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Q1 1 marks Product & Quotient Rules View

Given that $y = \frac { ( 1 - 3 x ) ^ { 2 } } { 2 x ^ { \frac { 3 } { 2 } } }$, which one of the following is a correct expression for $\frac { d y } { d x }$ ?
A $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
B $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } + \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
C $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
D $- \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } + \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
E $\quad - \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
F $- \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$

Q2 1 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

$PQRS$ is a rectangle.
The coordinates of $P$ and $Q$ are $( 0,6 )$ and $( 1,8 )$ respectively.
The perpendicular to $PQ$ at $Q$ meets the $x$-axis at $R$.
What is the area of $PQRS$ ?
A $\frac { 5 } { 2 }$
B $4 \sqrt { 10 }$
C 20
D $8 \sqrt { 10 }$
E 40

Q3 1 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View

The first term of a geometric progression is $2 \sqrt { 3 }$ and the fourth term is $\frac { 9 } { 4 }$ What is the sum to infinity of this geometric progression?
A $- 2 ( 2 - \sqrt { 3 } )$
B $4 ( 2 \sqrt { 3 } - 3 )$
C $\frac { 16 ( 8 \sqrt { 3 } + 9 ) } { 37 }$
D $\frac { 4 ( 2 \sqrt { 3 } - 3 ) } { 7 }$
$\mathbf { E } \frac { 4 ( 2 \sqrt { 3 } + 3 ) } { 7 }$
F $\quad 2 ( 2 + \sqrt { 3 } )$
G $4 ( 2 \sqrt { 3 } + 3 )$

Q4 1 marks Standard trigonometric equations Evaluate trigonometric expression given a constraint View

The following question appeared in an examination:
Given that $\tan x = \sqrt { 3 }$, find the possible values of $\sin 2 x$.
A student gave the following answer:
$$\begin{aligned} & \tan x = \sqrt { 3 } \text { so } x = 60 ^ { \circ } \text { and } 2 x = 120 ^ { \circ } \\ & \text { therefore } \sin 2 x = \frac { \sqrt { 3 } } { 2 } \end{aligned}$$
Which one of the following statements is correct?
A $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, and this is fully supported by the reasoning given in the student's answer.
B $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
C $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.
D $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
E $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.

Q5 1 marks Proof True/False Justification View

Consider the following three statements:
$1 \quad 10 p ^ { 2 } + 1$ and $10 p ^ { 2 } - 1$ are both prime when $p$ is an odd prime.
2 Every prime greater than 5 is of the form $6 n + 1$ for some integer $n$.
3 No multiple of 7 greater than 7 is prime.
The result $91 = 7 \times 13$ can be used to provide a counterexample to which of the above statements?
A none of them
B 1 only
C 2 only
D 3 only
E 1 and 2 only
F 1 and 3 only
G 2 and 3 only
H 1, 2 and 3

Q6 1 marks Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View

A sequence $u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots$ is defined as follows:
$$\begin{aligned} & u _ { 0 } = 1 \\ & u _ { n } = \int _ { 0 } ^ { 1 } 4 x u _ { n - 1 } d x \quad \text { for } n \geqslant 1 \end{aligned}$$
What is the value of $u _ { 1000 }$ ?
A $2 ^ { 1000 }$
B $4 ^ { 1000 }$
C $\frac { 4 } { 1000 ! }$
D $\frac { 4 } { 1001 ! }$
$\mathbf { E } \quad \frac { 2 ^ { 1000 } } { 1000 ! }$
F $\frac { 4 ^ { 1000 } } { 1000 ! }$
G $\frac { 2 ^ { 1000 } } { 1001 ! }$
$\mathbf { H } \frac { 4 ^ { 1000 } } { 1001 ! }$

Q7 1 marks Exponential Functions True/False or Multiple-Statement Verification View

The graphs of two functions are shown here:

$y = a ^ { x }$ is shown with a solid line, where $a$ is a positive real number
$y = f ( x )$ is shown with a dashed line

Which of the following statements $( \mathbf { 1 } , \mathbf { 2 } , \mathbf { 3 } , \mathbf { 4 } )$ could be true?
$1 f ( x ) = b ^ { x }$ for some $b > a$
$2 f ( x ) = b ^ { x }$ for some $b < a$
$3 f ( x ) = a ^ { k x }$ for some $k > 1$
$4 f ( x ) = a ^ { k x }$ for some $k < 1$
A $\mathbf { 1 }$ only
B 2 only
C 3 only
D 4 only
E 1 and 3 only
F 1 and 4 only
G 2 and 3 only
H 2 and 4 only

Q8 1 marks Laws of Logarithms Compare or Order Logarithmic Values View

Which one of the following numbers is smallest in value?
A $\quad \log _ { 2 } 7$
B $\left( 2 ^ { - 3 } + 2 ^ { - 2 } \right) ^ { - 1 }$
C $2 ^ { ( \pi / 3 ) }$
D $\frac { 1 } { 4 ( \sqrt { 2 } - 1 ) ^ { 3 } }$
E $\quad 4 \sin ^ { 2 } \left( \frac { \pi } { 4 } \right)$

Q9 1 marks Proof True/False Justification View

Consider the following attempt to prove this true theorem:
Theorem: $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$ has no solutions with $a , b$ and $c$ positive integers.
Attempted proof:
Suppose that there are positive integers $a , b$ and $c$ such that $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$.
I We have $a ^ { 3 } = c ^ { 3 } - b ^ { 3 }$.
II $\quad$ Hence $a ^ { 3 } = ( c - b ) \left( c ^ { 2 } + c b + b ^ { 2 } \right)$.
III It follows that $a = c - b$ and $a ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$, since $a \leqslant a ^ { 2 }$ and $c - b \leqslant c ^ { 2 } + c b + b ^ { 2 }$.
IV Eliminating $a$, we have $( c - b ) ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
V Multiplying out, we have $c ^ { 2 } - 2 c b + b ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
VI Hence $3 c b = 0$ so one of $b$ and $c$ is zero.
But this is a contradiction to the original assumption that all of $a , b$ and $c$ are positive. It follows that the equation has no solutions.
Comment on this proof by choosing one of the following options:
A The proof is correct
B The proof is incorrect and the first mistake occurs on line I.
C The proof is incorrect and the first mistake occurs on line II.
D The proof is incorrect and the first mistake occurs on line III.
E The proof is incorrect and the first mistake occurs on line IV.
F The proof is incorrect and the first mistake occurs on line V.
G The proof is incorrect and the first mistake occurs on line VI.

Q10 1 marks Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

$f ( x )$ is a function defined for all real values of $x$.
Which one of the following is a sufficient condition for $\int _ { 1 } ^ { 3 } f ( x ) d x = 0$ ?
A $f ( 2 ) = 0$
B $f ( 1 ) = f ( 3 ) = 0$
C $f ( - x ) = - f ( x )$ for all $x$
D $f ( x + 2 ) = - f ( 2 - x )$ for all $x$
E $\quad f ( x - 2 ) = - f ( 2 - x )$ for all $x$

Q11 1 marks Areas by integration View

The function $f ( x )$ is increasing and $f ( 0 ) = 0$.
The positive constants $a$ and $b$ are such that $a < b$.
The area of the region enclosed by the curve $y = f ( x )$, the $x$-axis and the lines $x = a$ and $x = b$ is denoted by $R$.
The function $g ( x )$ is defined by $g ( x ) = f ( x ) + 2 f ( b )$.
Which of the following is an expression for the area enclosed by the curve $y = g ( x )$, the $x$-axis and the lines $x = a$ and $x = b$ ?
A $\quad R + ( b - a ) f ( b )$
B $R + 2 ( b - a ) f ( b )$
C $\quad R + 2 f ( b ) - f ( a )$
D $R + 2 f ( b )$
E $\quad R + ( f ( b ) ) ^ { 2 }$
F $\quad R + ( f ( b ) ) ^ { 2 } - ( f ( a ) ) ^ { 2 }$
G $\quad R + 2 ( f ( b ) - f ( a ) ) f ( b )$

Q12 1 marks Trig Graphs & Exact Values View

The diagram shows the graphs of $y = \sin 2 x$ and $y = \cos 2 x$ for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
Which one of the following is not true?
A $\cos 2 x < \sin 2 x < \tan x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
B $\cos 2 x < \tan x < \sin 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
C $\sin 2 x < \cos 2 x < \tan x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
D $\sin 2 x < \tan x < \cos 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
E $\tan x < \sin 2 x < \cos 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
F $\tan x < \cos 2 x < \sin 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

Q13 1 marks Proof View

The positive real numbers $a \times 10 ^ { - 3 } , b \times 10 ^ { - 2 }$ and $c \times 10 ^ { - 1 }$ are each in standard form, and
$$\left( a \times 10 ^ { - 3 } \right) + \left( b \times 10 ^ { - 2 } \right) = \left( c \times 10 ^ { - 1 } \right)$$
Which of the following statements (I, II, III, IV) must be true?
$$\begin{array} { l l } \text { I } & a > 9 \\ \text { II } & b > 9 \\ \text { III } & a < c \\ \text { IV } & b < c \end{array}$$
A I only
B II only
C I and II only
D I and III only
E I and IV only
F II and III only
G II and IV only
H I, II, III and IV

Q14 1 marks Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

The diagram below shows the graph of $y = x ^ { 2 } - 2 b x + c$. The vertex of this graph is at the point $P$.
Which one of the following could be the graph of $y = x ^ { 2 } - 2 B x + c$, where $B > b$ ?

Q15 1 marks Sequences and series, recurrence and convergence View

The function $f$ is defined on the positive integers as follows:
$$f ( 1 ) = 5 , \text { and for } n \geqslant 1 : \quad \begin{array} { l l } f ( n + 1 ) = 3 f ( n ) + 1 & \text { if } f ( n ) \text { is odd } \\ & f ( n + 1 ) = \frac { 1 } { 2 } f ( n ) \end{array} \text { if } f ( n ) \text { is even }$$
The function $g$ is defined on the positive integers as follows:
$$\begin{array} { l l } g ( 1 ) = 3 , \text { and for } n \geqslant 1 : \quad & g ( n + 1 ) = g ( n ) + 5 \\ & \text { if } g ( n ) \text { is odd } \\ g ( n + 1 ) = \frac { 1 } { 2 } g ( n ) & \text { if } g ( n ) \text { is even } \end{array}$$
What is the value of $f ( 1000 ) - g ( 1000 )$ ?
A - 6
B - 5
C 1
D 2
E 4
F 8

Q16 1 marks Proof View

Consider the following statement:
(*) If $f ( x )$ is an integer for every integer $x$, then $f ^ { \prime } ( x )$ is an integer for every integer $x$.
Which one of the following is a counterexample to (*)?
A $f ( x ) = \frac { x ^ { 3 } + x + 1 } { 4 }$
B $f ( x ) = \frac { x ^ { 4 } + x ^ { 2 } + x } { 2 }$
C $f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x } { 2 }$
D $f ( x ) = \frac { x ^ { 4 } + 2 x ^ { 3 } + x ^ { 2 } } { 4 }$

Q17 1 marks Proof View

A set $S$ of whole numbers is called stapled if and only if for every whole number $a$ which is in $S$ there exists a prime factor of $a$ which divides at least one other number in $S$.
Let $T$ be a set of whole numbers. Which of the following is true if and only if $T$ is not stapled?
A For every number $a$ which is in $T$, there is no prime factor of $a$ which divides every other number in $T$.
B For every number $a$ which is in $T$, there is no prime factor of $a$ which divides at least one other number in $T$.
C For every number $a$ which is in $T$, there is a prime factor of $a$ which does not divide any other number in $T$.
D For every number $a$ which is in $T$, there is a prime factor of $a$ which does not divide at least one other number in $T$.
E There exists a number $a$ which is in $T$ such that there is no prime factor of $a$ which divides every other number in $T$.
F There exists a number $a$ which is in $T$ such that there is no prime factor of $a$ which divides at least one other number in $T$.
G There exists a number $a$ which is in $T$ such that there is a prime factor of $a$ which does not divide any other number in $T$.
H There exists a number $a$ which is in $T$ such that there is a prime factor of $a$ which does not divide at least one other number in $T$.

Q18 1 marks Laws of Logarithms Solve a Logarithmic Inequality View

Consider the following problem:
Solve the inequality $\left( \frac { 1 } { 4 } \right) ^ { n } < \left( \frac { 1 } { 32 } \right) ^ { 10 }$, where $n$ is a positive integer.
A student produces the following argument:
$$\begin{array} { r l r } \left( \frac { 1 } { 4 } \right) ^ { n } & < \left( \frac { 1 } { 32 } \right) ^ { 10 } & \\ \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) ^ { n } & < \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) ^ { 10 } & ( \mathrm { I } ) \\ n \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) & < 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) & \downarrow ( \mathrm { II } ) \\ n < \frac { 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) } { \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) } & \downarrow ( \mathrm { III } ) \\ n < \frac { 10 \times 5 } { 2 } = 25 & \downarrow ( \mathrm { IV } ) \\ 1 \leqslant n \leqslant 24 & \downarrow ( \mathrm {~V} ) \end{array}$$
Which step (if any) in the argument is invalid?
A There are no invalid steps; the argument is correct
B Only step (I) is invalid; the rest are correct
C Only step (II) is invalid; the rest are correct
D Only step (III) is invalid; the rest are correct
E Only step (IV) is invalid; the rest are correct
F Only step (V) is invalid; the rest are correct

Q19 1 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View

Which one of the following is a sufficient condition for the equation $x ^ { 3 } - 3 x ^ { 2 } + a = 0$, where $a$ is a constant, to have exactly one real root?
A $a > 0$
B $a \leqslant 0$
C $\quad a \geqslant 4$
D $a < 4$
$\mathbf { E } \quad | a | > 4$
$\mathbf { F } \quad | a | \leqslant 4$
G $\quad a = \frac { 9 } { 4 }$
$\mathbf { H } \quad | a | = \frac { 3 } { 2 }$

Q20 1 marks Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View

I have forgotten my 5-character computer password, but I know that it consists of the letters $\mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e }$ in some order. When I enter a potential password into the computer, it tells me exactly how many of the letters are in the correct position.
When I enter abcde, it tells me that none of the letters are in the correct position. The same happens when I enter cdbea and eadbc.
Using the best strategy, how many further attempts must I make in order to guarantee that I can deduce the correct password?
A None: I can deduce it immediately
B One
C Two
D Three
E More than three

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