PENRITH HIGH SCHOOL 2015 HSC TRIAL EXAMINATION M athematics E xtension 1 General Instructions: β’ Reading time β 5 minutes β’ Working time β 2 hours β’ Write using black or blue pen Black pen is preferred β’ Board-approved calculators may be used β’ A table of standard integrals is provided at the back of this paper β’ In questions 11 β 14, show relevant mathematical reasoning and/or calculations β’ Answer each question on a new sheet of paper Total marksβ70 Pages 3β5 10 marks β’ Attempt Questions 1β10 β’ Allow about 15 minutes for this section Pages 6β9 60 marks β’ Attempt Questions 11β14 β’ Allow about 1 hours 45 minutes for this section Student Number: Teacher Name: This paper MUST NOT be removed from the examination room Assessor: T Bales SECTION II SECTION I 1
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PENRITH HIGH SCHOOL 2015 HSC TRIAL EXAMINATION
Mathematics Extension 1
General Instructions: β’ Reading time β 5 minutes β’ Working time β 2 hours β’ Write using black or blue pen
Black pen is preferred β’ Board-approved calculators may
be used β’ A table of standard integrals is
provided at the back of this paper β’ In questions 11 β 14, show
10 marks β’ Attempt Questions 1β10 β’ Allow about 15 minutes for this section Pages 6β9 60 marks β’ Attempt Questions 11β14 β’ Allow about 1 hours 45 minutes for this section
Student Number: Teacher Name:
This paper MUST NOT be removed from the examination room
Assessor: T Bales
SECTION II
SECTION I
1
Section I 10 marks Attempt Questions 1β10 Allow about 15 minutes for this section Use the provided multipleβchoice answer sheet for Questions 1β10 1 For π₯π₯ > 1, πππ₯π₯ β lnπ₯π₯ is:
(A) = 0
(B) > 0
(C) < 0
(D) = ππ
2 Consider the function ππ(π₯π₯) given in the graph below: f(x) βππ 0 ππ π₯π₯ Which domain of the function f(x) above is valid for the inverse function to exist?: (A) π₯π₯ > 0
(B) βππ < π₯π₯ < ππ
(C) 0 < π₯π₯ < ππ
(D) π₯π₯ < 0
3 What is the acute angle between the lines 2π₯π₯ β π¦π¦ β 7 = 0 and 3π₯π₯ β 5π¦π¦ β 2 = 0 ?
(A) 4β24β²
(B) 32β28β²
(C) 57β32β²
(D) 85β36β²
3
4 A particle is moving in a straight line with velocity π£π£ ππ/π π and acceleration ππ ππ/π π 2. Initially the particle started moving to the left of a fixed point ππ. The particle is noticed to be slowing down during the course of the motion from 0 to π΄π΄. It turns around at π΄π΄, keeps speeding up for the rest of the course of motion, passing ππand π΅π΅ and continues. The particle never comes back. Take left to be the negative direction.
A ππ B π₯π₯
During the course of the particleβs motion from ππ to π΄π΄, which statement of the following is correct?
(A) π£π£ > 0 and ππ > 0
(B) π£π£ > 0 and ππ < 0
(C) π£π£ < 0 and ππ > 0
(D) π£π£ < 0 and ππ < 0
5 A particle is moving in a straight line with velocity, π£π£ = 11+π₯π₯
ππ/π π where π₯π₯ is the displacement of the particle from a fixed point ππ. If the particle was observed to have reached the position π₯π₯ = β2 ππ at a certain moment of time, then this particle:
(A) will definitely reach the position π₯π₯ = 1ππ (B) may reach the position π₯π₯ = 1ππ (C) will never reach the position π₯π₯ = 1ππ (D) will come to rest before reaching π₯π₯ = 1ππ
6 If the rate of change of a function π¦π¦ = ππ(π₯π₯) at any point is proportional to the value of the function at that point then the function π¦π¦ = ππ(π₯π₯) is a:
(A) Polynomial function (B) Trigonometric function (C) Exponential function (D) Quadratic function
4
7 Let πΌπΌ and π½π½ be any two acute angles such that πΌπΌ < π½π½. Which of the following statements is correct ?
10 Using the substitution, π’π’ = 1 + βπ₯π₯, find the value of β« 1
οΏ½1+βπ₯π₯οΏ½21βπ₯π₯πππ₯π₯4
1 is:
(A) 65
(B) 1
3
(C) 2
3
(D) 3
2
5
Section II 60 marks Attempt Questions 11β14 Allow about 1 hour and 45 minutes for this section Begin each question on a new sheet of paper. Extra sheets of paper are available. In questions 11β14, your responses should include relevant mathematical reasoning and/or calculations Question 11 (15 marks) Start a new sheet of paper. (a) (π₯π₯ + 1) and (π₯π₯ β 2) are factors of π΄π΄(π₯π₯) = π₯π₯3 β 4π₯π₯2 + π₯π₯ + 6. Find the third factor. 2 (b) Find the coordinates of the point ππ which divides the interval π΄π΄π΅π΅ internally in the ratio 2: 3 3 with π΄π΄(β3, 7) and π΅π΅(15,β6)
(c) Solve the inequality 1
4π₯π₯β1< 2, graphing your solution on a number line. 3
(d) Use the method of mathematical induction to prove that, for all positive integers π π : 3 12 + 32 + 52 + 72 + β―+ (2π π β 1)2 = ππ
3(2π π β 1)(2π π + 1)
(e) T P β’B β’Q A PT is the common tangent to the two circles which touch at T. PA is the tangent to the smaller circle at Q, intersecting the larger circle at points π΅π΅ and π΄π΄ as shown. i) State the property which would be used to explain why ππππ2 = πππ΄π΄ Γ πππ΅π΅ 1
ii) If ππππ = ππ,πππ΄π΄ = π π πππ π ππ πππ΅π΅ = ππ, prove that nrm
n r=
β 3
Proceed to next page for question (12)
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Question 12 (15 marks) Start a new sheet of paper. (a) The equation π π π π π π π₯π₯ = 1 β 2π₯π₯ has a root near π₯π₯ = 0.3. Use one application of Newtonβs 3 methods to find a better approximation, giving your answer correct to 2 decimal places. (b) Five couples sit at a round table. How many different seating arrangements are possible if: i) there are no restrictions? 1 ii) each person sits next to their partner? 2
(c) In the expansion of (4 + 2π₯π₯ β 3π₯π₯2) οΏ½2 β π₯π₯5οΏ½6, find the coefficient of π₯π₯5 3
(d) i) Write the binomial expansion for (1 + π₯π₯)ππ 2
ii) Using part (i), show that ( )3 10 0
11 31
nn n kk
kx dx C
k+
=+ =
+ββ« 2
iii) Hence show that 1 1
0
1 13 4 11 1
nn k n
kk
Ck n
+ +
=
= β+ +β 2
Proceed to next page for question (13) 7
Question 13 (15 marks) Start a new sheet of paper. (a) Use the substitution π’π’ = π₯π₯
ππ+ ππ. You do not need to prove this).
(b) Give the exact value for 3
23 3
dxxβ +β« 3
(c) The distinct points P, Q have parameters ππ = ππ1 and ππ = ππ2 respectively on the parabola π₯π₯ = 2ππ,π¦π¦ = ππ2. The equations of the tangents to the parabola at P and Q respectively are given by: π¦π¦ β ππ1π₯π₯ + ππ12 = 0 and π¦π¦ β ππ2π₯π₯ + ππ22 = 0 (You do not need to prove these) i) Show that the equation of the chord ππππ π π π π 2π¦π¦ β (ππ1 + ππ2)π₯π₯ + 2ππ1ππ2 = 0 2 ii) Show that M, the point of intersection of the tangents to the parabola at P and Q, 2 has coordinates οΏ½ππ1 + ππ2 , ππ1ππ2οΏ½.
iii) β) Prove that for any value of ππ1, except ππ1 = 0, there are exactly two values 3 of ππ2 for which M lies on the parabola π₯π₯2 = β4π¦π¦. π½π½) Find these two values of ππ2 in terms of ππ1. 2
Proceed to next page for question (14)
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Question 14 (15 marks) Start a new sheet of paper. (a) A particle ππ is moving in simple harmonic motion on the π₯π₯ axis, according to the law π₯π₯ = 4π π π π π π 3ππ where π₯π₯ is the displacement of ππ in centimetres from ππ at time ππ seconds. i) State the period and amplitude of the motion. 2 ii) Find the first time when the particle is 2cm to the positive side of the origin and 2 itβs velocity at this time. iii) Find the greatest speed and greatest acceleration of ππ 3 (b) π¦π¦ ππ ππ ππ ππ π₯π₯ Q A projectile is fired from ππ, with speed πππππ π β1, at an angle of elevation of ππ to the horizontal. After ππ seconds, its horizontal and vertical displacements from ππ (as shown) are π₯π₯ metres and π¦π¦ metres,
repectively. i) Prove that π₯π₯ = πππππππππ π ππ and π¦π¦ = β1
ii) Show that the time taken to reach ππ is given by ππ = 2ππππππππππππ
2
iii) The projectile falls to ππ, where its angle of depression from ππ is ππ. 3 Prove that, in its flight from ππ to ππ, ππ is the half-way point in terms of time.
