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Papers (30)
2025
pgmath 18
2024
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2023
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2022
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2021
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2020
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2019
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2018
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2017
ugmath 15
2016
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2015
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2014
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2013
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2012
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2011
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2010
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2013 ugmath

12 maths questions

QA1 5 marks Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
For sets $A$ and $B$, let $f : A \rightarrow B$ and $g : B \rightarrow A$ be functions such that $f ( g ( x ) ) = x$ for each $x$. For each statement below, write whether it is TRUE or FALSE. a) The function $f$ must be one-to-one.
Answer: $\_\_\_\_$ b) The function $f$ must be onto.
Answer: $\_\_\_\_$ c) The function $g$ must be one-to-one.
Answer: $\_\_\_\_$ d) The function $g$ must be onto.
Answer: $\_\_\_\_$
QA3 5 marks Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $S$ be a circle with center $O$. Suppose $A , B$ are points on the circumference of $S$ with $\angle A O B = 120 ^ { \circ }$. For triangle $A O B$, let $C$ be its circumcenter and $D$ its orthocenter (i.e., the point of intersection of the three lines containing the altitudes). For each statement below, write whether it is TRUE or FALSE. a) The triangle $A O C$ is equilateral.
Answer: $\_\_\_\_$ b) The triangle $A B D$ is equilateral.
Answer: $\_\_\_\_$ c) The point $C$ lies on the circle $S$.
Answer: $\_\_\_\_$ d) The point $D$ lies on the circle $S$.
Answer: $\_\_\_\_$
QA5 5 marks Combinations & Selection Selection with Group/Category Constraints View
There are 8 boys and 7 girls in a group. For each of the tasks specified below, write an expression for the number of ways of doing it. Do NOT try to simplify your answers. a) Sitting in a row so that all boys sit contiguously and all girls sit contiguously, i.e., no girl sits between any two boys and no boy sits between any two girls
Answer: b) Sitting in a row so that between any two boys there is a girl and between any two girls there is a boy
Answer: c) Choosing a team of six people from the group
Answer: d) Choosing a team of six people consisting of unequal number of boys and girls
Answer:
QA6 5 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Calculate the following integrals whenever possible. If a given integral does not exist, state so. Note that $[ x ]$ denotes the integer part of $x$, i.e., the unique integer $n$ such that $n \leq x < n + 1$. a) $\int _ { 1 } ^ { 4 } x ^ { 2 } d x$
Answer: $\_\_\_\_$ b) $\int _ { 1 } ^ { 3 } [ x ] ^ { 2 } d x$
Answer: $\_\_\_\_$ c) $\int _ { 1 } ^ { 2 } \left[ x ^ { 2 } \right] d x$
Answer: $\_\_\_\_$ d) $\int _ { - 1 } ^ { 1 } \frac { 1 } { x ^ { 2 } } d x$
Answer: $\_\_\_\_$
QA7 5 marks Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Let $A , B , C$ be angles such that $e ^ { i A } , e ^ { i B } , e ^ { i C }$ form an equilateral triangle in the complex plane. Find values of the given expressions. a) $e ^ { i A } + e ^ { i B } + e ^ { i C }$
Answer: $\_\_\_\_$ b) $\cos A + \cos B + \cos C$
Answer: $\_\_\_\_$ c) $\cos 2 A + \cos 2 B + \cos 2 C$
Answer: $\_\_\_\_$ d) $\cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C$
Answer: $\_\_\_\_$
QA8 5 marks Geometric Probability View
Consider the quadratic equation $x ^ { 2 } + b x + c = 0$, where $b$ and $c$ are chosen randomly from the interval $[ 0,1 ]$ with the probability uniformly distributed over all pairs $( b , c )$. Let $p ( b ) =$ the probability that the given equation has a real solution for given (fixed) value of $b$. Answer the following questions by filling in the blanks. a) The equation $x ^ { 2 } + b x + c = 0$ has a real solution if and only if $b ^ { 2 } - 4 c$ is
Answer: $\_\_\_\_$ b) The value of $p \left( \frac { 1 } { 2 } \right)$, i.e., the probability that $x ^ { 2 } + \frac { x } { 2 } + c = 0$ has a real solution is
Answer: $\_\_\_\_$ c) As a function of $b$, is $p ( b )$ increasing, decreasing or constant?
Answer: $\_\_\_\_$ d) As $b$ and $c$ both vary, what is the probability that $x ^ { 2 } + b x + c = 0$ has a real solution?
Answer: $\_\_\_\_$
QA10 5 marks Curve Sketching Asymptote Determination View
Let $$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$ where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones. a) How many vertical asymptotes does the graph of $f$ have?
Options: $n$ \quad less than $n$ \quad more than $n$ \quad impossible to decide
Answer: $\_\_\_\_$ b) What can you deduce about the value of $n$?
Options: $n < 4$ \quad $n = 4$ \quad $n > 4$ \quad impossible to decide
Answer: $\_\_\_\_$ c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?
Options: $x = 0$ \quad $x = 1$ \quad $x = n - 1$ \quad $x = n$
Answer: $\_\_\_\_$ d) How many inflection points does the graph of $f$ have in the region $x < 0$?
Options: none\quad 1\quad more than 1\quad impossible to decide
(Hint: Sketching is better than calculating.)
Answer: $\_\_\_\_$
QB1 15 marks Sine and Cosine Rules Multi-step composite figure problem View
In triangle $ABC$, the bisector of angle $A$ meets side $BC$ in point $D$ and the bisector of angle $B$ meets side $AC$ in point $E$. Given that $DE$ is parallel to $AB$, show that $\mathrm{AE} = \mathrm{BD}$ and that the triangle $ABC$ is isosceles.
QB2 15 marks Differential equations Solving Separable DEs with Initial Conditions View
A curve $C$ has the property that the slope of the tangent at any given point $( x , y )$ on $C$ is $\frac { x ^ { 2 } + y ^ { 2 } } { 2 x y }$. a) Find the general equation for such a curve. Possible hint: let $z = \frac { y } { x }$. b) Specify all possible shapes of the curves in this family. (For example, does the family include an ellipse?)
QB3 15 marks Number Theory Congruence Reasoning and Parity Arguments View
A positive integer $N$ has its first, third and fifth digits equal and its second, fourth and sixth digits equal. In other words, when written in the usual decimal system it has the form $xyxyxy$, where $x$ and $y$ are the digits. Show that $N$ cannot be a perfect power, i.e., $N$ cannot equal $a ^ { b }$, where $a$ and $b$ are positive integers with $b > 1$.
QB5 20 marks Stationary points and optimisation Prove an inequality using calculus-based optimisation View
Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$.
QB6 20 marks Sequences and Series Functional Equations and Identities via Series View
Define $f _ { k } ( n )$ to be the sum of all possible products of $k$ distinct integers chosen from the set $\{ 1,2 , \ldots , n \}$, i.e., $$f _ { k } ( n ) = \sum _ { 1 \leq i _ { 1 } < i _ { 2 } < \ldots < i _ { k } \leq n } i _ { 1 } i _ { 2 } \ldots i _ { k }$$ a) For $k > 1$, write a recursive formula for the function $f _ { k }$, i.e., a formula for $f _ { k } ( n )$ in terms of $f _ { \ell } ( m )$, where $\ell < k$ or ($\ell = k$ and $m < n$). b) Show that $f _ { k } ( n )$, as a function of $n$, is a polynomial of degree $2k$. c) Express $f _ { 2 } ( n )$ as a polynomial in variable $n$.