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Higher Mathematics

Exam Revision 2

1.[SQA] Three lines have equations 2x+ 3y− 4 = 0, 3x− y− 17 = 0 and x− 3y− 10 = 0.Determine whether or not these lines are concurrent. 4

Part Marks Level Calc. Content Answer U1 OC1

4 C NC CGD, G8 1996 P1 Q14

2.[SQA] Find the equation of the median AD of triangle ABC where the coordinates of A,B and C are (−2, 3) , (−3,−4) and (5, 2) respectively. 3


3 C CN G3, G7 1995 P1 Q5

hsn.uk.net Page 1Questions marked ‘[SQA]’ c© SQA

All others c© Higher Still Notes

Higher Mathematics

3.[SQA] The diagram shows a sketch of thefunction y = f (x) .

(a) Copy the diagram and on it sketchthe graph of y = f (2x) . 2

(b) On a separate diagram sketch thegraph of y = 1− f (2x) . 3

= f( )

(2, 8)(–4, 8)

O x

x

y

y


(a) 2 B CN A3 sketch 2009 P1 Q23

(b) 3 B CN A3 sketch

•1 ic: scaling parallel to x-axis•2 ic: annotate graph

•3 ss: correct order for refl(x) and trans•4 ic: start to annotate final sketch•5 ic: complete annotation

•1 sketch and one of (0, 0), (1, 8),(−2, 8)

•2 remaining points

•3 reflect in x-axis then verticaltranslation

•4 sketch and one of (0, 1), (1,−7),(−2,−7)

•5 remaining points

4.[SQA] Functions f and g , defined on suitable domains, are given by f (x) = 2x andg(x) = sin x+ cos x .

Find f(

g(x))

and g(

f (x))

. 4


4 C NC A4 1997 P1 Q3



Higher Mathematics

5.[SQA] Show that x2 + 8x+ 18 can be written in the form (x+ a)2 + b .

Hence or otherwise find the coordinates of the turning point of the curve withequation y = x2+ 8x+ 18. 3


3 C NC A5, A6 1994 P1 Q11

6.[SQA] Functions f and g are given by f (x) = 3x+ 1 and g(x) = x2 − 2.(a) (i) Find p(x) where p(x) = f (g(x)) .

(ii) Find q(x) where q(x) = g( f (x)) . 3

(b) Solve p′(x) = q′(x) . 3


(a) 3 C CN A4 3(x2− 2) + 1, (3x+ 1)2− 2 2009 P2 Q2(b) 3 C CN C1 x = − 12

•1 ss: substitute for g(x) in f (x)•2 ic: complete•3 ic: sub. and complete for q(x)

•4 ss: simplify•5 pd: differentiate•6 pd: solve

•1 f (x2 − 2)•2 3(x2 − 2) + 1•3 (3x+ 1)2 − 2

•4 p(x) = 3x2 − 5, q(x) = 9x2 + 6x− 1•5 p′(x) = 6x, q′(x) = 18x+ 6•6 x = − 12



Higher Mathematics

7.[SQA] The point P(−2, b) lies on the graph of the function f (x) = 3x3 − x2 − 7x+ 4.(a) Find the value of b . 1

(b) Prove that this function is increasing at P. 3


(a) 1 C NC A6 1995 P1 Q10

(b) 3 C NC C7

8.[SQA] If f (x) = kx3 + 5x− 1 and f ′(1) = 14, find the value of k . 3


3 C NC C1, A6 1994 P1 Q2

9.[SQA] Find the coordinates of the turning points of the curve with equationy = x3 − 3x2− 9x+ 12 and determine their nature. 8


8 C CN C8, C9 max. at (−1, 17) and min.at (3,−15)

2009 P2 Q1

•1 ss: know to differentiate•2 pd: differentiate•3 ss: set derivative to zero•4 pd: factorise•5 pd: solve for x•6 pd: evaluate y-coordinates•7 ss: know to, and justify turningpoints

•8 ic: interpret result

•1 dydx = · · · (1 term correct)•2 3x2 − 6x− 9•3 dydx = 0•4 3(x+ 1)(x− 3)•5 x = −1 or x = 3•6 y = −17 or y = −15•7 x · · · −1 · · · · · · 3 · · ·

dy/dx + 0 − − 0 +•8 max. at (−1, 17) and min. at (3,−15)



Higher Mathematics

10.[SQA] A sequence is defined by the recurrence relation un = 0·9un−1 + 2, u1 = 3.(a) Calculate the value of u2 . 1

(b) What is the smallest value of n for which un > 10? 1

(c) Find the limit of this sequence as n→ ∞ . 2


(a) 1 C CR A10 1994 P1 Q9

(b) 1 C CR A14

(c) 2 C CR A13

11.[SQA]

(a) On the same diagram, sketch the graphs of y = log10 x and y = 2− x where0 < x < 5.

Write down an approximation for the x -coordinate of the point of intersection. 3

(b) Find the value of this x -coordinate, correct to 2 decimal places. 3


(a) 3 C CR A2 1991 P2 Q4

(b) 1 C CR A26

(b) 2 A/B CR A26



Higher Mathematics

12.[SQA]

(a) The function f is defined by f (x) = x3 − 2x2− 5x+ 6.The function g is defined by g(x) = x− 1.Show that f

(

g(x))

= x3 − 5x2+ 2x+ 8. 4

(b) Factorise fully f(

g(x))

. 3

(c) The function k is such that k(x) =1

f(

g(x)) .

For what values of x is the function k not defined? 3


(a) 4 C NC A4 1990 P2 Q6

(b) 3 C NC A21

(c) 2 C NC A1



Higher Mathematics

13.[SQA] The diagram shows part of the graph of thecurve with equation y = 2x3− 7x2 + 4x+ 4.(a) Find the x -coordinate of the maximumturning point. 5

(b) Factorise 2x3 − 7x2 + 4x+ 4. 3

(c) State the coordinates of the point A andhence find the values of x for which2x3− 7x2 + 4x+ 4 < 0. 2

O x

y

A

(2, 0)

y = f (x)


(a) 5 C NC C8 x = 13 2002 P2 Q3

(b) 3 C NC A21 (x− 2)(2x+ 1)(x− 2)(c) 2 C NC A6 A(− 12 , 0), x < − 12

•1 ss: know to differentiate•2 pd: differentiate•3 ss: know to set derivative to zero•4 pd: start solving process of equation•5 pd: complete solving process

•6 ss: strategy for cubic, e.g. synth.division

•7 ic: extract quadratic factor•8 pd: complete the cubic factorisation

•9 ic: interpret the factors•10 ic: interpret the diagram

•1 f ′(x) = . . .•2 6x2 − 14x+ 4•3 6x2 − 14x+ 4 = 0•4 (3x− 1)(x− 2)•5 x = 13

•6· · · 2 −7 4 4

· · · · · · · · ·· · · · · · · · · 0

•7 2x2 − 3x− 2•8 (x− 2)(2x+ 1)(x− 2)

•9 A(− 12 , 0)•10 x < − 12

14.[SQA] Express x3 − 4x2 − 7x+ 10 in its fully factorised form. 4


4 C NC A21 1998 P1 Q2



Higher Mathematics

15. (a) (i) Show that (x− 4) is a factor of x3 − 5x2 + 2x+ 8.(ii) Factorise x3 − 5x2+ 2x+ 8 fully.(iii) Solve x3 − 5x2+ 2x+ 8 = 0. 6

(b) The diagram shows the curve with equation y = x3 − 5x2 + 2x+ 8.

Q R

y = x3 – 5

2 + 2 + 8

P O x

xxy

The curve crosses the x -axis at P, Q and R.

Determine the shaded area. 6


(a) 6 C CN A21, A22 −1, 2, 4 2012 P1 Q21(b) 6 C CN C16, C12 323



Higher Mathematics

16.[SQA] Functions f (x) = sin x , g(x) = cos x and h(x) = x+ π4 are defined on a suitableset of real numbers.

(a) Find expressions for:

(i) f (h(x)) ;

(ii) g(h(x)) . 2

(b) (i) Show that f (h(x)) = 1√2sin x+ 1√

2cos x .

(ii) Find a similar expression for g(h(x)) and hence solve the equationf (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π . 5


(a) 2 C NC A4 (i) sin(x + π4 ), (ii)cos(x+ π4 )

2001 P1 Q7

(b) 5 C NC T8, T7 (i) proof, (ii) x = π4 ,3π4

•1 ic: interpret composite functions•2 ic: interpret composite functions

•3 ss: expand sin(x+ π4 )•4 ic: interpret•5 ic: substitute•6 pd: start solving process•7 pd: process

•1 sin(x+ π4 )•2 cos(x+ π4 )

•3 sin x cos π4 + cos x sinπ

4 and

complete

•4 g(h(x)) = 1√2cos x− 1√

2sin x

•5 ( 1√2sin x+ 1√

2cos x)− ( 1√

2cos x− 1√

2sin x)

•6 2√2sin x

•7 x = π4 ,3π4 accept only radians

17.[SQA] Solve the equation cos 2x◦ + 5 cos x◦ − 2 = 0, 0 ≤ x < 360. 5


1 C CR T10 1994 P1 Q15

4 A/B CR T10



Higher Mathematics

18.[SQA] Find the values of t , where 0 < t < 2π , for which 4 cos(

2t− π4)

has its maximumvalue. 4


4 C NC T7 1989 P1 Q15

19. (a)[SQA] Find the equation of AB, theperpendicular bisector of the linejoing the points P(−3, 1) andQ(1, 9) . 4

(b) C is the centre of a circle passingthrough P and Q. Given that QC isparallel to the y-axis, determine theequation of the circle. 3

(c) The tangents at P and Q intersect atT.

Write down

(i) the equation of the tangent at Q

(ii) the coordinates of T. 2

Ox

y

A

B

C

Q(1, 9)

P(−3, 1)


(a) 4 C CN G7 x+ 2y = 9 2000 P2 Q2

(b) 3 C CN G10 (x− 1)2 + (y− 4)2 = 25(c) 2 C CN G11, G8 (i) y = 9, (ii) T(−9, 9)

•1 ss: know to use midpoint•2 pd: process gradient of PQ•3 ss: knowhow to find perp. gradient•4 ic: state equ. of line

•5 ic: interpret “parallel to y-axis”•6 pd: process radius•7 ic: state equ. of circle

•8 ic: interpret diagram•9 ss: know to use equ. of AB

•1 midpoint = (−1, 5)•2 mPQ = 9−11−(−1)•3 m⊥ = − 12•4 y− 5 = − 12(x− (−1))

•5 yC = 4 stated or implied by •7•6 radius = 5 or equiv.stated or implied by •7

•7 (x− 1)2 + (y− 4)2 = 25

•8 y = 9•9 T= (−9, 9)



Higher Mathematics

20.[SQA]


(a) 2 C CR G16 1998 P2 Q1

(b) 5 C CR G28

(c) 2 C CR CGD



Higher Mathematics

21.[SQA]

(a) By writing sin 3x as sin(2x+ x) , show that sin 3x = 3 sin x− 4 sin3 x . 4

(b) Hence find∫

sin3 x dx . 4


(a) 2 C NC T8, T8 1995 P2 Q9

(a) 2 A/B NC T8, T8

(b) 4 A/B NC C23

22.[SQA] Given f (x) = cos2 x− sin2 x , find f ′(x) . 3


1 C NC C21 1999 P1 Q19

2 A/B NC C21, C20



Higher Mathematics

23.[SQA] The results of an experiment give rise to the graph shown.

(a) Write down the equation of the line interms of P and Q . 2

O

P

Q

1·8

−3

It is given that P = loge p and Q = loge q .

(b) Show that p and q satisfy a relationship of the form p = aqb , stating thevalues of a and b . 4


(a) 2 A/B CR G3 P = 0·6Q+ 1·8 2000 P2 Q11(b) 4 A/B CR A33 a = 6·05, b = 0·6

•1 ic: interpret gradient•2 ic: state equ. of line

•3 ic: interpret straight line•4 ss: know how to deal with x ofx log y

•5 ss: know how to express number aslog

•6 ic: interpret sum of two logs

•1 m = 1·83 = 0·6•2 P = 0·6Q+ 1·8

Method 1

•3 loge p = 0·6 loge q+ 1·8•4 loge q0·6•5 loge 6·05•6 p = 6·05q0·6

Method 2ln p = ln aqb

•3 ln p = ln a+ b ln q•4 ln p = 0·6 ln q+ 1·8 stated or impliedby •5 or •6

•5 ln a = 1·8•6 a = 6·05, b = 0·6



Higher Mathematics

24.[SQA] The size of the human population, N , can be modelled using the equationN = N0e

rt where N0 is the population in 2006, t is the time in years since 2006,and r is the annual rate of increase in the population.

(a) In 2006 the population of the United Kingdomwas approximately 61 million,with an annual rate of increase of 1·6%. Assuming this growth rate remainsconstant, what would be the population in 2020? 2

(b) In 2006 the population of Scotland was approximately 5·1 million, with anannual rate of increase of 0·43%.Assuming this growth rate remains constant, how long would it take forScotland’s population to double in size? 3


(a) 2 B CR A30, A34 76 million 2009 P2 Q6

(b) 3 A CR A30, A34 t = 161·2 years

•1 ic: substitute into equation•2 pd: evaluate exponential expression

•3 ic: interpret info and substitute•4 ss: convert expo. equ. to log. equ.•5 pd: process

•1 61e0·016×14•2 76 million

•3 10·2 = 5·1e0·0043t•4 0·0043t = ln 2•5 t = 161·2 years

[END OF QUESTIONS]



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