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Q1 1 marks Indices and Surds Solving Equations Involving Surds View

Given that
$$\frac { 1 } { \sqrt { x } - 6 } - \frac { 1 } { \sqrt { x } + 6 } = \frac { 3 } { 11 }$$
what is the value of $x$ ?
A $2 \sqrt { 15 }$ B $4 \sqrt { 5 }$ C $5 \sqrt { 2 }$ D $\sqrt { 58 }$ E 50 F 58 G 60 H 80

Q2 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Evaluate
$$\int _ { 9 } ^ { 16 } \left( \frac { 1 } { \sqrt { x } } + \sqrt { x } \right) ^ { 2 } \mathrm {~d} x - \int _ { 9 } ^ { 16 } \left( \frac { 1 } { \sqrt { x } } - \sqrt { x } \right) ^ { 2 } \mathrm {~d} x$$
A 0 B 2 C 4 D 7 E 14 F 28 G 75 H 175

Q3 1 marks Exponential Functions True/False or Multiple-Statement Verification View

Consider the claim: For all positive real numbers $x$ and $y$,
$$\sqrt { x ^ { y } } = x ^ { \sqrt { y } }$$
Which of the following is/are a counterexample to the claim? I $x = 1 , y = 16$ II $x = 2 , y = 8$ III $x = 3 , y = 4$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q4 1 marks Proof True/False Justification View

A student attempts to answer the following question. What is the largest number of consecutive odd integers that are all prime? The student's attempt is as follows: I There are two consecutive odd integers that are prime (for example: 17, 19). II Any three consecutive odd integers can be written in the form $n - 2 , n , n + 2$ for some $n$. III If $n$ is one more than a multiple of 3 , then $n + 2$ is a multiple of 3 . IV If $n$ is two more than a multiple of 3 , then $n - 2$ is a multiple of 3 . V The only other possibility is that $n$ is a multiple of 3 . VI In each case, one of the integers is a multiple of 3 , so not prime. VII Therefore the largest number of consecutive odd integers that are all prime is two.
Which of the following best describes this attempt?
A It is completely correct. B It is incorrect, and the first error is on line I. C It is incorrect, and the first error is on line II. D It is incorrect, and the first error is on line III. E It is incorrect, and the first error is on line IV. F It is incorrect, and the first error is on line V. G It is incorrect, and the first error is on line VI. H It is incorrect, and the first error is on line VII.

Q5 1 marks Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View

Consider the two statements R: $\quad k$ is an integer multiple of $\pi$
$$\mathrm { S } : \quad \int _ { 0 } ^ { k } \sin 2 x \mathrm {~d} x = 0$$
Which of the following statements is true? A $R$ is necessary and sufficient for $S$. B R is necessary but not sufficient for $S$. C R is sufficient but not necessary for $S$. D $R$ is not necessary and not sufficient for $S$.

Q6 1 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View

Consider the following equation where $a$ is a real number and $a > 1$ :
$$( * ) \quad a ^ { x } = x$$
Which of the following equations must have the same number of real solutions as $( * )$ ? I $\quad \log _ { a } x = x$ II $\quad a ^ { 2 x } = x ^ { 2 }$ III $a ^ { 2 x } = 2 x$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q7 1 marks Straight Lines & Coordinate Geometry Slope and Angle Between Lines View

The graph of the line $a x + b y = c$ is drawn, where $a , b$ and $c$ are real non-zero constants. Which one of the following is a necessary but not sufficient condition for the line to have a positive gradient and a positive $y$-intercept?
A $\frac { c } { b } > 0$ and $\frac { a } { b } < 0$ B $\frac { c } { b } < 0$ and $\frac { a } { b } > 0$ C $a > b > c$ D $a < b < c$ E $\quad a$ and $c$ have opposite signs F $\quad a$ and $c$ have the same sign

Q8 1 marks Proof True/False Justification View

A student draws a triangle that is acute-angled or obtuse-angled but not right-angled. The student counts the number of straight lines that divide the triangle into two triangles, at least one of which is right-angled.
Which of the following statements is/are true? I The student can draw a triangle for which there is exactly 1 such straight line. II The student can draw a triangle for which there are exactly 2 such straight lines. III The student can draw a triangle for which there are exactly 3 such straight lines.
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q9 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View

Consider the following statement about a pentagon P: (*) If at least one of the interior angles in P is $108 ^ { \circ }$, then all the interior angles in P form an arithmetic sequence.
Which of the following is/are true? I The statement (*) II The contrapositive of (*) III The converse of (*)
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q10 1 marks Completing the square and sketching Sign analysis of quadratic coefficients and expressions from a graph View

Here is an attempt to solve the inequality $x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$ by completing the square:
$$x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$$
I if and only if $x ^ { 4 } - 2 x ^ { 2 } + 1 < 4$ II if and only if $\left( x ^ { 2 } - 1 \right) ^ { 2 } < 4$ III if and only if $- 2 < x ^ { 2 } - 1 < 2$ IV if and only if $x ^ { 2 } - 1 < 2$ V if and only if $x ^ { 2 } < 3$ VI if and only if $- \sqrt { 3 } < x < \sqrt { 3 }$
Which of the following statements is true? A The argument is completely correct. B The first error occurs in line I. C The first error occurs in line II. D The first error occurs in line III. E The first error occurs in line IV. F The first error occurs in line V. G The first error occurs in line VI.

Q11 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View

In this question, $k$ is a positive integer. Consider the following theorem: If $2 ^ { k } + 1$ is a prime, then $k$ is a power of $2 . \quad ( * )$ Which of the following statements, taken individually, is/are equivalent to (*)? I If $k$ is a power of 2 , then $2 ^ { k } + 1$ is prime. II $\quad 2 ^ { k } + 1$ is not prime only if $k$ is not a power of 2 . III A sufficient condition for $k$ to be a power of 2 is that $2 ^ { k } + 1$ is prime.

	Statement I is equivalent to (*)	Statement II is equivalent to (*)	Statement III is equivalent to (*)
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

Q12 1 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

In this question, $p$ is a real constant. The equation $\sin x \cos ^ { 2 } x = p ^ { 2 } \sin x$ has $n$ distinct solutions in the range $0 \leq x \leq 2 \pi$ Which of the following statements is/are true?
I $n = 3$ is sufficient for $p > 1$ II $n = 7$ only if $- 1 < p < 1$
A none of them B I only C II only D I and II

Q13 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View

Let $x$ be a real number. Which one of the following statements is a sufficient condition for exactly three of the other four statements?
A $x \geq 0$ B $x = 1$ C $x = 0$ or $x = 1$ D $x \geq 0$ or $x \leq 1$ E $\quad x \geq 0$ and $x \leq 1$

Q14 1 marks Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Three lines are given by the equations:
$$\begin{aligned} & a x + b y + c = 0 \\ & b x + c y + a = 0 \\ & c x + a y + b = 0 \end{aligned}$$
where $a$, $b$ and $c$ are non-zero real numbers. Which one of the following is correct? A If two of the lines are parallel, then all three are parallel. B If two of the lines are parallel, then the third is perpendicular to the other two. C If two of the lines are parallel, then the third is parallel to $y = x$. D If two of the lines are parallel, then the third is perpendicular to $y = x$. E If two of the lines are perpendicular, then all three meet at a point. F If two of the lines are perpendicular, then the third is parallel to $y = x$. G If two of the lines are perpendicular, then the third is perpendicular to $y = x$.

Q15 1 marks Arithmetic Sequences and Series View

The base 10 number 0.03841 has the value
$$0 \times 10 ^ { - 1 } + 3 \times 10 ^ { - 2 } + 8 \times 10 ^ { - 3 } + 4 \times 10 ^ { - 4 } + 1 \times 10 ^ { - 5 } = 0.03841$$
Similarly, the base 2 number 0.01101 has the value
$$0 \times 2 ^ { - 1 } + 1 \times 2 ^ { - 2 } + 1 \times 2 ^ { - 3 } + 0 \times 2 ^ { - 4 } + 1 \times 2 ^ { - 5 } = \frac { 13 } { 32 }$$
What is the value of the recurring base 2 number $0 . \dot { 0 } 01 \dot { 1 } = 0.001100110011 \ldots$ ? A $\frac { 1 } { 3 }$ B $\frac { 1 } { 5 }$ C $\frac { 1 } { 15 }$ D $\frac { 2 } { 15 }$ E $\frac { 4 } { 15 }$ F $\frac { 3 } { 16 }$ G $\frac { 5 } { 16 }$ H $\frac { 6 } { 31 }$

Q16 1 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View

A sequence is defined by:
$$\begin{aligned} u _ { 1 } & = a \\ u _ { 2 } & = b \\ u _ { n + 2 } & = u _ { n } + u _ { n + 1 } \quad \text { for } n \geq 1 \end{aligned}$$
where $a$ and $b$ are positive integers. The highest common factor of $a$ and $b$ is 7 . Which of the following statements must be true? I $u _ { 2023 }$ is a multiple of 7 II If $u _ { 1 }$ is not a factor of $u _ { 2 }$, then $u _ { 1 }$ is not a factor of $u _ { n }$ for any $n > 1$ III The highest common factor of $u _ { 1 }$ and $u _ { 5 }$ is 7
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q17 1 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

The ceiling of $x$, written $[ x ]$, is defined to be the value of $x$ rounded up to the nearest integer. For example: $\quad \lceil \pi \rceil = 4 , \quad \lceil 2.1 \rceil = 3 , \quad \lceil 8 \rceil = 8$ What is the value of the following integral?
$$\int _ { 0 } ^ { 99 } 2 ^ { \lceil x \rceil } d x$$
A $2 ^ { 99 }$ B $\quad 2 ^ { 99 } - 1$ C $2 ^ { 99 } - 2$ D $2 ^ { 100 }$ E $\quad 2 ^ { 100 } - 1$ F $\quad 2 ^ { 100 } - 2$

Q18 1 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View

The equation $x ^ { 4 } + b x ^ { 2 } + c = 0$ has four distinct real roots if and only if which of the following conditions is satisfied?
A $b ^ { 2 } > 4 c$ B $b ^ { 2 } < 4 c$ C $c > 0$ and $b > 2 \sqrt { c }$ D $c > 0$ and $b < - 2 \sqrt { c }$ E $\quad c < 0$ and $b < 0$ F $\quad c < 0$ and $b > 0$

Q19 1 marks Factor & Remainder Theorem True/False or Multiple-Statement Evaluation View

In this question, $f ( x )$ is a non-constant polynomial, and $g ( x ) = x f ^ { \prime } ( x )$ $f ( x ) = 0$ for exactly $M$ real values of $x$. $g ( x ) = 0$ for exactly $N$ real values of $x$. Which of the following statements is/are true? I It is possible that $M < N$ II It is possible that $M = N$ III It is possible that $M > N$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Q20 1 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f$ be a polynomial with real coefficients. The integral $I _ { p , q }$ where $p < q$ is defined by
$$I _ { p , q } = \int _ { p } ^ { q } ( f ( x ) ) ^ { 2 } - ( f ( | x | ) ) ^ { 2 } \mathrm {~d} x$$
Which of the following statements must be true? $1 I _ { p , q } = 0$ only if $0 < p$ $2 f ^ { \prime } ( x ) < 0$ for all $x$ only if $I _ { p , q } < 0$ for all $p < q < 0$ $3 \quad I _ { p , q } > 0$ only if $p < 0$
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

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