NESA Number: _______________________

Teacher: Mrs Giuliani

Ms Miller

Ms Rutter

Meriden School

2019
Preliminary
Yearly Examination

Mathematics Extension 1
General Instructions Total Marks - 70
• Reading time – 5 minutes Section I (10 marks)
• Working time – 2 hours • Attempt questions 1-10
• Write using blue or black pen. • 10 multiple-choice questions worth 1 mark each
• NESA approved calculators • Allow about 15 minutes for this section
may be used
• All necessary working should Section II (60 marks)
be shown.
• Attempt all questions
• Allow about 1 hour 45 minutes for this section

Mark: /70
Section I

10 marks
Attempt Questions 1 – 10
Allow about 15 minutes for this section

Use the multiple-choice answer sheet for Questions 1 – 10.

Select the alternative A, B, C or D that best answers the question. Fill in the response oval
completely.

Sample: 2 + 4= (A) 2 (B) 6 (C) 8 (D) 9

(A) (B) (C) (D)

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new
answer.

(A) (B) (C) (D)

If you change your mind and have crossed out what you consider to be the correct answer, then
indicate the correct answer by writing the word correct and drawing an arrow as follows.
correct

(A) (B) (C) (D)

Marks

1. What is the Cartesian form of the parametric equations x = 2t + 1 and y = 8t 2 − 1? 1

(A) y = 4 x2 − 8x + 3

(B) y = 2 x2 − 4 x + 1

(C) y = 8 x 2 − 16 x + 8

(D) y = 16 x + 7

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2. Let  ,  and  be the roots of x3 + px 2 + q = 0. 1

1 1 1
What is + + in terms of p and q?
  
(A) pq

(B) − pq

p
(C) −
q

p
(D)
q

The displacement, in centimetres, of a particle is given by x = ( 2t − 3) , where t

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3. 1
is time in seconds. What is the velocity of the particle when t = 2?
(A) −1 cm/s

(B) 1 cm/s

(C) 5 cm/s

(D) 10 cm/s

, what are the equations of the asymptotes of f −1 ( x ) ?

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4. If f ( x ) = 1 + 1
x−2
(A) Vertical asymptote is x = 1 and horizontal asymptote is y = 2.

(B) Vertical asymptote is x = 3 and horizontal asymptote is y = 2.

(C) Vertical asymptote is x = 2 and horizontal asymptote is y = 1.

(D) Vertical asymptote is x = 2 and horizontal asymptote is y = 3.

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5. It is known that ( x + 2 ) is a factor of the polynomial P ( x ) and that 1
P ( x ) = ( x 2 + x + 1)  Q ( x ) + ( 2 x + 3) for some polynomial Q ( x ) . Which of the
following statements is correct?
1
(A) Q ( −2 ) = −
3

(B) Q ( −2 ) = −1

1
(C) Q ( −2 ) =
3

(D) Q ( −2 ) = 1


6. If t = tan , which of the following expressions is equivalent to 1
2
4sin  + 3cos  + 5?

2 (t + 2)
2

(A)
1− t2

(t + 4)
2

(B)
1− t2

2 (t + 2)
2

(C)
1+ t2

(t + 4)
2

(D)
1+ t2

 
7. Which of the following is a correct expression for tan  x +  ? 1
 4
cos x + sin x
(A)
cos x − sin x
cos x + 2sin x
(B)
cos x − sin x
cos x + sin x
(C)
cos 2 x − sin x
cos x − sin x
(D)
cos x − sin x

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8. Graph One: Graph Two: 1

Graph One drawn above shows y = f ( x ) . Graph Two shows a related graph.
Which one of the following equations would give Graph Two?

(A) y= f (x)

(B) y2 = f ( x)

(C) y= f ( x)

(D) y = f ( x)

9. Which of the following statements is true? 1

If f ( x ) = sin x for 0  x   then f ( x ) exists.

−1
(A)

If f ( x ) = x for all real x then f ( x ) exists.

2 −1
(B)

If f ( x ) = mx for all real x then f ( x ) exists for any real value of m.

−1
(C)

(D) f −1 ( x ) does not exist for any of the above.

Which graph below represents y = sin ( cos x ) ?

−1
10. 1

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 (A)

(B)

(C)

(D)

End of Section I
Section II – 60 Marks

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Attempt Questions 11 - 28
Question 11 (4 marks) Marks

(i) In the space provided, sketch the graph of y = 2 x − 6 . 2

(ii) Using your graph, solve the inequality 2 x − 6  x. 2

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Question 12 (3 marks)

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 x +1
Solve  2. 3
x−2

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Question 13 (2 marks)

4 3
If cos  = and    2 , find the exact value of sin 2 . 2
5 2

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Question 14 (2 marks)

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 x3 + 2 x 2 − 3x + 1
Use polynomial division to express in the form 2
x+2
P ( x ) = A( x )  Q ( x ) + R ( x ).

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Question 15 (3 marks)

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Consider the function y = 2sin −1 ( 3 x ) .

(i) State its domain and range using interval notation. 2

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(ii) Hence, sketch y = 2sin −1 ( 3x ) in the space provided. 1

Question 16 (4 marks)

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Solve cos ( 2 x ) = sin ( 2 x ) − 1 for x in the domain [0, 2 ]. 4

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Question 17 (3 marks)

3 8
If A and B are both reflex angles, and given cos A = and tan B = , find the exact value 3
5 6
of sin ( A − B ) .

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Question 18 (2 marks)

Find the Cartesian equation of the curve with parametric equations x = cos t and 2
y = 4 + sin t in the domain [ , 2 ].

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Question 19 (3 marks)

The quartic polynomial P ( x ) = x 4 + 11x3 + 42 x 2 + 68 x + 40 is known to have a zero of 3

multiplicity 3. Factorise P ( x ) completely.

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Question 20 (5 marks)

1− t2
(i) Prove that cos 2 = , where t = tan  . 2
1+ t2

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Question 20 continues on page 15

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 
(ii) Hence, find tan . 3
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Question 21 (4 marks)

(i) If sin (17 x ) = sin (12 x + 5 x ) , give another expression for sin (17 x ) . 1

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(ii) If sin ( 7 x ) = sin (12 x − 5 x ) , give another expression for sin ( 7 x ) . 1

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(iii) Hence, express sin 5 x cos12 x as a difference of two terms. 2

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Question 22 (2 marks)

P ( x ) is an odd polynomial of degree 3. It has ( x + 4 ) as a factor, and when it is divided by 2

( x − 3) the remainder is 7. Find P ( x ) .

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Question 23 (5 marks)

2
The diagram shows the graph of f ( x ) = − + 1.
x +1

In the space provided, sketch the following graphs. Show all key features.
1 2
(i) y = .
f ( x)

Question 26 continues on page 19

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(ii) y = f ( x ) . 1

(iii) y 2 = f ( x ) . 2

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Question 24 (3 marks)

If one root of x3 + px 2 + qx + r = 0 equals the sum of the two other roots, show that 3
p + 8r = 4 pq.
3

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Question 25 (3 marks)

A spherical balloon is being inflated and its radius is increasing at a constant rate of 3
3cm/min. At what rate is its volume increasing when the radius of the balloon is 5 cm?
Leave your answer in exact form.

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Question 26 (6 marks)

A full rain water tank is drained so that at time t minutes, the volume V of water in litres is
given by:
2
 t 
V = 400 1 −  for 0  t  50.
 50 

(i) How much water is initially in the tank? 1

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(ii) After how many minutes is the tank half full? Round your answer to the 3
nearest minute.

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Question 26 continues on page 23

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 (iii) At what rate is the water draining when t = 40? 2

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Question 27 (3 marks)

 3  3 3
If m = sin −1  −  − cos −1  −  , find an exact expression for cos −1   in terms of m. 3
 5  5 5
Simplify your answer fully.

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Question 28 (3 marks)

B  A    
Show that 2sin( A) cos 2   + 2 cos 2   sin( B) = sin( A + B) + cos  A −  + cos  B −  . 3
2 2  2  2
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End of Paper

25
Section II extra writing space
If you use this space, clearly indicate which question you are answering.

26
Section II extra writing space
If you use this space, clearly indicate which question you are answering.

27
Section II extra writing space
If you use this space, clearly indicate which question you are answering.

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 Student Number:_______________________

MULTIPLE-CHOICE ANSWER SHEET

Meriden School
Strathfield

2019
MATHEMATICS EXTENSION 1

YEAR 11 ASSESSMENT TASK 3

1. A B C D

2. A B C D

3. A B C D

4. A B C D

5. A B C D

6. A B C D

7. A B C D

8. A B C D

9. A B C D

10. A B C D /10
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