Caringbah 2020 Extension 2 Solutions PDF

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Extension 2 Mathematics Paper ...

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Caringbah High School Year 12 2020 Mathematics Extension 2 HSC Course Assessment Task 4 General Instructions • Reading time – 10 minutes • Working time – 3 hours • Write using black pen • NESA approved calculators may be used • A reference sheet is provided • In Questions 11-16, show relevant mathematical reasoning and/or calculations

Total marks – 100 Section I 10 marks Attempt Questions 1-10 Mark your answers on the answer sheet provided. You may detach the sheet and write your name on it. Allow about 15 minutes for this section Section II 90 marks Attempt Questions 11-16 Write your answers in the numbered answer booklets provided. Ensure your name or student number is clearly visible. Allow about 2 hours and 45 minutes for this section.

Marker’s Use Only Section I Q 1-10

/10

Section II Q11

/15

Q12

/15

Q13

/15

Total

Q14

/15

Q15

/15

Q16

/15

/100

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

1.

2.

Which of the following is true for all complex numbers z? (A)

Im z =

(B)

Im z

(C)

Im z =

(D)

Im z

If

z −z 2i z z

2 z−z 2 z z

2i

 f ( x)sin x dx = − f ( x)cos x +  3x

2

cos x dx ,

which of the following could be f (x)?

3.

(A)

3x2

(B)

x3

(C)

–x3

(D)

–3x2

Which of the following is x (A) (B) (C) (D)

−

1 4 − x− 3 x+ 1

5

2 x 1 3

5

2 x 1 3

x

x −

3x 11 3 x 1 expressed in partial fractions?

1 4 + x− 3 x+ 1

–2–

4.

5.

Given that x and y are real numbers, which of the following statements is true? (A)

y(x : x = y)

(B)

y(x : x  y)

(C)

y(x : x  y)

(D)

y(x : x = − y)

What is the magnitude of the vector cos i + sin  j + tan  k where 0  

6.

7.

(A)

1

(B)

cosec

(C)

cot

(D)

sec

 2

?

Consider the statement x2 = 9  x = 3 . Which of the following statements is the contrapositive? (A)

x  3  x2  9

(B)

x2  9  x  3

(C)

x = 3  x2 = 9

(D)

x  3  x2  9

The points A, B and C are collinear where OA = i + j , OB = 2i − j + k , OC = 3i + aj + bk . What are the values of a and b? (A)

a = −3, b = −2

(B)

a = 3, b = −2

(C)

a = −3, b = 2

(D)

a = 3, b = 2

–3–

8.

9.

A particle is moving in simple harmonic motion about a fixed point O on a line. At time t seconds, its displacement from O is given by x = 2cos t metres. What is the time taken by the particle to travel the first 100 metres of its motion? (A)

20 seconds

(B)

25 seconds

(C)

50 seconds

(D)

100 seconds

A particle is projected horizontally with speed gh ms −1 from the top of a tower of height h metres. It moves under gravity where the acceleration due to 2 gravity is g ms− . At what angle to the horizontal will the particle hit the ground?

1 2

(A)

tan − 1

(B)

tan −

(C)

tan− 1 2

(D)

tan − 1 2

1

1 2

2

10.

i

The equation z 5 = 1 has roots 1, ,  2 ,  3,  4 where  = e 5 . What is the value of (1 −  )(1 − 2 )(1 − 3 )(1−  4 ) ?

(A)

−5

(B)

−4

(C)

4

(D)

5

End of Section I

–4–

Section II 90 marks Attempt Questions 11–16 Allow about 2 hours 45 minutes for this section Answer each question in the appropriate writing booklet. Extra writing booklets are available. For questions in Section II, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (15 marks) Use the Question 11 Writing Booklet. (a)

A particle is moving in a straight line. At time t seconds it has displacement x metres from a fixed point O on the line, where x is given by x =1 +cos 2t + sin 2t (i)

Express x in the form x = 1 + a cos(2t −  ) for constants a  0 and 0   

(ii)

(b)

 2

2

.

Find correct to 2 decimal places the average speed of the particle during the time it takes to first reach O.

3

In an Argand diagram the point P represents z1 = 3 + 2i , the point Q represents 12 − 5i z2 = and O is the origin. z 3 represents the centre C of the circle passing z1 through P, Q and O. (i)

Express z 2 in the form a + ib where a and b are real.

(ii)

Show that POQ =

(iii)

Express z 3 in the form a + ib where a and b are real.

 2

2 1

.

Question 11 continues on page 6

–5–

2

Question 11 (continued)

(c)

ABC is an acute angled triangle. The altitudes BE and CF intersect at O. The line AO produced meets BC at D. Relative to O the position vectors of A, B and C are a, b and c respectively.

(i)

Show that b. (c − a ) = 0 and c. (b − a ) = 0

(ii)

Hence show that AD ⊥ BC

(iii)

What geometrical property of the triangle has been proved?

2 2

End of Question 11

–6–

1

Question 12 (15 marks) Use the Question 12 Writing Booklet. 2

2

(a)

 x dx Find  2 .  x +1

(b)

(i)

Use the substitution u = x2 − 4 to show that  x dx = x 2 − 4 + c .  2  x −4

2

8

(ii)

 x ln( x2 − 4) dx . Hence find the exact value of  x2 − 4 

3

5

(c)

(i)

(ii)

(d)

x Find the parametric equations of the line l =  y  passing through z    4   A = (2, − 1, 3) which is parallel to v =  3  .  − 2  

Show that the point B = (10, 5, − 1) lies on this line.

If a = 3 i + 2 j + 2 k and b = −i + j + k ,

2

2

2

find a unit vector perpendicular to both a and b .

(e)

It is given that a  0 and b  0 are real numbers. Consider the statement   1 a  b, log 1 = loga b . a b   Prove that the statement is true. End of Question 12

–7–

2

Question 13 (5 marks) Use the Question 13 Writing Booklet. 

(a)

(

i

)

It is given that z = 2e12 is a root of the equation z 4 = a 1 + 3 i , where a is real.

(i)

Express 1 + 3 i in the form rei where r  0 and −    .

2

(ii)

Find the value of a.

1

(iii)

Find the other 3 roots of the equation in the form re where r  0 and −     .

i

3

2 3

(b)

(i)

Use the substitution t = tan

x 1 dx = ln 3 . to show that    sin x 2

2



3

2 3

(ii)

−x x dx =  dx . Use the substitution u =  − x to show that     sin x  sin x 



3

3

2 3

(iii)

2 3

x dx. Hence find the value of    sin x

2

2



3

(c)

Let  be the complex number satisfying  3 = 1 and Im( )  0 . The cubic polynomial P( z ) = z3 + az 2 + bz + c , has zeros 1, −  and −  .

Find the values for a, b and c in P(z). End of Question 13

–8–

3

Question 14 (15 marks) Use the Question 14 Writing Booklet.

(a)

Consider the equation z 5 + 1 = 0 . (i)

Draw a sketch of the roots of z 5 + 1 = 0 on an Argand Diagram.

1

(ii)

Factor z 5 + 1into irreducible factors with real coefficients.

2

(iii)

Deduce that cos

(iv)

Write a quadratic equation with integer coefficients which has roots  3 cos and cos . 5 5  3 Hence find the value of cos and cos as surds. 5 5



3 1 = and 5 5 2  3 −1 cos cos = . 5 5 4 + cos

2

2

0

(b)

(c)

If I n =

x

n

1 + x dx n = 0, 1, 2,....

−1

1

(i)

Find the value of I 0

(i)

Show that I n =

(ii)

Hence find the value of I 2 .

−2 n I n 1 for n = 1, 2,3,... 2n + 3 −

Use proof by contradiction to show that log2 5 is irrational.

End of Question 14

–9–

3

2

2

Question 15 (15 marks) Use the Question 15 Writing Booklet.

(a)

1  2 for any real number x  0 . x (DO NOT PROVE THIS RESULT) Recall that x +

(i)

1 1 1 Prove that (a + b + c)( + + )  9 a b c for any real numbers a  0, b  0, c  0 .

(ii)

Prove that

a b c 3 + +  b + c c + a a +b 2

2

3

for any real numbers a  0, b  0, c  0 by using the result in (i) and transformations of the form a → a +b. (b)

A particle is moving in a straight line from a fixed point O on the line, so that at time t seconds it has displacement x metres, a velocity v ms −1 and an acceleration 1 x of a ms − 2 given by a = e 2 . Initially the particle is at O and moving with a speed of 2 ms −1 while slowing down. (i)

1 x Show that v = −2e 4 .

2

(ii)

Find expression for x, v and a in terms of t.

3

(iii)

Describe the subsequent motion of the particle.

1

Question 15 continues on page 11

– 10 –

Question 15 continues (c)

In the Argand diagram above, the points A and B represent z 1 and z 2 respectively. AOB =

 2

and OA = OB .

(i)

Express z 2 in terms of z 1 .

1

(ii)

Copy the diagram and on it draw the locus L1 of points satisfying

1

z − z1 = z − z 2 . (iii)

On your diagram draw the locus L 2 of points satisfying arg( z − z2 ) = arg z1 .

1

(iv)

Find in terms of z 1 the complex number representing the point of intersection of L1 and L2 .

1

End of Question 15

– 11 –

Question 16 (15 marks) Use the Question 16 Writing Booklet.

ln 3  e2 x dx   ex + 1 0

4

(a)

Find

(b)

A particle of mass m kg falls from rest under the influence of gravity g in a medium where the resistance to motion is mkv when the particle has velocity v ms −1 . (i)

Draw a diagram showing the forces acting on the particle.

1

(ii)

− v ) where Show that the equation of motion of the particle is = −1 V ms is the terminal velocity of the particle in this medium, and x metres is the distance fallen in t seconds.

2

(iii)

Find in terms of V and k the time T seconds for the particle to attain 50% of its terminal velocity, and the distance fallen in this time.

5

(iv)

What percentage of its terminal velocity will the particle have attained in time 2T seconds? Sketch a graph of v against t showing this information.

2

(v)

If the particle has reached 87.5% of its terminal velocity in 15 seconds, find the value of k.

1

End of Paper

– 12 –...