NATIONAL QUALIFICATIONS

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1 H Mathematics Higher Paper Practice Paper A Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters

2 FORMULAE LIST Circle: The equation The equation g f c represents a circle centre ( g, f ) and radius ( a) ( b) r represents a circle centre (, ) a b and radius r. g f c. Scalar Product : a. b a b cos, where is the angle between a and b. or a b a. bab ab ab, where a a and b. b a b Trigonometric formulae: sin(ab) sinacosbcosasinb cos(a B) cos A cos B sin A sin B sin A sin Acos A cos A cos A sin A cos A sin A Table of standard derivatives : f ( ) f ( ) sin a acos a cos a asin a Table of standard integrals : f ( ) f ( ) d sin a cos a C a cosa sin a C a Page

3 SECTION A ALL questions should be attempted.. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient 7. What is the value of k?. A circle has equation 8. What is the radius of this circle?. What is the derivative of with respect to? 5. Find d. 6. If 7 is written in the form ( p) q, find the value of q. 7. A sequence is generated b the recurrence relation u 8u 6. n n What is the limit of this sequence as n? 8. A circle with centre (,5) passes through the point (, 7). What is the equation of the circle? 9. The vectors p and q with components k p k and q are perpendicular. What is the value of k?. Identif the nature of the roots of the equation 8. Page

4 . What is the value of 5 7 cos tan?. Given that log 8 p, find the value of p.. Find () 5 d. K and L are the points with coordinates (,, ) and (,, 5) respectivel. If KM KL, find the coordinates of M. 5. h ( ). 8 For what values of is h ( ) undefined? 6. Here are two statements about the graph with equation a b, shown opposite. () a ; () is alwas increasing Which of these statements are true? a b 7. The diagram shows part of the graph of a cubic. What is the equation of this graph? 8. Given that log log 5, epress in terms of. Page 5

5 9. If p.( pq ) 8 and p, find the value of p. q. The diagram shows part of the curve with equation plog ( k). What is the value of p (6, ) End of Section A Page 6

6 SECTION B ALL questions should be attempted.. Triangle PQR has vertices P(, 5), Q(7, ) and R(, 5), as shown. P Q Marks R (a) Find the equation of the median RM. (b) Find the equation of the altitude AP. (c) Find the coordinates of the point of intersection of RM and AP.. Find the stationar points on the curve given b 9 and determine their nature. 7. (a) Functions f and g are defined on suitable domains b f g ( ) 5 and ( ) Find f ( g ( )). (b) Sketch the curve with equation f( g( )). π. (a) Show that 6 sin cos sin cos. (b) Epress sin cos in the form ksin( a) where k and a π (c) Hence, or otherwise, solve sin cos, where π. 6 π. End of question paper Page 7

7 H Mathematics Higher Paper Practice Paper A Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

8 ALL questions should be attempted. Marks. K is the point (,, ), L(5,, 7) and M(7,, ). (a) Write down the components of KL and KM. (b) Calculate the size of angle LKM. 5. (a) (i) Show that ( ) is a factor of f ( ). (ii) Hence factorise f ( ) full. (b) Solve ( ) ( ).. (a) Find the equation of the tangent to the parabola with equation 6 at the point (, 6). (b) Show that this line is also a tangent to the circle with equation 5 5. In the right angled triangle shown in Diagram, tan a. (a) Find the eact values of (i) cos a ; Diagram (ii) cos a. a In the right angled triangle shown in Diagram, tan b. b (b) Find the eact value of sin( a b). 5 Diagram Page

9 Marks 5. Solve log( ) log( 5), The diagram below shows part of the graph of p qsin r. p qsin r. (a) Write down the values of p, q and r. The graph of p qsin r. has a minimum turning point at A and a maimum turning point at B. B A p qsin r. (b) Calculate the shaded area in the diagram above. 7 Page

10 Marks 7. Cobalt 6 is used in food irradiation and decas to Nickel 6, which is a stable substance. kt Cobalt 6 decas according to the law m m e, where m is the initial mass t of Cobalt 6 present and m is the mass remaining after t ears. t The time taken for half the mass of Cobalt 6 to deca to Nickel 6 is 5 ears. (a) Find the value of k, giving our answer correct to significant figures. (b) In a sample of Cobalt 6 what percentage has decaed to Nickel 6 after ears? 8. A rectangular park measures metres b metres. A path connecting the two entrances, at opposite corners of the park, is to be laid through the park as shown. Entrance Park metres metres metres metres Entrance The cost per metre of laing the path through the park is twice the cost, per metre, of laing the path along the perimeter. (a) Show that the total cost of laing this path can be modelled b C ( ) (b) Find the value of which would minimise the cost of laing the path. 6 End of Question Paper Page 5

11 H Mathematics Higher Paper Practice Paper B Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 7

12 SECTION A ALL questions should be attempted.. Given that f ( ) 5, find f ().. Find ( )( ) d.. P and Q have coordinates (,, ) and (,, 5). What is the distance between P and Q?. If 8 is epressed in the form ( p) q, what is the value of q? 5. Here are two statements about the equation 5. () The roots are equal. () The roots are rational. Which of the following is true? A Neither statement is correct. B Onl statement () is correct. C Onl statement () is correct. D Both statements are correct. 6. Find all the values of in the interval for which cos. 7. S is the point with coordinates (,, ), T(,, 5) and U(5,, 7). Find the ratio in which T divides SU. d 8. Given that sin( ), find. d Page

13 9. The angle between the line shown in the diagram and the -ais is. What is the gradient of the line?. Given that log 9, what is the value of a? a π. What is the maimum value of 5 9 sin?. Find ( ) d.. The graph shown in the diagram has equation of the form cos( p) q. State the values of p and q? 5 cos( p) q. Given that h ( ) 6, what is the largest possible domain for h? 5. Vector t has components Find the value of k.. u is a unit vector such that u kt, where k. Page

14 Page 6. The diagram shows the graph of ( ). f Which diagram below shows the graph of ( )? f A B C D

15 7. The equation of the parabola shown is of the form k( ). What is the value of k? (, ) k( ) 8. Simplif log log ( ). 9. What is the solution to 5?. If v t and the rate of change of v with respect to t at t k, k is 6, find the value of k. End of Section A Page 5

16 SECTION B ALL questions should be attempted. Marks. A circle with equation 6 9 has centre C. C (a) Write down the coordinates of the centre C and find the length of the radius of this circle. A second circle with equation ( ) ( 7) 6 has centre C. C C (b) (i) Find the distance between the centres C and C. (ii) Hence find the minimum distance between the circumferences of the two circles. Page 6

17 Marks. A is the point with coordinates (,, ), B(,, ) and C(,, ). (a) Epress AB and AC in component form. (b) Find the size of angle BAC.. Solve sin 5cos for. 5. The diagram shows part of the quartic with equation g( ). There are stationar points at, and a. g( ) a On separate diagrams sketch the graph of (a) g( ). (b) g( ). 5. Find the values of for which the function f ( ) 5 is decreasing P is the point with coordinates (, 6) and Q is(, ). Find the locus of points which are equidistant from both P and Q. End of question paper Page 7

18 H Mathematics Higher Paper Practice Paper B Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

19 ALL questions should be attempted. Marks. (a) A sequence is defined b the recurrence relation u u 6, u. n n Determine the values of u, u and u. (b) Wh does this sequence have a limit as n? (c) A second sequence, generated b v pv, has the same limit as the sequence in (a). n Find the value of p. n. A function f is defined on the set of real numbers b f ( ) 7. (a) Show that ( ) is a factor of f ( ), and hence factorise f ( ) full. The graph shown has equation of the form 7. 7 S T (b) Calculate the shaded area labelled S. (c) Find the total shaded area.. D has coordinates (7,,) and F is (,,5). (a) Find the coordinates of E which divides DF in the ratio :. G has coordinates (6,, 5). (b) Show that EG is perpendicular to DF. Page

20 Marks. P, Q and R have coordinates (,6), (8, ) and (, 8) respectivel. (a) Show that PQ is perpendicular to QR. (b) Hence find the equation of the circle which passes through P, Q and R. 5. Two functions f and g are defined on the set of real numbers b (a) Find (i) f ( g ( )); f ( ) k and g( ) k, where k (ii) g( f( )). (b) Find the value of k for which f ( g ( )) g( f( )) has equal roots A closed wooden bo, in the shape of a cuboid, is constructed from a sheet of wood of area 6 cm. The base of the bo measures cm b cm. The height of the bo is h cm. cm h cm cm (a) Assuming the thickness of the sides of the bo are negligible, show that the volume (in cubic centimetres) of the bo is given b V( ) (b) (i) Calculate the value of for which this volume is a maimum. (ii) Find the maimum volume of the bo. 7 Page

21 Marks 7. Whilst carring out an eperiment a scientist gathered some data. The table shows the data collected b the scientist The variables and, in the table, are connected b a relationship of the form ae b. Find the values of a and b Solve cossin for 8 End of Question Paper Page 5

22 H Mathematics Higher Paper Practice Paper C Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

23 SECTION A ALL questions should be attempted.. A sequence is defined b the recurrence relation What is the value of u? u u 5, u 6 n n. Here are two statements about the line with equation 8. () This line is parallel to a line with gradient () This line cuts the ais at the point (, 8). Which of these statements is true?.. Functions f and g are defined on suitable domains b Find an epression for f( g( )). f( ) 5 and g( ).. A curve has equation 5. What is the gradient of the tangent at the point where? 5. A circle with centre (,) passes through the point (5, ). What is the equation of the circle? 6. Find d. 7. g ( ) 7. What is the remainder when g( ) is divided b ( )? R 8. Vectors u and v are shown in the diagram below. Page Q S v u T u P

24 QR ST Find PQ in terms of u and v. 9. P and Q are the points with coordinates (,,5) and (,,) respectivel. If PR PQ, find the coordinates of R.. What is the eact value of sin 5 cos?. Find 5cos() d.. Given that log log log 8, epress in terms of.. Given that d d sin, find.. If 56 is written in the form p q ( ), what is the value of p? 5. Solve tan for 6. The diagram shows the graph with equation log ( a ). b (, ) log ( a) b What are the values of a and b? 7. What is the nature of the roots of the quadratic equation 5? 8. The diagram shows part of the graph of cubic with equation g( ). The graph has turning points at and. Page 5

25 Sketch the graph of 9. Solve 85.. The diagram shows part of the curve f( ). (, 7) L The curve passes through the points K(, ) and L(, 7). Which of the following represents the equation of the curve? (, ) K A B C e D End of Section A Page 6

26 SECTION B ALL questions should be attempted. Marks. A function f is defined b f ( ), where is a real number. (a) Find the coordinates of the points where the curve with equation f( ) crosses the and aes. (b) Find the stationar points on the curve f( ) and determine their nature. 6 (c) (i) Sketch the curve f( ). (ii) Hence solve.. Two sequences are generated b the recurrence relations un un 8 v kv n The two sequences approach the same limit as n. n (a) Evaluate this limit. (b) Hence determine the value of k.. Given that sin a and 5 values of : 5 π π b, sin b, where a and find the eact (a) sin( a b); (b) tan( a b).. In the triangle opposite a b units a b c Find a.( abc ) 6 End of question paper Page 7

27 H Mathematics Higher Paper Practice Paper C Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

28 ALL questions should be attempted.. (a) Given that ( ) is a factor of k 6, Marks find the value of k. (b) Hence, or otherwise, solve k 6.. OABC,DEFG is a rectangular prism as show. z D 7 5 G C E F B 8 A OA is 8 units long, OC is 5 units and OD is 7 units. (a) Write down the coordinates of B and G. (b) Calculate the size of angle BEG. 6. A circle, centre C, has equation. (a) Find the centre C and radius of this circle. (b) (i) Show that the point P(5, ) lies on the circumference of the circle. (ii) Find the equation of radius CP. (c) Find the equation of the chord which passes through (7,) and is perpendicular to radius CP.. Solve cos cos 6 for 6 Page

29 5. (a) Diagram shows part of the graph with equation 5 8. Marks 5 8 Calculate the shaded area. Diagram 5 (b) Given that p ( 5 8) d 5 8 find the total shaded area in diagram. p Diagram 6. Find the smallest integer value of c for which g c ( ) ( )( ) has onl one real root (a) Write sin 5cos in the form ksin( a), where k and a. (b) Sketch the graph of sin 5cos for Page

30 Marks 8. For a particular radioactive isotope, the mass of the original isotope remaining, m grams, after time t ears is given b m of the isotope. 8t m e where m is the original mass (a) If the original mass is g, find the mass of the original isotope remaining after ears. The half life of the isotope is the time taken for half the original mass to deca. (b) Find the half life of this isotope. 9. Find sin d sin. 5 6 End of Question Paper Page 5

31 H Mathematics Higher Paper Practice Paper D Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8 Page

32 SECTION A ALL questions should be attempted.. The midpoint of the line joining G(,,7) to H(5,, p) is M( q,, ). What are the values of p and q?. Given that f( ), find f ( ). 5. If 7 is written in the form ( a) r, find the value of r.. A straight line passes through the points (, ) and (, ). What is the equation of the line? 5. Functions f and g are defined on the set of real numbers b What is the value of g( f ( ))? f g ( ) and ( ) 5 6. The vectors with components 7 What is the value of t? 5 and t are perpendicular. 7. The diagram shows a right angled triangle with sides, and. What is the value of cos? 8. Find 6 d 9. For what value of k does the equation k have equal roots? Page

33 . DE and EF 5 have components and respectivel. Given that D has coordinates (,, ), what are the coordinates of F?. What is the maimum value of 7π 9. Find ( 5) d. 8 sin?. How man solutions does the equation( 7cos )(tan9) have in the interval find f. Given that f( ) sin, The diagram shows the line ST with equation. The angle between ST and the positive direction of the ais is Find an epression for A tan B tan tan C tan D log 6. What is the value of? log 8 7. The diagram shows a sketch of the curve with equation What are the values of a and k? k( )( )( a) (, 5) 5 8. Here are two statements about the function f ( ). () The largest possible domain is. () The range is f( ). Which of these statements is true? 9. Given that Page 5

34 , for f( ), for, for Sketch a curve to represent f( )?. If 5 a, find an epression for. End of Section A Page 6

35 SECTION B ALL questions should be attempted. Marks. A(, ), B(, ) and C(, 8) are the vertices of triangle ABC shown in the diagram. C A B (a) Write down the equation of the altitude from C. (b) Find the equation of the perpendicular bisector of BC. (c) Find the point of intersection of the lines found in (a) and (b).. P is the point (,, ), Q is (5,, ) and R is (7,,). (a) Show that P, Q and R are collinear. (b) Find the ratio in which Q divides PR. Page 7

36 Marks. Find the equation of the tangent to the curve with equation at the point where. 6. (a) Given that f( ) and ( ) is a factor of f ( ), find a formula for f ( ). (b) Hence factorise f ( ) full. (c) Solve f ( ). b 5. The graph illustrates the law a. The straight line joins the points (, ) and (, ). log Find the values of a and b. log End of question paper Page 8

37 National Higher Mathematics Sample Questions D Paper Calculator

38 ALL questions should be attempted. Marks. A sequence is defined b recurrence relation u ku 6, u. n n (a) Given that u 8, find the value of k. (b) (i) Wh does this sequence tend to a limit as n? (ii) Find the value of this limit.. f p q ( ). Given that ( ) is a factor of f ( ), and the remainder when f ( ) is divided b( ) is 9, find the values of p and q. 5. Securit guards are watching a parked car, via two CCTV cameras, in a supermarket car park. With reference to a suitable set of aes, the car is at C(5,,) and the cameras are at positions A(,6, ) and B(7, 9, 5) as shown. (,6, ) A B(7, 9, 5) C(5,, ) Calculate the size of angle ACB. 7 Page

39 Marks. Part of the graphs of and 5 are shown opposite. S 5 The curves intersect at the points S and T. T (a) Find the coordinates of S and T. (b) Find the shaded area enclosed between the two curves A circle with centre C has equation 6 5. (a) Write down the coordinates of the centre and calculate the length of the radius of this circle. C A second circle with centre C has a diameter twice that of the circle with centre C. C lies on the circumference of this second circle. The line joining C and C is parallel to the ais. C C (b) Find the equation of the circle with centre C. Page

40 Marks 6. A manufacturer of eecutive desks estimates that the weekl cost, in, of making desks is given bc ( ) Each eecutive desk sells for. (a) Show that the weekl profit made from making desks is given b ( ) 6 8 P (b) (i) How man desks would the manufacturer have to make each week in order to maimise his profit? (ii) What would his annual profit be? 8 7. The number of bacteria, b, in a culture after t hours is given b b kt b e where b is the original number of bacteria present. (a) The number of bacteria in a culture increases from 8 to in hours. Find the value of k correct to significant figures. (b) How man bacteria, to the nearest hundred, are present after a further hours? 8. (a) Epress cos 5sin in the form k cos( a ), where k and a 9. (b) (i) Hence write cos 5sin in the form R cos( b ), where R and b 9. (ii) Solve cos 5sin 5 in the interval 6. End of Question Paper Page 5

41 H Mathematics Higher Paper Practice Paper E Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

42 SECTION A ALL questions should be attempted.. K and L have position vectors and respectivel. What is the magnitude of KL?. If f( ) 7, find f ( ).. Find d. A function f is defined on the set of real numbers b f( ) 5. Find an epression for f( f( )). 5. Evaluate. sin cos 6. A circle with centre (,) passes through the point (,). What is the equation of the circle? 7. f ( ) 5. What is the remainder when f( ) is divided b ( )? 8. The diagram shows the part of the graph of the cubic f( ). Sketch the graph of f( )? Page

43 9. The graph shown in the diagram has equation p sin( q). What are the values of p and q?. A sequence is generated b the recurrence relation u 7 u. n n If u 5, what is the value of u?. For what value of k does the equation k 6 have equal roots?. Find ( 7) d.. Given that and () 5, f ( ) 6 f find a formula for f( ) in terms of.. What are the coordinates of the centre of the circle with equation 685? 5. The diagram shows part of the graph of a cubic function. 6 What is the equation of this graph? Page 5

44 6. The diagram shows part of the graph of the cubic f( ). There are stationar points at and. Sketch the graph of f( )? 7. If 8 is epressed in the form ( p) q, what is the value of q? 8. If log t log 5, find the value of t. 9. If p find the rate of change of p with respect to when.. Find the solutions for 8? End of Section A Page 7

45 SECTION B ALL questions should be attempted. Marks. A line joins the points P(, ) and Q(, 7). Find the equation of the perpendicular bisector of PQ. P Q. Show that the line with equation is a tangent to the circle with equation 5 and find the coordinates of the point of contact of the tangent and circle. 6. The diagram shows a right angled triangle with height units, base unit and an angle of p. (a) Find the eact values of: (i) cos p ; (ii) cos p. p (b) B writing p p p, find the eact value of cos p.. A function f is defined b f( ), where. Find the maimum and minimum values of f. 5 Page 8

46 5. (a) Epress cos sin in the form kcos( a), where k Marks and a 6. (b) Find: (i) the maimum value of sin cos ; (ii) a value of where this maimum value occurs in the interval 6. End of question paper Page 9

47 H Mathematics Higher Paper Practice Paper E Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8

48 ALL questions should be attempted. Marks. (a) A line, l, passes through the points A(,) and B(5, ). B l The line makes an angle of a with the positive direction on the ais. A a Find the value of a. (b) A second line, l, with equation, crosses the line in (a). The angle between the two lines is b, as shown. A Find the value of b. l a b B l. The rectangular based pramid D,OABC has vertices A(6,,), B(6,8,) and D(,, 7). z D(,, 7) C B(6,8,) A(6,,) (a) (i) Write down the coordinates of C. (ii) Epress AC and AD in component form. (b) Calculate the size of angle CAD. 5 Page

49 Marks. (a) (i) Show that ( ) is a factor of 6. (ii) Hence factorise 6 full. The line with equation intersects the curve with equation at the points A, B and C. 6 C B A 6 (b) Find the coordinates of the points A and C. The area between the curve and the line from A to C is shaded in the diagram below. C B A 6 (c) Calculate the total shaded area shown in the diagram. 7 Page

50 Marks. Solve cossin for 6 5. A new hour anti biotic is being tested on a patient in hospital. It is know, that over a hour period, the amount of anti biotic remaining in the bloodstream is reduced b 8%. On the first da of the trial, an initial 5 mg dose is given to a patient at 7 a.m. (a) After hours and just prior to the second dose being given, how much anti biotic remains in the patient s bloodstream? The patient is then given a further 5 mg dose at 7 a.m. and at this time each subsequent morning thereafter. (b) A recurrence relation of the form u au b can be used to model this course of treatment. n Write down the values of a and b. It is also known that more than 5 mg of the drug in the bloodstream results in unpleasant side effects. (c) Is it safe to administer this anti biotic over an etended period of time? n 6. The diagram shows part of the graph of cos( ). cos( ) Find the equation of the tangent at the point T, where. T 7 7. Solve log ( ) log (),. 5 Page 5

51 Marks 8. A circle has the following properties: The ais and the line are tangents to the circle. The circle passes through the points (,) and (,8). The centre lies in the first quadrant. Find the equation of this circle. 6 End of Question Paper Page 6