Maths245 views·Updated Jun 14, 2026·2 pages

2019 Higher Maths Paper 1 Solutions

Beth Taylor@bethtaylor_gqoz

These maths problems cover some of the trickiest A-level topics...

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$12. functions $f$ and $g$ are defined by $f(x) = \frac{1}{\sqrt{x}}$ where $x \ge 0$ $g(x) = 5-x$ a) determine expression for $f(g(x))$$

Composite Functions and Trigonometry

Composite functions might look intimidating, but they're just functions within functions. When you see f(g(x)), you substitute the entire g(x) expression into f(x). So if f(x) = 1/√x and g(x) = 5-x, then f(g(x)) = 1/√ $5-x$ .

The key trick is spotting when these composite functions are undefined. Since you can't take the square root of a negative number, 5-x must be greater than 0, which means x must be less than 5.

For right-angled triangles, remember SOHCAHTOA is your best mate. When finding cos p or cos q, you're looking for adjacent/hypotenuse. Use Pythagoras' theorem to find missing sides first - it's usually the stepping stone to solving trigonometric ratios.

Top tip: Always check your triangle measurements make sense using Pythagoras before calculating trig ratios!

The compound angle formula sin $p+q$ = sin p cos q + cos p sin q is essential for A-level success. Once you've found the individual trig ratios, substitute them carefully and simplify step by step.

of 2

$12. functions $f$ and $g$ are defined by $f(x) = \frac{1}{\sqrt{x}}$ where $x \ge 0$ $g(x) = 5-x$ a) determine expression for $f(g(x))$$

Logarithms and Advanced Trigonometry

Logarithms become much easier when you remember the key rules. The most useful ones are: log a + log b = log(ab) and n log a = log $a^n$ . These let you combine and split logarithmic expressions to solve equations efficiently.

When solving logarithmic equations, get everything on one side first, then use the definition that if log_b(x) = n, then x = b^n. This converts your logarithmic equation into a simple algebraic one you can solve normally.

Double angle trigonometry problems often use the identity sin 2x = 2 sin x cos x. Look for opportunities to factorise - you'll often get expressions like cos x(something) = 0, which gives you multiple solutions to consider.

Remember: Always check your trigonometric solutions fit within the given range, and don't forget there might be multiple solutions!

The beauty of these advanced topics is that they build on simpler concepts you already know. Master the basic identities and logarithm rules, and you'll find even complex problems break down into manageable steps.

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Maths245 views·Updated Jun 14, 2026·2 pages

2019 Higher Maths Paper 1 Solutions

Beth Taylor@bethtaylor_gqoz

Add to Google sources

These maths problems cover some of the trickiest A-level topics that often pop up in exams - composite functions, trigonometry, and logarithms. Mastering these will give you a solid foundation for tackling complex mathematical relationships and solving multi-step problems with...

of 2

$12. functions $f$ and $g$ are defined by $f(x) = \frac{1}{\sqrt{x}}$ where $x \ge 0$ $g(x) = 5-x$ a) determine expression for $f(g(x))$$

Composite Functions and Trigonometry

The key trick is spotting when these composite functions are undefined. Since you can't take the square root of a negative number, 5-x must be greater than 0, which means x must be less than 5.

Top tip: Always check your triangle measurements make sense using Pythagoras before calculating trig ratios!

of 2

$12. functions $f$ and $g$ are defined by $f(x) = \frac{1}{\sqrt{x}}$ where $x \ge 0$ $g(x) = 5-x$ a) determine expression for $f(g(x))$$

Logarithms and Advanced Trigonometry

Remember: Always check your trigonometric solutions fit within the given range, and don't forget there might be multiple solutions!