TMUA Specimen Paper 1

The sum of the two values of $x$ that satisfy the simultaneous equations $x-3y+1=0$ and $3x^2-7xy=5$ is

The number of solutions in the interval $0\le \theta \le 4\pi$ of the equation ${\sin }^2\theta +3\cos \theta =3$ is

The perpendicular bisector of the line segment joining the points $(2,-6)$ and $(5,4)$ cuts the $x$-axis at the point with $x$-coordinate

The complete set of values of $x$ for which $(x^2-1)(x-2)>0$ is

Given that $y=-{\log }_{10}(1-x)$ for $x<1$, find $x$ in terms of $y$.

It is given that $x+2$ is a factor of $x^3+4c{x}^2+x(c+1{)}^2-6$. The sum of the possible values of $c$ is

A bag contains $n$ red balls, $n$ yellow balls, and $n$ blue balls. One ball is selected at random and not replaced. A second ball is then selected at random and not replaced. Each ball is equally likely to be chosen. The probability that the two balls are not the same colour is

Given that $a^x{b}^{2x}{c}^{3x}=2$, where $a$, $b$, and $c$ are positive real numbers, then $x=$

The roots of the equation $2x^2-11x+c=0$ differ by 2. The value of $c$ is

The curve $y=\cos x$ is reflected in the line $y=1$ and the resulting curve is then translated by $\frac{\pi }{4}$ units in the positive $x$-direction. The equation of this new curve is

The sum of the roots of the equation $2^{2x}-8\times 2^x+15=0$ is

The cross-section of a triangular prism is an equilateral triangle with side $2x$ cm. The length of the prism is $d$ cm. Let the total surface area of the prism be $T$ cm${}^2$. Given that the volume of the prism is $T$ cm${}^3$, which one of the following is an expression for $d$ in terms of $x$?

How many real roots does the equation $x^4-4x^3+4x^2-10=0$ have?

$a$, $b$, $x$, and $y$ are real and positive. $a$ and $b$ are constants. $x$ and $y$ are related. A graph of $\log y$ against $\log x$ is drawn. For which one of the following relationships will this graph be a straight line?

The smallest possible value of ${\int }_0^1(x-a{)}^{2}dx$ as $a$ varies is

Given that $c$ and $d$ are non-zero integers, the expression $\frac{10^{c-2d}\times 20^{2c+d}}{8^c\times 125^{c+d}}$ is an integer if

For what values of the non-zero real number $a$ does the quadratic equation $a{x}^2+(a-2)x=2$ have real distinct roots?

The angle $x$ is measured in radians and is such that $0\le x\le \pi$. The total length of any intervals for which $-1\le \tan x\le 1$ and $\sin 2x\ge 0.5$ is

A geometric series has first term $4$ and common ratio $r$, where $0<r<1$. The first, second, and fourth terms of this geometric series form three successive terms of an arithmetic series. The sum to infinity of the geometric series is

The coefficient of $x^2$ in the expansion of $(4-x^2)\left[(1+2x+3x^2{)}^6-(1+4x^3{)}^5\right]$ is