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QB1 10 marks Circles Circle Identification and Classification View

Solve the following two independent problems.
(i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers.
If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .
Option A: must be one-to-one and must be onto. Option B: must be one-to-one but need not be onto. Option C: need not be one-to-one but must be onto. Option D: need not be one-to-one and need not be onto. Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$.
(ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)

QB2 10 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View

Solve the following two independent problems.
(i) A mother and her two daughters participate in a game show. At first, the mother tosses a fair coin.
Case 1: If the result is heads, then all three win individual prizes and the game ends. Case 2: If the result is tails, then each daughter separately throws a fair die and wins a prize if the result of her die is 5 or 6. (Note that in case 2 there are two independent throws involved and whether each daughter gets a prize or not is unaffected by the other daughter's throw.)
(a) Suppose the first daughter did not win a prize. What is the probability that the second daughter also did not win a prize?
(b) Suppose the first daughter won a prize. What is the probability that the second daughter also won a prize?
(ii) Prove or disprove each of the following statements.
(a) $2 ^ { 40 } > 20!$
(b) $1 - \frac { 1 } { x } \leq \ln x \leq x - 1$ for all $x > 0$.

QB3 10 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View

You are supposed to create a 7-character long password for your mobile device.
(i) How many 7-character passwords can be formed from the 10 digits and 26 letters? (Only lowercase letters are taken throughout the problem.) Repeats are allowed, e.g., 0001a1a is a valid password.
(ii) How many of the passwords contain at least one of the 26 letters and at least one of the 10 digits? Write your answer in the form: (Answer to part i) $-$ (something).
(iii) How many of the passwords contain at least one of the 5 vowels, at least one of the 21 consonants and at least one of the 10 digits? Extend your method for part ii to write a formula and explain your reasoning.
(iv) Now suppose that in addition to the lowercase letters and digits, you can also use 12 special characters. How many 7-character passwords are there that contain at least one of the 5 vowels, at least one of the 21 consonants, at least one of the 10 digits and at least one of the 12 special characters? Write only the final formula analogous to your answer to part iii. Do NOT explain.

QB4 10 marks Proof Existence Proof View

Show that there is no polynomial $p ( x )$ for which $\cos ( \theta ) = p ( \sin \theta )$ for all angles $\theta$ in some nonempty interval.
Hint: Note that $x$ and $| x |$ are different functions but their values are equal on an interval (as $x = | x |$ for all $x \geq 0$). You may want to show as a first step that this cannot happen for two polynomials, i.e., if polynomials $f$ and $g$ satisfy $f ( x ) = g ( x )$ for all $x$ in some interval, then $f$ and $g$ must be equal as polynomials, i.e., in each degree they must have the same coefficient.

QB5 10 marks Standard Integrals and Reverse Chain Rule Convergence and Estimation of Improper Integrals View

Define a function $f$ as follows: $f ( 0 ) = 0$ and, for any $x > 0$, $$f ( x ) = \lim _ { L \rightarrow \infty } \int _ { \frac { 1 } { x } } ^ { L } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$$ (or, in simpler notation, the improper integral $\int _ { \frac { 1 } { x } } ^ { \infty } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$).
(i) Show that the definition makes sense for any $x > 0$ by justifying why the limit in the definition exists, i.e., why the improper integral converges.
(ii) Find $f ^ { \prime } \left( \frac { 1 } { \pi } \right)$ if it exists. Clearly indicate the basic result(s) you are using.
(iii) Using the hint or otherwise, find $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( h ) - f ( 0 ) } { h }$, i.e., the right hand derivative of $f$ at $x = 0$. We can take the limit only from the right hand side because $f ( x )$ is undefined for negative values of $x$. Hint: Break $f ( h )$ into two terms by using a standard technique with an appropriate choice. Then separately analyze the resulting two terms in the derivative.

QB6 10 marks Number Theory Combinatorial Number Theory and Counting View

$n$ and $k$ are positive integers, not necessarily distinct. You are given two stacks of cards with a number written on each card, as follows.
Stack A has $n$ cards. On each card a number in the set $\{ 1 , \ldots , k \}$ is written. Stack B has $k$ cards. On each card a number in the set $\{ 1 , \ldots , n \}$ is written. Numbers may repeat in either stack. From this, you play a game by constructing a sequence $t _ { 0 } , t _ { 1 } , t _ { 2 } , \ldots$ of integers as follows. Set $t _ { 0 } = 0$. For $j > 0$, there are two cases: If $t _ { j } \leq 0$, draw the top card of stack $A$. Set $t _ { j + 1 } = t _ { j } +$ the number written on this card. If $t _ { j } > 0$, draw the top card of stack $B$. Set $t _ { j + 1 } = t _ { j } -$ the number written on this card. In either case discard the taken card and continue. The game ends when you try to draw from an empty stack. Example: Let $n = 5 , k = 3$, stack $A = 1,3,2,3,2$ and stack $B = 2,5,1$. You can check that the game ends with the sequence $0,1 , - 1,2 , - 3 , - 1,2,1$ (and with one card from stack $A$ left unused).
(i) Prove that for every $j$ we have $- n + 1 \leq t _ { j } \leq k$.
(ii) Prove that there are at least two distinct indices $i$ and $j$ such that $t _ { i } = t _ { j }$.
(iii) Using the previous parts or otherwise, prove that there is a nonempty subset of cards in stack $A$ and another subset of cards in stack $B$ such that the sum of numbers in both the subsets is same.

Q1 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View

Consider the two equations numbered [1] and [2]:
$$\begin{aligned} \log _ { 2021 } a & = 2022 - a \\ 2021 ^ { b } & = 2022 - b \end{aligned}$$
(a) Equation [1] has a unique solution.
(b) Equation [2] has a unique solution.
(c) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a = b$.
(d) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a + b$ is an integer.

Q2 4 marks Number Theory Congruence Reasoning and Parity Arguments View

A prime $p$ is an integer $\geq 2$ whose only positive integer factors are 1 and $p$.
(a) For any prime $p$ the number $p ^ { 2 } - p$ is always divisible by 3.
(b) For any prime $p > 3$ exactly one of the numbers $p - 1$ and $p + 1$ is divisible by 6.
(c) For any prime $p > 3$ the number $p ^ { 2 } - 1$ is divisible by 24.
(d) For any prime $p > 3$ one of the three numbers $p + 1 , p + 3$ and $p + 5$ is divisible by 8.

Q3 4 marks Sine and Cosine Rules Ambiguous case and triangle existence/uniqueness View

We want to construct a triangle ABC such that angle A is $20.21 ^ { \circ }$, side AB has length 1 and side BC has length $x$ where $x$ is a positive real number. Let $N ( x ) =$ the number of pairwise noncongruent triangles with the required properties.
(a) There exists a value of $x$ such that $N ( x ) = 0$.
(b) There exists a value of $x$ such that $N ( x ) = 1$.
(c) There exists a value of $x$ such that $N ( x ) = 2$.
(d) There exists a value of $x$ such that $N ( x ) = 3$.

Q4 4 marks Roots of polynomials Existence or counting of roots with specified properties View

Consider polynomials of the form $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$ where $a , b , c$ are integers. Name the three (possibly non-real) roots of $f ( x )$ to be $p , q , r$.
(a) If $f ( 1 ) = 2021$, then $f ( x ) = ( x - 1 ) \left( x ^ { 2 } + s x + t \right) + 2021$ where $s , t$ must be integers.
(b) There is such a polynomial $f ( x )$ with $c = 2021$ and $p = 2$.
(c) There is such a polynomial $f ( x )$ with $r = \frac { 1 } { 2 }$.
(d) The value of $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$ does not depend on the value of $c$.

Q5 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number.
(a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$.
(b) There is a unique complex number $z$ such that $P ( z ) = 3$.
(c) If $| z | \neq 1$, then $P ( z )$ is infinite.
(d) $P \left( e ^ { i } \right)$ is infinite.

Q6 4 marks Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

A stationary point of a function $f$ is a real number $r$ such that $f ^ { \prime } ( r ) = 0$. A polynomial need not have a stationary point (e.g. $x ^ { 3 } + x$ has none). Consider a polynomial $p ( x )$.
(a) If $p ( x )$ is of degree 2022, then $p ( x )$ must have at least one stationary point.
(b) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ must have at least 2020 stationary points.
(c) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ can have at most 2020 stationary points.
(d) If $r$ is a stationary point of $p ( x )$ AND $p ^ { \prime \prime } ( r ) = 0$, then the point $( r , p ( r ) )$ is neither a local maximum nor a local minimum point on the graph of $p ( x )$.

Q7 4 marks Simultaneous equations View

Given three distinct positive constants $a , b , c$ we want to solve the simultaneous equations
$$\begin{aligned} a x + b y & = \sqrt { 2 } \\ b x + c y & = \sqrt { 3 } \end{aligned}$$
(a) There exists a combination of values for $a , b , c$ such that the above system has infinitely many solutions $( x , y )$.
(b) There exists a combination of values for $a , b , c$ such that the above system has exactly one solution $( x , y )$.
(c) Suppose that for a combination of values for $a , b , c$, the above system has NO solution. Then $2 b < a + c$.
(d) Suppose $2 b < a + c$. Then the above system has NO solution.

Q8 4 marks Vector Product and Surfaces View

Given two distinct nonzero vectors $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ in 3 dimensions, define a sequence of vectors by
$$\mathbf { v } _ { n + 2 } = \mathbf { v } _ { n } \times \mathbf { v } _ { n + 1 } \left( \text { so } \mathbf { v } _ { 3 } = \mathbf { v } _ { 1 } \times \mathbf { v } _ { 2 } , \mathbf { v } _ { 4 } = \mathbf { v } _ { 2 } \times \mathbf { v } _ { 3 } \text { and so on } \right) .$$
Let $S = \left\{ \mathbf { v } _ { n } \mid n = 1,2 , \ldots \right\}$ and $U = \left\{ \left. \frac { \mathbf { v } _ { n } } { \left| \mathbf { v } _ { n } \right| } \right\rvert\, n = 1,2 , \ldots \right\}$. (Note: Here $\times$ denotes the cross product of vectors and $| \mathbf { v } |$ denotes the magnitude of the vector $\mathbf { v }$. The vector $\mathbf { 0 }$ with 0 magnitude, if it occurs in $S$, is counted. But in that case of course the $\mathbf { 0 }$ vector is not considered while listing elements of $U$.)
(a) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 2.
(b) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 3.
(c) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 4.
(d) Suppose that for some $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$, the set $S$ is infinite. Then the set $U$ is also infinite.

Q9 4 marks Small angle approximation View

$$f ( x ) = \frac { x } { x + \sin x } \quad \text { and } \quad g ( x ) = \frac { x ^ { 4 } + x ^ { 6 } } { e ^ { x } - 1 - x ^ { 2 } }$$
(a) Limit as $x \rightarrow 0$ of $f ( x )$ is $\frac { 1 } { 2 }$.
(b) Limit as $x \rightarrow \infty$ of $f ( x )$ does not exist.
(c) Limit as $x \rightarrow \infty$ of $g ( x )$ is finite.
(d) Limit as $x \rightarrow 0$ of $g ( x )$ is 720.

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