#Question
1What is the significance of prime numbers? 
2Do you know where the multiplication sign came from?
3Why do you play table tennis?  
4Whatever got you into pole dancing?  
5if your friends were here now instead of you, what would they say about you?
6Prove by contradiction that 2(a)^2 - b^2 is true only if a and b are both odd?   
7Differentiate y = x with respect to x^2?   
8You are given that y = t^t and x = cost. What is the value of dy/dx?   
9If a round table has n people sitting around it, what is the probability of person A sitting exactly k seats away from person B?   
10If (cos(x))^2 = 2sin(a), what are the intervals of values of a in the interval 0 ≤ a ≤ pi so that this equation has a solution?  
11What makes a tennis ball spin as it's travelling through the air?
12If we have 25 people, what is the likelihood that at least one of them is born each month of the year?
13Do you know where the multiplication sign came from
14Integrate arctan x!
15If you could have half an hour with any mathematician past or present, who would it be?
16Integrate from 0 to infinity the following: Int[xe^(-x^2)]dx and Int[(x^3)e^(-x^2)]dx. 
17Sketch y^2 = x^3 - x. 
18Sketch Y = (x^4 - 7x^2 + 12)/(x^4 - 4x^2 +4). 
19By considering (6 + (37^0.5))^20 + previous expression, show this second expression is very close to an integer. 
20Observation about (6 - (37^0.5))^20 being very small. 
21If you have n non-parallel lines in a plane, how many points of intersection are there? 
22If each face of a cube is coloured with one of 6 different colours, how many ways can it be done? 
23What is the square root of i?
24Sketch y=cos(1/x). 
25Differentiate x^x and (x^0.5)^(x^0.5). 
26Sketch y=(lnx)/x. 
27Compare the integrals between the values e and 1: a) int[ln(x^2)]dx; b) int[(lnx)^2]dx and c) int[lnx]dx. Which is largest?. 
28Sketch y=ln(x) explaining its shape. 
29Prove by contradiction that when z^2 = x^2 + y^2 has whole number solutions that x and y cannot both be odd. 
30How many solutions to kx=e^x for different values of k?
31Integrate xlog(x).
32If a cannon is pointed straight at a monkey in a tree, and the monkey lets go and falls towards the ground at the same instant the cannon is fired, will the monkey be hit? Describe any assumptions you make.
33Integrate 1/(x^2) between -1 and 1. Describe any difficulties in doing this?
34Is it possible to cover a chess-board with dominoes, when two corner squares have been removed from the chessboard and they are (a) adjacent corners, or conversely, (b) diagonally opposite.
35Differentiate x^x, then sketch it.
36Show that if n is an integer, n^3 - n is divisible by 6.  Comments
37There's a torus/ring doughnut shaped space station with 2 spacemen on a spacewalk standing diametrically oppositie each other. Can then ask a variety of questions such as if spaceman A wants to throw a spanner to spaceman B, what angle and speed should they choose (stating any assumptions made, e.g. that gravity = 0)?  Comments
38Draw the graph of tan(x).
39Draw the graph of y=x.
40Prove the two graphs have only one point in common in the interval [-pi/2, pi/2].
41Differentiate tan(x).
42Integrate log(x).
43How many zeros at the end of 30!.
44Prove that every odd number can be represented as a difference of two squares.
45How many numbers satisfy conditions: a) they have 3 digits b) all digits are even c) digits are distinct
46If a+1/a is an integer, prove that so is a^2+1/a^2.
47Sketch y=x^2/(x^2-2x-3)
48Sketch y=x^x. (that obviously involved logarithmic differentiation).
49Simple mechanics problem with a bead on a wire sliding down at some angle.
50Binomial coefficients and Pascal’s triangle 
51Draw graphs of :y=sin(1/x) ; y=x*sin(1/x) ; y=(x^3)*sin(1/x)
52A mechanics question based on M1. Bead on wire – how long to fall from A to B.
53Draw graph of y=e^(something)
54Prove for a 3 digit number that if the sum of the digits is a multiple of 3, the 3 digit number is a multiple of 3.
55Do the same as above, but in hexadecimal
56Do same as above, but for multiples of 5 
57Shown a model of a dodecahedron. Without counting work out:How many edges and how many vertices
58String tied inside dodecahedron between vertices, forming a cube. How many possible positions could there be for a cube inside the dodecahedron?
59Show that if x is a prime number greater than 3, x^2-1 is divisible by 24
60Drawing graphs
61Number theory
62Differentiate x^x 
63What is the square root of i?
64If I had a cube and six colours and painted each side a different colour, how many (different) ways could I paint the cube? 
65What about if I had n colours instead of 6? 
66How can I prove this expression is always an integer? 
67What is the largest integer value of the denominator for which this expression is always an integer?
68Logarithms on graphs and points of intersection
69Proof by induction
70There is a square, in which is a circle. the circle meets the square in four places so that the sides of the square form tangents to the circle. if the circumference of the circle increases at a rate of 6cm/s, by what rate does the perimeter of the square increase?
71A question on infinate probability
72If I go from A to B at 40mph, and back from B to A at 60mph, assuming there is no time taken in the turn, is my average speed more or less that 50mph? 
73One concerning binary numbers and multiple routes of a path
74Proof by contradiction
75The pigeon hole theory
76Why is there a shortage of mathematicians today?
77Which is more important: representing your college or completing your maths assignments?
78Is maths a universal language?
79I was asked to decide whether a certain type of arithmetic (a form of summation of columns of numbers that the interviewer had made up) was commutative
80How many digits there were in the binary representations of certain numbers
81How I would test a hypothesis involving numbers and shapes on playing cards
82What shape is obtained when a cube is cut along the plane bisecting two opposite vertices
83Various questions on AC current
84Sketch the curve (y^2-2)^2+(x^2-2)^2=2
853 girls and 4 boys were standing in a circle. What is the probability that two girls are together but one is not with them. 
86Is there such number N that 7 divided N^2-3? 
87Prove 1+1/2+1/3+…+1/1000<10.
88What is the integral of x squared multipied by the cosine of x cubed? 
89How many squares can be made from a grid of ten by ten dots (ignore diagonal squares)?
90Things like finding integer solutions to m^n = n^m
91Solve a^b=b^a for all real a and b. 
92If you could spend 30 minutes with any mathematician past or present, who would it be?
93Draw lnx/x and use this to solve a^b=b^a
94Some question involving three straight lines and the maximum point of their minimums.
95A divisibility proof using induction. Prove that 7^n-4^n is divisible by 3.
96The questions asked all involved the function f(1)=1 f(2n)=f(n) f(2n+1)=f(n)+1
97The final question was to work out f(n).
98Sketch a graph of logx and deduce from it which was bigger, e^pi or pi^e.
99Asked to draw a graph with some inequalities
100Sketching and integrating sin(1/x)
101Questions on curve sketching
102What is the last digit of 17^23?
103How many solutions does mod x+1 = mod x-1 have?
104How many soltuions does e^x^2=x have?
105An eulerian graph question.
106Basic differentiation/integration questions.
107Talk about an intersting piece of maths.
108A modular arithmetic question
109Draw y^2 = sinx
110Prove some things relating to the graph.
111Prove some basic algebraic things.
112Factorials,
113Number theory
114Sketch the graph of y = x^3 – x. How does it change if y = |x^3 – x| ?
115How many different types, and numbers, of rotations on axis of symmetry can you have for a cube?
116Prove n^3 – n is always divisible by 3.
117Prove that 3 > pi > 4 , considering the area of a circle compared to that of a square, and an n-sided shape placed inside the circle.
118Find which is the larger of e^pi and pi^e.
119Prove that x/y + y/x >= 2 when xy > 0.
120If cosx – ax – b = 0 has exactly two solutions, show that sinx = -a has at least one solution.
121Pi notation
122Binomials
123How many 0’s in 100!
124Prove that the angle at the centre of a circle is twice that at the circumference
125Using three colours, how many ways are there in which you can colour three equal portions of a disc 
126Integrate 1/(9 +x^2)
127Draw y=e^x, then draw y=kx, then draw a graph of the numbers of solutions of x against x for e^x=kx, and then find the value of k where there is only 1 solution 
128Four cards with A 2 3 K written on them, odd number implies that there is a vowel on the other side of the card, how many cards do I need to turn in order to disprove the statement
129Prove n^2 (mod 7) = (n+7)^2 (mod 7) 
130She then took out a Rubik’s cube and held it by two diagonally opposite vertices and rotated it till it reached the same position, by how many degrees did it take a turn
131Asked me to write down several different type of series that converge, is there a statement that if true the sequence must converge? 
132Integrate e^x * x^2 between limits of 1 and 0. Draw this graph.
133Prove why, when x is odd, x^2-1 is divisible by 8.
134Integrate x^-2 between limits of 1 and -1
135 Differentiating x^x 
136This old probability question about the car being behind one of 3 doors
137Write down 3 digits, then write the number again next to itself, eg: 145145. They then asked why is it divisible by 13.
138Sketch x^3 – a^2x
139Prove n^3 – n is divisible by three
140Show 3
141How many solutions and why |x|+|x-1|=0
142For what values of x is this true (x^2+1)/(x^2-1)<1 skecth it.
143Find the general formula for the sum of interior angles of a polygon.
144Solve dy/dx=(y+x)/2x
145Sketch y=x^2 – x^4
146Sketch y^2=x^2 – x^4
147Prove (n^2 – 1) is divisible by 8 when n is odd.
148Stuff about odd no.s being written as the difference between two squares.
149Integrate xsinx
150You are given a triangle with a fixed perimeter but you want to maximise the area. What shape will it be? Prove it. 
151You are given a quadrilateral with fixed perimeter and you want to maximise the area. What shape will it be? Prove it. 
152Draw a triangle, form inequality that the sum of any two sides is greater than or equal to the third side.
153Now draw a quadrilateral, draw in diagonals.
154Deduce that sum of the diagonals is greater than the sum of two opposite sides.
155There are a collection of points on a plane. Join them together to make a circuit. Uncrossed circuits can be made by finding crossings, and simply uncrossing them. Repeat until uncrossed.
156Deduce that there exists a shortest circuit – there are a finite number of points, hence there are a finite number of circuits. A finite set has a smallest member, hence there exists a shortest circuit. This circuit will not have any crossings, since the the length of the corresponding uncrossed circuit (created by uncrossing the crossing) would be longer (using result found above).
157Integration of some trigometric functions i.e. sin^2(x)cos^3(x)
158When f(x+y) = f(x)f(y), prove f(0) = 1 where f is a non-zero, real valued function.
159Sketch a graph of y=x^3 + ax + b
160What would be the relationship between a and b in which this curve would have 3 roots, 2 roots, 1 root.
161What are the possible remainders when a square number is divided by 4? Can the number 4000003 be written as the sum of 2 square numbers
162A sector is cut from a circle and a cone is made from the remaining material by pulling the two freshly cut edges together. What should the angle of the sector be in order to maximise the volume of the cone.
163Draw the graph of (ln x)/x. Find solutions of the equation a^b = b^a.
164With just two i/o commands, draw a line and turn a pixel on/off, how would you draw a circle on a finite matrix of pixels?
165What is the area between two circles, radius one, that go through each other’s centres?
166If every term in a sequence S is defined by the sum of every item before it, give a formula for the nth term.
167Why are there no pythagorean tripples in which both x and y are odd?
168Suggest a function that would describe this curve
169Integrate from first principles to find the volume of a sphere either side of an intersection with a line.
170 I was asked to optimize an algorith to find the highest, 3, 7, 15 then n numbers.
171 The conditions in wich a cubic equation has two, one or no solutions. 
172Then using calculus, to maximize the area of a rectangle given a fixed perimeter, then volume of a cuboid.
173Graph x cubed and x to five. 
174Integrate x to -2 from -1 to 1
175A question on flotation 
176A question about a ladder being leant on a wall and slowly collapsing down. What is the locus of the point half-way along the ladder?