Advanced Higher Curriculum Applications in Algebra and Calculus Assessment Standard: 1.1 Applying algebraic skills to the binomial theorem and to complex numbers. Binomial Coefficient formula Ex 1 3 questions Expanding function of the form (x + y)n Ex 2 1 question Expanding function of the form (x + y)n using binomial expansion theory Ex 3 4 question Performing Algebraic operation on complex GPS Document Numbers.

1.1 Binomial Expansion Exercise 1 “Binomial Coefficient Formula”

1. Evaluate the following: (a) 5! (b) 3! (c) 7! (d) 0! (e) 3!

5! (f) 2!5!

7! (g) 3!

1!2! (h) 2!

3!4!÷ 3!2!4!

2. Evaluate the following coefficients:

(a) 52

⎛⎝⎜

⎞⎠⎟

(b) 72

⎛⎝⎜

⎞⎠⎟

(c) 51

⎛⎝⎜

⎞⎠⎟

(d) 40

⎛⎝⎜

⎞⎠⎟

(e) 52

⎛⎝⎜

⎞⎠⎟+ 5

3⎛⎝⎜

⎞⎠⎟

(f) 83

⎛⎝⎜

⎞⎠⎟− 4

2⎛⎝⎜

⎞⎠⎟ (g) 3 9

2⎛⎝⎜

⎞⎠⎟

3. Solve the following equations:

(a) n2

⎛⎝⎜

⎞⎠⎟= 10 (b) n

2⎛⎝⎜

⎞⎠⎟= 21 (c) n

2⎛⎝⎜

⎞⎠⎟= 36

Exercise 2 “Expanding function of the form (x + y)n ”

1. Write down expressions for the expanded form of the following functions:

(a) x +1( )5 (b) 1− x( ) 4 (c) a + 2b( ) 3 (d) 2a − b( )5 (e) 2x − 3y( )5 (f) x + 1

x⎛⎝⎜

⎞⎠⎟5 (g) a − 2b( )6 (h) 2a + b( ) 7

Exercise 3 “Expanding function of the form (x + y)n using binomial expansion theory”

1. Find the following using the Binomial Expansion:-

(a) 3+ x( ) 3 (b) 5 + 2x( ) 3 (c) 3+ x( ) 4 (d) 2 − x( ) 3 (e) x + 2y( ) 3 (f) 2x − 3y( ) 3 (g) 2

x+ 3x2⎛

⎝⎜⎞⎠⎟4 (h) 2 − 3x2( )5 (i) 2x + 3

x⎛⎝⎜

⎞⎠⎟5

   

2. Expand 2 + x( )5 and use your expansion to find (a) 2.15 (b) 1.95

3. Expand 2 + x( ) 7 in ascending powers of x up to and including the term in x3 . Hence evaluate 2.17 correct to 6 significant figures.

4. Use the Binomial Theorem to evaluate the following to the stated

degree of accuracy.

(a) 1.014 correct to 5 decimal places. (b) 0.9885 correct to 7 significant figures. (c) 0.9910 correct to 10 decimal places. (d) 1.9910 correct to 4 significant figures.

   

“Performing Algebraic operation on complex numbers” All the aspects of Complex Numbers assessable are cover in the GPS document

Assessment Standard: 1.2 Applying algebraic skills to sequences and series. Arithmetic Series: General terms and Sum to. Ex 1 questions Geometric Series: General terms and Sum to. Ex 2 questions Using Maclaurin series expansions for functions Ex 3 questions

1.2 Algebra Skills in sequence and series

Exercise 1 “General form for a term in an Arithmetic Sequence” “Sum to nth term of an Arithmetic Sequence”

1. Find the formula for the nth term of each of the following sequences and find the requested term: (a) 3, 11, 19,………….; u19 (b) 8, 5, 2, …………….; u15 (c) 7, 6.5, 6,……………; u12

2. Find the number of terms in the following sequences: (a) 2, 4, 6,………….46 (b) 50, 47, 44, …………14 (c) 2, -9, -20,……………-130

3. State the values of a and d in each of the following series and find the requested Sn . (a) 4 + 10 + 16…………. S10 (b) 15 + 13 + 11………… S16 (c) 20 + 13 + 6………… S15

4. For each of the Arithmetic Sequences, find Sn indicated.

(a) u2 = 15 , u5 = 21 ………….S10 (b) u4 = 18 , d = −5 …………. S16 (c) u3 = 7 , u12 = 61 …………. S15

5. S10 = 120 , S20 = 840 find S30 ?

6. u15 = 7 , S9 = 18 , find a , d and u20 ?

7. How many terms of the Arithmetic series 28 + 24 + 20…does it take to give a sum of zero?

8. The sixth term of an Arithmetic sequence is twice the third term. If

the first term is 3, find d and the tenth term.

9. How many terms of the Arithmetic series 1 + 3 + 5…. to give a sum of 1521?

Exercise 2 “General form for a term in an Geometric Sequence” “Sum to nth term of an Geometric Sequence” 1. Find the common ratio for each of these Geometric Series. (a) 1, 3, 9, 27…… (b) 12, 6, 3, 1.5……. (c) 7, 0.7, 0.07….. (d) 18, 54, 162….. (e) 2.25, 1.5, 1….. (f) 1

4 , 18

, 116

……

(g) 1, -1, 1, -1…. (h) 1, -2, 4, -8….. 2. Write down the first 4 terms of these Geometric Series:

(a) un = 3

(n−1) (b) un = 3(−2)(n−1)

(c) un = 612

⎛⎝⎜

⎞⎠⎟(n−1)

3. Find the required term in the following Geometric Series

(a) 1, 2, 4….; u5 (b) 2, 6, 18….; u6 (c) 4, 12, 36…..; u6 (d) 2, 20, 200….; u5 (e) 1, -2, 4…..; u6 (f) 6, 3, 3

2 …..; u7

4. Find the formula for the nth term of the Geometric sequence:-

(a) 1, 2, 4…. (b) 3, 6, 12…. (c) 2, -6, 18….. (d) 9, 3, 1…. (e) 4, 2, 1….. (f) 1

2 , 14

, 18

, 116

….

5. Find the common ratio and the 5th term of these Geometric Sequences: (a) a = 6 , u3 = 24 (b) a = 50 , u4 = 400 (c) a = 36 , u2 = −12

6. Find the sum of each of the following Geometric Series and simplify the answer as far as possible.

(a) 1 + 2 + 4… to 8 terms. (b) 2 + 6 + 18…to 6 terms. (c) 2 - 4 + 8… to 5 terms. (d) 2 - 6 + 18… to 5 terms. (e) 1 + 1

2 + 1

4 … to 6 terms. (f) 1 + 1

3 + 1

9 … to 5 terms.

(g) 1 + x + x2 … to "n" terms. (h) 1 - y + y2 … to "n" terms.

7. Find "n" if :

(a) 3+ 32 + 33........3n = 363 (b) 2 + 22 + 23........2n = 510

 

Exercise 3 “Sum to infinity for a Geometric Sequence”  Find the common ratio and state if it has a limit for the sum to “n” terms: (a) 1 + 1

3 + 1

9…… (b) 1 + 2 + 4……

(c) 4 + 1 + 1

4 ….. (d) 8 + 4 + 2…..

(e) 1 - 5 + 25…. (f) 10 - 9 + 8.1… (g) 1 - 1

2 + 1

4 …. (h) 2 + 2

3 + 8

9 …

Exercise 4 “ 1

1− r as the sum to infinity for sequence 1+ r + r2..”

Express the following as an increasing power of x : 1. 1

1+ 2x 2. 1

1− 3x

3. 1

1+ x2

4. 12 + 4x

5. 1

3− x 6. 1

2 − 3x

 

.As  far        

       

           

  Exercise 5 “Maclaurin Series to represent functions”  

Expand the following functions in ascending powers of x as far as the power indicated:

1. f (x) = cos x as far as x6 2. f (x) = tan x as far as x3 3. f (x) = sin−1 x as far as x3 4. ln(1− x) as far as x4 5. f (x) = e3x as far as x4 6. ln(1+ 2x) as far as x5 7. f (x) = sin 3x as far as x5 8. f (x) = tan2x as far as x4

Exercise 6 “Maclaurin Series to represent composite functions”

1. Expand esin x as far as x4 .

2. Expand ln(1+ sin x) as far as x4 .

3. Expand ex sin x as far as x5 .  

4. Expand ln(1+ ex ) as far as x4 .

Assessment Standard: 1.3 Applying algebraic skills to summation and mathematical proof. Application of Summation Formulae Ex 1 questions Proof by Induction Ex 2 questions

   

1.3 Application of summation Formula

1r=1

n

∑ = n r = 12n(n +1)

r=1

n

∑ r2 = 16n(n +1)(2n +1)

r=1

n

∑ r3 = 14n2 (n +1)2

r=1

n

∑

Exercise 1 “Using summation notation for the sum to “n” terms”

Using the common summations above identify an implicit expression in “n” for:  

1. r +1r=1

n

∑ 2. r2 −1r=1

n

∑ 3. 13r2 + r −1

r=1

n

∑

4. r(r +1r=1

n

∑ ) 5. 4r2r=1

n

∑ + 2 6. r3 + 2r2 −1r=1

n

∑

Exercise 2 “Evaluate the following sequences”

1. r +1r=1

5

∑ 2. r2 −1r=1

7

∑ 3. 13r2 + r −1

r=1

3

∑

4. r(r +1r=1

9

∑ ) 5. 4r2r=1

7

∑ + 2 6. r3 + 2r2 −1r=1

12

∑

7. r(r +1r=3

6

∑ ) 8. 4r2r=5

7

∑ + 2 9. r3 + 2r2 −1r=9

12

∑

Exercise 3 “Using Proof by Induction – summation series”

1. Prove by induction that r2 = 16n(n +1)(2n +1)

r=1

n

∑ for all n∈N .

2. Prove by induction that r3 = 14n2 (n +1)2

r=1

n

∑ for all n∈N .

3. Prove by induction that r(r +1) = 13n(n +1)(n + 2)

r=1

n

∑ for all n∈N .

4. Prove by induction that r(r +1)(r + 2) = 14n(n +1)(n + 2)

r=1

n

∑ (n + 3) for all n∈N .

Exercise 2 “Using Proof by Induction – divisibilty” 1. Show that 4n + 6n −1 is divisible by 9 for all n ≥1 . 2. Show that n3 + 3n2 −10n is divisible by 3 for all n ≥1 . 3. Show that 7n + 4n +1n is divisible by 6 for all n ≥1 .

Exercise 3 “Using Proof by Induction – S∞ ”      

1. Prove that Sn of the series 11× 2

+ 12 × 3

+ 13× 4

+ ....= nn +1

for all n ≥1 .

2. Prove that Sn of the series 11× 3

+ 13× 5

+ 15 × 7

+ ....= n2n +1

for all n ≥1 .

3. Prove that Sn of the series 11× 2 × 3

+ 12 × 3× 4

+ 13× 4 × 5

+ ....= 14− 12(n +1)(n + 2)

for all n ≥1 .  

Assessment Standard: 1.4 Applying algebraic and calculus skills to properties of functions. Finding the vertical asymptotes of rational functions Ex 1 questions Finding the non-vertical asymptotes of rational functions Ex 2 questions Sketching the graph of a rational function including Ex 3 questions appropriate analysis of stationary points Sketch the following curves. Ex 4 questions Inverse Functions, odd and even functions, modulus of functions of functions modulus of functions Ex 5 questions

Exercise 1 “Vertical asymptotes of functions” Find all the vertical asymptotes of these functions:

1. f (x) = 4x − 2

2. f (x) = 3x −1x2 + 2x − 3

3. f (x) = 12x2 − 2x − 3

4. f (x) = x + 4x − 2

5. f (x) = x2

4 − x2 6. f (x) = x(x +1)

(x −1)(x + 2)

7. f (x) = (x − 4)(x −1)(x − 2)

8. f (x) = x2 + 3x −1

9. f (x) = x3

x2 + 3 10. f (x) = x

x2 + 4

Exercise 2 “Finding the non-vertical asymptotes of Rational functions” 1. f (x) = 4

x − 2 2. f (x) = 3x −1

x2 + 2x − 3

3. f (x) = 12x2 − 2x − 3

4. f (x) = x + 4x − 2

5. f (x) = x2

4 − x2 6. f (x) = x(x +1)

(x −1)(x + 2)

7. f (x) = (x − 4)(x −1)(x − 2)

8. f (x) = x2 + 3x −1

9. f (x) = x3

x2 + 3 10. f (x) = x

x2 + 4

Exercise 3 “Sketching Rational Functions”

1. f (x) = 4x − 2

2. f (x) = x − 2x −1

3. f (x) = x2 + x − 2x2 + x − 6

4. f (x) = x2 + 2x + 5x +1

5. f (x) = x +1x2 + 2x + 5

6. f (x) = 2x2

x2 −1

7. f (x) = x2 −10x + 9x2 +10x + 9

8. f (x) = 2x2 − 3x − 3

x2 − 3x + 2

Exercise 4 “Inverse Functions, odd and even functions, modulus of functions of functions modulus of functions”

1. Write down the equations of the inverse of the following functions:

(a) f (x) = 2x (b) f (x) = 2 − x (c) f (x) = 2x

(d) f (x) = 2x (e) f (x) = 1− 2x (f) f (x) = ln(x − 2) 2. Evaluate:

(a) sin−1( 32) (b) tan−1( 1

3) (c) tan−1(1)

(d) sin−1(12) (e) cos−1(− 3

2) (f) tan−1( 3)

3. Sketch the graph of f (x) and f (x) . (a) f (x) = x + 2 (b) f (x) = 5 − 2x (c) f (x) = x2 − 2x − 3 (d) f (x) = 3x − x2

(e) f (x) = x3 +1 (f) f (x) = 1x− 2

4. Which of the following functions are odd, even or neither? (a) f (x) = (x + 4)(x − 2) (b) f (x) = 3x2 + 5 (c) f (x) = 2x − x3 (d) f (x) = sin2x

Assessment Standard: 1.5 Applying algebraic and calculus skills to properties of functions. Applying differentiation to rectilinear motion. Ex 1 questions Applying differentiation to Optimisation. Ex 2 questions

   

   

2002 Question 3

2002 Question 4

2002 Question 5

2002 Question 6

2002 Question 9

2002 Question 15

2003 Question A

2003 Question A5

2003 Question A10

2003 Question B6

2004 Question 1

2004 Question 3

2004 Question 5

2004 Question 9

   

2004 Question 11

2004 Question 14

2005 Question 4

2005 Question 9

2005 Question 3

  2006 Question 2  

2005 Question 13

2005 Question 14

2005 Question 14

  2006 Question 6  

  2006 Question 8  

  2006 Question 11