Abbotsleigh 2016 4U Trials & Solutions PDF

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HSC TRIAL PAPERS...

Description

Student’s Name: Student Number: Teacher’s Name:

A BBO TSLEIG H

2016 HIGHER SCHOOL CERTIFICATE

Assessment 4 Trial Examination

Mathematics Extension 2 General Instructions

Total marks - 100

x Reading time – 5 minutes

x Attempt Sections I and II.

x Working time – 3 hours x Write using black pen.

Section I

Pages 3 - 6

x Board-approved calculators may be used. x A reference sheet is provided. x All necessary working should be shown in every question. x Make sure your HSC candidate Number is on the front cover of each booklet. x Start a new booklet for Each Question. x Answer the Multiple Choice questions on the answer sheet provided. x If you do not attempt a whole question, you must still hand in the Writing Booklet, with the words 'NOT ATTEMPTED' written clearly on the front cover.

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

Section II

Pages 7 - 13

90 marks x Attempt Questions 11 – 16. x Allow about 2 hr and 45 minutes for this section. x All questions are of equal value.

Outcomes to be assessed: Mathematics Extension 2 A student: HSC : E1 Appreciates the creativity, power and usefulness of Mathematics to solve a broad range of problems. E2 Chooses appropriate strategies to construct arguments and proofs in both concrete and abstract settings E3 Uses the relationship between algebraic and geometric representations of complex numbers and of conic sections E4 Uses efficient techniques for the algebraic manipulation required in dealing with questions such as those involving conic sections and polynomials E6 Combines the ideas of algebra and calculus to determine the important features of the graphs of a wide variety of functions E7 Uses the techniques of slicing and cylindrical shells to determine volumes E8 Applies further techniques of integration, including partial fractions, integration by parts and recurrence formulae, to problems E9 Communicates abstract ideas and relationships using appropriate notation and logical argument Mathematics Extension 1 A student: Preliminary course: PE1 Appreciates the role of mathematics in the solution of practical problems PE2 Uses multi-step deductive reasoning in a variety of contexts PE3 Solves problems involving inequalities, polynomials, circle geometry and parametric representations PE4 Uses the parametric representation together with differentiation to identify geometric properties of parabolas PE5 Determines derivatives that require the application of more than one rule of differentiation PE6 Makes comprehensive use of mathematical language, diagrams and notation for communicating in a wide variety of situations HSC course: HE1 Appreciates interrelationships between ideas drawn from different areas of mathematics HE2 Uses inductive reasoning in the construction of proofs HE3 Uses a variety of strategies to investigate mathematical models of situations involving projectiles HE4 Uses the relationship between functions, inverse functions and their derivatives HE5 Applies the chain rule to problems including those involving velocity and acceleration as functions of displacement HE6 Determines integrals by reduction to a standard form through a given substitution HE7 Evaluates mathematical solutions to problems and communicates them in an appropriate form

Ext2 Task 4 2016

-2-

SECTION I 10 marks Attempt Questions 1 – 10 Use the multiple-choice answer sheet Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely. 2 + 4 =

Sample

(A)

2

(A)

(B) 6

(C)

(B)

(C)

8

(D)

9

(D)

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer. (B) (C) (D) (A) If you change your mind and have crossed out what you consider to be the correct answer, then indicate this by writing the word correct and drawing an arrow as follows. correct

(B)

(A)

(C)

(D)

_______________________________________________________________________________________

1.

What is the eccentricity of the ellipse

(A)

7 16

(B)

7 4

x2 y 2  9 16

1?

(C)

9 16

S

2.

´6 3 µ sin x dx is equal in value to: ¶0 S

(A)

´6 2 µ sin x  sin x cos x dx ¶0 S

(C)

1´ 6 3 µ S sin x dx 2 ¶ 6

Ext2 Task 4 2016

S

´6 S  sin 3 x dx (B) µ 6 ¶0 S

´6 (D) µ cos3 x dx ¶0

-3-

(D)

7 9

3.

Diagram A shows the complex number z represented in the Argand plane.

Im(z)

Im(z)

Re(z)

Re(z)

Diagram A

Diagram B

Diagram B shows:

(A)

4.

z

(B)

(C) 2z

2iz

(D) z 2

Which of the following is the equation of the circle below? Im(z)

0

(A)

 z  2  z  2

(C)

 z  2i   z  2 i 

Ext2 Task 4 2016

4

4

2

4

Re(z) R

(B)

 z  2  z  2

4

(D)

 z  2  z  2

4

-4-

5.

6.

z and ware two complex numbers. Which of the the following statements is always TRUE? (A)

z  w t zw

(B)

z  w d zw

(C)

z  w d zw

(D)

zw  z t w

If D , E and J are the roots of the equation x3  3x  4 0. Then the cubic equation with roots D 2 , E 2 and J 2 is:

(A)

8 x3  9 x  4 0

(B)

x3  9 x2  12 x  4 0

(C)

x 3  6 x 2  9 x  16 0

(D)

8 x 3  4 x 2  9 x  16 0

7. y

x 2 1

y

P ( x, y )

0

1

x

2

The region bounded by the x-axis, the curve y

x 2  1 and the line x

about the y-axis. The slice at P( x, y ) on the curve is perpendicular to the axis of rotation. What is the volume of G V of the annular slice formed?





(A)

S 5  y2 G y

(B)

S 4  y 2 1 G y

(C)

S  4  x2  G x

(D)

S  2  x 2 G x

Ext2 Task 4 2016

-5-

2 is rotated

8.

What is the acute angle between the asymptotes of the hyperbola

(A)

S 3

(B)

S 4

(C)

S 6

(D)

S 2

x2 2 y 1? 3

9. Which of the following is the range of the function f ( x ) sin 1 x  tan 1 x ?

(B) S d y d S

(A) S  y  S

(C)

3S 3S dyd 4 4

S S dy d 2 2

(D)

S

10. Using the substitution x

´ ¶0

S  y , the definite integral µ x sin x dx

will simplify to:

S

´ (B) µ sin x dx ¶0

(A)

0

(C)

S´ µ sin x dx 2 ¶0

S

(D)

S2

End of Section I

Ext2 Task 4 2016

-6-

4

SECTION II Total Marks – 90 Attempt Questions 11 - 16 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11-16, your responses should include relevant mathematical reasoning and/or calculations. Question 11 (15 marks) Use a SEPARATE writing booklet.

(a) Given that z1

(b)

(c)

§z · 3  4i , find the value of Re ¨ 1 ¸ in x  iy form. © z2 ¹

5  2i and z2

2

(i)

Show that the square roots of 35  12i are r(1  6i ) .

2

(ii)

Hence solve z  (5  4i) z  11  7i

2

(i)

Express z1

(ii)

Given z 2

(iii)

Hence express  z1z 2  in the form x  iy , where x and y are real numbers.

(d) (i) (ii)

2

0.

1  3i in modulus argument form.

§ §S · § S ·· 3¨ cos ¨ ¸  i sin ¨ ¸ ¸ , find the value of z1 z 2 in © ©6¹ © 6¹¹ modulus argument form. 3

On an Argand diagram sketch arg(z  2) arg z  Describe the locus.

S 2

.

2

2

2 1

End of Question 11

Ext2 Task 4 2016

2

-7-

Question 12 (15 marks) Use a SEPARATE writing booklet.

(a)

(i)

(ii)

(b)

(i)

(ii)

Express

3 x 1 a bx  c in the form  2 . 2 ( x 1)( x 1) x1 x 1

2

3x  1 ´ dx . Hence find µ 2 ¶ (x 1)(x 1)

If t

tan

T 2

, show that dT

By using the substitution t

(c) The diagram shows y

2

2 dt . 1 t2

1 S

T

´3 tan , show that. µ secT dT 2 ¶0





ln 2  3 .

3

f  x which is an odd function. There is a turning point at 1,1 . y 1

x

-1 Draw a separate sketch of each of the following graphs. Use about one third of a page for each graph. Show all significant features. (i)

y

1 . f x 

(ii)

y

f

(iii)

y

e

(iv)

y

2

 x . f x

1

.

2

f  x u sin1 x (show the coordinates of the endpoints). End of Question 12

Ext2 Task 4 2016

-8-

2

Question 13 (15 marks) Use a SEPARATE writing booklet.

(a)

Given that the roots of the equation 4 x 3  24x 2  45x  26 0 form an arithmetic

3

sequence, solve the equation.

(b)

The polynomial P( x) 12 x 3  44 x 2  5x  100 has a double root.

2

Factorise P( x) over the real number system.

(c)

The hyperbola H has equation has equation

x2 y2  a2  b2 b2

x2 y2  a2 b2

1 and eccentricity e , while the ellipse E

1.

x2 y2 1 and hence that E has equation 2 2  2 ae b e

1.

(i)

Show that E has eccentricity

(ii)

Show that E passes through one focus of H, and H passes through one focus

2

2

of E .

(iii)

Sketch H and E on the same diagram, labelling the foci S, S’ of H and T, T’

2

of E, and the directrices of H and E. Give the coordinates of the foci and the equations of the directrices in terms of a and e .

(iv)

If H and E intersect at P in the first quadrant.

2



2 § 2 a e 1 ¨ Show that the coordinates of P is ae 2 , ¨ e  1 1  e2 ©

(v)

¸ ¹

Show that the acute angle D between the tangents to the curves at P satisfies tan D

1· § 2 ¨e  ¸ . © e¹ End of Question 13

Ext2 Task 4 2016

 ·¸.

-9-

2

Question 14 (15 marks) Use a SEPARATE writing booklet.

(a)

(b)

The equation x3  2 x  1 0 has roots D , E , J . Evaluate D 3  E 3 J 3 .

For n

In

0,1, 2,.... let

³

2

S 4

tan n T dT .

0

(c)

1 ln 2 . 2

(i)

Show that I1

(ii)

Use integration by parts to show that, for n t 2, In  In 2

2

1 . n 1

The area enclosed by the curve y ( x  2) 2 and the line y 4 is rotated around the y  axis . Use the method of cylindrical shells to find the volume formed.

4

3

y

y

4

x

(d)

(i)

On the same number plane diagram sketch the curves

y | x | 2 and y

(ii)

4  3x  x 2 .

Hence or otherwise solve the inequality

| x | 2 !0 . 4  3 x  x2

End of Question 14 Ext2 Task 4 2016

2

- 10 -

2

Question 15 (15 marks) Use a SEPARATE writing booklet.

(a)

§ c· § c· The points P ¨ cp , ¸ and Q ¨cq , ¸ lie on the rectangular hyperbola xy q¹ p¹ © ©

c2 .

The chord PQ meets the x-axis at C. O is the centre of the hyperbola and R is the midpoint of PQ.

(b)

(i)

Draw a sketch showing all information.

1

(ii)

Find the equation of the chord PQ .

2

(iii)

Find the coordinates of C.

1

(iv)

Find the coordinates of R.

1

(v)

Show that OR

2

(i)

Show that sin  A  B sin A  B

(ii)

Use the method of Mathematical Induction to show that for all positive

RC.

2 cos Asin B .

1

4

integers n :

cos x  cos 2 x  cos 3x  ....  cos nx

(iii)

1 1 sin §¨ n  ·¸ x  sin x 2¹ 2 © 1 2sin x 2

Hence show that:

3

cos 2 x  cos 4 x  cos 6 x  ...  cos16 x 8 cos9 x cos 4 x cos 2 x cos x .

End of Question 15

Ext2 Task 4 2016

- 11 -

Question 16 (15 marks) Use a SEPARATE writing booklet.

(a)

(i) Show that the area enclosed between the parabola x 2

4ay and its latus rectum

2

2

is

8a units 2 . 3

A solid figure (as shown below) has the ellipse

x2 y2  16 4

1 as its base in the xy plane.

Cross-sections perpendicular to the x-axis are parabolas with latus rectums in the xy plane.

2

(ii) Show that the area of the cross-section at x = h is

16  h 6

(iii) Hence, find the volume of this solid.

3

2

Question 16 continues on the next page.

Ext2 Task 4 2016

units2 .

- 12 -

Question 16 continued. (b)

y NOT TO SCALE

T V 2 sin 2 T m g

x





V 2 1 3 m 4g

dam

A man ascending in a hot air balloon throws a set of car keys to his wife who is on the ground. The keys are projected at a constant velocity of V ms  1 at an angle of T to the horizontal, 0q  T  90q , and from a point

2 2 V sin T m vertically above the ground. g

The edge of a dam closest to where the balloon took off, lies horizontally from the point of projection. The dam is





V 2 1 3 m 4g

V2 m wide. 2g

The position of the keys at time t seconds after they are projected is given by:

x Vt cosT and

(i)

V 2 sin 2 T  gt 2  Vt sin T  g 2

Show that the Cartesian equation of the path of the keys is given by :

y

(ii)

y

V 2 sin 2 T  gx 2 sec2 T   T x tan g 2V 2

Show that the horizontal range of the keys on the ground is given by:

x





2g

Find the values of T for which the keys will NOT land in the dam. END OF PAPER

Ext2 Task 4 2016

3

V 1  3 sin 2T 2

(iii)

1

- 13 -

4

Extension 2 Mathematics Task 4 Trial Examination 2016 Solutions: Question

Working 2

1

Solution

2

x y  9 16

1

sin ce b ! a ? a 2 b 2 (1 e 2 )

B

9 16(1 e2 ) 7 e2 16 7 4

e

?B

S

2

´6 3 µ sin x dx ¶0 S

´6 2 µ sin x.sin x dx ¶0 S

´6 2 µ sin x 1 cos x  dx ¶0

A

S

´6 2 µ sin x  sin x cos x dx ¶0 3

z 2 double the argument

?A

?D

D

Im(z)

4

0

2

4

Re(z) R

Centre 0, 2  and radius of 2





? z  2  z  2

4

(x  iy  2)(x  iy  2) 4 (x  iy )(x  iy )  2(x  iy )  2(x  iy )  4 4 2 2 x  y  4x 4 4

(x  2) 2  y 2

22

?B

B

5

B

C

( z  w)

w

( z  w) A D

z 0 The length of any two sides of a triangle must exceed the length of the third side. Therefore

' OAB

z  w t z w

' OAC

z  w d z w

' AOB z  w d z  w ' AOB z  w  z t w

6

If replace x with

?D

x

 x   3 x   4 3

x x 3 x

0

4 C

x  x  3  4 x x  3

2

16

2 x x  6x  9  16

x 3  6x 2  9x 16 0

7

y V ?V

x2  1

? x2

?C

y2  1

S 22  x 2 G y S 4  ( y 1)  G y 2

B

?B

8

? tan T

tan T

tan T ?T

9

1 x 3

Asymptotes y r

m1  m2 1  m1m 2 § 1 · § 1 · ¨ 3 ¸ ¨ 3 ¸ © ¹ © ¹ 1 1 § ·§ · 1 ¨ ¸¨  ¸ 3 3 © ¹© ¹ 3 3 60q

S

A

?A

3

Domain : 1 d x d1 sin1 ( 1)  tan 1 ( 1) d y d sin  1 (1)  tan  1(1)

S S S S  d yd  2 4 2 4 3S 3S dyd 4 4

10

S

C

?C

0

´ ´ µ x sin x dx µ (S  y)sin (S  y).  dy ¶0 ¶S S

´ µ S sin (S  y)  y sin( S  y) dy ¶0 S

´ µ S sin y  ysin y dy ¶0 S

´ µ S sin x  x sin x dx ¶0 S

´ ? 2µ x sin x dx ¶0 S

´ ? µ x sin x dx ¶0

S

´ S µ sin x dx ¶0 S

S´ µ sin x dx 2 ¶0

?C

C

Question 11(a)

Working

Solution

§ z1 · § 5  2 i · ¨ ¸ ¨  ¸ © z1 ¹ © 3 4 i ¹ § 5 2i 3 4i · ¨  u  ¸ © 3 4i 3 4i ¹

2

§ 15  20i  6i  8i 2 · ¨ ¸ 2 9  16i © ¹ § 7  26i · ¨ ¸ © 25 ¹ § 7  26i · 7 ? Re ¨ ¸ © 25 ¹ 25

11(b)(i)

 35  12i

a  ib

 35  12i