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1
BAULKHAM HILLS HIGH SCHOOL
TRIAL 2014 YEAR 12 TASK 4
Mathematics Extension 1 General Instructions Reading time – 5 minutes Working time – 120 minutes Write using black or blue pen Board-approved calculators may be used Show all necessary working in
Questions 11-14 Marks may be deducted for careless or
badly arranged work
Total marks – 70 Exam consists of 11 pages. This paper consists of TWO sections. Section 1 – Page 2-4 (10 marks) Questions 1-10 Attempt Question 1-10 Section II – Pages 5-10 (60 marks) Attempt questions 11-14 Table of Standard Integrals is on page 11
2
Section I - 10 marks Use the multiple choice answer sheet for question 1-10
1. Given the equation 10 , what is the value of given that 3.6and 5.
(A) 0.717
(B) 0.204
(C) 0.204
(D) 0.717
2.
In the diagram above, is a tangent and is a chord produced to . The value of is
(A) 12
(B) 2√3
(C) 4√2
(D) 4√6
3. How many distinct permutations of the letter of the word " " are possible in a straight line when the word begins and ends with the letter (A) 12
(B) 180
(C) 360
(D) 720
A
P Q T4 8
Not to scale
x
3
4. The coordinates of the point that divides the interval joining 7,5 and 1, 7 externally in the ratio 1: 3 are (A) 10,8
(B) 10,11
(C) 2,8
(D) 2,11
5. What is the domain and range of 2 cos ?
(A) : , : 0 2
(B) : , : 0 2
(C) : , : 0
(D) : , : 0
6. Which of the following is the general solution of 3 tan 1 0, where is an integer?
(A)
(B)
(C) 2
(D) 2
7.
The displacement of a particle moving in simple harmonic motion is given by 3 cos where is the time in seconds. The period of oscillation is: (A)
(B)
(C) 2
(D) 3
4
8.
and are parallel chords in a circle, which are 10cm apart. , 14cm and 12cm.
Find the diameter of the circle to 1 decimal place
(A) 4.4cm
(B) 8.2cm
(C) 14.8cm
(D) 16.5cm
9.
The domain of log 4 5 is
(A) 4 5
(B) 4, 5
(C) 4 5
(D) 4, 5
10. Which of the following represents the derivate of sin ?
(A) √
(B) √
(C) √
(D) √
End of Section 1
A B
C D
R
T
O
5
Section II – Extended Response All necessary working should be shown in every question.
Question 11 (15 marks) - Start on the appropriate page in your answer booklet
Marks
a) Evaluate cos 4
3
b) Findlog
, using the substitution log
2
c) Prove the identity 1 sin 2 cos 2
cos sin2 cos
2
d) Solve for
41
3
3
e)
(i) Show that a root of the continuous function ln 1 lies between 0.8 and 0.9.
(ii) Hence use the halving the interval method to find the value of the root correct to 1 decimal place.
1 1
f) i Find tan tan
ii Hence sketch tan tan for 2 2
2 1
End of Question 11
6
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
Marks
a) When a polynomial is divided by 4 the remainder is 2 3. What is the remainder when is divided by 2
2
b) Inthegivendiagram, and aretangentsand , , arecollinear.
Copyortracethediagramintoyourwritingbooklet.Provethatthepoints , , , areconcyclic.
3
P
Q
R
S
T
Not to scale
7
Question 12 continues on the following page
Question 12 (continued)
c)
Points P 2 , and 2 , lies on the parabola 4 . The chord subtends a right angle at the origin.
(i) Prove 4
(ii) Find the equation of the locus of , the midpoint of .
2 3
d) Findthecoefficientof intheexpressionof
2
e) Prove by mathematical induction
1 2 2 2 3 2 ⋯ 2 1 2 2
for positive integers 1
3
End of Question 12
x
y
P
Q
M
O
x2
= 4ay (2ap, ap2)
(2aq, aq2)
Not to Scale
8
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
Marks
a) (i) Express √3 sin cos in the form sin where 0 and 0 .
(ii) Hence state the least value of √3 sin cos and the smallest positive value
of for this least value to occur.
2
2
b) In the cubic equation 3 2 4 5 0 the sum of the roots is equal to twice their product. Find the values of .
3
c) Find the number of arrangements of the letters of the word if there are 3 letters between and .
2
d) Below is the graph of a function
Copy the diagram in your booklet, and on the same set of axes sketch a possible graph for
.
2
e) It is estimated that the rate of increase in the population of a particular species of bird is given by the equation
where and are positive constants.
(i) Verify that for any positive constant , the expression
satisfies the above differential equation.
(ii) What can be deduced about as increases?
3 1
End of Question 13
-1 1 2 3
2
y
x
9
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a)
Water is poured into a conical vessel at a constant rate of 24cm /s. The depth of water is cm at any time seconds.
(i) Show that the volume of water is given by .
(ii) Find the rate at which the depth of water is increasing when 16cm.
(iii) Hence find that rate of increase of the area of surface of the liquid when 16.
1
2
1
b) The acceleration of a particle is given by the equation 8 1 ,where is the
displacement in centimetres from a fixed point , after seconds. Initially the particle is moving from with speed 2cm/s in a negative direction.
(i) Prove the general result .
(ii) Hence show that the speed is given by 2 1 cm/s.
(iii) Find an expression for in terms of .
2
2
2
Question 14 continues on the following page
30cm
30cmx
cmh
Not to Scale
10
Question 14 (continued)
c) A projectile is fired from the origin with velocity with an angle of elevation , where . YOU MAY ASSUME
cos ,12
sin
Where and y are the horizontal and vertical displacements from , seconds after firing
(i) Show the equation of flight can be expressed as
tan14
1 tan where2
(ii) Show that a point , can be hit by firing at 2 different angles and provided 4 .
(iv) Show that no point above the -axis can be hit by firing at 2 different angles
and satisfying both and .
2
2
1
End of Paper.
11
STANDARD INTEGRALS
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