Scottish Highers · Higher Mathematics

Expressions, Functions and Logarithmic Equations

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

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A recurrence relation is defined by $u_{n + 1} = 0.4 u_{n} + 18$ with $u_{0} = 50$ . What is the limit of this sequence as $n \to \infty$ ?
Solve the logarithmic equation: $lo g_{2} (x + 3) - lo g_{2} (x - 1) = 2$ .
Functions $f$ and $g$ are defined on suitable domains by $f (x) = 2 x - 5$ and $g (x) = x^{2} + 1$ . Find the composite expression $g (f (x))$ .
The graph of $y = k a^{x}$ passes through $(1, 6)$ and $(3, 24)$ . What are the values of $k$ and $a$ ( $a > 0$ )?
For what values of $p$ does the equation $x^{2} - (p + 2) x + 4 = 0$ have equal real roots?
Given that $f (x) = k sin (x) + cos (2 x)$ , find all values of $k$ such that $f^{'} (x) = 0$ has a solution at $x = \frac{π}{6}$ .

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