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Papers (30)
2025
pgmath 18
2024
pgmath 4 ugmath 20
2023
pgmath 12 ugmath 16
2022
pgmath 18 ugmath_22may 16 ugmath_23may 16
2021
pgmath 11 ugmath 15
2020
pgmath 18 ugmath 16
2019
pgmath 20 ugmath 15
2018
pgmath 4 ugmath 6
2017
ugmath 15
2016
pgmath 10 ugmath 16
2015
pgmath 5 ugmath 13
2014
ugmath 9
2013
pgmath 11 ugmath 12
2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
pgmath 4 ugmath 19
2018 ugmath

6 maths questions

QA1 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
Consider an equilateral triangle $ABC$ with altitude 3 centimeters. A circle is inscribed in this triangle, then another circle is drawn such that it is tangent to the inscribed circle and the sides $AB$, $AC$. Infinitely many such circles are drawn; each tangent to the previous circle and the sides $AB$, $AC$. Find the sum of the areas of all these circles.
QA4 4 marks Integration by Substitution Substitution Combined with Symmetry or Companion Integral View
Compute the following integral $$\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{~d}x}{(\sqrt{\sin x} + \sqrt{\cos x})^{4}}.$$
QA5 4 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
List in increasing order all positive integers $n \leq 40$ such that $n$ cannot be written in the form $a^{2} - b^{2}$, where $a$ and $b$ are positive integers.
QA6 4 marks Complex numbers 2 Solving Polynomial Equations in C View
Consider the equation $$z^{2018} = 2018^{2018} + i$$ where $i = \sqrt{-1}$.
(a) How many complex solutions does this equation have?
(b) How many solutions lie in the first quadrant?
(c) How many solutions lie in the second quadrant?
QA7 4 marks Roots of polynomials Determine coefficients or parameters from root conditions View
Let $x^{3} + ax^{2} + bx + 8 = 0$ be a cubic equation with integer coefficients. Suppose both $r$ and $-r$ are roots of this equation, where $r > 0$ is a real number. List all possible pairs of values $(a, b)$.
QB5 15 marks Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
An alien script has $n$ letters $b_{1}, \ldots, b_{n}$. For some $k < n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered sacred if:
i) no letter appears twice and,
ii) if a letter $b_{i}$ appears in the word then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_{1}$ and $b_{0} = b_{n}$.)
For example, if $n = 7$ and $k = 3$ then $b_{1}b_{3}b_{6}, b_{3}b_{1}b_{6}, b_{2}b_{4}b_{6}$ are sacred 3-words. On the other hand $b_{1}b_{7}b_{4}, b_{2}b_{2}b_{6}$ are not sacred. What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.