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QA1 4 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View

Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$.
Statements
(1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$.
(2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$.
(3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$.
(4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.

QA2 4 marks Independent Events View

Any two events $X$ and $Y$ are called mutually exclusive when the probability $P(X$ and $Y) = 0$ and they are called exhaustive when $P(X$ or $Y) = 1$. Suppose $A$ and $B$ are events and the probability of each of these two events is strictly between 0 and 1 (i.e., $0 < P(A) < 1$ and $0 < P(B) < 1$).
Statements
(5) $A$ and $B$ are mutually exclusive if and only if not $A$ and not $B$ are exhaustive. (6) $A$ and $B$ are independent if and only if not $A$ and not $B$ are independent. (7) $A$ and $B$ cannot be simultaneously independent and exhaustive. (8) $A$ and $B$ cannot be simultaneously mutually exclusive and exhaustive.

QA3 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let $A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 11 & 22 & k \end{array}\right]$ 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
(9) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $BA =$ the $3 \times 3$ identity matrix. (10) There is a unique $k$ such that determinant of $A$ is 0. (11) 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. (12) 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 $q = 10p$.

QA4 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View

Consider the following conditions on a function $f$ whose domain is the closed interval $[0,1]$. (For any condition involving a limit, at the endpoints, use the relevant one-sided limit.) I. $f$ is differentiable at each $x \in [0,1]$. II. $f$ is continuous at each $x \in [0,1]$. III. The set $\{f(x) \mid x \in [0,1]\}$ has a maximum element and a minimum element.
Statements
(13) If I is true, then II is true. (14) If II is true, then III is true. (15) If III is false, then I is false. (16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.)

QA5 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View

Statements
(17) Let $a = \frac{1}{\ln 3}$. Then $3^a = e$. (18) $\sin(0.02) < 2\sin(0.01)$. (19) $\arctan(0.01) > 0.01$. (20) $4\int_0^1 \arctan(x)\, dx = \pi - \ln 4$.

QA6 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View

Let $$f(x) = \frac{1}{|\ln x|}\left(\frac{1}{x} + \cos x\right)$$
Statements
(21) As $x \rightarrow \infty$, the sign of $f(x)$ changes infinitely many times. (22) As $x \rightarrow \infty$, the limit of $f(x)$ does not exist. (23) As $x \rightarrow 1$, $f(x) \rightarrow \infty$. (24) As $x \rightarrow 0^+$, $f(x) \rightarrow 1$.

QA7 4 marks Chain Rule Iterated/Nested Exponential Differentiation View

Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$. (26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$. (27) $\int_0^1 f_3(x)\, dx = 1$. (28) The derivative of $f_{123}$ at $x = 1$ is 1.

QA8 4 marks Permutations & Arrangements Counting Functions with Constraints View

Let $N = \{1,2,3,4,5,6,7,8,9\}$ and $L = \{a,b,c\}$.
Statements
(29) Suppose we arrange the 12 elements of $L \cup N$ in a line such that all three letters appear consecutively (in any order). The number of such arrangements is less than $10! \times 5$. (30) More than half of the functions from $N$ to $L$ have the element $b$ in their range. (31) The number of one-to-one functions from $L$ to $N$ is less than 512. (32) The number of functions from $N$ to $L$ that do not map consecutive numbers to consecutive letters is greater than 512. (e.g., $f(1) = b$ and $f(2) = a$ or $c$ is not allowed. $f(1) = a$ and $f(2) = c$ is allowed. So is $f(1) = f(2)$.)

QA9 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.
Statements
(33) If $|z| = 1$, then $f(z)$ must be real. (34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$. (35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$. (36) If $f(z)$ is a real number, then $f(z)$ must be positive.

QA10 4 marks Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View

Suppose that cards numbered $1, 2, \ldots, n$ are placed on a line in some sequence (with each integer $i \in [1,n]$ appearing exactly once). A move consists of interchanging the card labeled 1 with any other card. If it is possible to rearrange the cards in increasing order by doing a series of moves, we say that the given sequence can be rectified.
Statements
(37) The sequence 912345678 can be rectified in 8 moves and no fewer moves. (38) The sequence 134567892 can be rectified in 8 moves and no fewer moves. (39) The sequence 134295678 cannot be rectified. (40) There exists a sequence of 99 cards that cannot be rectified.

QB1 11 marks Sine and Cosine Rules Multi-step composite figure problem View

[11 points] Given $\triangle XYZ$, the following constructions are made: mark point $W$ on segment $XZ$, point $P$ on segment $XW$ and point $Q$ on segment $YZ$ such that
$$\frac{WZ}{YX} = \frac{PW}{XP} = \frac{QZ}{YQ} = k$$
Extend segments $QP$ and $YX$ to meet at the point $R$ as shown. Prove that $XR = XP$.
Hint (use this or your own method): A suitable construction may help in calculations.

QB2 11 marks Number Theory Lattice Points and Geometric Number Theory View

[11 points] In the XY plane, draw horizontal and vertical lines through each integer on both axes so as to get a grid of small $1 \times 1$ squares whose vertices have integer coordinates.
(i) Consider the line segment $D$ joining $(0,0)$ with $(m,n)$. Find the number of small $1 \times 1$ squares that $D$ cuts through, i.e., squares whose interiors $D$ intersects. (Interiors consist of points for which both coordinates are non-integers.) For example, the line segment joining $(0,0)$ and $(2,3)$ cuts through 4 small squares, as you can check by drawing.
(ii) Now $L$ is allowed to be an arbitrary line in the plane. Find the maximum number of small $1 \times 1$ squares in an $n \times n$ grid that $L$ can cut through, i.e., we want $L$ to intersect the interiors of maximum possible number of small squares inside the square with vertices $(0,0)$, $(n,0)$, $(0,n)$ and $(n,n)$.

QB3 14 marks Stationary points and optimisation Find critical points and classify extrema of a given function View

[14 points] For a positive integer $n$, let $f(x) = \sum_{i=0}^{n} x^i = 1 + x + x^2 + \cdots + x^n$. Find the number of local maxima of $f(x)$. Find the number of local minima of $f(x)$. For each maximum/minimum $(c, f(c))$, find the integer $k$ such that $k \leq c < k+1$.
Hints (use these or your own method): It may be helpful to (i) look at small $n$, (ii) use the definition of $f$ as well as a closed formula, and (iii) treat $x \geq 0$ and $x < 0$ separately.

QB4 14 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View

[14 points] Let $\mathbb{R}_+$ denote the set of positive real numbers. For a continuous function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, define $A_r =$ the area bounded by the graph of $f$, X-axis, $x = 1$ and $x = r$ $B_r =$ the area bounded by the graph of $f$, X-axis, $x = r$ and $x = r^2$. Find all continuous $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ for which $A_r = B_r$ for every positive number $r$. Hints (use these or your own method): Find an equation relating $f(x)$ and $f(x^2)$. Consider the function $xf(x)$. Suppose a sequence $x_n$ converges to $b$ where all $x_n$ and $b$ are in the domain of a continuous function $g$. Then $g(x_n)$ must converge to $g(b)$. E.g., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.

QB5 15 marks Proof Existence Proof View

[15 points] Two distinct real numbers $r$ and $s$ are said to form a good pair $(r, s)$ if
$$r^3 + s^2 = s^3 + r^2$$
(i) Find a good pair $(a, \ell)$ with the largest possible value of $\ell$. Find a good pair $(s, b)$ with the smallest possible value $s$. For every good pair $(c, d)$ other than the two you found, show that there is a third real number $e$ such that $(d, e)$ and $(c, e)$ are also good pairs.
(ii) Show that there are infinitely many good pairs of rational numbers.
Hints (use these or your own method): The function $f(x) = x^3 - x^2$ may be useful. If $(r, s)$ is a good pair, can you express $s$ in terms of $r$? You may use that there are infinitely many right triangles with integer sides such that no two of these triangles are similar to each other.

QB6 15 marks Number Theory Congruence Reasoning and Parity Arguments View

[15 points] Solve the following. You may do (i) and (ii) in either order.
(i) Let $p$ be a prime number. Show that $x^2 + x - 1$ has at most two roots modulo $p$, i.e., the cardinality of $\{n \mid 1 \leq n \leq p$ and $n^2 + n - 1$ is divisible by $p\}$ is $\leq 2$. Find all primes $p$ for which this set has cardinality 1.
(ii) Find all positive integers $n \leq 121$ such that $n^2 + n - 1$ is divisible by 121.
(iii) What can you say about the number of roots of $x^2 + x - 1$ modulo $p^2$ for an arbitrary prime $p$, i.e., the cardinality of
$$\left\{n \mid 1 \leq n \leq p^2 \text{ and } n^2 + n - 1 \text{ is divisible by } p^2\right\}?$$
You do NOT need to repeat any reasoning from previous parts. You may simply refer to any such relevant reasoning and state your conclusion with a brief explanation.

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