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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
2010 ugmath

19 maths questions

Q1 4 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Find all $x \in [ - \pi , \pi ]$ such that $\cos 3 x + \cos x = 0$.
Q2 4 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
A polynomial $f ( x )$ has integer coefficients such that $f ( 0 )$ and $f ( 1 )$ are both odd numbers. Prove that $f ( x ) = 0$ has no integer solutions.
Q4 4 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
Show that there is no infinite arithmetic progression consisting of distinct integers all of which are squares.
Q5 4 marks Number Theory Modular Arithmetic Computation View
Find the remainder given by $3 ^ { 89 } \times 7 ^ { 86 }$ when divided by 17.
Q6 4 marks Proof Direct Proof of a Stated Identity or Equality View
Prove that $$\frac { 2 } { 0 ! + 1 ! + 2 ! } + \frac { 3 } { 1 ! + 2 ! + 3 ! } + \cdots + \frac { n } { ( n - 2 ) ! + ( n - 1 ) ! + n ! } = 1 - \frac { 1 } { n ! }$$
Q7 4 marks Laws of Logarithms Prove a Logarithmic Identity View
If $a , b , c$ are real numbers $> 1$, then show that $$\frac { 1 } { 1 + \log _ { a ^ { 2 } b } \frac { c } { a } } + \frac { 1 } { 1 + \log _ { b ^ { 2 } c } \frac { a } { b } } + \frac { 1 } { 1 + \log _ { c ^ { 2 } a } \frac { b } { c } } = 3$$
Q8 4 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
If 8 points in a plane are chosen to lie on or inside a circle of diameter 2 cm then show that the distance between some two points will be less than 1 cm.
Q9 4 marks Proof Direct Proof of a Stated Identity or Equality View
If $f ( x ) = \frac { x ^ { n } } { n ! } + \frac { x ^ { n - 1 } } { ( n - 1 ) ! } + \cdots + x + 1$, then show that $f ( x ) = 0$ has no repeated roots.
Q10 4 marks Trigonometric equations in context View
Given $\cos x + \cos y + \cos z = \frac { 3 \sqrt { } 3 } { 2 }$ and $\sin x + \sin y + \sin z = \frac { 3 } { 2 }$ then show that $x = \frac { \pi } { 6 } + 2 k \pi , y = \frac { \pi } { 6 } + 2 \ell \pi , z = \frac { \pi } { 6 } + 2 m \pi$ for some $k , \ell , m \in \mathbf { Z }$.
Q11 4 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
Using the fact that $\sqrt { n }$ is an irrational number whenever $n$ is not a perfect square, show that $\sqrt { 3 } + \sqrt { 7 } + \sqrt { 21 }$ is irrational.
Q12 4 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
In an isoceles $\triangle \mathrm { ABC }$ with A at the apex the height and the base are both equal to 1 cm. Points $\mathrm { D } , \mathrm { E }$ and F are chosen one from each side such that BDEF is a rhombus. Find the length of the side of this rhombus.
Q13 4 marks Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
If $b$ is a real number satisfying $b ^ { 4 } + \frac { 1 } { b ^ { 4 } } = 6$, find the value of $\left( b + \frac { i } { b } \right) ^ { 16 }$ where $i = \sqrt { - 1 }$.
Q14 8 marks Number Theory Congruence Reasoning and Parity Arguments View
Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 100 }$ be 100 positive integers. Show that for some $m , n$ with $1 \leq m \leq n \leq 100 , \sum _ { i = m } ^ { n } a _ { i }$ is divisible by 100.
Q15 8 marks Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
In $\triangle \mathrm { ABC } , \mathrm { BE }$ is a median, and O the mid-point of BE. The line joining A and O meets BC at D. Find the ratio $\overline { \mathrm { AO } } : \overline { \mathrm { OD } }$ (Hint: Draw a line through E parallel to AD.)
Q16 8 marks Number Theory Combinatorial Number Theory and Counting View
(a) A computer program prints out all integers from 0 to ten thousand in base 6 using the numerals $0,1,2,3,4$ and 5. How many numerals it would have printed?
(b) A 3-digit number $abc$ in base 6 is equal to the 3-digit number $cba$ in base 9. Find the digits.
Q17 8 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
(a) Show that the area of a right-angled triangle with all side lengths integers is an integer divisible by 6.
(b) If all the sides and area of a triangle were rational numbers then show that the triangle is got by 'pasting' two right-angled triangles having the same property.
Q18 8 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Prove that $\int _ { 1 } ^ { b } a ^ { \log _ { b } x } d x > \ln b$ where $a , b > 0 , b \neq 1$.
Q19 8 marks Circles Area and Geometric Measurement Involving Circles View
Let $C _ { 1 } , C _ { 2 }$ be two circles of equal radii $R$. If $C _ { 1 }$ passes through the centre of $C _ { 2 }$ prove that the area of the region common to them is $\frac { R ^ { 2 } } { 6 } ( 4 \pi - \sqrt { 27 } )$.
Q20 8 marks Straight Lines & Coordinate Geometry Collinearity and Concurrency View
Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ and $b _ { 1 } , b _ { 2 } , \ldots , b _ { n }$ be two arithmetic progressions. Prove that the points $\left( a _ { 1 } , b _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } \right) , \ldots , \left( a _ { n } , b _ { n } \right)$ are collinear.