TMUA 2019 Paper 1

$\text{f}(x)$ is a quadratic function in $x$. The graph of $y=\text{f}(x)$ passes through the point $(1,-1)$ and has a turning point at $(-1,3)$. Find an expression for $\text{f}(x)$.

Find the complete set of values of the real constant $k$ for which the expression $$x^2+kx+2x+1-2k$$ is positive for all real values of $x$.

Find the coefficient of $x$ in the expression: $$(1+x{)}^0+(1+x{)}^1+(1+x{)}^2+(1+x{)}^3+\cdots +(1+x{)}^{79}+(1+x{)}^{80}$$

The sequence $x_n$ is given by: $$x_1=10$$ $$x_{n+1}=\sqrt{x_n}\text{for }n\ge 1$$ What is the value of $x_{100}$? [Note that $a^{b^c}$ means $a^{(b^c)}$]

S is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the first 3 terms of S. The $7^{\text{th}}$ term of S is 360. Find the $1^{\text{st}}$ term of S.

The circles with equations $$(x+4{)}^2+(y+1{)}^2=64\text{and}(x-8{)}^2+(y-4{)}^2=r^2\text{where }r>0$$ have exactly one point in common. Find the difference between the two possible values of $r$.

A curve has equation $$y=(2q-x^2)(2qx+3)$$ The gradient of the curve at $x=-1$ is a function of $q$. Find the value of $q$ which minimises the gradient of the curve at $x=-1$.

The function $\text{f}$ is such that $0<\text{f}(x)<1$ for $0\le x\le 1$. The trapezium rule with $n$ equal intervals is used to estimate ${\int }_0^1\text{f}(x)\mathrm{d}x$ and produces an underestimate. Using the same number of equal intervals, for which one of the following does the trapezium rule produce an overestimate?

$p$ is a positive constant. Find the area enclosed between the curves $y=p\sqrt{x}$ and $x=p\sqrt{y}$.

Evaluate $${\int }_{-1}^3|x|(1-x)\mathrm{d}x$$

Find the sum of the real values of $x$ that satisfy the simultaneous equations: $${\log }_3(x{y}^2)=1$$ $$({\log }_{3}x)({\log }_{3}y)=-3$$

It is given that $$\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{24\pi (t-1)}{(1+\sqrt{t})}\text{for }t\ge 1$$ and $V=7$ when $t=1$. Find the value of $V$ when $t=9$.

Find the maximum value of $$4^{\sin x}-4\times 2^{\sin x}+\frac{17}{4}$$ for real $x$.

$x$ satisfies the simultaneous equations $$\sin 2x+\sqrt{3}\cos 2x=-1$$ and $$\sqrt{3}\sin 2x-\cos 2x=\sqrt{3}$$ where $0^{\circ }\le x\le 360^{\circ }$. Find the sum of the possible values of $x$.

Find the real non-zero solution to the equation $$\frac{2^{(9^x)}}{8^{(3^x)}}=\frac{1}{4}$$

Given that $$2{\int }_0^1\text{f}(x)\mathrm{d}x+5{\int }_1^2\text{f}(x)\mathrm{d}x=14$$ and $${\int }_0^1\text{f}(x+1)\mathrm{d}x=6,$$ find the value of $${\int }_0^2\text{f}(x)\mathrm{d}x$$

Find the fraction of the interval $0\le \theta \le \pi$ for which the inequality $$\left(\sin (2\theta )-\frac{1}{2}\right)\left(\sin \theta -\cos \theta \right)\ge 0$$ is satisfied.

Find the shortest distance between the curve $y=x^2+4$ and the line $y=2x-2$.

Find the value of $$\sum _{k=0}^{90}\sin (10+90k{)}^{\circ }$$

What is the complete range of values of $k$ for which the curves with equations $$y=x^3-12x$$ and $$y=k-(x-2{)}^2$$ intersect at three distinct points, of which exactly two have positive $x$-coordinates?