mathematics - specialised · mathematics – specialised page 3 candidate instructions you must...

Pages: 16 Questions: 30 ©Copyright for part(s) of this examination may be held by individuals and/or organisations other than the Tasmanian Qualifications Authority.

Tasmanian Certificate of Education

MATHEMATICS - SPECIALISED

Senior Secondary

Subject Code: MTS315109

External Assessment

2009

Time: Three Hours

On the basis of your performance in this examination, the examiners will provide results on the following criteria taken from the syllabus statement: Criterion 3 Demonstrate an understanding of finite and infinite sequences and

series. Criterion 4 Demonstrate an understanding of matrices and linear transformations. Criterion 5 Use differential calculus and apply integral calculus to areas and

volumes. Criterion 6 Use techniques of integration and solve differential equations. Criterion 7 Demonstrate an understanding of complex numbers. T

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Mathematics – Specialised

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CANDIDATE INSTRUCTIONS You MUST ensure that you have addressed ALL of the externally assessed criteria on this examination paper. This examination paper has FIVE sections. You must answer ALL questions. It is suggested that you spend approximately 36 minutes on each section. The 2009 Mathematics – Specialised Formula Sheet can be used throughout the examination. No other printed material is allowed into the examination. The presentation of answers and the statement of arguments leading to answers will be considered when determining your result on each criterion. You are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching and an approved scientific and/or graphics calculator (memory may be retained). Answer each section in a separate answer booklet.


SECTION A – SEQUENCES AND SERIES

This section assesses Criterion 3. Use a separate answer booklet for this section. Question 1 State, without proof, the convergence or divergence of each of the following sequences (if a sequence is convergent state its limit):

(i)

€

3n −12n +1

;

(ii)

€

−1( )3n−1

2n +1

;

(iii)

€

3n2 −12n +1

.

(4 marks) Question 2

Given the sequence

€

23× 5

, 45 × 7

, 87 × 9

, ... (i) state the next term in the sequence; (1 mark)

(ii) state the nth term of the sequence. (3 marks)

Question 3

Simplify

€

2r +1

−2

r + 2

r=4

n+1∑ . (4 marks)

Question 4 Using series summation techniques find an expression, in terms of n, for:

€

4 + 4 + 2 + 8 + 9 + 6 +12 +16 +18 + ... to 3n terms. (The answer does not need to be factorised.) (6 marks)

Section A continues opposite.


Section A (continued) Question 5 (i) Prove, by mathematical induction, that

€

1+ 2 12

+ 3

12

2

+ 4 12

3

+ ... to n terms = 4 − n + 22n−1

. (6 marks)

(ii) Hence determine the sum to infinity of

€

1+ 2 12

+ 3

12

2

+ 4 12

3

+ ... (1 mark)

Question 6

Give a formal proof of the convergence of the sequence

€

4 − nn × 4n−1

. (7 marks)


SECTION B – MATRICES & LINEAR TRANSFORMATIONS

This section assesses Criterion 4. Use a separate answer booklet for this section. Question 7

Find the point whose image is (1, 1) under the linear transformation

€

L : x, y( )→ 2x − 3y, x + y( ). (3 marks) Question 8

Given the matrix,

€

A =c 20 1

.

(i) Find all values of c such that the matrix A is non-singular. (1 mark)

(ii) Show that det

€

A−1( ) =

€

det A( )−1 for the values of c in part (i). (3 marks)

Question 9

The region enclosed between the two curves

€

y = 2x −1 and

€

y = x2 on

€

0, 1[ ] undergoes the transformation

€

L : x, y( )→ x − y, 2x + 2y( ) . Find the area of the transformed region correct to 2 decimal places. (5 marks) Question 10 (i) A curve in the co-ordinate plane undergoes the following series of transformations: • a shear parallel to the x axis by a factor of 2, followed by • a dilation by a factor of 3 parallel to the y axis, followed by • a reflection in the line

€

y = −x . Find the composite transformation. (5 marks) (ii) Find the image of

€

y2 = x under this composite transformation. (3 marks)

Section B continues opposite.


Section B (continued) Question 11 Find the non-singular 2x2 matrix A such that

€

BAD+ DAD = BD2 where

€

B =1 23 4

and

€

D =2 34 8

. (6 marks)

Question 12

Given

€

A =a bc d

which is such that the sum of the elements of each row equals 5.

If

€

A*

=b ad c

then prove that the sum of the elements in each row of the matrix

€

AA* equals 25. (6 marks)


SECTION C – DIFFERENTIAL CALCULUS, AREAS AND

VOLUMES

This section assesses Criterion 5. Use a separate answer booklet for this section. Question 13 Use the Trapezoidal Rule to estimate the area under the graph of

€

y = x2 on [1, 2], taking four intervals. (4 marks) Question 14

Find

€

dydx

in each of the following cases:

(i) y = artan 3x, (2 marks) (ii) y = 4arcos x. (2 marks) Question 15 Write down the nature of the point

€

c, f c( )( ) for each of the following: (i)

€

f '(c) = 0 and

€

f "(c) < 0, (1 mark) (ii)

€

f '(c) = 0 and

€

f "(c) = 0, (1 mark) (iii)

€

f '(c −ε) < 0, f '(c) < 0, f '(c + ε) < 0 and

€

f "(c) = 0 where

€

ε is a very small positive number. (2 marks) Question 16

If

€

xy2 = log10 y , find

€

dydx. (6 marks)

Section C continues opposite.


Section C (continued) Question 17 (i) Given

€

y = ln(sin x) show that

€

y '= cot x. (1 mark)

(ii) Hence find the exact area between the x axis and the curve

€

y = −cot x on

€

π6, 2π3

. (6 marks)

Question 18 Using first and second derivatives, find and classify any stationary points and points of inflection of the function

€

y = ln x − x . Exact values are to be given. (7 marks)


SECTION D – INTEGRAL CALCULUS

This section assesses Criterion 6. Use a separate answer booklet for this section. Question 19 Integrate each of the following functions with respect to x: (i)

€

2sin2x cos2x , (3 marks) (ii)

€

tan x sec2 x . (3 marks) Question 20

Solve the differential equation:

€

dydx

= x2y. (3 marks)

Question 21 Evaluate the indefinite integral

€

2xe3xdx∫ . (3 marks) Question 22

Solve

€

x dydx

= y +x2

1+ x2 for y. (5 marks)

Question 23

By using partial fractions, integrate

€

4x − 4x3 + x2 + x +1

with respect to x. (7 marks)

Section D continues opposite.


Section D (continued) Question 24 Tim, the scientist, ordered a bowl of pumpkin soup from the university restaurant. He burnt his lips on the first mouthful. As he had his thermometer with him he found the temperature of the soup to be 60˚C (far too hot to consume) and the air temperature in the restaurant to be 21˚C. Two minutes later he found the temperature of the soup to be 50˚C. He wished to know how much longer he needed to wait for the temperature of the soup to cool to 37˚C, at which time he believed it would be ready to be consumed. He was aware of Newton’s Law of Cooling which states that ‘the rate at which a body cools is proportional to the excess of its temperature above that of its surroundings’. (i) Write down a differential equation linking the temperature T of the bowl of soup,

with the time elapsed t since he took the first temperature reading. (2 marks) (ii) Solve the differential equation for T and determine, to the nearest minute, how much longer he needed to wait for the temperature of the soup to reach 37˚C. (6 marks)


SECTION E – COMPLEX NUMBERS

This section assesses Criterion 7. Use a separate answer booklet for this section. Question 25

If

€

Z = cosπ7

+ isin π7

and

€

W = cosπ9

+ isin π9

then simplify

€

Z2W expressing

the result in polar co-ordinate form. (4 marks) Question 26

Solve, using algebra, the equation

€

1− 2Z( ) 2Z

+ i

Z2 + Z +1( ) = 0 for Z.

(Give answers in the form

€

x + iy where x and y are real numbers). (5 marks) Question 27

Prove that

€

Z2 Z2

Z( )4 =1 where

€

Z = r cosθ + isinθ( ) and

€

Z ≠ 0 . (4 marks)

Question 28

€

1+ 2i is a zero of a polynomial in Z with real coefficients. Determine a real quadratic factor of this polynomial. (4 marks) Question 29 (i) Solve

€

W 3 = 8 , where W is a complex number, writing the solutions in exact Cartesian form. (3 marks)

(ii) Hence solve

€

Z + 3i( )3 − 8Z3 = 0 for Z. (4 marks)

Section E continues opposite.


Section E (continued) Question 30 (i) Sketch the region in the Argand plane representing complex numbers Z such that

€

Z − 2i ≤ 2( ) Im(Z + 2 − 6i) >24

Re Z − 4 + 8i( )

. (5 marks)

(ii) Justify whether, or not, the representation of the complex number

€

− 3 + 3i is on the boundary of this region. (3 marks)


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