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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
2024 ugmath

20 maths questions

QB1 10 marks Areas Between Curves Maximize or Optimize Area View
(a) Draw a qualitatively accurate sketch of the unique bounded region R in the first quadrant that has maximum possible finite area with boundary described as follows. R is bounded below by the graph of $y = x^2 - x^3$, bounded above by the graph of an equation of the form $y = kx$ (where $k$ is some constant), and R is entirely enclosed by the two given graphs, i.e., the boundary of the region R must be a subset of the union of the given two graphs (so R does not have any points on its boundary that are not on these two graphs). Clearly mark the relevant point(s) on the boundary where the two given graphs meet and write the coordinates of every such point.
(b) Consider the solid obtained by rotating the above region R around $Y$-axis. Show how to find the volume of this solid by doing the following: Carefully set up the calculation with justification. Do enough work with the resulting expression to reach a stage where the final numerical answer can be found mechanically by using standard symbolic formulas of algebra and/or calculus and substituting known values in them. Do not carry out the mechanical work to get the final numerical answer.
QB2 15 marks Indefinite & Definite Integrals Accumulation Function Analysis View
(a) Find the domain of the function $g(x)$ defined by the following formula. $$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$ Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations.
(b) $g(1000)$.
(c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope.
(d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope.
(e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.
QB3 15 marks Proof Direct Proof of an Inequality View
(a) For non-negative numbers $a, b, c$ and any positive real number $r$ prove the following inequality and state precisely when equality is achieved. $$a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r(c-a)(c-b) \geq 0$$ Hint: Assuming $a \geq b \geq c$ do algebra with just the first two terms. What about the third term? What if the assumption is not true?
(b) As a special case obtain an inequality with $a^4 + b^4 + c^4 + abc(a+b+c)$ on one side.
(c) Show that if $abc = 1$ for positive numbers $a, b, c$, then $$a^4 + b^4 + c^4 + a^3 + b^3 + c^3 + a + b + c \geq \frac{a^2+b^2}{c} + \frac{b^2+c^2}{a} + \frac{c^2+a^2}{b} + 3.$$
QB4 10 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
Find all solutions of the following equation where it is required that $x, k, y, n$ are positive integers with the exponents $k$ and $n$ both $> 1$. $$20x^k + 24y^n = 2024$$
QB5 15 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
(a) Find all complex solutions of $z^6 = z + \bar{z}$.
(b) For an integer $n > 1$, how many complex solutions does $z^n = z + \bar{z}$ have?
Q2 2 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A test developed to detect Covid gives the correct diagnosis for $99\%$ of people with Covid. It also gives the correct diagnosis for $99\%$ of people without Covid. In a city $\frac{1}{1000}$ of the population has Covid.
If the probability is $x\%$, then your answer should be the integer closest to $x$. E.g., for probability $\frac{1}{3} = 33.33\ldots\%$, you should type 33 as your answer. For probability $\frac{2}{3}$ you should type 67 as your answer.
What is the probability that a randomly selected person tests positive? (We assume that in our random selection every person is equally likely to be chosen.) [2 points]
Q3 2 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
A test developed to detect Covid gives the correct diagnosis for $99\%$ of people with Covid. It also gives the correct diagnosis for $99\%$ of people without Covid. In a city $\frac{1}{1000}$ of the population has Covid.
If the probability is $x\%$, then your answer should be the integer closest to $x$. E.g., for probability $\frac{1}{3} = 33.33\ldots\%$, you should type 33 as your answer. For probability $\frac{2}{3}$ you should type 67 as your answer.
Suppose that a randomly selected person tested positive. What is the probability that this person has Covid? [2 points]
Q4 1 marks Factor & Remainder Theorem Remainder by Linear Divisor View
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find the remainder when $p(x)$ is divided by $x+1$. [1 point]
Q5 2 marks Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Let $z_1, z_2, z_3, z_4, z_5, z_6$ be the six complex roots of $p(x)$. Evaluate $\sum_{i=1}^{6} z_i^2$. [2 points]
Q6 2 marks Sign Change & Interval Methods View
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find an integer $n$ with the least possible absolute value such that $p(x)$ has a real root between $n$ and $n+1$. Write this number along with your reason as per the given instruction. [2 points]
Instruction for (6): Write two numbers separated by a comma: value of $n$, number of the theorem below that justifies this answer. E.g., if you think that $n=5$ because of the factor theorem, then type $\mathbf{5,1}$ as your answer with no space, full stop or any other punctuation.
  1. Factor theorem
  2. Mean value theorem
  3. Intermediate value theorem
  4. Fundamental theorem of algebra
  5. Fundamental theorem of calculus
Q7 3 marks Number Theory Modular Arithmetic Computation View
Two mighty frogs jump once per unit time on the number line as described in the question.
The first frog is at $x = 2^i$ at time $t = i$. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]
Q8 3 marks Number Theory Combinatorial Number Theory and Counting View
Two mighty frogs jump once per unit time on the number line as described in the question.
The second frog starts at $x=0$ and jumps $i+1$ steps to the right just after $t=i$, so that at times $t=0,1,2,3,\ldots$ this frog is at positions $x=0,1,3,6,\ldots$ respectively. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]
Q9 1 marks Partial Fractions View
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024\}$. [1 point]
Options:
  1. The empty set
  2. A singleton set
  3. A line
  4. A pair of lines
  5. A circle
  6. A plane perpendicular to $\mathbf{u}$
  7. A plane parallel to $\mathbf{u}$
  8. An infinite cone
  9. A finite cone
  10. A sphere
  11. None of the above
Q10 2 marks Complex Numbers Arithmetic Geometric Interpretation and Triangle/Shape Properties View
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points]
Options:
  1. The empty set
  2. A singleton set
  3. A line
  4. A pair of lines
  5. A circle
  6. A plane perpendicular to $\mathbf{u}$
  7. A plane parallel to $\mathbf{u}$
  8. An infinite cone
  9. A finite cone
  10. A sphere
  11. None of the above
Q11 2 marks Measures of Location and Spread View
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024(\mathbf{v} \cdot \mathbf{v})\}$. [2 points]
Options:
  1. The empty set
  2. A singleton set
  3. A line
  4. A pair of lines
  5. A circle
  6. A plane perpendicular to $\mathbf{u}$
  7. A plane parallel to $\mathbf{u}$
  8. An infinite cone
  9. A finite cone
  10. A sphere
  11. None of the above
Q12 2 marks Number Theory Divisibility and Divisor Analysis View
An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$.
Write the number of even positive integers that are factors of $2024^2$. [2 points]
Q13 3 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$.
Write the number of ordered pairs $(a,b)$ of positive integers such that $a^2 - b^2 = 2024^2$. If there are infinitely many such pairs, write the word infinite as your answer. [3 points]
Q14 1 marks Combinations & Selection Lattice Path Counting View
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,4)$ and $(2,3)$. [1 point]
Q15 2 marks Combinations & Selection Lattice Path Counting View
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,2)$ and $(3,4)$. [2 points]
Q16 3 marks Combinations & Selection Lattice Path Counting View
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ such that neither of the two points $(1,2)$ and $(3,4)$ occurs on the path, i.e., the path must miss both of the points $(1,2)$ and $(3,4)$. [3 points]