TMUA 2019 Paper 1

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

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

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

The sequence \(x_n\) is given by: \(x_1 = 10\) and \(x_{n+1} = \sqrt{x_n}\) for \(n \geq 1\). 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 7th term of S is 360. Find the 1st term of S.

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

A curve has equation \(y = (2q - x^2)(2q x + 3)\). 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 f is such that \(0 < f(x) < 1\) for \(0 \leq x \leq 1\). The trapezium rule with n equal intervals is used to estimate \(\displaystyle\int_{0}^{1} f(x) d x\) 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 \(\displaystyle\int_{-1}^3 |x|(1 - x) d x\)

Find the sum of the real values of x that satisfy the simultaneous equations: \(\log_3 (x y^2) = 1\) and \((\log_3 x)(\log_3 y) = -3\)

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

Find the maximum value of \(4^{\sin x} - 4 \times 2^{\sin x} + \dfrac{17}{4}\) 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 \(0^{\circ} \leq x \leq 360^{\circ}\). Find the sum of the possible values of x.

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

Given that \(2 \displaystyle\int_{0}^{1} f(x) d x + 5 \displaystyle\int_{1}^{2} f(x) d x = 14\) and \(\displaystyle\int_{0}^{1} f(x+1) d x = 6\), find the value of \(\displaystyle\int_{0}^{2} f(x) d x\)

Find the fraction of the interval \(0 \leq \theta \leq \pi\) for which the inequality \((\sin(2 \theta) - \dfrac{1}{2})(\sin \theta - \cos \theta) \geq 0\) is satisfied.

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

Find the value of \(\displaystyle\sum_{k=0}^{90} \sin(10 + 90 k)^{\circ}\)

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