TMUA 2016 Paper 1

It is given that the expansion of $(ax+b{)}^3$ is $8x^3-p{x}^2+18x-3\sqrt{3}$, where $a$, $b$ and $p$ are real constants. What is the value of $p$?

The expression $3x^3+13x^2+8x+a$, where $a$ is a constant, has $(x+2)$ as a factor. Which one of the following is a complete factorisation of the expression?

A line is drawn normal to the curve $y=\frac{2}{x^2}$ at the point on the curve where $x=1$. This line cuts the $x$-axis at $P$ and the $y$-axis at $Q$. The length of $PQ$ is

The sequence $a_n$ is defined by the rule: $$a_n=(-1{)}^n-(-1{)}^{n-1}+(-1{)}^{n+2}\text{for }n\ge 1.$$ Find the value of $\sum _{n=1}^{39}{a}_n$.

What is the total area enclosed between the curve $y=x^2-1$, the $x$-axis and the lines $x=-2$ and $x=2$?

P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P is 30

60

Find the maximum angle $x$ in the range $0^{\circ }\le x\le 360^{\circ }$ which satisfies the equation $${\cos }^2(2x)+\sqrt{3}\sin (2x)-\frac{7}{4}=0.$$

The line segment joining the points $(3,3)$ and $(7,5)$ is a diameter of a circle. This circle is translated by 3 units in the negative $x$-direction, then reflected in the $x$-axis, and then enlarged by a scale factor of 4 about the centre of the resulting circle. The equation of the final circle is

How many solutions does the equation $x\tan x=1$ have in the interval $-2\pi \le x\le 2\pi$?

The real roots of the equation $4^{2x}+12=2^{2x+3}$ are $p$ and $q$, where $p>q$. The value of $p-q$ can be expressed as

A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the circumferences of both ends of the cylinder are in contact with the sphere. The diagram shows a planar cross section through the centre of the sphere and cylinder. Find, in cubic centimetres, the maximum possible volume of the cylinder.

How many real roots does the equation $3x^5-10x^3-120x+30=0$ have?

The terms of an infinite series S are formed by adding together the corresponding terms in two infinite geometric series, T and U. The first term of T and the first term of U are each 4. In order, the first three terms of the combined series S are $8$, $3$, and $\frac{5}{4}$. What is the sum to infinity of S?

The least possible value of the gradient of the curve $y=(2x+a)(x-2a{)}^2$ at the point where $x=1$, as $a$ varies, is

Given the simultaneous equations $${\log }_{10}2+{\log }_{10}(y-1)=2{\log }_{10}x$$ $${\log }_{10}(y+3-3x)=0,$$ the values of $y$ are

It is given that $y=(1+2\cos x)\cos 2x$ for $0<x<\pi$. The complete set of values of $x$ for which $y$ is negative is

The function $\frac{1-x}{\sqrt[3]{x^2}}$ is defined for all $x\ne 0$. The complete set of values of $x$ for which the function is decreasing is

The coefficient of $x^3$ in the expansion of $(1+2x+3x^2{)}^6$ is equal to twice the coefficient of $x^4$ in the expansion of $(1-a{x}^2{)}^5$. Find all possible values of the constant $a$.

The diagram shows a square-based pyramid with base $PQRS$ and vertex $O$. All the edges of the pyramid are of length 20 metres. Find the shortest distance, in metres, along the outer surface of the pyramid from $P$ to the midpoint of $OR$.