National 5 (Scotland) · National 5 Mathematics

Expressions, Formulae and Relationships

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

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Simplify $72$ .
Express $\frac{6}{3}$ with a rational denominator, in simplest form.
Simplify $\frac{( x ^{3} ) ^{4}}{x ^{5}}$ .
Evaluate $2 7^{\frac{2}{3}}$ .
Write $0.000 047$ in scientific notation.
Calculate $(3.2 \times 1 0^{5}) \times (2.5 \times 1 0^{- 2})$ , giving the answer in scientific notation.
Simplify $25 \times 310$ .
Expand and simplify $(2 x - 3) (x + 5)$ .
Factorise fully $3 x^{2} - 12$ .
Factorise $2 x^{2} + 7 x + 3$ .
Write $x^{2} + 6 x + 7$ in the form $(x + p)^{2} + q$ .
Simplify $\frac{x ^{2} - 9}{x ^{2} + 5 x + 6}$ .
Write $\frac{2}{x} + \frac{3}{x + 1}$ as a single fraction.
A sector of a circle has radius 12 cm and angle $7 5^{\circ}$ . Find the arc length, in terms of $π$ .
A cone has base radius 6 cm and perpendicular height 8 cm. Find its volume in terms of $π$ .
A sphere has radius 6 cm. What is its volume in terms of $π$ ?
A square-based pyramid has a base of side 10 cm and a perpendicular height of 9 cm. Find its volume.
Solve $3 x - 7 > 11$ .
Solve $- 2 x \leq 10$ .
Solve the simultaneous equations $2 x + y = 11$ and $x - y = 1$ .
Make $a$ the subject of $v^{2} = u^{2} + 2 a s$ .
Find the equation of the straight line through $(- 1, 3)$ and $(4, - 7)$ .
$f (x) = 3 x^{2} - 2 x$ . Find $f (- 2)$ .
Solve $x^{2} - 5 x - 14 = 0$ .
Use the quadratic formula to solve $2 x^{2} + 3 x - 4 = 0$ , giving answers to 2 decimal places.
The equation $x^{2} - 6 x + k = 0$ has equal roots. Find $k$ .
How many real roots does $2 x^{2} - 3 x + 5 = 0$ have?
What are the coordinates of the turning point of $y = (x - 2)^{2} - 9$ ?
A ladder 6.5 m long leans against a vertical wall with its foot 2.5 m from the wall. How high up the wall does it reach?
Two triangles are similar. In the smaller, two sides are 4 cm and 6 cm. The side corresponding to the 6 cm side in the larger triangle is 15 cm. What is the length of the side corresponding to the 4 c…
AB is a diameter of a circle and C is a point on the circumference. Angle $B A C = 3 4^{\circ}$ . Find angle $A B C$ .
A tangent from point P touches a circle with centre O at T. $O T = 5$ cm and $T P = 12$ cm. How far is P from O?
A chord of length 16 cm is drawn in a circle of radius 10 cm. How far is the chord from the centre?
The length of a rectangle is $(x + 4)$ cm and its width is $x$ cm. Its area is 60 cm $^{2}$ . Find $x$ .

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