girraween high school 2018 year 12 trial higher school certificate · 2020-04-25 · girraween high...
TRANSCRIPT
Girraween High School
2018 Year 12 Trial Higher School Certificate
Mathematics Extension 1
Time allowed: Two (2) Hours
(Plus 5 minutes reading time)
Instructions
• Attempt all questions.
• For Questions 1-10, shade the circle for the letter corresponding to the correct answer on
your answer sheet.
• For Questions 11-15, start each question on a new page. Each question should be clearly
labelled.
• All necessary working must be shown for Questions 11-15.
• Marks may be deducted for careless or badly arranged work.
• Board approved calculators may be used.
• A Mathematics reference sheet is provided.
• All diagrams are NOT TO SCALE.
Section l
10 marks
Attempt Questions 1-10
Allow about 15 minutes for this section
Question l
The point C which divides the interval between A(14, -20) and B(-4,1)
in the ratio 4: -1 is:
(A) (-10,8)
Question 2
2x Jim .
3 =
x~o sin x 3
(A)-2
Question 3
(B) (20, -27)
2 (B) -
3
(C) (2, -6)
1 (C)-
3
(D) (8,-13)
(D) 3
The number of different ways of arranging the letters of the word MOORABOOL
in a circle is
(A)362880 (B) 40 320 (C) 15 120 (D) 1680
Question 4
The remainder when ax3 + x 2 + x + 11 is divided by (2x + 3) is 5. The value
of a is:
(A) 1 (B) 2 (C) 3 (D) 4
Question 5
The co-efficient of x2 in the expansion of [ 2x + :2f1 is
Question 6
cos5xcos2x - sin5xsin2x =
(A) cos3x (B) sin3x (C) cos?x (D) sin?x
Exa111i11atio11 co11ti1111es 011 the followi11g page
Question 7
J sin23x.dx =
(A) ;sin3 3x + C 1 1 1 1
(B) - x - - cos6x + C (C) - x - - cos6x + C 2 2 2 12
(D) !:.x _ 2.sin6x + C 2 12
Question 8
Which of the graphs below is ofy = x 3 - 6x2 + 9x - 4?
(A)
y
x
(C)
y
x
Question 9
The general solution of sin2x = - !:. is 2
(B)
(D)
(A)x=nrr+c-1rx~ 6
(B) x = nrr + (-1r x !!_ 2 12
(D) x = nrr + c-1r+1 x !!_ 2 12
Question 10
Ify = sin-1 (Sx) dy = 'dx
1 5 (A) v1-2sx2 (B) V1-25x2
y
x
y
x
(C) x = nrr + (-1r+1 x ~ 6
Examination continues 011 the following page
Section II
69 marks
Attempt Questions 11-15
Allow about 1 hour and 45 minutes for this section
Start all answers on a separate page in your answer booklet.
In Questions 11-15 your responses should include all relevant mathematical reasoning and/ or
calculations.
Question 11 (16 Marks)
5 1 (a) Solve for x: -- > - -
3x-z 4
(b) Find the acute angle between the lines y = 2x - 1 and x + 3y = 2
(c) Find the exact value of cos 75°.
(d) If a:, fJ and y are the roots of the polynomial equation
1 1 1 2x3 - x 2 + 3x - 4 = 0, find the value of-+ - + -
a{3 ay {3y
n: .
(e) Use the substitution u = cosx to find J.04 sm x . dx
l+cosx
(f) Use the substitution x = u 2 to find J 1 r;::;. dx
l+vx
Question 12 (12 Marks)
(a) Use the method of mathematical induction to prove
Marks
3
3
2
2
3
3
2 + 6 + 20+ ... +(n2 + n) = ~ (n + l)(n + 2) for all positive integers n. 4
Question 12 continues 011 the followi11g page
Question 12 (continued)
(b) In the diagram below, line PTQ is a tangent to the circle TABC at T.
BT bisects AC at D and AT= CT. LQTC =a and LCTB = /J.
(see diagram)
(i) Copy the diagram in to your answer booklet and state why LT AC = a.
(ii) Prove LATP =a.
(iii) Prove LATE= /J.
(iv) Prove TB is a diameter of the circle TABC.
Question 13 (14 Marks)
(a) Sketch the graph ofy = Zcos-1 G), stating its domain and range.
(b) The probability that it will rain on a certain day in August in Sydney
is~. Assuming that the probability that it will rain on one day is independent 9
of the probability that it will rain on the next day:
Marks
1
2
2
3
Marks
3
(i) Find the probability that it will rain on 3 days in the next week. 2
(ii) Find the probability that it will rain on at least 2 days in the next week. 2
Question 13 continues 011 the following page
Question 13 (continued)
(c) A particle is moving so that its position at time tis given by
x = 2sin3t + 2../3cos3t.
Marks
(i) By showing that x = -n2x, n a real number, prove that the particle is l
moving with simple harmonic motion.
(ii) By expressing the particle's position at time tin the fonn 3
x = Rcos(3t - a), find the period and amplitude of the motion.
(iii) Find the first time that the particle reaches x = 0 and find its velocity 3
and acceleration then.
Question 14 (14 Marks)
(a) A particle is moving with simple hannonic motion about x = 0 with
acceleration x = -n2x and amplitude a. (Note that in this question a refers
to the amplitude of the motion).
(i) Show that v 2 = n2 (a2 - x2).
(ii) For this particle, v = 10 when x = 5 and v = 5 when x = 7.
Find the period of the motion.
Question 14 co11ti11ues 011 the .following page
2
3
Question 14 (continued)
(b) A howitzer with a muzzle velocity of200m/s is located lOOm
up the side of a hill. It is firing shells at an angle of a to the horizontal.
(see diagram)
DIAGRAM NOT TO SCALE
HILL 100m
WALL
(i) Given that x = 0 and ji = -g, show that x = 200tcosa and
y = - ~ gt 2 + 200tsina + 100.
2
(ii) Show that y = - --2.!.._ sec 2 a + xtana + 100. 80 000
(iii) The shell is being aimed at the base of a wall 3000m away horizontally.
At which two angles must the shell be fired in order to hit this target?
(Let g = 10m/s2).
Exami11atio11 continues 011 the followi11g page
Marks
4
2
3
Question 15 (13 Marks)
(a) P(Zap, ap 2) and Q (Zaq, aq 2
) are variable points on the parabola
x 2 = 4ay. A is the intersection of the tangent to the parabola at P and the
line through Qparallel to the axis of the parabola, while B is the intersection
of the tangent to the parabola at Q and the line through P parallel to the axis
of the parabola. (see diagram)
The equation of the tangent to the parabola at P is y = px - ap 2
(Do NOT prove this!).
(i) Prove that PQAB is a parallelogram.
(ii) lfp > q, find the area of the parallelogram in tenns ofp and q.
Question 15 continues on the following page
Marks
3
2
Question 15 (continued)
(b) An aeroplane flying due East at 900km/h at a constant altitude
is seen by an observer on the ground on a bearing of 290°T at an
angle of elevation of a. One minute later it is sighted by the observer
at an angle of elevation of 35° and on a bearing of OS0°T. If the
aeroplane flies from A to B, the observer is at P and C and D are on
the ground directly beneath A and B (see diagram)
A-----------fi>B
D
p
(i) Show that AB = 15000m and LCPD = 120°.
(ii) Show that the height of the aeroplane above the ground is given by
BD = lSOOOsin20°tan35°
sin60°
(iii) Find the angle of elevation (a) when the aeroplane is first seen
(answer to the nearest minute).
END OF EXAMINATION
Marks
2
3
3
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