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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
2014 ugmath

9 maths questions

QA3 3 marks Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Given a real number $x$, define $g ( x ) = x ^ { 2 } e ^ { x }$ if $x \geq 0$ and $g ( x ) = x e ^ { - x }$ if $x < 0$.
(A) The function $g$ is continuous everywhere.
(B) The function $g$ is differentiable everywhere.
(C) The function $g$ is one-to-one.
(D) The range of $g$ is the set of all real numbers.
QA4 4 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
Find the slope of a line L that satisfies both of the following properties: (i) L is tangent to the graph of $y = x ^ { 3 }$. (ii) L passes through the point $( 0, 200 )$.
QA5 4 marks Combinations & Selection Combinatorial Probability View
A regular 100-sided polygon is inscribed in a circle. Suppose three of the 100 vertices are chosen at random, all such combinations being equally likely. Find the probability that the three chosen points form vertices of a right angled triangle.
QA6 4 marks Number Theory Divisibility and Divisor Analysis View
What is the smallest positive integer $n$ for which $\frac { 50 ! } { 24 ^ { n } }$ is not an integer?
QA7 4 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x ) = ( x - a ) ( x - b ) ^ { 3 } ( x - c ) ^ { 5 } ( x - d ) ^ { 7 }$, where $a , b , c , d$ are real numbers with $a < b < c < d$. Thus $f ( x )$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. Consider $f ^ { \prime } ( x )$, i.e. the derivative of $f ( x )$. Find the following, if you can: (i) the number of real roots of $f ^ { \prime } ( x )$, counting multiplicities, (ii) the number of distinct real roots of $f ^ { \prime } ( x )$.
QA8 4 marks Factor & Remainder Theorem Remainder by Quadratic or Higher Divisor View
Let $f ( x ) = 7 x ^ { 32 } + 5 x ^ { 22 } + 3 x ^ { 12 } + x ^ { 2 }$. (i) Find the remainder when $f ( x )$ is divided by $x ^ { 2 } + 1$. (ii) Find the remainder when $x f ( x )$ is divided by $x ^ { 2 } + 1$. In each case your answer should be a polynomial of the form $a x + b$, where $a$ and $b$ are constants.
QA9 4 marks Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { 13 }$ be real numbers and let $A$ be the average of the complex numbers $e ^ { i \theta _ { 1 } } , e ^ { i \theta _ { 2 } } \ldots , e ^ { i \theta _ { 13 } }$, where $i = \sqrt { - 1 }$. As the values of $\theta$'s vary over all 13-tuples of real numbers, find (i) the maximum value attained by $| A |$, (ii) the minimum value attained by $| A |$.
QA10 4 marks Sine and Cosine Rules Ambiguous case and triangle existence/uniqueness View
In each of the following independent situations we want to construct a triangle $ABC$ satisfying the given conditions. In each case state how many such triangles $ABC$ exist up to congruence.
(i) $AB = 30 \quad BC = 95 \quad AC = 55$
(ii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad \angle C = 55 ^ { \circ }$
(iii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad BC = 55$
(iv) $\angle A = 30 ^ { \circ } \quad AB = 95 \quad BC = 55$
QA11 4 marks Radians, Arc Length and Sector Area View
Let $A _ { n } =$ the area of a regular $n$-sided polygon inscribed in a circle of radius 1 (i.e., vertices of this regular $n$-sided polygon lie on a circle of radius 1). (i) Find $A _ { 12 }$. (ii) Find $\left\lfloor A _ { 2014 } \right\rfloor$, i.e., the greatest integer $\leq A _ { 2014 }$.