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St George Girls - Past paper
Applied Mathematics (STABEX) (MATH4406)
University of Sydney
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I Student Number:
I Teacher:
St George Girls High School
Mathematics Extension 2
2 0 2 0 Trial HSC Examination
General · Reading time - 10 minutes Instructions Working Time - 3 hours - Write using black pen Calculators approved by NESA may be used
- A reference sheet is provided
- For questions in Section I, use the multiple-choice answer sheet provided
Total marks:
100
- For questions in Section II: o Start each question in a new writing booklet o Show relevant mathematical reasoning and/or calculations o Extra writing booklets are provided if needed o Marks may not be awarded for incomplete or poorly presented solutions
Section I - 10 marks (pages 3 -6) Attempt Questions 1 - 10 Allow about 15 minutes for this section
Section II - 90 marks (pages 7 -12) Attempt Questions 11 - 16
- Allow about 2 hours and 45 minutes for this section
Q1-Q Q Q Q Q Q Q
/ / / / / / / Total / %
Trial HSC Examination - Mathematics Extension 2 - 2020
Section I
10 marks Attempt Questions 1-
On the multiple choice answer sheet circle the letter corresponding to the most correct answer for questions 1-IO.
- In modular-argument form, the complex number i - 1 is :
(A) ,/zcis ( - ~)
(B) '12 (3:)
(C) Zeis ( -
5 4
n)
(D) Zeis (3:)
- Which of the following is a primitive of tansec 2 x? 3 x.
(A)! 2 cot 2 x (B) _! 2 cot 2 x (C)! 4 cot 4 x (D) _! 4 cot 4 x
- A lawyer for a person under investigation for a bank robbery states
IF MY CLIENT WAS NOT IN THE SUBURB AT THE TIME OF THE ROBBERY, THE ROBBER CANNOT BE MY CLIENT.
Which of the following statements is logically equivalent to the statement made by the lawyer?
(A) If my client was in the suburb, then he was the robber.
(B) If my client was the robber, then my client was in the suburb.
(C) If my client was not the robber, then he was not in the suburb.
(D) If my client was not the robber, then my client was in the suburb.
Trial HSC Examination - Mathematics Extension 2 - 2020
- If -----S -- ___ A_+ --_ 8 - , t h en A an dB h aveva J ues o: f (2x + 1)(2 - x) 2x + 1 2 - x
(A) A= -1, B = 2
(B) A= 1, B = -
(C) A=2,B=-
(D) A= 2, B = 1
27'. 7. The equation z 5 = 1 has roots 1, w, w 2 , w 3 , w 4 where w =es'. What is the value of (1 - w)(l - w 2 )(1-w 3 )(1-w 4 )?
(A) -5 (B) -
- The negation of 'P and not Q' is :
(A) 'not Q or not P'.
(B) 'Q or notP'.
(C) 'Q and notP'.
(D) 'not Q and notP'.
(C) 4 (D) 5
Trial HSC Examination - Mathematics Extension 2 - 2020
- The lines f1 and f2 are given by the equations
Where s E !Rl., t E IRl. and a is a constant.
Given that f 1 and f 2 are perpendicular what are the values of a?
(A) a= 1, a= 3
(B) a= -1, a= 3
(C) a= 1, a= -
(D) a= -1, a= -
- Without evaluating the integrals, which one of the following integrals is greater than zero?
(A) " J!;i: 4
" (C) J!;i: 4
1 + cosx xs
tan -- 3 x d X xS
dx (B)
Trial HSC Examination - Mathematics Extension 2 - 2020
Question 12 (15 marks) Begin a new booklet
(a) (i) Find the numbers A, Band C such that:
1-x A Bx+c
####### -----=-+--(l+x)(l+x2) - l+x 1+x 2
rl 1-x (ii) Hence find the exact value of J,o ( )( 2 ) dx. l+x l+x
(b) The vertices of a triangle ABC are defined by the position vectors
(i) Show that cosLBAC = ~
(ii) Find the exact area of triangle ABC.
( c) Consider the point P ( 2 , 1 , - 6) and the line L with equation
Find the shortest distance from the point P to the line L.
( d) Prove by contradiction that if n is a positive integer then
-,,/Sn + 6 is always irrational.
2 3 3 2 2 3
Trial HSC Examination - Mathematics Extension 2 - 2020
Question 13 (15 marks) Begin a new booklet
(a) The points P and Q have vector positions:
- OP= -2£ + 14j - 5~ and - OQ = - £ + 12j - 2~
(i) Show that the equation of the line f 1 that passes through Pand Qis
7:. 1 = (-2 + s) i + (14 - 2s)j + (-5 + 3s)k where sis a real number. 2
(ii) Consider the line f 2 with equation
r 2 = ( 2 + at) i + (27 + (a+ l)t)j + ( 1 +(a+ 2)t)k
~ - -
where a is a constant and t is a real number. The line f 2 intersects f 1
at the point R. Find the coordinates of R.
(b) Evaluate f: u. + 1 du.
(c) (i) For a complex number z, shade the region of the Argand Plane where
the inequalities
3
3
lz + 1 + ii < 1 and - ?:. 4 -< arg (z + 1 + i) < - ?:. 4 hold simultaneously. 3
(ii) The complex number z satisfies lz + 1 + ii = 1.
What is the smallest distance that z can be from the real number -2 on
an Argand diagram?
(d) Use contrapositive proof to prove that if x 2 (y 2 - 4y) is odd then
the integers x and y are odd.
1
3
Trial HSC Examination - Mathematics Extension 2 - 2020
Question 15 (15 marks) Begin a new booklet
( a) A sequence is given by the recurrence relation
####### u 1 = 7, Un+l = 2un + 3 for n <:: 1.
Prove by induction that the general formula for the sequence is Un = 5(2n) - 3.
(b) (i) Use De Moivre's Theorem to express cos58 and sin58 in terms of
4
powers of sin8 and cos8. 2
(c)
(ii) Write an expression for tan58 in terms oft, where t = tan8. 1
(iii) By solving tan58 = 0, deduce that: tan; tan
2 ; tan
3 : tan
4 ; - 5. 3
A
ABC is an acute angled triangle. The altitudes BE and CF intersect at 0. The line AO produced meets BC at D. Relative to O the position vectors of A, B and C are g, !! and r; respectively.
####### (i) Show that Q • (£ - g) = 0 and r; · (Q - g) = 0.
(ii) Hence show that AD . BC.
(iii) What geometrical property of the triangle has been proved?
2
2
1
Trial HSC Examination - Mathematics Extension 2 - 2020
Question 16 (15 marks) Begin a new booklet
(a) Use mathematical induction to prove that 4n+1 + 6n is divisible by 10 when n is even.
1 (b) Recall that x + - X ~ 2 for any real number x > 0. (DO NOT PROVE THIS RESULT)
####### (i) Prove that (a+ b + c) (~ + ~ + ~) > 9
a b C
for any real numbers a > 0, b > 0, c > 0.
(ii) Hence prove that (~ + _!: + c) ~)-
b+c c+a a+b 2 for any real numbers a> 0, b > 0, c > 0.
(c) The function F(p) is defined as F(p) = lim J, 0 t xP- 1 e-x dx, for p > 0. · Moo
(i) Show that F (1) = 1.
(ii) Use integration by parts to show F(p + 1) = pF(p).
(iii) Hence find F(n) for integers n > 1.
End of Examination
3 2 3 2 3 2
St George Girls - Past paper
Course: Applied Mathematics (STABEX) (MATH4406)
University: University of Sydney
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