Revision Materials Applications Unitmaths.lanark.s-lanark.sch.uk/wp-content/uploads/2017/12/AH-Revision...Advanced Higher Mathematics – Applications of Algebra and Calculus Unit

MATHEMATICS

Advanced Higher

Revision Materials

Revision Materials

Applications Unit

Applications of Algebra and Calculus

NATIONAL

QUALIFICATIONS

http://www.google.co.uk/url?sa=i&rct=j&q=south+lanarkshire+council+logo&source=images&cd=&cad=rja&uact=8&docid=CPztyDuwoEJ_xM&tbnid=XTnaVdkSQHQfCM:&ved=0CAUQjRw&url=https://www.civica.co.uk/articles/217-South-Lanarkshire-Council-drives-efficiencies-with-case-management-upgrade-from-Civica&ei=9bngU7y9M8Ky0QWqm4HACw&bvm=bv.72197243,d.d2k&psig=AFQjCNFXmTQpeBma7hiudBmpA0AOeOIpfQ&ust=1407322984600190

Advanced Higher Mathematics – Applications of Algebra and Calculus Unit Assessment Preparation - Further Practice Questions

Page 2

Summations

(Arithmetic series) nS n a n d 1

2 12

(Geometric series) n

n

a rS

r

1

1

( ) ( )

, ,n n n

r r r

n n nn n n nr r r

2 22 3

1 1 1

1 2 11 1

2 6 4

Binomial theorem

n

n n r r

r

na b a b

r

0

where !

!( )!

n

r

n nC

r r n r

Maclaurin expansion ( ) ( ) ( )

( ) ( ) ( ) ...! ! !

ivf x f x f xf x f f x

2 3 40 0 00 0

2 3 4

De Moivre’s theorem

(cos sin ) cos sinp pr i r p i p

Vector product

ˆsina a a a a a

a a ab b b b b b

b b b

i j k

a×b a b n i j k2 3 1 3 1 2

1 2 3

2 3 1 3 1 2

1 2 3

Matrix Transformation

Anti-clockwise rotation through an angle, about the origin,O cos sin

sin cos

Standard derivatives Standard integrals

f x f x f x f x dx

sin x1

x 2

1

1

sec ax2 tan( )ax c

a

1

cos x1

x

2

1

1

a x2 2

1 sin

xc

a

1

tan x1 x 2

1

1

a x2 2

1 11

tanx

ca a

tan x sec x2 1

x

ln x c

cot x 2cosec x axe axe ca

1

sec x sec tanx x

cosec x -cosec cotx x

ln x x

1

xe xe


Page 3

1.1 Applying Algebraic skills to the binomial theorem

Expanding expressions using the binomial theorem

1. Expand the following using the Binomial Theorem:

(a) (2a+1)5 (b) (4 + 3m)6 (c) (5k -1)4 (d) (2 + h)7 (e) (4x -1)5

1.1 Applying Algebraic skills to complex numbers

Performing algebraic operations on complex numbers

2. Express the following in the form a+ ib

(a) c+ d , c- d , cd and c

d, where c = k + i and d = 2 - 3i

(b) 2a+ 3b, 3a-b , a

b and ab , where a = 2m+ i and b = 4 - 3i

(c) x + 4y , 2x - y, xy and y

x, where x = 3- 2i and y = w+ 2i

(d) 3v1 - v2 + 2v3, v3 - 3v2

, v1

v3

and v1v2, where v1 = 4 - 3i, v2 = h+ i and v3 = 3- i

(e) 4w1 + 3w2 -w3, w1 - 3w3

, w2

w3

and w1w3 , where w1 = 3g- 4i, w2 =1+ i and w3 = 2 - i

1.2 Applying Algebraic skills to sequences and series

Finding the general term and summing arithmetic sequences

3. An arithmetic sequence is given by: 6, 27, 48….. Find (a) the 40th term of the sequence (b) the sum of the first 40 terms

4. An arithmetic sequence is given by: 3, 22, 41….. Find

(a) the 30th term of the sequence (b) the sum of the first 30 terms

5. An arithmetic sequence is given by: 5, 22, 39….. Find (a) the 50th term of the sequence (b) the sum of the first 50 terms

6. An arithmetic sequence is given by: 2, 29, 56….. Find



Finding the general term and summing geometric sequences

7. A geometric sequence is given by: 6, 18, 54….. Find (a) the 15th term of the sequence (b) the sum of the first 15 terms

8. A geometric sequence is given by: 3, 6, 12….. Find

(a) the 20th term of the sequence (b) the sum of the first 20 terms 9. A geometric sequence is given by: 1, 7, 49….. Find

(a) the 11th term of the sequence (b) the sum of the first 11 terms 10. A geometric sequence is given by: 5, 20, 100….. Find



Page 4


Using the Maclaurin series expansion to find a stated number of terms of the power series for a simple function

11. Find the first four terms of the Maclaurin series for:

(a) f (x) = e2x (b) f (x) = sin(3x) (c) f (x) = ln(1+ 3x) (d) f (x) = cos(2x)

(e) f (x) = 6x6 + 5x5 - 4x4 + 3x3 - 2x2 - x+ 7 (f) f (x) = 7x6 - 6x5 + 5x4 - 4x3 + 3x2 - 2x+1

1.3 Applying algebraic skills to summation

Know and use sums of certain series and other straightforward results

12. Evaluate

(a) 5nn=1

40

å (b) 4bb=1

60

å (c) 5k + 3k=1

20

å (d) 2m - 3m=1

19

å

(e) 4r2

r=1

50

å (f) 2h2

h=1

40

å (g) p3

p=1

80

å (h) 4 f 3

f =1

50

å

1.3 Applying summation formulae

use mathematical induction to prove summation formulae

13. Use proof by induction to show that, for all n ³1, nÎN

(a) rr=1

n

å =n(n +1)

2 (b) 6r

r=1

n

å = 3n(n +1) (c) 8rr=1

n

å = 4n(n+1)

(d) 2r -1= n2

r=1

n

å

(e) (6r -1)r=1

n

å = n(3n + 2) (f) 8r - 5 = n(4n -1)r=1

n

å

1.4 Applying algebraic and calculus skills to properties of functions

Find the asymptotes of a rational function

14 f (x) =x2 - x - 5

x - 3 , xÎR :x ¹ 3

For the graph y = f (x)

(a) Give the equation of the vertical asymptote.

(b) Show that there is a non-vertical asymptote and state the equation.


Page 5

15 f (x) =x2 + x -17

x - 4 , xÎR : x ¹ 4


(a) Give the equation of the vertical asymptote.

(b) Determine the equation of the non-vertical asymptote.

16 f (x) =x2 - 6x + 9

x - 5 , xÎR : x ¹ 5


(a) Write down the equation of the vertical asymptote.


17 f (x) =x2 - 2x -19

x - 6 , xÎR :x ¹ 6




18 f (x) =x2 - 5x -13

x - 7 , xÎR : x ¹ 7




1.4 Applying algebraic and calculus skills to properties of functions

Sketch related functions (Modulus or Inverse Functions, Functions Differentiated, Translations and Reflections)

19 Given that f (x) = sin(2x) , sketch the graph of 3 f (x) where 0 £ x ³ p

20 Given that f (x) = sin(4x) , sketch the graph of 5 f (x) where 0 £ x ³ p

21 Given that , sketch the graph of 2 f (x)-1 where 0 £ x ³ p

22 Given that f (x) = cos(2x) , sketch the graph of 3 f (x)+ 2 where 0 £ x ³ p

23 Let f (x) be a function with a maximum turning point at (a,14) and a minimum turning

point at (b,-3), 0 < a < b . The graph crosses the y-axis at (0,-1).

(a) Sketch the graph f (x) on the interval 0 £ x ³ b

(b) Hence sketch the graph of g(x) = f (x)- 3 on the interval 0 £ x ³ b


Page 6

24 Let f (x) be a function with a minimum turning point at (p,-8) and a maximum turning

point at (q,5), p < q < 0. The graph crosses the y-axis at (0,0).

(a) Sketch the graph f (x) on the interval p £ x ³ 0

(b) Hence sketch the graph of h(x) = f (x)+ 2 on the interval p £ x ³ 0

25 A particle moves in a straight line such that its displacement, s metres, from a fixed point O

on the line after t seconds, is given by

s = t 3 + t , t ³ 0

Find the velocity of the particle after 3 seconds.

26 The equation of rectilinear motion of a particle after time, t seconds, is given by

d =10t - et where d is the displacement of the particle after t seconds.

Find the velocity of this particle after 2 seconds.

27 A particle moves a distance s metres in t seconds.

The distance travelled by the particle is given by

s = 2t 3 -23

2t 2 + 3t + 5

Find the acceleration of the particle after 4 seconds.

28 A particle moves along a straight line such that its displacement, s metres,

from a fixed point O on the line after t seconds is given by

s = 2t 3 - 21t2 + 60t t ³ 0

Find the initial acceleration of the particle.

29 The turning effect, T , of a power boat, is given by the formula

T = 8cosxsin2 x , 0 < x <p

2

where x is the angle (in radians) between the rubber and the central line of the boat.

Find the size of x which maximizes the turning effect.

30 The net of a closed box is to be cut from a square piece of card of length 30 units.

The volume of the box formed, is given by

V = x(30 - 2x)2 where x is the length of the side of the squares to be cut from each corner.

Determine the size of the square of side x to be cut out to ensure maximum volume.


Page 7

Answers:

1 (a) 32a5 + 80a4 + 80a3 + 40a2 +10a+1

(b) 4096 +18432m+ 34560m2 + 34560m3 +19440m4 + 5832m5 + 729m6

(c) 625k4 - 500k3 +150k2 - 20k +1

(d) 128 + 448h+ 672h2 + 560h3 + 280h4 + 84h5 +14h6 +h7

(e) 1024x5 -1280x4 + 640x3 -160x2 + 20x-1

2 (a) c+ d = 2+ k - 2i , c- d = k - 2+ 4i, cd = 2k + 3+ (2 - 3k)i, c

d=

2k - 3

13+

(3k + 2)

13i

(b) 2a+ 3b = 4m+12 - 7i , 3a-b = 6m- 4 + 6i, ab = 8m+ 3+ 2(2 - 3m)i ,

a

b=

4m - 3

25+

2(3m + 2)

25i

(c) x+ 4y = 3+ 4w+ 6i , 2x- y = 6 -w- 6i, xy = 3w+ 4 + 2(3-w)i, y

x=

3w- 4

13+

2(w+ 3)

13i

(d) 3v1 - v2 + 2v3 =18 - h-12i , v3 - 3v2 = 3- 3h- 4i, v1

v3

=3

2-

1

2i , v1v2 = 4h+ 3+ (4 - 3h)i

(e) 4w1 + 3w2 -w3 =12g+1-12i ,w1 - 3w3 = 3g- 6 - i , w2

w3

=1

5+

3

5i , w1w3 = 6h- 4 - (3h+ 8)i

3 (a) U40 = 825 (b) S40 =16620 4 (a) U30 = 554 (b) S30 = 8355

5 (a) U50 = 838 (b) S50 = 21075 6 (a) U40 =1055 (b) S40 = 63420

7 (a) U15 = 28697814 (b) S15 = 43046718 8 (a) U20 =1572864 (b) S20 = 3145725

9 (a) U11 = 282475249 (b) S11 = 329554457 10 (a) U13 = 83886080 (b) S13 =111848105

11 (a) f (x) = 1- 2x2 +2x4

3-

4x6

45 (b) f (x) = 3x -

9x3

2+

81x5

40-

729x7

1680

(c) f (x) = x -3x2

2+ 3x3 -

27x4

4 (d) f (x) = 1- 2x2 +

2x4

3-

4x6

45

(e) f (x) = 7 - x - 2x2 + 3x3 (f) f (x) =1- 2x+ 3x2 - 4x3

12 (a) 4100 (b) 7320 (c) 1110 (d) 323 (e) 171 700 (f) 44 280 (g) 10 497 600 (h) 6 502 500

13 (a) Assume true for r =n(n +1)

2r=1

n

å and Consider n = k +1

LHS = r = r + (k +1)r=1

k

år=1

k+1

å =k(k +1)

2+ (k +1) =

k(k +1)+ 2(k +1)

2=

(k +1)(k + 2)

2

RHS = k(k +1)

2Þ

(k +1)(k +1+1)

2=

(k +1)(k + 2)

2

LHS = RHS, so true for n = k +1

Using r =n(n +1)

2r=1

n

å , Check for n =1, LHS = 1=1 , RHS = 1(1+1)

2=

2

2= 1

LHS = RHS, so true statement for n =1. Hence, if true for n = k implies true for

n = k +1, and since true for n =1 then by induction, statement is true for all nÎN .


Page 8

Answers:

13 (b) Assume true for 6r = 3k(k +1)r=1

k


LHS = 6r = 6r + 6 ´ (k +1)r=1

k

år=1

k+1

å = 3k(k +1)+ 6(k +1) = (k +1)(3k + 6) = 3(k +1)(k + 2)

RHS = 3k(k +1)Þ 3(k +1)(k +1+1) = 3(k +1)(k + 2), LHS = RHS, so true for n = k +1

Using 6r = 3n(n +1)r=1

n

å , Check for n =1, LHS = 6 ´1= 6 , RHS = 3´1(1+1) = 6

LHS = RHS, so statement true for n =1. Hence, if true for n = k implies true for

n = k +1, and since true for n =1 then by induction, statement true for all nÎN .

13 (c) Assume true for 8r = 4k(k +1)r=1

k


LHS = 8r = 8r + 8(k +1)r=1

k

år=1

k+1

å = 4k(k +1)+ 8(k +1) = 4(k +1)(k + 2)

RHS = 4(k +1)(k +1+1)Þ 4(k +1)(k + 2), LHS = RHS, so true for n = k +1

Using 8r = 3n(n +1)r=1

n

å , Check for n =1, LHS = 8 ´1= 8 , RHS = 4 ´1(1+1) = 8



13 (d) Assume true for (2r -1) = k2

r=1

k


LHS = (2r -1) = (2r -1)+ 2(k +1)r=1

k

år=1

k+1

å -1= k2 + 2k +1= (k +1)2

RHS = (k +1)2 LHS = RHS, so true for n = k +1

Using (2r -1) = n2

r=1

n

å , Check for n =1, LHS = 2 ´1-1=1 , RHS = 1´1=1



13 (e) Assume true for (6r -1) = k(3k + 2)r=1

k


LHS = (6r -1) = (6r -1)+ 6(k +1)-1r=1

k

år=1

k+1

å = k(3k + 2)+ 6(k +1)-1= 3k2 + 2k + 6k + 6 -1= (3k + 5)(k +1)

RHS = (k +1)´ (3(k +1)+ 2) = (k +1)(3k + 5), LHS = RHS, so true for n = k +1

Using (6r -1) = n(3n + 2)r=1

n

å , Check for n =1, LHS = 6 ´1-1= 5 , RHS = 1(3´1+ 2) = 5




Page 9

13(f) Assume true for (8r - 5) = k(4k -1)r=1

k


LHS = (8r - 5) = (8r - 5)+ 8(k +1)- 5r=1

k

år=1

k+1

å = k(4k -1)+ 8(k +1)- 5 = 4k2 - k + 8k + 8 - 5 = (4k + 3)(k +1)

RHS = (k +1)´ (4(k +1)-1) = (k +1)(4k + 3), LHS = RHS, so true for n = k +1

Using (8r - 5) = n(4n -1)r=1

n

å , Check for n =1, LHS = 8 ´1- 5 = 3 , RHS = 1(4 ´1-1) = 3



14 (a) vertical asymptote, x = 3 (b) non-vertical asymptote, y = x + 2




18 (a) vertical asymptote, x = 5 (b) non-vertical asymptote, y = x -1

x

y

3

19 20

21 22

5

x

y

5

x

y

3

1

x

y

1


Page 10

23 (a) 23 (b)

0 > a > b 0 > a > b 24 (a) 24 (b) p < q < 0

p < q < 0

25 v = 3t2 +1, v = 28ms-1 26 v =10 - et , t = ln(10)

27a =12t - 23, a = 25ms-2 28 a =12t - 42, a = -42ms-2

29 T (x) =16cos2 xsin x- 8sin3 x, x = 0.96 30 v '(x) = (30 - 2x)2 - 4x(30 - 2x), Max when x = 5

(a, 11)

x

y

11

6

4

(b, 6)

(a, 14)

x

y

(b, -3)

-1

x

y

(p, -8)

(q, 3)

x

y

(p, 6)

(q, 5)

2

Documents

Revision Materials Applications Unitmaths.lanark.s-lanark.sch.uk/wp-content/uploads/2017/12/AH-Revision...Advanced Higher Mathematics – Applications of Algebra and Calculus Unit