Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Find $\displaystyle \int_{\Gamma} \left(x+y \right)\;dx+\left(y-x\right)\;dy,\;\Gamma$ is the line from $(0,0)$ to $(a,b)$.

$x_n=\frac{an^2+b}{cn^2+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| < 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$. Determine whether the sequence is increasing, decreasing or neither.

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\hat{t}$ for $t$. 

Given two propositions in mathematics using quantifiers, choose the corresponding negation of the proposition. For example, the negation of: $\displaystyle \exists a \in \mathbb{R^+},\;\exists b \in \mathbb{N},\;\exists c \in \mathbb{N}\;\left[(c \lt b+1) \land \left(\frac{1}{2^n} \geq 3a\right)\right]$

The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$.

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$.

Question on $\displaystyle{\lim_{n\to \infty} a^{1/n}=1}$. Find least integer $N$ s.t.  $\ \left |1-\left(\frac{1}{c}\right)^{b/n}\right| \le10^{-r},\;n \geq N$

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

$x_n=\frac{an+b}{cn+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| \le 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Calculate definite integrals: $\int_0^\infty\;e^{-ax}\,dx$, $\int_1^2\;\frac{1}{x^{b}}\,dx$, $\; \int_0^{\pi}\;\cos\left(\frac{x}{2n}\right)\,dx$

Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Questions on right-angled triangles asking for the calculation of angles using inverse-trigonometrical functions.

A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always obtuse. A secondary application is finding the area of a triangle.

Two questions testing the application of the Sine Rule when given two angles and a side. In this question the triangle is obtuse. In one question, the two given angles are both acute. In the second, one of the angles is obtuse.

A question testing the application of the Sine Rule when given two sides and an angle.  In this question the triangle is obtuse and the first angle to be found is obtuse.

Modular arithmetic. Find the following numbers modulo the given number $n$. Three examples to do.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

This question tests the students ability to calculate the area of different 2D shapes given the units and measurements required. The formulae for the areas are available if required but students are encouraged to try to remember them themselves.

The shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.

Author of gif: Picknick
https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

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Finding unknown sides/angles in right-angled triangles.  6 different combinations of unknowns are included in this single question. Makes my previous questions redundant

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