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QA1 4 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View

The three sides of triangle $a < b < c$ are in arithmetic progression (AP) with common difference $d = b - a = c - b$. Denote the angles opposite to sides $a , b , c$ respectively by $A , B , C$.
Statements
(1) $d$ must be less than $a$.
(2) $d$ can be any positive number less than $a$.
(3) The numbers $\sin A , \sin B , \sin C$ are in AP.
(4) The numbers $\cos A , \cos B , \cos C$ are in AP.

QA2 4 marks Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View

You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$.
Statements
(5) There is exactly one such triangle T that is equilateral. (6) There are exactly two such triangles $T$ that are right angled isosceles. (7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles. (8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.

QA3 4 marks Matrices Matrix Entry and Coefficient Identities View

$M$ is a $3 \times 3$ matrix with integer entries. For $M$ we have (Sum of column 2) $= 4 \times$ (sum of column 1). (Sum of column 3) $= 4 \times$ (sum of column 2). (Sum of row $2) = 6 +$ (sum of row $1$). (Sum of row $3) = 6 +$ (sum of row 2).
Statements
(9) The sum of all the entries in $M$ must be divisible by 21. (10) None of the row sums is divisible by 7. (11) One of the column sums must be divisible by 7. (12) None of the column sums is divisible by 6.

QA4 4 marks Trig Graphs & Exact Values View

Statements
(13) As $x \rightarrow - \infty$ the function $\cos \left( e ^ { x } \right)$ tends to a finite limit. (14) As $x \rightarrow \infty$ the function $\cos \left( e ^ { x } \right)$ changes sign infinitely many times. (15) As $x \rightarrow \infty$, the function $\sin ( \ln ( x ) )$ tends to a finite limit. (16) $\sin ( \ln ( x ) )$ changes sign only finitely many times as $x$ goes towards 0 from 1.

QA5 4 marks Number Theory Combinatorial Number Theory and Counting View

Statements
(17) $\sqrt [ 4 ] { 4 } < \sqrt [ 5 ] { 5 }$. (18) $\log _ { 10 } 11 > \log _ { 11 } 12$. (19) $\frac { \pi } { 4 } < \sqrt { 2 - \sqrt { 2 } }$. (20) $( 2022 ! ) ^ { 2 } > 2022 ^ { 2022 }$.

QA6 4 marks Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $f ( x ) = \left| \frac { \sin x } { x } \right| ^ { 1.001 }$ for $x \neq 0$ and $f ( 0 ) = L$ such that $f$ is continuous. Let $I ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
Statements
(21) $L = 1.001$ (22) $I ( 0.001 ) > 0.001$. (23) As $x \rightarrow \infty$ the limit of $I ( x )$ is greater than 1001 (possibly $\infty$). (24) The function $I ( x )$ is NOT differentiable at infinitely many points.

QA7 4 marks Number Theory GCD, LCM, and Coprimality View

Statements
(25) There is a unique natural number $n$ such that $n ^ { 2 } + 19 n - n ! = 0$. (26) There are infinitely many pairs $( x , y )$ of natural numbers satisfying $$( 1 + x ! ) ( 1 + y ! ) = ( x + y ) ! .$$ (27) For any natural number $n$, consider GCD of $n ^ { 2 } + 1$ and $( n + 1 ) ^ { 2 } + 1$. As $n$ ranges over all natural numbers, the largest possible value of this GCD is 5. (28) If $n$ is the largest natural number for which 20! is divisible by $80 ^ { n }$, then $n \geq 5$.

QA8 4 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

Let $a$ be a point in the domain of a continuous real valued function $f$. One says that $f$ has a flex point at $a$ if we can find a small interval $(a - \epsilon , a + \epsilon)$ in the domain of $f$ such that the following happens: (i) for all $x$ in the open interval $(a - \epsilon , a)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and, (ii) for all $x$ in the open interval $(a , a + \epsilon)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and opposite to the sign of $f ^ { \prime \prime } ( x )$ in $(a - \epsilon , a)$.
Statements
(29) If $f$ is a cubic polynomial with a local maximum at $x = p$ and a local minimum at $x = q$, then $f$ has a unique flex point at $x = \frac { p + q } { 2 }$. (30) If $f ^ { \prime \prime } ( a ) = 0$ then $f$ must have a flex point at $a$. (31) Let $f ( x ) = x ^ { 2 }$ for $x \geq 0$ and $f ( x ) = - x ^ { 2 }$ for $x < 0$. Then $f$ has no flex points. (32) If $f ^ { \prime }$ is monotonic on an open interval $I$, then $f$ cannot have a flex point in $I$.

QA9 4 marks Probability Definitions Verifying Statements About Probability Properties View

Suppose $A$, $B$ and $C$ are three events and $P ( A ) = a , P ( B ) = b , P ( C ) = c$ are known. Let $P ( A \cup B \cup C ) = p$. The statements below are about whether we can find the value of $p$ if we know some additional information. (Note: $\cup$ is the same as OR. Similarly $\cap$ is the same as AND.)
Statements
(33) We can find the value of $p$ if we know that at least one of $a , b , c$ is 1. (34) We can find the value of $p$ if we know that at least one of $a , b , c$ is 0. (35) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are mutually exclusive. (36) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are independent and we know the value of $P ( A \cap B \cap C )$.

QA10 4 marks 3x3 Matrices Determinant of Parametric or Structured Matrix View

Let $A = \left[ \begin{array} { c c c } 1 & 2 & 3 \\ 10 & 20 & 31 \\ 11 & 22 & k \end{array} \right]$ and $\mathbf { v } = \left[ \begin{array} { l } x \\ y \\ z \end{array} \right]$, where $k$ is a constant and $x , y , z$ are variables.
Statements
(37) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $B A =$ the $3 \times 3$ identity matrix. (38) There is a unique $k$ such that determinant of $A$ is 0. (39) The set of solutions $( x , y , z )$ of the matrix equation $A \mathbf { v } = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ is either a line or a plane containing the origin. (40) If the equation $A \mathbf { v } = \left[ \begin{array} { c } p \\ q \\ r \end{array} \right]$ has a solution, then it must be true that $q = 10 p$.

QB1 12 marks Combinations & Selection Counting Arrangements with Run or Pattern Constraints View

[12 points] Let $N = \{ 1,2,3,4,5,6,7,8,9 \}$ and $L = \{ a , b , c \}$.
(i) Suppose we arrange the 12 elements of $L \cup N$ in a line such that no two of the three letters occur consecutively. If the order of the letters among themselves does not matter, find the number such arrangements.
(ii) Find the number of functions from $N$ to $L$ such that exactly 3 numbers are mapped to each of $a , b$ and $c$.
(iii) Find the number of onto functions from $N$ to $L$.

QB2 12 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View

[12 points] Let $f$ be a function from natural numbers to natural numbers that satisfies
$$\begin{aligned} & f ( n ) = n - 2 \quad \text { for } n > 3000 \\ & f ( n ) = f ( f ( n + 5 ) ) \quad \text { for } n \leq 3000 \end{aligned}$$
Show that $f ( 2022 )$ is uniquely decided and find its value.

QB3 14 marks Sine and Cosine Rules Multi-step composite figure problem View

[14 points] In $\triangle A B C , \angle B A C = 2 \angle A C B$ and $0 ^ { \circ } < \angle B A C < 120 ^ { \circ }$. A point $M$ is chosen in the interior of $\triangle A B C$ such that $B A = B M$ and $M A = M C$. Prove that $\angle M C B = 30 ^ { \circ }$.
Hint (use this or your own method): Draw a suitable segment $C D$ of appropriate length making an appropriate angle with $C A$.

QB4 14 marks Roots of polynomials Determine coefficients or parameters from root conditions View

[14 points] We want to find a nonzero polynomial $p ( x )$ with integer coefficients having the following property.
$$\text { Letting } q ( x ) : = \frac { p ( x ) } { x ( 1 - x ) } , \quad q ( x ) = q \left( \frac { 1 } { 1 - x } \right) \text { for all } x \notin \{ 0,1 \}$$
(i) Find one such polynomial with the smallest possible degree.
(ii) Find one such polynomial with the largest possible degree OR show that the degree of such polynomials is unbounded.

QB5 14 marks Composite & Inverse Functions Derivative of an Inverse Function View

[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$.
(i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers.
(ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.

QB6 14 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View

[14 points] Suppose an integer $n > 1$ is such that $n + 1$ is not a multiple of 4 (i.e., such that $n$ is not congruent to $3 \bmod 4$). Prove that there exist $1 \leq i < j \leq n$ such that the following is a perfect square.
$$\frac { 1 ! 2 ! \cdots n ! } { i ! j ! }$$
Hint (use this or your own method): Make cases and first treat the case $n = 4k$.

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