TMUA 2023 Paper 1

Given that $${\int }_0^1(ax+b)\mathrm{d}x=1$$ and $${\int }_0^{1}x(ax+b)\mathrm{d}x=1$$ find the value of $a+b$.

The graphs of $y=x^2+5x+6$ and $y=mx-3$, where $m$ is a constant, are plotted on the same set of axes.

Given that the graphs do not meet, what is the complete range of possible values of $m$?

For any integer $n\ge 0$, $${\int }_n^{n+1}f(x)\mathrm{d}x=n+1$$Evaluate $${\int }_0^{3}f(x)\mathrm{d}x+{\int }_1^{3}f(x)\mathrm{d}x+{\int }_2^{3}f(x)\mathrm{d}x+{\int }_4^{3}f(x)\mathrm{d}x+{\int }_5^{3}f(x)\mathrm{d}x$$

Evaluate $$\sum _{n=0}^{\infty }\frac{\sin \left(n\pi +\frac{\pi }{3}\right)}{2^n}$$

The following shape has two lines of reflectional symmetry.

$MNOP$ is a square of perimeter 40 cm. The vertices of rectangle $RSTU$ lie on the edge of square $MNOP$. $MR$ has length $x$ cm.

What is the largest possible value of $x$ such that $RSTU$ has area $20{\text{cm}}^2$?

In the simplified expansion of $(2+3x{)}^{12}$, how many of the terms have a coefficient that is divisible by 12?

$P(x)$ and $Q(x)$ are defined as follows: $$P(x)=2^x+4$$ $$Q(x)=2^{(2x-2)}-2^{(x+2)}+16$$Find the largest value of $x$ such that $P(x)$ and $Q(x)$ are in the ratio $4:1$, respectively.

A triangle $XYZ$ is called *fun* if it has the following properties: $$\text{angle }YXZ=30^{\circ }$$ $$XY=\sqrt{3}a$$ $$YZ=a$$ where $a$ is a constant.

For a given value of $a$, there are two distinct *fun* triangles $S$ and $T$, where the area of $S$ is greater than the area of $T$.

Find the ratio $$\text{area of }S:\text{area of }T$$

How many solutions are there to $$(1+3\cos 3\theta {)}^2=4$$ in the interval $0^{\circ }\le \theta \le 180^{\circ }$?

The trapezium rule with 4 strips is used to estimate the integral $${\int }_{-2}^2\sqrt{4-x^2}\mathrm{d}x$$What is the positive difference between the estimate and the exact value of the integral?

It is given that $f(x)=x^2-6x$.

The curves $y=f(kx)$ and $y=f(x-c)$ have the same minimum point, where $k>0$ and $c>0$.

Which of the following is a correct expression for $k$ in terms of $c$?

How many solutions are there to the equation $$\frac{2^{{\tan }^{2}x}}{4^{{\sin }^{2}x}}=1$$ in the range $0\le x\le 2\pi$?

Point $P$ lies on the circle with equation $(x-2{)}^2+(y-1{)}^2=16$.

Point $Q$ lies on the circle with equation $(x-4{)}^2+(y+5{)}^2=16$.

What is the maximum possible length of $PQ$?

The function $$f(x)=\frac{2}{3}{x}^3+2m{x}^2+n,m>0$$ has three distinct real roots.

What is the complete range of possible values of $n$, in terms of $m$?

The difference between the maximum and minimum values of the function $f(x)=a^{\cos x}$, where $a>0$ and $x$ is real, is 3.

Find the sum of the possible values of $a$.

A right-angled triangle has vertices at $(2,3)$, $(9,-1)$ and $(5,k)$.

Find the sum of all the possible values of $k$.

A circle $C_n$ is defined by $$x^2+y^2=2n(x+y)$$ where $n$ is a positive integer.

$C_1$ and $C_2$ are drawn and the area between them is shaded. Next, $C_3$ and $C_4$ are drawn and the area between them is shaded. This process continues until 100 circles have been drawn.

What is the total shaded area?

You are given that $$S=4+\frac{8k}{7}+\frac{16k^2}{49}+\frac{32k^3}{343}+\cdots +4{\left(\frac{2k}{7}\right)}^n+\cdots$$The value for $k$ is chosen as an integer in the range $-5\le k\le 5$.

All possible values for $k$ are equally likely to be chosen.

What is the probability that the value of $S$ is a finite number greater than 3?

The solution to the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=|-6x|\text{for all }x$$ is $y=f(x)+c$, where $c$ is a constant.

Which one of the following is a correct expression for $f(x)$?

The diagram shows the graph of $y=f(x)$.

The function $f$ attains its maximum value of 2 at $x=1$, and its minimum value of $-2$ at $x=-1$.

Find the difference between the maximum and minimum values of $(f(x){)}^2-f(x)$.