Mathematics Paper 1 (NonCalculator)


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1 H National Qualifications CFE Higher Mathematics  Specimen Paper A Duration hour and 0 minutes Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full credit will be given onl to solutions which contain appropriate working. State the units for our answer where appropriate. Write our answers clearl in the answer booklet provided. In the answer booklet ou must clearl identif the question number ou are attempting. Use blue or black ink. Before leaving the examination room ou must give our answer booklet to the Invigilator; if ou do not ou ma lose all the marks for this paper. Pegass 05 CFE Higher Specimen Paper A
2 FORMULAE LIST Circle: The equation x + + gx+ f+ c 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( x a) + ( b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: ( A± B) ( A± B) sin cos sina cosa sin Acos B± cos Asin B cos Acos Bm sin Asin B sin Acos A cos A sin A cos A sin A Scalar Product: a. b a b cosθ, where θ isthe anglebetween a and b. or a.b a b + a b + a b where a a a a and b b b b Table of standard derivatives: f ( x) sin ax cosax f ( x) acosax asin ax Table of standard integrals: f ( x) f ( x) dx sin ax cos ax cos ax + a sin ax + a C C Pegass 05 CFE Higher Specimen Paper A
3 Attempt ALL questions Total marks 60. Given that g( x) x ( x ), find g '( x), giving our answer with positive indices. x. Find the points of intersection of the curve with equation 6x x and the line with equation x Given that a i + j + k and b i j + k, find the value of cos θ, in its simplest form, where θ is the angle between a and b. 5. Find the range of values for k, given that the equation x + 8 kx has no real roots and k R. 5. (a) Find the gradient of a line which is perpendicular to x 5. (b) What angle does this perpendicular line makes with the positive direction of the x axis? 6. (a) Find an equivalent expression for cos(x + 0). (b) Hence, or otherwise, determine the exact value of cos (a) Show that ( x ) is a factor of x x x+ 6 (b) Hence, factorise x x x+ 6 full. Pegass 05 CFE Higher Specimen Paper A
4 8. Two functions, defined on suitable domains, are given as f ( x) x and g( x) x. (a) Find an expression, in its simplest form, for f ( g( x)). (b) B expressing f ( g( x)) in the form a ( x+ p) + q, state the turning point of the graph of f ( g( x)). (c) Sketch the graph of f ( g( x)) showing clearl its turning point and intercept with the axis. 9. A(, ), B( 7, 6 ) and C( 7, ) are the vertices of a triangle. (a) B using gradients, show clearl that triangle ABC is rightangled at A. (b) M is the midpoint of AC. Find the equation of the median BM. π 0. Find the rate of change of the function cos x when x.. The diagram shows part of the graph of f (x). It has stationar points at (0, 0) and (, 6). O x (, 6) ' Make a sketch of f ( x). Pegass 05 CFE Higher Specimen Paper A
5 . The diagram below shows part of the graph of sin x+, for 0 x π, and the line with equation. A O π π x Find the coordinates of the point A.. For what value(s) of x is the function + 75x x decreasing? [ END OF QUESTION PAPER ] Pegass 05 CFE Higher Specimen Paper A
6 H National Qualifications CFE Higher Mathematics  Specimen Paper A Duration hour and 0 minutes Mathematics Paper (Calculator) Total marks 70 Attempt ALL questions. You ma use a calculator. Full credit will be given onl to solutions which contain appropriate working. State the units for our answer where appropriate. Write our answers clearl in the answer booklet provided. In the answer booklet ou must clearl identif the question number ou are attempting. Use blue or black ink. Before leaving the examination room ou must give our answer booklet to the Invigilator; if ou do not ou ma lose all the marks for this paper. Pegass 05 CFE Higher Specimen Paper A
7 FORMULAE LIST Circle: The equation x + + gx+ f+ c 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( x a) + ( b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: ( A± B) ( A± B) sin cos sina cosa sin Acos B± cos Asin B cos Acos Bm sin Asin B sin Acos A cos A sin A cos A sin A Scalar Product: a. b a b cosθ, where θ isthe anglebetween a and b. or a.b a b + a b + a b where a a a a and b b b b Table of standard derivatives: f ( x) sin ax cosax f ( x) acosax asin ax Table of standard integrals: f ( x) f ( x) dx sin ax cos ax cos ax + a sin ax + a C C Pegass 05 CFE Higher Specimen Paper A
8 Attempt ALL questions Total marks 70. Three points have coordinates A(, 0, ), B(5, 6, 6) and D(, 8, ). A(,0, ) C E B(5, 6, 6) D(, 8, ) (a) Establish the coordinates of C if AB is parallel and equal in length to CD. (b) The point E divides the line AB in the ratio :. Find the coordinates of E. (c) Hence prove that CE is perpendicular to AB.. Two unique sequences are defined b the following recurrence relations U pu + 6 and U p U 9, where p is a constant. n+ n n+ n+ (a) If both sequences have the same limit, find the value of p. (b) For both sequences U 0 00, find the difference between their first terms.. A function is defined as: f ( x) x x + x. (a) Find the stationar values of the function and the corresponding values of x. Determine the nature of each stationar point. 7 (b) Establish the equation of the tangent to the curve f (x) at the point where x. Pegass 05 CFE Higher Specimen Paper A
9 . The diagram, which is not drawn to scale, shows part of a graph of log x against log. The straight line has a gradient of and passes through the point (0, ). log m O log x Find an equation connecting x and. 5. Given that log (x) log ( x) + find a relationship connecting x and. 6. The circle, centre S, has as its equation x T( p, ) is a point of tangenc. O R x S T( p, ) (a) Find the value of p, the xcoordinate of T. (b) Write down the coordinates of S, the centre of the circle. (c) Find the equation of the tangent through T and hence state the coordinates of R. (d) Establish the equation of the circle which passes through the points S, T and R. Pegass 05 CFE Higher Specimen Paper A
10 7. On a third stage burn the space shuttle uses fuel from a secondar tank. The mass of fuel left in the tank after a burn of t seconds can be calculated using the formula 0 0 t M t M e 0, where 0 fuel remaining after t seconds. M is the initial paload in tonnes and M t is the Calculate, to the nearest second, how long the shuttle takes to use up 60% of the fuel in its tank From a square sheet of metal of side 0 centimetres, equal squares of side x centimetres are removed from each corner. The sides are then folded up and sealed to form an open cuboid. 0cm 0cm x x (a) Show that the volume of this resulting cuboid is given b V ( x) x 0x + 900x. (b) If the cuboid is to have maximum possible volume, what size of square should be removed from each corner? 5 (c) How man litres of water would this particular cuboid hold? Pegass 05 CFE Higher Specimen Paper A
11 9. A funtion is defined as f ( θ ) cos θ where 0 θ π. Part of the graph of f (θ ) is shown. cos θ (a) State the value of a in radians. (b) Calculate the shaded area in square units. o a θ 0. (a) o o o Express cos x + sin x + in the form k cos( x α) + where 0 < α< 60 and k > 0. (b) Hence solve the equation f ( x) 0 for 00 < x < 60. [ END OF QUESTION PAPER ] Pegass 05 CFE Higher Specimen Paper A
12 CFE Higher Specimen Paper A Marking Scheme Paper Give mark for each Illustration(s) for awarding each mark ans: + ( marks) 5 x brings power up x ( x x ) prepares to differentiate x x differentiates + x 5 write with positive indices + 5 x Note: mark can onl be awarded when differentiating a negative power. ans: (, 7); (, 6); (5, 5) (5 marks) equates equations x+ 0 6x x ; sets to zero x 6x + x+ 0 0 factorises full ( x + )( x )( x 5) 0 calculates x  coordinates x ; ; 5 5 calculates  coordinates 5 (, 7); (, 6); (5, 5) ans: (5 marks) states components of both vectors a ; b finds scalar product a. b finds magnitude of both vectors a b substitutes into formula 7 cos θ 5 simplifies 5 cos θ ans: 8 < x < 8 ( marks) knows condition for non real roots b ac< 0 for non real roots stated or implied at simplifies b ac k..8< 0 factorises ( p + 8)( p 8) < 0 correct range 8 < x < 8 Pegass 05 CFE Higher Specimen Paper A
13 Give mark for each Illustration(s) for awarding each mark 5(a) ans: ( marks) rearranges equation to find gradient x 5 states perpendicular gradient (b) ans: 5 o or π ( marks) using tan θ m tan θ calculates angle θ 5 or 6(a) ans: cosx o cos0 o sinx o sin0 o ( mark) π θ correct expansion cosx o cos0 o sinx o sin0 o (b) ans: ( marks) an expression equivalent to cos75 o cos(5 + 0) o or equivalent correct exact values substituted establishes d 7(a) ans: ( marks) knows to use x completes snthetic division recognition of zero remainder (x ) is a factor as remainder is zero (b) ans: (x )(x )(x + ) ( marks) identif quotient ( x )(x + x ) factorises full (x )(x )(x + ) Alternative methods of showing (x ) is a factor, such as long division, inspection and evaluating are acceptable. 6 Pegass 05 CFE Higher Specimen Paper A
14 Give mark for each Illustration(s) for awarding each mark 8(a) ans: x² 8x + 7 ( marks) correct substitution f ( g( x) ( x) expanding brackets ( x + x ) simplifing x 8x+ 7 (b) ans: (x )² ; (, ) ( marks) identif common factor (x² x) stated or implied at complete the square (x )². process for q (x )² state turning point (, ) (c) ans: graph drawn ( marks) correct shape parabola with minimum T.P. annotation, including axis intercept T.P. at ((, ) and intercept at (0, 5) 9(a) ans: proof ( marks) finds gradient of AB m AB finds gradient of AC m AC recognises wh right angled at A since m m (b) ans: 5 + x 7 ( marks) AB AC finds midpoint of AC midpoint AC (, 8) finds gradient of BM m BM 5 substitutes into equation of straight line 8 ( x+ ) triangle is right angled at A 0 ans: 9 ( marks) 8 starts to differentiate 9cos x... completes differentiation... sin x substitutes π π 9cos sin evaluates 9 9 ( ) 8 Pegass 05 CFE Higher Specimen Paper A
15 Give mark for each Illustration(s) for awarding each mark ans: graph drawn ( marks) correct shape correct intercepts with x  axis f (x) 0 x π ans: A (, ) ( marks) equates line & curve, reorganises sin x + ;sin x finds values for x 7π π x, 6 6 finds values for x π x, states coordinates of A π A (, ) ans: x < 5 or x > 5 ( marks) knows to differentiate 75 x knows derivative < 0 75 x < 0 attempts to solve for x graph drawn (or an acceptable method) answer x < 5 or x > 5 Total: 60 marks Pegass 05 CFE Higher Specimen Paper A
16 CFE Higher Specimen Paper A Marking Scheme Paper Give mark for each Illustration(s) for awarding each mark (a) ans: C(7,,) ( mark) states coordinates of C stepping out from B to A is the same from D to C or b vector algebra (b) ans: E(,,) ( marks) for using ratio AE EB (or equivalent) introducing vector algebra e a ( b e) 9 introducing correct column vectors & e b+ a e e 6 working to answer (c) ans: proof ( marks) for scalar product 0 If perp. then EC. AB 0 (or equiv.) 6 for two correct column vectors EC, AB 6 calculating scalar product thus proof EC. AB (6) + ( 6) + ( )() 0 (a) ans: p 0 5 ( marks) gives expression for both limits 6 9 L ; L p p equates limits 6 9 p p starts to solve 6 6 p 9 9 p; 6 p 9 p+ 0 solves and discards ( p )( p ) 0; p 0 5 or p (b) ans: ( marks) finds st term for one RR U (00) finds st term for other RR U ( ) (00) + 6 calculates difference in terms 56 Pegass 05 CFE Higher Specimen Paper A
17 Give mark for each Illustration(s) for awarding each mark (a) ans: 7, x,max :, x,min (7 marks) knowing to differentiate and equate to 0 ' f ( x) x x+ 0 ' making f ( x) 0..0 solving x or substituting into f (x) ) ( ) + or ( () () 5 evaluating 5, 7 6 setting up table of values 6 or second derivative f ' ( x) shape identifies correct nature of each value 7 max at, ) ; min at (, ) ( 7 (b) ans: 5x 9 (marks) knows how to find gradient + m finds point on tangent () f + subs into ( b) m( x a) & arrange ( ) 5( x ). ans: x ( marks) n knowing original form kx n original data in form kx (stated or implied) gradient is power power gradient n finding k intercept, log k, k final equation x 5 ans: 8x ( marks) for bringing up power of log ( ) x... (or equivalent) for taking logs to the same side log (x) log ( x) for knowing to divide to combine log (8x ) changing to index form for answer 8x Pegass 05 CFE Higher Specimen Paper A
18 Give mark for each Illustration(s) for awarding each mark 6(a) ans: p 6 ( marks) subs into equation of circle p² solves for p p² 6; p 6 (b) ans: (0, 8) ( mark) states centre of circle (0, 8) (c) ans: x ; R(, 0) ( marks) finds gradient of ST m ST ⅔ finds gradient of tangent m tan subs into equation of straight line + ( x 6) [or equivalent] finds coords of point R x 0; x (, 0) (d) ans: (x 7)² + ( + )² 65 ( marks) finds midpoint of SR (centre of circle) centre of circle (7, ) finds radius 65 subs into equation of circle ( x 7) + ( + ) 65 7 ans: seconds (5 marks) for solving exponential to 0 0 0t e 0 (or equivalent) taking logs of both sides 0 0t log log 0 releasing the power 0 0 t log log 0 knowing log e e 0 0t loge 0 5 making t the subject + calc. answer 5 log e 0 t 0 5 seconds 0 0 8(a) ans: proof ( marks) gives expression for length and breadth ( 0 x) subs into formula and starts to simplif x( 0 x) completes simplification to answer x (900 0x+ x ) (b) ans: x 5 (5 marks) knows to make derivative 0 V '( x) 0 takes derivative x 0x factorises ( x 5)( x 5) 0 solves and discards x 5 5 justifies answer 5 nature table or nd derivative (c) ans: litres ( mark) calculates volume cm litres Pegass 05 CFE Higher Specimen Paper A e e e e e e
19 Give mark for each Illustration(s) for awarding each mark 9(a) ans: a π ( marks) solving cos θ to zero cos θ 0 answer π θ, θ (b) ans: units ( marks) setting up integral A π 0 π cos θ d θ integrating sin θ [sin θ ] 0 substituting limits π A ( sin ( )) (sin 0) answer A units π 0(a) ans: cos(x 0) o + ( marks) finds k k 9 + finds tan α tan α finds α α 0 (b) ans: 0 o ( marks) equates to 0 cos(x 0) o + 0 simplifies cos(x 0) o finds values x 80 o ; 0 o discards 0 o Total: 70 marks Pegass 05 CFE Higher Specimen Paper A
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