TMUA · Paper 1: Applications of Mathematical Knowledge

Functions

16 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 16 questions

$f (x) = 2 x + 3$ and $g (x) = x^{2}$ . What is the sum of the solutions of $f (g (x)) = g (f (x))$ ?
$f (x) = \frac{x + 1}{x - 2}$ for $x \neq = 2$ . Which expression is $f^{- 1} (x)$ ?
The point $(3, 5)$ lies on the graph of $y = f (x)$ . Which point must lie on the graph of $y = f (2 x - 1)$ ?
The function $f$ is defined by $f (x) = x^{2} - 4 x + 7$ for $x \geq 2$ . What is the range of $f$ ?
If $x + y = 5$ and $x y = 6$ , what is the value of $x^{2} + y^{2}$ ?
The circle $x^{2} + y^{2} = 25$ and the line $y = x + 1$ intersect at two points. What is the sum of the $y$ -coordinates of these points?
What is the sum of all real solutions of $(x^{2} - 3 x)^{2} - 2 (x^{2} - 3 x) - 8 = 0$ ?
What is the minimum value of $x^{2} + 6 x + 11$ ?
What is the sum of the squares of all the real roots of $x^{4} - 5 x^{2} + 4 = 0$ ?
How many real solutions does $x^{4} + x^{2} + 1 = 0$ have?
$a$ and $b$ are positive integers with $a^{2} - b^{2} = 47$ . What is the value of $a$ ?
What is the smallest integer $k$ for which $x^{2} - 4 x + k > 0$ for every real value of $x$ ?
The line $y = m x + 1$ is a tangent to the curve $y = x^{2} + 2 x + 5$ . What is the positive value of $m$ ?
The quadratic $x^{2} - p x + 12$ has two positive integer roots which differ by $1$ . What is $p$ ?
What is the largest value of $\frac{1}{x ^{2} - 4 x + 7}$ for real $x$ ?
If $x^{2} - 5 x + 1 = 0$ , what is the value of $x^{3} + \frac{1}{x ^{3}}$ ?

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