TMUA 2017 Paper 1

Given that $$\frac{dy}{dx}=3x^2-\frac{2-3x}{x^3},x\ne 0$$ and $y=5$ when $x=1$, find $y$ in terms of $x$.

The function $f$ is given by $$f(x)={\left(\frac{2}{x}-\frac{1}{2x^2}\right)}^2(x\ne 0).$$ What is the value of $f^{\prime \prime }(1)$?

A line $l$ has equation $y=6-2x$. A second line is perpendicular to $l$ and passes through the point $(-6,0)$. Find the area of the region enclosed by the two lines and the $x$-axis.

When $(3x^2+8x-3)$ is multiplied by $(px-1)$ and the resulting product is divided by $(x+1)$, the remainder is 24. What is the value of $p$?

$S$ is the complete set of values of $x$ which satisfy both the inequalities $$x^2-8x+12<0\text{and}2x+1>9.$$ The set $S$ can also be represented as a single inequality. Which one of the following single inequalities represents the set $S$?

A tangent to the circle $x^2+y^2=144$ passes through the point $(20,0)$ and crosses the positive $y$-axis. What is the value of $y$ at the point where the tangent meets the $y$-axis?

The first three terms of an arithmetic progression are $p$, $q$ and $p^2$ respectively, where $p<0$. The first three terms of a geometric progression are $p$, $p^2$ and $q$ respectively. Find the sum of the first 10 terms of the arithmetic progression.

Find the complete set of values of $x$, with $0\le x\le \pi$, for which $$(1-2\sin x)\cos x\ge 0.$$

A circle has equation $x^2+y^2-18x-22y+178=0$. A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle. What is the area of the hexagon?

A curve $C$ has equation $y=f(x)$ where $$f(x)=p^3-6p^{2}x+3p{x}^2-x^3$$ and $p$ is real. The gradient of the normal to the curve $C$ at the point where $x=-1$ is $M$. What is the greatest possible value of $M$ as $p$ varies?

The sequence $x_n$ is defined by the rules $$x_1=7,x_{n+1}=\frac{23x_n-53}{5x_n+1}.$$ The first three terms in the sequence are 7, 3, 1. What is the value of $x_{100}$?

The polynomial function $f(x)$ is such that $f(x)>0$ for all values of $x$. Given ${\int }_2^{4}f(x)dx=A$, which one of the following statements must be correct?

In the expansion of $(a+bx{)}^5$ the coefficient of $x^4$ is 8 times the coefficient of $x^2$. Given that $a$ and $b$ are non-zero positive integers, what is the smallest possible value of $a+b$?

The solution of the simultaneous equations $$2^x+3\times 2^y=3$$ $$2^{2x}-9\times 2^{2y}=6$$ is $x=p$, $y=q$. Find the value of $p-q$.

It is given that $f(x)=-2x^2+10$. Consider the following three curves: (1) $y=f(x)$, (2) $y=f(x+1)$, (3) the curve $y=f(x+1)$ reflected in the line $y=6$. The trapezium rule is used to estimate the area under each of these three curves between $x=0$ and $x=1$. State whether the trapezium rule gives an overestimate or underestimate for each of these areas.

The functions $f$ and $g$ are given by $f(x)=3x^2+12x+4$ and $g(x)=x^3+6x^2+9x-8$. What is the complete set of values of $x$ for which one of the functions is increasing and the other decreasing?

The two functions $F(n)$ and $G(n)$ are defined as follows for positive integers $n$: $$F(n)=\frac{1}{n}{\int }_0^n(n-x)dx,G(n)=\sum _{r=1}^{n}F(r).$$ What is the smallest positive integer $n$ such that $G(n)>150$?

The graph of $y={\log }_{10}x$ is translated in the positive $y$-direction by 2 units. This translation is equivalent to a stretch of factor $k$ parallel to the $x$-axis. What is the value of $k$?

The set of solutions to the inequality $x^2+bx+c<0$ is the interval $p<x<q$, where $b$, $c$, $p$ and $q$ are real constants with $c<0$. In terms of $p$, $q$ and $c$, what is the set of solutions to the inequality $x^2+bcx+c^3<0$?

The lengths of the sides $QR$, $RP$ and $PQ$ in triangle $PQR$ are $a$, $a+d$ and $a+2d$ respectively, where $a$ and $d$ are positive and such that $3d>2a$. What is the full range, in degrees, of possible values for angle $PRQ$?