H I G H E R S T I L L. Extended Unit Tests Higher Still Higher Mathematics. (more demanding tests covering all levels)

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1 M A T H E M A T I C S H I G H E R S T I L L Higher Still Higher Mathematics Extended Unit Tests 00-0 (more demanding tests covering all levels) Contents Unit Tests (at levels A, B and C) Detailed marking schemes Pegasys Educational Publishing Pegasys 00

2 DINGWALL ACADEMY MATHEMATICS Higher Grade Extended Unit Test - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used.. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 00

3 Section A In this section the correct answer to each question is given by one of the alternatives A, B, C or D. Indicate the correct answer by writing A, B, C or D opposite the number of the question. Rough working may be done on the paper provided. marks will be given for each correct answer.. A sequence is defined by u n = u n + with u =. The value of u is A. 7 B. C. 9 D. 7. Here are two statements about the line ST where S is the point (, ) and T the point (7, ). () The length of ST is 5 units. () The gradient of ST is 4. Which of the following is true? A. Neither statement is correct B. Only statement () is correct C. Only statement () is correct D. Both statements are correct. The gradient of the tangent to the curve with equation y x 4 x at the point (, 8) is A. 0 B. 6 C. 4 D. Pegasys 00

4 4. Two functions, f and g, are defined on suitable domains as f ( x) x and g ( x) x. The value of g f ( )) is ( A. 9 B. 7 C. 7 D The point A(, 4) lies on the graph with equation y f (x). The graph of the related function y f ( x ) is drawn. The coordinates of the image of point A are A. ( 4, 7) B. ( 4, ) C. (0, 7) D. (0, ) End of Section A Pegasys 00

5 Section B ALL QUESTIONS SHOULD BE ATTEMPTED In this section credit will be given for all correct working. 6. (a) The line L passes through the point ( 5, ) and makes an angle of 45º with the positive direction the x - axis. y Find the equation of L L (b) The line L is perpendicular to L and passes through the point with coordinates ( 6, ). ( 5, ) Find the coordinates of P, the point of intersection of the lines L and L. 4 0 x P L ( 6, ) 7. The graph shows part of the graph of y = f(x). It crosses the x and y axes at (, 0), (, 0) and (0, 4) as shown. Sketch the graph of the related function y y f ( x ) x O 4 8. A function is defined as g ( x) x ( x ). (a) Find the points where the graph of g(x) cuts the x and y axes. (b) Find the stationary points of this function g and determine the nature of each. 6 Pegasys 00

6 9. Given that f ( x) ( x ), evaluate f (4) 4 x 0. Two functions are defined on suitable domains as f ( x) x and g ( x) x. x Find an expression, in its simplest form, for g ( f ( x)).. For what value(s) of x is the function + 75x x decreasing? 4. A farmer intends to use 0 metres of fencing to form the perimeter a rectangular enclosure. The width of the rectangle will be x metres. (a) Show that the area that can be enclosed is given by A( x) x(60 x) (b) Calculate the maximum area that can be enclosed, justifying your answer. 4 END OF QUESTION PAPER Pegasys 00

7 Higher Grade Unit Tests 00/0 Marking Scheme - UNIT Give mark for each Illustration(s) for awarding each mark A B A 4 C Award marks for each correct answer 0 marks 5 D 6(a) ans: y x 7 ( marks) substitutes values m, (a,b) = ( 5, ) forms equation y ( x 5) simplifies equation y x 7 6(b) ans: P(8, ) (4 marks) finds L y x 9 equates lines x 9 x 7 finds x coordinate x = 8 4 finds y coordinate 4 y = 7 ans: graph drawn ( marks) correct shape of graph graph through (, ) and (, ) graph through (, ) y (,) O (, ) x (, ) 8(a) ans: (0, 0), (, 0) ( marks) makes g(x) = 0 x ( x ) 0 finds values of x x 0, x 0 so (0, 0),(, 0) substitutes x = 0 (0, 0) 8(b) ans: max at (0,0) min at (, 4) (6 marks) knows g'(x) = 0 g'(x) = 0 at SP finds derivative x 6x solves for x x(x ) = 0, x = 0, 4 finds corresponding y values 4 0, 4 5 sets up table of values (or nd derivative) 5 table of values or takes nd derivative 6 states nature of SPs 6 max at (0,0); min at (, 4) Pegasys 00

8 Give mark for each Illustration(s) for awarding each mark 9 ans: (4 marks) 8 prepares to differentiate x x differentiates x... differentiates a negative power... x 4 substitutes for x ans: x ( marks) x substitutes ( x ) x removes brackets x x states answer x x ans: x < 5 or x > 5 (4 marks) knows to differentiate 75 x knows derivative < 0 75 x < 0 attempts to solve for x graph drawn (or any acceptable method) 4 answer 4 x < 5 or x > 5 Total: 40 marks Pegasys 00

9 Give mark for each Illustration(s) for awarding each mark (a) ans: proof ( mark) finds area proves expression for area (b) ans: 900m (4 marks) differentiates A' ( x) 60 x solves for x x = 0 justifies answer sets up table of values or uses nd deriv. 4 finds maximum area 4 maximum = 0 0 = 900m Total: 45 marks Pegasys 00

10 Higher Still - 00/0 MATHEMATICS Higher Grade Extended Unit Test - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used.. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 00

11 FORMULAE LIST Circle: The equation x y gx fy c 0 represents a circle centre ( g, f ) and radius g f c. The equation ( x a) ( y b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: sin A B cos A B sina cosa sin Acos B cos Asin B cos Acos B sin Asin B sin Acos A cos A sin A cos A sin A Pegasys 00

12 Section A In this section the correct answer to each question is given by one of the alternatives A, B, C or D. Indicate the correct answer by writing A, B, C or D opposite the number of the question. Rough working may be done on the paper provided. marks will be given for each correct answer.. The roots of a quadratic equation can be described as: I Real II Equal III Distinct IV Non-real Which of the above can be used to describe the roots of the equation x 4x 5 0? A. I and II B. I and III C. II and IV D. IV only. If sinα = 5, then the value of sinα is: A. B. C. D The circle with centre (, ) and radius 5 units has equation: A. ( x ) ( y ) 5 B. ( x ) ( y ) 5 C. ( x ) ( y ) 5 D. ( x ) ( y ) 5 Pegasys 00

13 4. When x 8x 7 is expressed in the form ( x p) q. The value of q is A. B. C. D. 5. dy A curve for which x dx passes through the point (, ). The equation of the curve is A. y x x B. y x x C. y x x 5 D. y x x End of Section A Pegasys 00

14 Section B ALL QUESTIONS SHOULD BE ATTEMPTED In this section credit will be given for all correct working. 6. Find the equation of the tangent to the circle with equation x y 4x 4y 9 0 at the point (4, 6) Solve the equation = cosxº + 4sinxº in the interval 0 x If x kx 5x 6 is exactly divisible by (x ), find k and hence fully factorise the expression Two curves with equations y ( x ) and as shown in the diagram. y 6x x meet at A and B y 6x x (a) Calculate the coordinates of A and B. (b) Find the area between the two curves. B i.e the shaded area in the diagram. 4 A y ( x ) 0. Find k, such that the line with equation y x k is a tangent to the curve y x 5x 5 and find the coordinates of the point of contact. Pegasys 00

15 y. A circle has equation ( x ) ( y 4) 5. The straight line with equation y x 5 cuts the circle at A and B. 0 B x Find the coordinates of A and B. 6. A END OF QUESTION PAPER Higher Grade Unit Tests 00/0 Marking Scheme - UNIT Pegasys 00

16 Give mark for each Illustration for awarding each mark D D D 4 A Award marks for each correct answer 0 marks 5 C 6 ans: x 4y 6 0 (4 marks) centre of circle centre is (, ) finds gradient of radius 8 m rad = 6 finds gradient of tangent m tan = 4 4 substitutes in equation of straight line 4 y 6 ( x 4) 4 7 ans: 0 o, 50 o (4 marks) substitutes for cos x ( sin x) 4sin x multiplies out and simplifies 4sin x 4sin x 0 solves for sinxº sinxº = 4 solves for x 4 x = 0 o, 50 o 8 ans: (x )(x +)(x + ) (4 marks) knows to use syn.div. evidence k uses x = in division 4 k 4k k k 4k 8 0 finds k k = 4 completes factorising 4 (x )(x +)(x + ) 5 6 Pegasys 00 Give mark for each Illustration(s) for awarding each mark

17 9(a) ans: A(, 7), B(5, 7) ( marks) equates lines 6x x x 4x solves for x x = 0 or 5 finds coordinates of A and B A(0, ), B(5, 7) 9(b) ans 4 sq units (4 marks) sets up integration and simplifies ( 0x x ) dx x x integrates 0 substitutes [ 5 5 5] [0] 4 4 evaluates 4 0 ans: k = 9, (, 6) (5 marks) equates equations to get x 6x k 0 finds b 4ac k = 0 finds k k = 9 4 uses k 4 x 6x 9 0 so x = 5 5 finds y y = 6 ans: A(, 7), B(5, 0) (6 marks) substitutes for y ( x ) [( x 5) 4] 5 expands x 4x 4 x x 5 0 simplifies x x factorises 4 ( x )( x 5) 0 5 finds x coords 5 x = or 5 6 y coords, states A and B 6 y = 7 or 0 so A(, 7) B(5, 0) Total: 40 marks Pegasys 00

18 Higher Still - 00/0 MATHEMATICS Higher Grade Extended Unit Test - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used.. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 00

19 FORMULAE LIST Scalar Product: a. b a b cosθ, where θ is theangle between a and b. or a.b a b a b a b where a a a a and b b b b Trigonometric formulae: sin A B cos A B sina cosa sin Acos B cos Asin B cos Acos B sin Asin B sin Acos A cos A sin A cos A sin A Table of standard derivatives: f ( x) sin ax cos ax f ( x) acos ax asin ax Table of standard integrals: f ( x) f ( x) dx sin ax cos ax a a cos ax sin ax C C Pegasys 00

20 . The point (7, ) lies on the graph of y log a ( x ). The value of a is Section A In this section the correct answer to each question is given by one of the alternatives A, B, C or D. Indicate the correct answer by writing A, B, C or D opposite the number of the question. Rough working may be done on the paper provided. marks will be given for each correct answer. A. B. C. 4 D. 5. A(,, ), P(0,, 6), and B(4, 5, 6) are three collinear points. P divides AB in the ratio A. : B. : C. : D. :4. 4 cos(x ) dx equals A. sin( x ) c B. 4 sin(x ) c C. 8 sin(x ) c D. sin(x ) c Pegasys 00

21 4. d dx ( 5 x 4 ) equals A. 4(5 x) B. 8(5 x) C. 6(5 x) D. 0 (5 x) 5 5. The vectors with components k and k k 4 4 are perpendicular. The value of k is: A. B. - C. D. - End of Section A Pegasys 00

22 Section B ALL QUESTIONS SHOULD BE ATTEMPTED In this section credit will be given for all correct working. 6. R and S are the points with coordinates (, 5, ) and (6,, ) respectively. (a) State the coordinates of the mid point of RS. (b) T divides RS in the ratio :. Find the coordinates of T 7. Triangle PNQ has coordinates P(5, 0, ) N(, 6, 0) and Q(8, 6, ). Find the size of angle PNQ Evaluate ( x ) dx -⅔ 9. Express cosxº + sinxº in the form kcos(x α)º 0 x 80 and hence solve the equation cosxº + sinxº = (a) Simplify: log a + log a 4 (b) Simplify: 5log 9 log 9 7 d. Find (cos x sin x), expressing your answer in terms of x and x dx. The number of germs present in a laboratory sample after t hours is given by the formula G( t) 6 t 50e. State how many germs were present in the sample initially, and find how many minutes the germs will take to double in number. 5 END OF QUESTION PAPER Higher Grade Unit Tests 00/0 Marking Scheme - UNIT Pegasys 00

23 Give mark for each Illustration(s) for awarding each mark A B A 4 B Award marks for each correct answer 0 marks 5 C 6(a) ans: (4,, 5) ( mark) answer (4,, 5) 6(b) ans: T(5,, 8) ( marks) 4 Finds RS RS= 8 4 Finds RT RT= 8 4 answer T(5,, 8) 7 ans: 8º (5 marks) finds NP and NQ NP= 6 ; NQ= finds scalar product = 0 6 finds magnitudes of NP and NQ 74 and 89 4 subs into formula 4 0 cos PNQ finds angle 5 8º Pegasys 00

24 Give mark for each Illustration(s) for awarding each mark 8 ans: 8 9 ( marks) integrates (x ) substitutes values ( ) ( ) 9 9 evaluates answer ans: 60 o, 80 o (6 marks) expands k cos( x ) o k cos xcos k sin xsin calculates k = k : k = calculates α = 60 o tan : 60 4 uses answer 4 cos(x 60) º = 5 evaluates (x 60) 5 60 o, 00 o 6 evaluates x 6 60 o, 80 o 0(a) ans: log a 48 ( mark) answer lo48 a 0(b) ans: ( marks) changes 5log 9 5 log 9 log 9 7 uses laws of logs 4 log 9 ( ) 7 answer log 9 9 ans: 6sinx sin x ( marks) differentiates st term sinx 6sin x differentiates nd term sin xcos x answer 6sinx sin x ans: 50, 6 minutes (5 marks) 50 equates values takes logs 4 finds t hours 5 answer t = 0, G(0) = e 6t or 6 t e 6tlne = ln 4 t = 0 4 hours hours = 6 minutes Total: 40 marks Pegasys 00