national mathematics qualifications higher prelim ...given that k is a constant of integration ......

Pegasys 2012

Read carefully

Calculators may NOT be used in this paper.

Section A - Questions 1 - 20 (40 marks)

Instructions for the completion of Section A are given on the next page.

For this section of the examination you should use an HB pencil.

Section B (30 marks)

1. Full credit will be given only where the solution contains appropriate working.

2. Answers obtained by readings from scale drawings will not receive any credit.

Mathematics

Higher Prelim Examination 2012/2013

Paper 1

Assessing Units 1 & 2

Time allowed - 1 hour 30 minutes

NATIONAL

QUALIFICATIONS

Pegasys 2012

Sample Question

A line has equation .14 xy

If the point )7,(k lies on this line, the value of k is

A 2

B 27

C 15

D 2

The correct answer is A 2. The answer A should then be clearly marked in pencil with a

horizontal line (see below).

Changing an answer

If you decide to change an answer, carefully erase your first answer and using your pencil, fill in the

answer you want. The answer below has been changed to D.

Read carefully

1 Check that the answer sheet provided is for Mathematics Higher Prelim 2012/2013 (Section A).

2 For this section of the examination you must use an HB pencil and, where necessary, an eraser.

3 Make sure you write your name, class and teacher on the answer sheet provided.

4 The answer to each question is either A, B, C or D. Decide what your answer is, then, using

your pencil, put a horizontal line in the space below your chosen letter (see the sample question below).

5 There is only one correct answer to each question.

6 Rough working should not be done on your answer sheet.

7 Make sure at the end of the exam that you hand in your answer sheet for Section A with the rest

of your written answers.

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FORMULAE LIST

Circle:

The equation 02222 cfygxyx represents a circle centre ),( fg and radius cfg 22 .

The equation ( ) ( )x a y b r 2 2 2 represents a circle centre ( a , b ) and radius r.

A

A

AAA

AAA

BABABA

BABABA

2

2

22

sin21

1cos2

sincos2cos

cossin22sin

sinsincoscoscos

sincoscossinsin

Scalar Product: .andbetweenangletheisθwhere,cosθ. bababa

or

ba .

3

2

1

3

2

1

332211

b

b

b

and

a

a

a

where babababa

Trigonometric formulae:

Table of standard derivatives:

Table of standard integrals:

)(xf )(xf

axaax cossin

axaax sincos

)(xf )(xf dx

axsin Caxa

cos1

axcos Caxa

sin1

Pegasys 2012

1. If 2

2)(

xxf , where 0x , then )2(f equals

A 1

B 21

C 32

D 21

2. When 123 xx is divided by )3( x the remainder is

A 35

B 7

C 19

D 13

3. P and Q have coordinates ( 1 , 2) and ( 3 , k ) respectively.

Both points lie on the curve with equation 22xy .

The gradient of the line PQ is

A 81

B 8

C 18

D 12

4. The exact value of 6

5tan is

A 3

1

B 3

C 1

D 3

1

SECTION A

ALL questions should be attempted

Pegasys 2012

5. A straight line passes through the point ( 0 , 4 ) and is parallel to the line with

equation 962 yx .

Its equation is

A 431 xy

B 43 xy

C 431 xy

D 462 yx

6. The limit of the sequence defined by the recurrence relation )1( 2

1 ppUU nn

where p is a constant and 10 p is

A p

B p1

C p1

D p1

1

7. A circle has as its equation .9)2()9( 22 yx

Which of the following statements is true?

A the circle intersects the x-axis at two distinct points.

B the circle does not intersect or touch either of the two axes.

C the circle touches the y-axis at a single point.

D the circle intersects the y-axis at two distinct points.

8. A curve for which xdx

dy6 passes through the point ( 1 , 5 ).

The particular solution is

A y = 56 x

B y = 23 2 x

C y = 13 2 x

D y = 6

Pegasys 2012

9. The point T( 0 , 6 ) lies on the circumference of the circle with centre C and

equation 0216222 yxyx .

The gradient of the radius CT is

A 3

B 31

C 3

D 9

10. The diagram below shows a square of side 4x centimetres.

The ratio of the shaded area to the unshaded area is

A 2 : 7

B 1 : 7

C 1 : 8

D 1 : 4

11. How many real roots does the equation 0)75)(1( 22 xxx have?

A 1

B 2

C 3

D 4

4x cm

x cm

Pegasys 2012

12. Part of the curve )(xfy is shown below.

Area P is 65 square units and area Q is

651 square units.

Which of the following is/are true?

(1) 2

165)( dxxf

(2) 4

2651)( dxxf

(3) 4

1322)( dxxf

A (1) only

B (2) only

C (1) and (2) only

D (1), (2) and (3)

13. The gradient of the tangent to the curve 2)3( xy at the point where 2x is

A 25

B 19

C 3232

D 10

O x

y

1 2 4

P

Q

)(xfy

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14. If )2)(1( pppn where p is a whole number, which of the following

must be true?

(1) n must be even

(2) n is a multiple of 3

(3) n is a multiple of 4

A (1) only

B (2) only

C (1) and (2) only

D (1) and (3) only

15. The diagram shows part of the graph of )(xfy as a full line and a part graph of a

related function as a broken line.

The equation of the related function is

A )(2 xfy

B 2)( xfy

C 2)( xfy

D )2( xfy

x

y )(xfy

( 4 , 6 )

O 2

( 4 , 4 )

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16. Given that 3)1( 31

x , then 1x equals

A 7

B 26

C 25

D 27

17. Given that kx 2cos , then x2sin equals

A )1(21 k

B )1(21 k

C 21 k

D )1(21 k

18. If the minimum value of axx 42 is 10, where Rx , then a is

A 6

B 4

C 14

D 10

19. Given that k is a constant of integration dxx 4

6equals

A kx

3

2

B kx

55

6

C kx

3

6

D kx

5

6 5

Pegasys 2012

20. Consider the triangle below.

The ratio of AB : AC is

A 1:2

B 2 : 1

C 3:1

D 1:3

30o 60

o

A

B C

[ END OF SECTION A ]

Pegasys 2012

21. Two functions are defined on suitable domains as

12)( xxf and k

xxh

)1(2)(

2 .

(a) Find ))(( xff in its simplest form. 2

(b) Hence show clearly that the equation ))(()( xffxh can be written as

0)23(42 2 kkxx 3

(c) Find the two possible values of k so that ))(()( xffxh has equal roots. 5

22. A( 1 , 2 ), B( 7 , 6 ) and C( 7 , 14 ) are the vertices of a triangle.

(a) Show clearly that triangle ABC is right-angled at A. 3

(b) M is the mid-point of AC. Find the equation of the median BM. 3

(c) If the median BM is extended it meets the line through C, with gradient 5, at P.

Find the coordinates of P. 4

23. A function is given as 144)( 3 xxxf .

(a) If 14)( pf , find the value of p if 0p . 2

(b) Prove that the function is increasing when p takes this value. 3

SECTION B


Pegasys 2012

24. The diagram shows the graph of bxay sin for 20 x .

(a) Write down the values of a and b (where a is a whole number). 2

(b) Determine the exact x-coordinates for the points P and Q.

Your answers must be accompanied by the appropriate working. 3

y

x O 2

3 bxay sin

P Q

[ END OF SECTION B ]

[ END OF QUESTION PAPER 1 ]

Pegasys 2012

Read carefully

1. Calculators may be used in this paper.

2. Full credit will be given only where the solution contains appropriate working.

3. Answers obtained from readings from scale drawings will not receive any credit.

Mathematics


Paper 2

Assessing Units 1 & 2

Time allowed - 1 hour 10 minutes

NATIONAL

QUALIFICATIONS

Pegasys 2012

FORMULAE LIST

Circle:

The equation 02222 cfygxyx represents a circle centre ),( fg and radius cfg 22 .

The equation ( ) ( )x a y b r 2 2 2 represents a circle centre ( a , b ) and radius r.

A

A

AAA

AAA

BABABA

BABABA

2

2

22

sin21

1cos2

sincos2cos

cossin22sin

sinsincoscoscos

sincoscossinsin

Scalar Product: .andbetweenangletheisθwhere,cosθ. bababa

or

ba .

3

2

1

3

2

1

332211

b

b

b

and

a

a

a

where babababa

Trigonometric formulae:

Table of standard derivatives:

Table of standard integrals:

)(xf )(xf

axaax cossin

axaax sincos

)(xf )(xf dx

axsin Caxa

cos1

axcos Caxa

sin1

Pegasys 2012

1. The graph of the curve with equation axxxy 332 23 crosses the x-axis at

the point ( 1 , 0

).

(a) Find the value of a. 2

(b) Hence write down the coordinates of the point at which this curve crosses

the y-axis. 1

(c) This curve also crosses the x-axis at a further two points.

Find algebraically the coordinates of these other two points. 4

2. Solve algebraically the following system of equations for 900 x .

xy 2cos3

xy cos101

5

3. A recurrence relation is defined as 14801 nn UU .

(a) Given that 200 U , find 3U . 2

(b) Express the difference between the limit of this sequence and 3U as a

percentage of the limit. 3

4. A function is defined on a suitable domain as 2

3 46)(

x

xxg

.

If a

gdxx1

)2()42( , find the value of a where 1a . 7


Pegasys 2012

5. Part of the graph of 593 23 xxxy is shown in the diagram below.

The curve crosses the x-axis at the points ( 1 , 0

) and ( 5 , 0 ) as shown.

(a) Find the coordinates of the stationary point A. 4

(b) Calculate the finite area enclosed between this curve and the x-axis. 5

6. The diagram below shows a circle, centre C, with equation .01241622 yxyx

The point Q( 4 , 6 ) lies on the circumference of the circle.

The line PQ is a tangent to the circle.

(a) Find the equation of the tangent PQ. 4

(b) Write down the coordinates of P. 2

(c) Establish the equation of the circle which passes through the points P, Q and C. 5

P

C

Q( 4 , 6 )

x

y

x

y

O

593 23 xxxy

1

A

5

Pegasys 2012

7. An underground storm drain has a cross-section in the shape of a rectangle with a

semi-circular base.

The rectangle part of the drain measures 2x metres by h metres as shown.

(a) Show clearly that if the perimeter of the drain is approximately 83 metres

then h can be expressed as

xxh 2191 . 2

(b) Hence show that the cross-sectional area, A, in terms of x, can be written as

22

21283)( xxxxA . 2

(c) Find the value of x which will produce the largest cross-sectional area.

Justify your answer. 5

8. Angle A is such that 5

1tan A and both A and 2A are acute.

(a) Show clearly that the exact value of A2sin is 35 . 3

(b) Given now that 322cos A , show clearly that )2sin(

3A has an exact

value of 53261 . 4

[ END OF QUESTION PAPER ]

2x

h

Pegasys 2012

Indicate your choice of answer with a

single mark as in this example

Mathematics


Paper 1 - Section A - Answer Sheet

NATIONAL

QUALIFICATIONS

NAME :

TEACHER :

CLASS :

You should use an HB pencil.

Erase all incorrect answers thoroughly.

Section A

Section B

40

30

Total (P1)

70

Total (P2)

60

Overall Total

130

%

Please make sure you have filled in all your details above before handing in this answer sheet.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A B C D

A B C D

Pegasys 2012

Higher Grade - Paper 1 2012/2013 ANSWERS - Section A

1 D

2 C

3 B

4 A

5 A

6 C

7 A

8 B

9 C

10 B

11 B

12 A

13 D

14 C

15 A

16 C

17 D

18 C

19 A

20 D

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A B C D

Pegasys 2012

Higher Grade Paper 1 2012/2013 Marking Scheme

21(a) ans: 4x – 3 (2 marks)

●1 substitutes ●

1 1)12(2))(( xxff

●2 simplifies

●

2

34))(( xxff

(b) ans: proof (3 marks)

●1 equates expressions ●

1 34

)1(2 2

xk

x

●2 cross multiplies ●

2

)34()1(2 2 xkx ;

●3 multiplies and rearranges to answer ●

3 kkxx 3422 2

(c) ans: k = –½ or 2 (5 marks)

●1 knows condition for equal roots ●

1 042 acb

[stated or implied]

●2 substitutes into b

2 – 4ac ●

2 0)23.(2.4)4( 2 kk

●3 simplifies ●

3 0162416 2 kk ;

●4 factorises ●

4 0)2)(12(8 kk

●5 solves for k ●

5

21k or 2

22(a) ans: proof (3 marks)

●1 knows to find gradients of AB and AC ●

1 evidence of finding gradients of two lines

●2 finds gradients of AB and AC ●

2

2

3;

3

2 ACAB mm

●3 communicates conclusion ●

3 since 1 ACAB mm angle A is right

(b) ans: 5y + x = 37 (3 marks)

●1 finds midpoint of AC ●

1 M(–3, 8)

●2 finds gradient of BM ●

2

5

1BMm

●3 substitute ●

3 )3(

5

18 xy or )7(

5

16 xy

(b) ans: P(–8, 9) (4 marks)

●1 finds equation of line through C ●

1 495)7(514 xyxy

●2 knows to use system of equations ●

2 evidence of scaling

●3 finds values for x and y ●

3 x = –8 ; y = 9

●4 states coordinates of P ●

4 P(–8, 9)

Give 1 mark for each Illustration(s) for awarding each mark

Pegasys 2012

23(a) ans: 2p (2 marks)

●1 equates expressions and takes terms to LHS ●

1 0414144 33 pppp

●2 factorises, solves and discards ●

2 20)4( 2 ppp


●1 knows to take derivative ●

1 43)(' 2 ppf

●2 substitutes p = 2 ●

2 84)2(3)2(' 2 f

●3 communicates ●

3 derivative > 0 so function is increasing

24(a) ans: a = 2; b = –√3 (2 marks)

●1 states value of a ●

1 a = 2

●2 states value of b

●

2 b = –√3

(b) ans: π/3; 2π/3 (3 marks)

●1 equates equation to zero ●

1 03sin2 x

●2 finds x – coordinate of P ●

2

3

●3 finds x – coordinate of Q ●

3

3

2


Total: 70 marks

Pegasys 2012

Higher Grade Paper 2 2012/2013 Marking Scheme

1(a) ans: a = 2 (2 marks)

●1 substitutes into equation ●

1 0)1(3)1(3)1(2 23 a

●2 solves for a ●

2 a = 2

(b) ans: (0, 2) (1 mark)

●1 states coordinates of y - intercept ●

1 (0, 2)

(c) ans: (2, 0); (½, 0) (4 marks)

●1 equates equation to zero ●

1 02332 23 xxx

●2 knows to use synthetic division ●

2

0112

224

23322

●3 factorises and solves for x ●

3 1,

2

1,2);1)(12)(2( xxxx

●4 states coordinates of two points ●

4 (2, 0); (½, 0)

2 ans: 70∙5o (5 marks)

●1 equates two equations ●

1 xx cos1012cos3

●2 replaces cos

2x

o ●

2 xx cos101)1cos2(3 2

●3 multiplies out and rearranges to trinomial ●

3 04cos10cos6 2 xx

●4 factorises

●

4 0)2)(cos1cos3(2 xx

●5 solves for other bracket and discards ●

5 70∙5

o

3(a) ans: 44∙4 (2 marks)

●1 strategy of finding consecutive terms ●

1 finds U1 and U2

●2 answer ●

2 finds U3

(b) ans: 36∙6% (3 marks)

●1 finds limit of sequence

●

1 70

20

14

L

●2 finds difference and percentage ●

2 %100

70

625;62544470

L

●3 answer ●

3 36∙6%


Pegasys 2012

4 ans: a = 2 (7 marks)

●1 finds integral ●

1 axx 1

2 4

●2 subs and simplifies ●

2 542 aa

●3 prepares to differentiate ●

3 246)( xxxg

●4 differentiates

●

4

3

86)('

xxg

●5 evaluates and equates to integral ●

5 7

8

86)2('g 7542 aa

●6 factorises ●

6 0)2)(6( aa

●7 discards and solves for a ●

7 a = 2

5(a) ans: A(3, –32) (4 marks)

●1 knows to take derivative ●

1 0

dx

dy for stationary point

●2 differentiates ●

2 0963 2 xx

●3 factorises and solves for x ●

3 3;0)1)(3(3 xxx

●4 substitutes and states coords of A ●

4 A(3, –32)

(b) ans: 108 units² (5 marks)

●1 uses correct limits ●

1

5

1.............

●2 knows how to find area ●

2 dxxxx 593 2

5

1

3

●3 integrates ●

3

5

1

23

4

52

9

4

x

xx

x

●4 substitutes limits ●

4

)5(5

2

)5(95

4

5 23

4

)1(5

2

)1(9)1(

4

)1( 23

4

●5 evaluates ●

5 -108 = 108 units²


Pegasys 2012

6(a) ans: 2y – x = 8 (4 marks)

●1 finds centre of circle ●

1 centre (8, –2)

●2 finds gradient of radius ●

2 mrad = –2

●3 finds gradient of tangent ●

3 mtan = ½

●4 substitutes into equation of line ●

4 y – 6 = ½(x – 4)

(b) ans: P(0, 4) (2 marks)

●1 realises x = 0 ●

1 2y – 0 = 8

●2 states coordinates of P ●

2 P(0, 4)

(c) ans: (x + 6)² + (y – 1)² = 25 (5 marks)

●1 realises angle PQC is right ●

1 angle PQC = 90

o

●2 knows PC is diameter ●

2 PC is diameter

●3 finds centre of circle ●

3 centre is (4, 1)

●4 finds radius ●

4 radius = 5 [(4, 1) (8, –2) or (0, 4)]

●5 subs into equation of a circle ●

5 (x – 4)² + (y – 1)² = 25


●1 finds expression for perimeter ●

1 8322 xhx

●2 rearranges to given answer ●

2 xxhxxh

2

191;2832


●1 finds expression for area ●

1 2

2

12)( xxhxA

●2 subs expression for h and rearranges ●

2 2

2

1)

2

191(2)( xxxxxA

222

2

1283)( xxxxxA

(c) ans: 0∙532m (5 marks)

●1 knows to make derivative = 0 ●

1 0)(' xA

●2 differentiates first 2 terms ●

2 3∙8 – 4x......

●3 differentiates last term ●

3 ..... – πx

●4 solves for x ●

4 πx + 4x = 3∙8; x = 0∙532

●5 justifies answer ●

5 table of values shown


Pegasys 2012


●1 uses expansion for A2sin ●

1 AAA cossin22sin

●2 finds values of Asin and Acos ●

2

6

1sin A and

6

5cos A

●3 subs and rearranges to answer ●

3

3

5

6

52

6

5

6

122sin A


●1 expands ●

1 AAA 2sin

3cos2cos

3sin2

3sin

●2 subs exact values ●

2 AA 2sin

2

12cos

2

3

●3 subs other values ●

3

3

5

2

1

3

2

2

3

●4 rearranges to given answer ●

4 )532(

6

1

6

532

6

5

6

32

Total: 60 marks


national mathematics qualifications higher prelim ...given that k is a constant of integration ......

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