product: a.b = a b cos q, where q is the angle between a and b a b or a.b = a b + a b + a b where a = a b and b=. a b Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A = sin A cos A cos A = cos A sin A = cos A = sin A ± sin A sin B Table of standard derivatives: f() f () sin a cos a a cos a a sin a Table of standard integrals: f() f ()d sin a a cos a + C cos a a sin a + C 06 Hodder & Stoughton Page two

11 Attempt ALL questions Total marks 70 MARKS. The diagram shows a cuboid OPQR,STUV relative to the coordinate aes. z V U (,, ) N y S T R Q (,, 0) M O P (, 0, 0) P is the point (, 0, 0), Q is (,, 0) and U is (,, ). M is the midpoint of OR. N is the point on UQ such that UN = (a) State the coordinates of M and N. UQ. (b) Epress VM and VN in component form. (c) Calculate the size of angle MVN.. (a) Given that + is a factor of + + k +, find the value of k. (b) Hence solve the equation + + k + = 0 when k takes this value. dy. Given that y = sin + cos, find. d. On a suitable set of real numbers, functions f, g and h are defined by Find an epression for f() =, g() = and h() = +. (a) f(g()) in its simplest form (b) h (). dy. The curve y = f() is such that = 6. The curve passes through the point d (, 9). Epress y in terms of. 06 Hodder & Stoughton Page three

12 6. On the first day of March, a bank loans a man 00 at a fied rate of interest of % per month. This interest is added on the last day of each month and is calculated on the amount due on the first day of the month. He agrees to make repayments on the first day of each subsequent month. Each repayment is 00, ecept for the smaller final amount which will pay off the loan. MARKS (a) The amount that he owes at the start of each month is taken to be the amount still owed just after the monthly repayment has 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. (b) Find the date and amount of the final payment. 7. (a) A chord joins the points A(, 0) and B(, ) on the circle as shown in the diagram. y B (, ) O A (, 0) Show that the equation of the perpendicular bisector of chord AB is + y =. (b) The point C is the centre of this circle. The tangent at the point A on the circle has the equation + y =. y C O A (, 0) Find the equation of the radius CA. (c) (i) Determine the coordinates of the point C. (ii) Find the equation of the circle. 06 Hodder & Stoughton Page four

13 8. (a) Epress cos + sin in the form k cos( a) where k > 0 and 0 a 90. (b) Hence solve the equation cos + sin = for MARKS 9. Variables and y are related by the equation y = k n. The graph of log y against log is a straight line through the points (0, ) and (, 7), as shown in the diagram. log y (0, ) (, 7) O log Find the values of k and n. 0. The value V (in million) of a cruise ship t years after launch is given by the formula V = e 0 06t. (a) What was its value when launched? (b) The owners decide to sell the ship once its value falls below 0 million. After how many years will it be sold?. An open cuboid measures internally units by units by h units and has an inner surface area of units. h (a) Show that the volume, V units, of the cuboid is given by V() = (6 ). (b) Find the eact value of for which this volume is a maimum. [END OF MODEL PAPER] 06 Hodder & Stoughton Page five

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15 HIGHER FOR CfE MATHEMATICS MODEL PAPER Paper (Non-Calculator) Generic Scheme. Question Give one mark for each. (a) Ans: y = find midpoint of AC find gradient of median find equation of median (b) Ans: + y = 9 find gradient of BC Illustrative Scheme (,) m median = y ( ) = ( ( )) m BC = Ma mark find perpendicular gradient find equation of altitude (c) Ans: (,) use valid approach solve for one variable find coordinates of point of intersection. Ans: C(7,7,8) set up vector equation epress c in terms of a and b. (a) evaluate components state coordinates Ans: use addition formula m perp = y = ( ( )) e.g. = + 9 = or y = (,) c b = (b a) c = b a c = C(7,7,8) sin c cos d + cos c sin d find eact length of hypotenuse in each triangle substitute into epansion and evaluate eact value of sin(c + d) 0 (b) (i) Ans: use double angle formula evaluate eact value of sin c sin c cos c = 06 Hodder & Stoughton

16 Question (ii) Generic Scheme. Give one mark for each Ans: proof use double angle formula Illustrative Scheme cos d sin d = ( ) ( ) 0 0 Ma mark complete steps leading to eact value of cos d. Ans: a =, b =, c = state value of a state value of b state value of c. (a) Ans: units know to integrate integrate substitute in limits = 8 0 = d + + () () + () + 0 evaluate area (b) Ans: units know how to find area below -ais d integrate and substitute in limits (( () () + () + ) ) evaluate total area 6. Ans: proof set equation of parabola equal to equation of line epress in standard form start proof continue proof communication + + = = 0 b ac = b ac = b ac < 0 no real roots no real roots line and parabola do not intersect 06 Hodder & Stoughton

17 7. Question Generic Scheme. Give one mark for each Ans: Illustrative Scheme Ma mark preparation for integration 0 ( + ) d integrate ( + ) substitute in limits evaluate integral 8. Ans: < and > differentiate know that f () > 0 factorise correct range 9. Ans: use law of logarithms use law of logarithms evaluate logarithm 0. Ans: π 8, π 8, π 8, 7π 8 solve for sin f () = > 0 6( + ) ( ) > 0 < and and > log + log 0 = log 00 log 00 log = log log = sin = solve for 0 π = π 8, π 6. solve for π π solve for 0 π Ans: 7 = π 6, 7π 6 = π 8, π 8, π 8, 7π 8 centre of smaller circle (,) use general equation of circle to determine value of k k = 6 use r = g + f c r = ( ) + ( ) + find radius of larger circle 7, since 7 > 06 Hodder & Stoughton

18 Question. (a) Ans: Generic Scheme. Give one mark for each y Illustrative Scheme Ma mark (, 8) (, 8) y = f() y = f() O scale parallel to -ais sketch and one point correct annotate graph other two points correct (b) Ans: y y = f() (0, ) O y = f() (, 7) (, 7) y = f() correct order for reflection and translation start to annotate final sketch complete annotation reflect in -ais, then vertical translation sketch and one final point correct the other two final points correct 06 Hodder & Stoughton

19 HIGHER FOR CfE MATHEMATICS MODEL PAPER Paper Question Generic Scheme. Give one mark for each. (a) Ans: M(0,,0), N(,,) find coordinates of M (0,,0) Illustrative Scheme Ma mark (b) find coordinates of N 0 Ans: VM =, VN = 0 epress VM in component form (,,) 0 epress VN in component form 0 (c) Ans: 76 7 or 9 rads know to use scalar product VM.VN cosmvn = VM VN find scalar product VM.VN = find magnitude of a vector VM = 0 find magnitude of a vector VN = 7 evaluate angle MVN. (a) Ans: k = start synthetic division 76 7 or 9 rads k complete synthetic division k 0 6 k k+6 k 0 (b) find valu e of k Ans: =,, k = start to factorise ( + )( + ) complete factorisation and solve equation ( + )( )( ) = 0 =,,. 06 Hodder & Stoughton dy Ans: cos sin d = differentiate first term start to differentiate second term complete differentiation cos sin cos sin

20 Generic Scheme. Question Give one mark for each. (a) Ans: (b) start composite process substitute in g() find f (g()) in simplest form Ans: + start to change subject to complete changing subject to epress function in terms of. Ans: y = + know to integrate integrate substitute in coordinates epress y in terms of 6. (a) Ans: u n+ = 0u n 00 start recurrence relation complete recurrence relation (b) Ans: st December, state u 0 and u find last positive and first negative term state date of final payment state amount of final payment 7. (a) Ans: proof find midpoint of AB find gradient of AB find perpendicular gradient show steps leading to equation of perpendicular bisector in given form (b) Ans: y = rearrange equation of tangent into the form y = m + c gradient of tangent gradient of radius equation of radius f( ) + Illustrative Scheme y = = y = y+ + y = 6 d y = + c 9 = ( ) ( ) + c y = + u n+ = 0u n... u n+ = u 0 = 00, u = 7 u 8 = 86 8, u 9 = 9 st December (,) m AB = m perp = y = ( ) y = + + y = y = + m tangent = m CA = y 0 = ( ) Ma mark 06 Hodder & Stoughton

21 Question (c) (i) Ans: (, ) Generic Scheme. Give one mark for each use valid approach find coordinates of centre of circle (ii) Ans: ( ) + (y ) = 0 find radius state equation of circle 8. (a) Ans: k =, a = 9 cos ( 9 ) use addition formula compare coefficients process k process a (b) Ans: equate wave function with solve for cos( 9) solve for 0 90 e.g. + y = y = (, ) Illustrative Scheme = 8 ( ) + ( 0) = 0 ( ) + (y ) = 0 kcos cosa + ksin sina k(cos cosa + sin sin a ) kcosa = and k sin a = 9 cos( 9) = cos( 9) = or Ma mark 06 Hodder & Stoughton

22 9. Question Generic Scheme. Give one mark for each Ans: k =, n= Method introduce logarithms to n y = k use laws of logarithms interpret intercept Illustrative Scheme Method n log y = log k stated eplicitly log y = nlog + log k stated eplicitly log k = or log y = Ma mark solve for k interpret gradient k = or n = Method state linear equation introduce logarithms use laws of logarithms use laws of logarithms interpret result 0. (a) Ans: million state value of ship at launch (b) Ans: 0 years interpret equation process equation take log to base e process for t. (a) Ans: proof form equation for surface area change subject to h show steps leading to given formula for volume Method log y = log log or... + log y = + y = log or y = log log log... log log y = or y = million e 0 06 t = 0 t e = t = ln( ) 0 ln( ) t = = 0 years 0 06 = 6h + 6 h = 6 V( ) = ( )( ) = ( 6 ) 06 Hodder & Stoughton

23 Question (b) Generic Scheme. Give one mark for each Ans: = prepare for differentiation V( ) = Illustrative Scheme Ma mark differentiate V ( ) = set derivative equal to 0 = 0 solve for valid eact value for justify nature of stationary points.... V ( ) Hodder & Stoughton