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QA1 4 marks Differentiation from First Principles View

Define the right derivative of a function $f$ at $x = a$ to be the following limit if it exists. $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( a + h ) - f ( a ) } { h }$, where $h \rightarrow 0 ^ { + }$ means $h$ approaches 0 only through positive values.
Statements
(1) If $f$ is differentiable at $x = a$ then $f$ has a right derivative at $x = a$.
(2) $f ( x ) = | x |$ has a right derivative at $x = 0$.
(3) If $f$ has a right derivative at $x = a$ then $f$ is continuous at $x = a$.
(4) If $f$ is continuous at $x = a$ then $f$ has a right derivative at $x = a$.

QA2 4 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

Suppose a rectangle $EBFD$ is given and a rhombus $ABCD$ is inscribed in it so that the point $A$ is on side $ED$ of the rectangle. The diagonals of $ABCD$ intersect at point $G$.
Statements
(5) Triangles $CGD$ and $DFB$ must be similar. (6) It must be true that $\frac { AC } { BD } = \frac { EB } { ED }$. (7) Triangle $CGD$ cannot be similar to triangle $AEB$. (8) For any given rectangle $EBFD$, a rhombus $ABCD$ as described above can be constructed.

QA3 4 marks Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View

This question is about complex numbers.
Statements
(9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant. (10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0. (11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions. (12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.

QA4 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Statements
(13) $\lim _ { x \rightarrow 0 } e ^ { \frac { 1 } { x } } = + \infty$. (14) The following inequality is true. $$\lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { 100 } } < \lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { \frac { 1 } { 100 } } }$$ (15) For any positive integer $n$, $$\int _ { - n } ^ { n } x ^ { 2023 } \cos ( n x ) \, dx < \frac { n } { 2023 }$$ (16) There is no polynomial $p ( x )$ for which there is a single line that is tangent to the graph of $p ( x )$ at exactly 100 points.

QA5 4 marks Indices and Surds Ordering and Comparing Surd or Numerical Values View

Statements
(17) $4 < \sqrt { 5 + 5 \sqrt { 5 } }$. (18) $\log _ { 2 } 11 < \frac { 1 + \log _ { 2 } 61 } { 2 }$. (19) $( 2023 ) ^ { 2023 } < ( 2023 ! ) ^ { 2 }$. (20) $92 ^ { 100 } + 93 ^ { 100 } < 94 ^ { 100 }$.

QA6 4 marks Sequences and Series Convergence/Divergence Determination of Numerical Series View

For a sequence $a _ { i }$ of real numbers, we say that $\sum a _ { i }$ converges if $\lim _ { n \rightarrow \infty } \left( \sum _ { i = 1 } ^ { n } a _ { i } \right)$ is finite. In this question all $a _ { i } > 0$.
Statements
(21) If $\sum a _ { i }$ converges, then $a _ { i } \rightarrow 0$ as $i \rightarrow \infty$. (22) If $a _ { i } < \frac { 1 } { i }$ for all $i$, then $\sum a _ { i }$ converges. (23) If $\sum a _ { i }$ converges, then $\sum ( - 1 ) ^ { i } a _ { i }$ also converges. (24) If $\sum a _ { i }$ does not converge, then $\sum i \tan \left( a _ { i } \right)$ cannot converge.

QA7 4 marks Number Theory Congruence Reasoning and Parity Arguments View

Statements
(25) To divide an integer $b$ by a nonzero integer $d$, define a quotient $q$ and a remainder $r$ to be integers such that $b = q d + r$ and $| r | < | d |$. Such integers $q$ and $r$ always exist and are both unique for given $b$ and $d$. (26) To divide a polynomial $b ( x )$ by a nonzero polynomial $d ( x )$, define a quotient $q ( x )$ and a remainder $r ( x )$ to be polynomials such that $b = q d + r$ and $\deg ( r ) < \deg ( d )$. (Here $b ( x )$ and $d ( x )$ have real coefficients and the 0 polynomial is taken to have negative degree by convention.) Such polynomials $q ( x )$ and $r ( x )$ always exist and are both unique for given $b ( x )$ and $d ( x )$. (27) Suppose that in the preceding question $b ( x )$ and $d ( x )$ have rational coefficients. Then $q ( x )$ and $r ( x )$, if they exist, must also have rational coefficients. (28) The least positive number in the set $$\left\{ \left( a \times 2023 ^ { 2020 } \right) + \left( b \times 2020 ^ { 2023 } \right) \right\}$$ as $a$ and $b$ range over all integers is 3.

QA8 4 marks Tree Diagrams Multi-Stage Sequential Process View

You play the following game with three fair dice. (When each one is rolled, any one of the outcomes $1,2,3,4,5,6$ is equally likely.) In the first round, you roll all three dice. You remove every die that shows 6. If any dice remain, you roll all the remaining dice again in the second round. Again you remove all dice showing 6 and continue.
Questions
(29) Let the probability that you are able to play the second round be $\frac { a } { b }$, where $a$ and $b$ are integers with $\gcd = 1$. Write the numbers $a$ and $b$ separated by a comma. (30) Let the probability that you are able to play the second round but not the third round be $\frac { c } { d }$ where $c$ and $d$ are integers with $\gcd = 1$. Write only the integer $c$ as your answer.

QA9 4 marks Vectors 3D & Lines Shortest Distance Between Two Lines View

Two lines $\ell _ { 1 }$ and $\ell _ { 2 }$ in 3-dimensional space are given by $$\ell _ { 1 } = \{ ( t - 9 , - t + 7 , 6 ) \mid t \in \mathbb { R } \} \quad \text{and} \quad \ell _ { 2 } = \{ ( 7 , s + 3 , 3 s + 4 ) \mid s \in \mathbb { R } \}.$$
Questions
(31) The plane passing through the origin and not intersecting either of $\ell _ { 1 }$ and $\ell _ { 2 }$ has equation $ax + by + cz = d$. Write the value of $| a + b + c + d |$ where $a, b, c, d$ are integers with $\gcd = 1$. (32) Let $r$ be the smallest possible radius of a circle that has a point on $\ell _ { 1 }$ as well as a point on $\ell _ { 2 }$. It is given that $r ^ { 2 }$ (i.e., the square of the smallest radius) is an integer. Write the value of $r ^ { 2 }$.

QA10 4 marks Implicit equations and differentiation Horizontal tangent point on implicit curve (single-step) View

Consider the part of the graph of $y ^ { 2 } + x ^ { 3 } = 15 x y$ that is strictly to the right of the $Y$-axis, i.e., take only the points on the graph with $x > 0$.
Questions
(33) Write the least possible value of $y$ among considered points. If there is no such real number, write NONE. (34) Write the largest possible value of $y$ among considered points. If there is no such real number, write NONE.

QB1 11 marks Number Theory Congruence Reasoning and Parity Arguments View

We want to find odd integers $n > 1$ for which $n$ is a factor of $2023 ^ { n } - 1$.
(a) Find the two smallest such integers.
(b) Prove that there are infinitely many such integers.

QB2 12 marks Sequences and Series Recurrence Relations and Sequence Properties View

Let $\mathbb { Z } ^ { + }$ denote the set of positive integers. We want to find all functions $g : \mathbb { Z } ^ { + } \rightarrow \mathbb { Z } ^ { + }$ such that the following equation holds for any $m, n$ in $\mathbb { Z } ^ { + }$. $$g ( n + m ) = g ( n ) + n m ( n + m ) + g ( m )$$ Prove that $g ( n )$ must be of the form $\sum _ { i = 0 } ^ { d } c _ { i } n ^ { i }$ and find the precise necessary and sufficient condition(s) on $d$ and on the coefficients $c _ { 0 } , \ldots , c _ { d }$ for $g$ to satisfy the required equation.

QB3 13 marks Roots of polynomials Location and bounds on roots View

Suppose that for a given polynomial $p ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c x + d$, there is exactly one real number $r$ such that $p ( r ) = 0$.
(a) If $a, b, c, d$ are rational, show that $r$ must be rational.
(b) If $a, b, c, d$ are integers, show that $r$ must be an integer.
Possible hint: Also consider the roots of the derivative $p ^ { \prime } ( x )$.

QB4 14 marks Permutations & Arrangements Linear Arrangement with Constraints View

There are $n$ students in a class and no two of them have the same height. The students stand in a line, one behind another, in no particular order of their heights.
(a) How many different orders are there in which the shortest student is not in the first position and the tallest student is not in the last position?
(b) The badness of an ordering is the largest number $k$ with the following property. There is at least one student $X$ such that there are $k$ students taller than $X$ standing ahead of $X$. Find a formula for $g _ { k } ( n ) =$ number of orderings of $n$ students with badness $k$.
Example: The ordering $64\,61\,67\,63\,62\,66\,65$ (the numbers denote heights) has badness 3 as the student with height 62 has three taller students (with heights 64, 67 and 63) standing ahead in the line and nobody has more than 3 taller students standing ahead.
Possible hints for (b): It may be useful to first count orderings of badness 1 and/or to find $f _ { k } ( n ) =$ the number of orderings of $n$ students with badness less than or equal to $k$.

QB5 15 marks Proof Proof That a Map Has a Specific Property View

Throughout this question every mentioned function is required to be a differentiable function from $\mathbb { R }$ to $\mathbb { R }$. The symbol $\circ$ denotes composition of functions.
(a) Suppose $f \circ f = f$. Then for each $x$, one must have $f ^ { \prime } ( x ) = $ \_\_\_\_ or $f ^ { \prime } ( f ( x ) ) = $ \_\_\_\_. Complete the sentence and justify.
(b) For a non-constant $f$ satisfying $f \circ f = f$, it is known and you may assume that the range of $f$ must have one of the following forms: $\mathbb { R } , ( - \infty , b ] , [ a , \infty )$ or $[ a , b ]$. Show that in fact the range must be all of $\mathbb { R }$ and deduce that there is a unique such function $f$. (Possible hints: For each $y$ in the range of $f$, what can you say about $f ( y )$? If the range has a maximum element $b$ what can you say about the derivative of $f$?)
(c) Suppose that $g \circ g \circ g = g$ and that $g \circ g$ is a non-constant function. Show that $g$ must be onto, $g$ must be strictly increasing or strictly decreasing and that there is a unique such increasing $g$.

QB6 15 marks Number Theory Combinatorial Number Theory and Counting View

Starting with any given positive integer $a > 1$ the following game is played. If $a$ is a perfect square, take its square root. Otherwise take $a + 3$. Repeat the procedure with the new positive integer (i.e., with $\sqrt { a }$ or $a + 3$ depending on the case). The resulting set of numbers is called the trajectory of $a$. For example the set $\{ 3, 6, 9 \}$ is a trajectory: it is the trajectory of each of its members.
Which numbers have a finite trajectory? Possible hint: Find the set $$\{ n \mid n \text{ is the smallest number in some trajectory } S \}.$$
If you wish, you can get partial credit by solving the following simpler questions.
(a) Show that there is no trajectory of cardinality 1 or 2.
(b) Show that $\{ 3, 6, 9 \}$ is the only trajectory of cardinality 3.
(c) Show that for any integer $k \geq 3$, there is a trajectory of cardinality $k$.
(d) Find an infinite trajectory.

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