Problems

Solution: \begin{align*} \text{N2}(radially): && T + mg \cos \theta &= m\frac{v^2}{r} \\ \Rightarrow && v^2-gc \cos \theta &\geq 0 \\ \text{COE}: && \frac12 m u^2 &= \frac12mv^2 + mgc\cos \theta \\ \Rightarrow && u^2 &= v^2 + 2gc\cos \theta \\ && u^2 &\geq gc \cos \theta+2gc\cos \theta \\ \Rightarrow && u^2 &\geq 3gc\cos \theta \end{align*} Therefore it will complete bounces with the string taught if it leaves the table with \(u^2 \geq 3gc\). After \(6\) bounces it will leave the table with speed \(\sqrt{6gc} > \sqrt{3gc}\) and after \(7\) bounces it will leave the table with speed \(\sqrt{\frac{3}{2} gc} < \sqrt{3 gc}\). When it leaves the table with speed \(\sqrt{\tfrac32 gc}\), the string will go slack when \begin{align*} && \tfrac32 gc &= 3gc \cos \theta \\ \Rightarrow && \cos \theta &= \frac{1}{2} \end{align*} ie at a height \(c\cos \theta = \frac12c\) above the table. Once the string goes slack, the particle travels under circular motion, \begin{align*} && u^2 &= \frac12 gc \\ \Rightarrow && u_\rightarrow &= \sqrt{\tfrac12 gc} \cos \theta \\ && u_{\uparrow} &= \sqrt{\tfrac12 gc} \sin \theta \\ \Rightarrow && s &= ut - \tfrac12 gt^2 \\ \Rightarrow && -\frac{c}{2} &= \sqrt{\tfrac12 gc} \frac{\sqrt{3}}{2} t - \tfrac12 g t^2 \\ \Rightarrow && t &= \sqrt{\frac{c}{g}} \left ( \frac{\sqrt{6}+\sqrt{22}}{4} \right) \\ \Rightarrow && s_{\rightarrow} &= \tfrac12 \sqrt{\tfrac12 gc} \sqrt{\frac{c}{g}} \left ( \frac{\sqrt{6}+\sqrt{22}}{4} \right) - \frac{\sqrt{3}}{2}c \\ &&&= \left ( \frac{\sqrt{11}-3\sqrt{3}}{8} \right)c \\ \end{align*} We need to establish whether this position is positive or negative (ie if we cross the centre line. Clearly \(\sqrt{11} < 3\sqrt{3}\) so we haven't crossed the centre line and we land closer to where we took off. Since it's the 7th take off, this is closer to \(B\).

Solution: This can either occur via \(N-2\) heads and \(1\) tail in the first \(N-1\) flips, followed by a tail, or \(N-2\) tails and \(1\) head in the first \(N-1\) flips, followed by another head, ie \begin{align*} \mathbb{P}(N = n) &= \underbrace{\binom{n-1}{1}}_{\text{ways to choose when the first tail occurs}}p^{n-2}q^2 + \underbrace{\binom{n-1}{1}}_{\text{ways to choose when the first head occurs}}q^{n-2}p^2 \\ &= (n-1)p^2q^2(p^{n-4}+q^{n-4}) \\ \\ \mathbb{E}(N) &= \sum_{n=4}^{\infty} n \cdot \mathbb{P}(N = n) \\ &= \sum_{n=4}^{\infty} n \cdot (n-1)p^2q^2(p^{n-4}+q^{n-4}) \\ &= \sum_{n=4}^{\infty} n \cdot (n-1)(p^{n-2}q^2+q^{n-2}p^2) \\ &= q^2\sum_{n=4}^{\infty} n(n-1)p^{n-2}+p^2\sum_{n=4}^{\infty} n(n-1)q^{n-2} \\ &= q^2\left ( \sum_{n=2}^{\infty} n(n-1)p^{n-2} -2 \cdot 1 - 3 \cdot 2 \cdot p\right)+p^2\left ( \sum_{n=2}^{\infty} n(n-1)q^{n-2} - 2-6q\right) \\ &= q^2\left ( 2(1-p)^{-3} -2 - 6 p\right)+p^2\left ( 2(1-q)^{-3} - 2-6q\right) \\ &= q^2\left ( 2q^{-3} -2 - 6 p\right)+p^2\left ( 2p^{-3} - 2-6q\right) \\ &= \frac{2}{q} - 2q^2 - 6pq^2+\frac{2}{p} -2p^2-6p^2q \\ &= \frac{2}{q}+\frac2p - 2(p^2+q^2) - 6pq \\ &= \frac{2}{pq} - 2((p+q)^2-2pq) - 6pq \\ &= \frac{2}{pq} - 2 -2pq \\ &= 2 \left (\frac1{pq} - 1 - pq \right) \end{align*}

Solution: There are \(87\) years between 1905 and 1992. Of those years, every 4th is a leap years, starting with 1908, and ending with 1992, so there are 22 leap years. There are 30 days between Apr 3 and May 3, 31 days between May 3 and Jun 3 and a further 23 days to the 26th. Therefore \(87 \times 365 + 22 \cdot 1 + 61 + 23\) total days. \(\pmod{7}\) this is \(3 \times 1 + 1 + 5 + 2 = 3\), therefore it is \(4\) week days before Friday, ie Monday.

Solution:

1. \((a+b+c)+(d+e+f)+(g+h+k) = 3n = 1 + 2 + \cdots + 9 = 45 \Rightarrow n = 15\).
2. Summing all rows, columns, diagonals through \(e\) we have \((a+e+k)+(b+e+h)+(c+e+g)+(d+e+f) = 45 + 3e = 60 \Rightarrow e = 5\).
3. Suppose that one of the corners is \(9\), then we need to find \(2\) ways to make \(6\) not using \(5\) and \(1\) (as \(5\) is in the middle and \(1\) diagonally opposite). Clearly this is not possible as the only remaining numbers are \(2,3,4\) and only \(2+4 = 6\). Therefore \(9\) cannot be in the corner or central squares, ie it's one of \(b,d,h,f\)
4. We must have \begin{array}{ccc} a & 9 & c\\ d & 5 & f\\ g & 1 & k \end{array} and so \(a\) or \(c = 4\). Once we place \(4\) by symmetry there will be another solution with \(a = 2\). So: \begin{array}{ccc} 4 & 9 & 2\\ d & 5 & f\\ g & 1 & k \end{array} we now see \(k\), then \(f\), then \(d\) then \(g\) must be determined, ie: \begin{array}{ccc} 4 & 9 & 2\\ 3 & 5 & 7\\ 8 & 1 & 6 \end{array} so our two solutions must be this and \begin{array}{ccc} 2 & 9 & 4\\ 7 & 5 & 3\\ 6 & 1 & 8 \end{array}
5. For each of the \(4\) possible placements of \(9\) there are two magic squares, so there are \(8\) possible magic squares, all related by reflection and rotation.

Solution:

1. \begin{align*} \int_{-\pi}^\pi |\sin x | \d x &= \int_{-\pi}^{0} - \sin x \d x + \int_0^\pi \sin x \d x \\ &= \left [\cos x \right]_{-\pi}^{0} +[-\cos x]_0^{\pi} \\ &= 1-(-1)+(1)-(-1) \\ &= 4 \end{align*}
2. \begin{align*} \int_{-\pi}^\pi \sin | x | \d x &= \int_{-\pi}^0 - \sin x \d x + \int_0^\pi \sin x \d x \\ &= 4 \end{align*}
3. \begin{align*} \int_{-\pi}^\pi x \sin x \d x &= \left [ -x \cos x \right]_{-\pi}^\pi + \int_{-\pi}^{\pi} \cos x \d x \\ &= \pi -(-\pi) + \left [\sin x \right]_{-\pi}^\pi \\ &= 2\pi \end{align*}
4. \begin{align*} \int_{-\pi}^{\pi} x^{10} \sin x \d x &\underbrace{=}_{x^{10}\sin x \text{ is odd}} 0 \end{align*}

Solution:

1. Multiplication by \(1+i\) rotates by \(45^{\circ}\) anticlockwise
   

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2. \(|z| \leq \sqrt{2}\), \(\textrm{Re}(z^2) \geq 0\) means \(\textrm{Re}{z} \geq \textrm{Im}{z}\)
   

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3. These are all points within \(1\) unit from a circle radius \(2\) units.
   

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Solution: \begin{align*} && p_0(x) &= (1-x)(1-x^2)(1-x^4) \\ &&&= (1-x)(1-x)(1+x)(1-x^2)(1+x^2) \\ &&&= (1-x)^2 (1+x)(1-x)(1+x)(1+x^2) \\ &&&= (1-x)^3 (1+x)^2 (1+x^2) \end{align*} \begin{align*} && p_0'(x) &= 3(1-x)^2(1+x)^2(1+x^2) + (1-x)^3 q(x) \\ \Rightarrow && p_0'(1) &= 3 \cdot 0 \cdots + 0 \cdots \\ &&&= 0 \\ && p_1'(x) &= p_0(x) + xp'_0(x) \\ \Rightarrow && p_1'(1) &= p_0(1) + 1\cdot p_0'(1) \\ &&&= 0 + 1 \cdot 0 \\ &&&= 0 \end{align*} Notice that \(p_0(x) = (1-x-x^2+x^3)(1-x^4) = 1-x-x^2+x^3-x^4+x^5+x^6-x^7\), so: \(p'_0(x) = -1-2x+3x^2-4x^3+5x^4+6x^5-7x^6 \Rightarrow p'_0(1) = 0 = -1 -2 -4 -7 + 3 + 5+6\). \((xp'_1(1))' = 0 = -1^2-2^2-4^2-7^2 + 3^2 + 5^2 + 6^2\). Consider \(q_0(x) = (1-x)(1-x^2)(1-x^4)(1-x^8)\), then \((1-x)^4\) is a factor, so in particular we know \(q_0(1), (xq_0(x))'|_{x=1} = 0,(x(xq_0(x))')'|_{x=1} = 0\), and so: \(q_0(x) = 1-x-x^2+x^3-x^4+x^5+x^6-x^7 - x^8+x^9+x^{10}-x^{11}+x^{12}-x^{13}-x^{14}+x^{15}\), and so: \(1^r+2^r+4^r+7^r+8^r+11^r+13^r+14^r = 3^r+5^r+6^r+9^r+10^r+12^r+15^r\) for \(r = 1,2,3\)

Solution: If \(g(0) \in \mathbb{Z} \Rightarrow b \in \mathbb{Z}\). If \(g(1) \in \mathbb{Z} \Rightarrow a+b \in \mathbb{Z} \Rightarrow a \in \mathbb{Z}\), therefore \(a \cdot n + b \in \mathbb{Z}\), in particular \(g(n) \in \mathbb{Z}\) for all integers \(n\). \(f(0) \in \mathbb{Z} \Rightarrow C \in \mathbb{Z}\), \(f(1) \in \mathbb{Z} = A+ B + C \in \mathbb{Z} \Rightarrow A+ B \in \mathbb{Z}\) \(f(-1) \in \mathbb{Z} = A- B + C \in \mathbb{Z} \Rightarrow A- B \in \mathbb{Z}\) \(\Rightarrow 2A, 2B \in \mathbb{Z}\) \begin{align*} f(n) &= An^2 + Bn + C \\ &= An^2-An + An+Bn + C \\ &= 2A \frac{n(n-1)}2 + (A+B)n + C \\ &\in \mathbb{Z} \end{align*} Consider \(g(x) = f(x + \alpha)\), therefore \(g(0), g(1), g(-1) \in \mathbb{Z} \Rightarrow g(n) \in \mathbb{Z} \Rightarrow f(n+\alpha) \in \mathbb{Z}\)