TMUA 2021 Paper 1

Two circles have the same radius. The centre of one circle is $(-2,1)$. The centre of the other circle is $(3,-2)$. The circles intersect at two distinct points. What is the equation of the straight line through the two points at which the circles intersect?

The curve $y=x^3-6x+3$ has turning points at $x=\alpha$ and $x=\beta$, where $\beta >\alpha$. Find $${\int }_{\alpha }^{\beta }{x}^3-6x+3\mathrm{d}x$$

An arithmetic progression and a convergent geometric progression each have first term $\frac{1}{2}$. The sum of the second terms of the two progressions is $\frac{1}{2}$. The sum of the third terms of the two progressions is $\frac{1}{8}$. What is the sum to infinity of the geometric progression?

Find the minimum value of the function $$2^{2x}-2^{x+3}+4$$

The function f is such that $$\mathrm{f}(mn)=\{\begin{array}{ll}\mathrm{f}(m)\mathrm{f}(n)& \text{if }mn\text{ is a multiple of }3\\ mn& \text{if }mn\text{ is not a multiple of }3\end{array}$$ for all positive integers $m$ and $n$. Given that $\mathrm{f}(9)+\mathrm{f}(16)-\mathrm{f}(24)=0$, what is the value of $\mathrm{f}(3)$?

The function f is given by $$\mathrm{f}(x)=\frac{\cos x+3}{7+5\cos x-{\sin }^{2}x}$$ Find the positive difference between the maximum and the minimum values of $\mathrm{f}(x)$.

The function f is such that $\mathrm{f}(0)=0$, and $x\mathrm{f}(x)>0$ for all non-zero values of $x$. It is given that $${\int }_{-2}^2\mathrm{f}(x)\mathrm{d}x=4\text{and}{\int }_{-2}^2|\mathrm{f}(x)|\mathrm{d}x=8.$$ Evaluate $${\int }_{-2}^0\mathrm{f}(|x|)\mathrm{d}x$$

The line $y=2x+3$ meets the curve $y=x^2+bx+c$ at exactly one point. The line $y=4x-2$ also meets the curve $y=x^2+bx+c$ at exactly one point. What is the value of $b-c$?

Find the area enclosed by the graph of $$|x|+|y|=1$$

Use the trapezium rule with 3 strips to estimate $${\int }_{\frac{1}{2}}^{2}2{\log }_{10}x\mathrm{d}x$$

The function f is given by $$\mathrm{f}(x)=x^{\frac{1}{7}}(x^2-x+1)$$ Find the fraction of the interval $0<x<1$ for which $\mathrm{f}(x)$ is decreasing.

The minimum value of the function $x^4-p^2{x}^2$ is $-9$, where $p$ is a real number. Find the minimum value of the function $x^2-px+6$.

The function f is such that, for every integer $n$, $${\int }_n^{n+1}\mathrm{f}(x)\mathrm{d}x=n+1.$$ Evaluate $$\sum _{r=1}^8\left({\int }_0^r\mathrm{f}(x)\mathrm{d}x\right)$$

This question uses radians. Find the number of distinct values of $x$ that satisfy the equation $$(x+1)(3-x)=2(1-\cos (\pi x))$$

The diagram shows the graph of $y=\mathrm{f}(x)$. The graph consists of alternating straight-line segments of gradient $1$ and $-1$ and continues in this way for all values of $x$. The function g is defined as $$\mathrm{g}(x)=\sum _{r=1}^{10}f\left(2^{r-1}x\right)$$ Find the value of $${\int }_0^1\mathrm{g}(x)\mathrm{d}x$$

Consider the expansion of $(a+bx{)}^n$. The third term, in ascending powers of $x$, is $105x^2$. The fourth term, in ascending powers of $x$, is $210x^3$. The fourth term, in descending powers of $x$, is $210x^3$. Find the value of ${\left(\frac{a}{b}\right)}^2$.

Which of the following sketches shows the graph of $$\sin (x^2+y^2)=\frac{1}{2}$$ where $x^2+y^2\le 8\pi$?

The curve with equation $$x=y^2-6y+11$$ is rotated $90^{\circ }$ clockwise about the point $P$ to give the curve $C$. $P$ has $x$-coordinate $-2$ and $y$-coordinate $3$. What is the equation of $C$?

The equation $${\sin }^2\left(4^{\cos \theta }\times 60^{\circ }\right)=\frac{3}{4}$$ has exactly three solutions in the range $0^{\circ }\le \theta \le x^{\circ }$. What is the range of all possible values of $x$?

Find the length of the curve with equation $$2{\log }_{10}(x-y)={\log }_{10}(2-2x)+{\log }_{10}(y+5)$$