Higher Supported Study

May 2015

Session 2

Paper 1

21 April 2023

Solve the equation for

Question 1

xxx cos6cossin2

xx cos62sin 3600 x

xxx cossin22sin

0cos6cossin2 xxx rearrange to = 0

0)3(sincos2 xx factorise

03sin0cos xx solve equations

270,90x solutionno

270,90x

21 April 2023

D, E and F have coordinates (10,- 8, -15), (1,-2,-3) and (-2,0,1) respectively.(a) (i) Show that D, E and F are collinear. (ii) Find the ratio in which E divides DF(b) G has coordinates (k,1,0). Given that DE is perpendicular to GE, find the value of k.)()( ia

0. GEDE

DE

12

6

9

de EF

4

2

3

efEF3

DE

so EF and DE are parallel

E is a common point

so D, E and F are collinear)()( iia 1:3: EFDE

)(bGE

3

3

1 k

ge 03618)1(9 k

639 k

7k

Question 2

21 April 2023

)(a )(b

The diagram shows a sketch of the function y = f (x)(a)Copy the diagram and on it sketch the graph of y = f (2x)(b) On a separate diagram sketch the graph of y = 1 - f (2x).

Question 3

21 April 2023

2

1gradient

Two variables, x and y, are connected by the law xay

The graph of log 4 y against x is a straight line passing throughthe origin and the point A(6,3). Find the value of a O x

log 4 y

A(6,3)

xy2

1log4

xy 2

1

4

xy )4( 21

xy 2

2a

Question 4

The diagram shows a sketch of and the tangent to the curve at x = 2 Show that the equation of the tangent to the curve at x = 2 is y = 5x - 8

xxxy 23 2

xxxxf 23 2)(

143)( 2 xxxf

1)2(4)2(3)2( 2 f

5)2( f

5m

2)2(2)2()2( 23 f

2)2( f

)2,2(at5m

)2(52 xy

1052 xy

85 xy

Question 5a

Find algebraically the coordinates of the point where this tangent meets the curve again

852 23 xxxx

0842 23 xxx

2 1 -2 -4 8

2 0 - 8

1 0 -4 0

0)4()2( 2 xx

0)2()2()2( xxx

2or2 x

)18,2(point

188)2(5 y

Question 5a

2100)2(10 22

Angle x is acute and is such that

(a) Show clearly that the exact value of is

102cos x

1027

2

10 98

2724998

27

xsin

10

27sin x

Question 6a

444 sincoscossin)(sin xxx

(b) Hence show that 80)(sin 4 x

2

1

10

2

2

1

10

27

210

227

8.010

8

210

28

Question 6b

21 April 2023

Question 7

Two identical circles touch at the point P(9,3) as shown in the diagram.

 One of the circles has equation

Find the equation of the other circle. x

y

O

P (9, 3)

)2,5(),( fg

cfg 22 17

)4,13(2circleofcentre

17)4()13( 22 yx

01241022 yxyx

21 April 2023

4log2log2log2 32

33

1134

1

x

x

Question 8

Solve the equation 12log21log 33 x

3log4log)1(log 333 x

3log4

1log 33

x

6,,2 pcpba

042 acb0)6(82 pp

124 p

04882 pp0)4)(12( pp

4,12 pp

0

y

x0

y

x

(–4, 0) (12, 0)

Question 9

)(a

remainder = 0so x + 1 is a factor

-1 1 0 -13 -12

-1 1 12

1 -1 -12 0

)(b

)12()1(

12132

3

xxx

xx

)4()3()1( xxx

Question 10

60sincos60cossin)60sin( xxx )(a xx cossin 2

321

)(b )6045sin(105sin 60sin45cos60cos45sin

23

21

21

21

22

3

221

22

31

Question 11

3)(sin4)( xxf

)(cos)(sin12)( 2 xxxf

xxxf cossin12)( 2

xxf cossin12)( 26

5

65

65

65

65 cossinsin12)( f )(12 2

321

21

233

Question 12

dxxx

x

22

3

2

1

2

3

dxxx 221

23

Cxx 1212

43

Question 13

Question 14

Sketch the graph of )(xhy