Maths National 5 Exam Facts and Formulae

A collection of important mathematical terms and their definitions for the National 5 Exam.

Area of a Rectangle

A = l × b

Perimeter of a Rectangle

P = 2 × (l + b) (Two times the sum of length and breadth)

Area of a Parallelogram

A = b × h

Area of a Kite/Rhombus

A = 1/2 × d1 × d2 (One-half times the product of the diagonals)

Area of a Trapezium

A = 1/2 (a + b) × h (One-half times the sum of the bases times the height)

Area of a Triangle

A = 1/2 × b × h

Area of a Circle

A = π × r²

Circumference of a Circle

C = π × d

Area of a Sector

Sector Area = x/360° × π × r²

Length of an Arc

Arc Length = x/360° × π × d

Indices Rule for Multiplication

a^m × a^n = a^(m+n)

Indices Rule for Division

a^m ÷ a^n = a^(m-n)

Indices Power Rule

a^m = a^(n×m)

Negative indices rule

a^(-m) = 1/a^m

Fractional Indices Rule

a^(m/n) = = n√a^m (REMEMBER flower power, square root)

Indices Power of 0

a^0 = 1

Surds Multiplication Rule

√a × √b = √(a × b)

Surds Division Rule

√a ÷ √b = √(a ÷ b)

Surds square roots squared

√a x √a = a

Square numbers from 1-15

1,4,9,16,25,36,49,64,81,100,121,144,169,196,225

Cube numbers 1-5

1, 8, 27, 64, 125

Gradient of a Straight Line

m = (y2−y1)/(x2−x1)

Equation of a Straight Line

y = mx + c

Volume of a Cuboid

V = l × b × h

Volume of a prism

V = A x h

Volume of a cylinder

V = π x r² x h (notice: area of circle)

Curved surface area of a cylinder

CSA = 2 x π x r x h

Order of priority in factorising

Highest common factor, difference of two squares, trinomial

Discriminant formula

b² - 4ac

Discriminant use

To determine the nature of the roots of a quadratic equation

Discriminate value indicating the nature of roots

Positive = 2 real, distinct roots

Zero = 2 real, equal roots

Negative = no real roots

Completing the square format

y = (x + a)² + b

How to find turning point from completed square format

When y = (x + a)² + b, TP(-a, b)

Length scale factor formula

new / original

Area scale factor

(Length scale factor)² aka (new / original)²

Volume scale factor

(Length scale factor)³ aka (new / original)³

Pythagorean Theorem

c² = a² + b²

Sum of Angles in a Triangle

180 degrees

Sum of angles in a Quadrilateral

360 degrees

Degrees in a complementary/right angle

90 degrees

Degrees in a supplementary angle/straight line

180 degrees

What is special about vertically opposite angles

Always equal

What is special about corresponding/F shape angles

Always equal

What is special about alternate/Z shape angles

Always equal

How to find a component vector

𝐴𝐵 = 𝑏 − 𝑎

How to find the magnitude of a 2D vector

If 𝑎 = (𝑥, 𝑦) then |𝑎| = √(𝑥² + 𝑦²)

How to find the magnitude of a 3D vector

If 𝑎 = (𝑥, 𝑦, 𝑧) then |𝑎| = √(𝑥² + 𝑦² + 𝑧²)

Scientific notation form

𝑎 × 10^𝑛 𝑤ℎ𝑒𝑟𝑒 0 < 𝑎 < 10 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

Range

Highest value - lowest value

Interquartile range

Q3 - Q1

Semi-interquartile range

(Q3 - Q1) / 2

5 figure summary

Highest, Q3, Q2/Median, Q1, Lowest

Mean Average

Sum of values divided by the number of values (x = Σx/n)

Mode average

Most common value

SoH represents

Sin x = Opposite/Hypotenuse

CaH represents

Cos x = Adjacent/Hypotenuse

ToA

Tan x = Opposite/Adjacent

Trigonometric Identities (2)

Tan x = sin x / cos x and sin² x + cos² x = 1.

Standard Deviation Wording for Mean

On average (describe trend in context) because the mean is higher/lower

Standard Deviation Wording for Standard Deviation

More/less consistent (more/less varied)

Converse of Pythagoras Wording (if yes)

Yes. As (8.1)² + (10.8)² = (13.5)², by the Converse of Pythagoras, the triangle contains a right angle.

Converse of Pythagoras Wording (if no)

No. Since (8.1)² + (10.8)² ≠ (13.5)², by the Converse of Pythagoras, the triangle does not contain a right angle.

What does the a represent in y = asinx?

Amplitude

What does the b represent in y = sinbx?

Period (how many waves in 360 [or 180 for tan])

What does the c represent in y = sinx + b?

Vertical shift (up or down the y axis)

What does the c represent in y = sin (x + b)?

Horizontal/phase shift (left or right)

How to find the amplitude of a trig graph?

½ the distance between peak(top) and trough(bottom)

How to find the period of a graph?

How many waves fit into 360 degrees for sine and cosine graphs or 180 degrees for tan graphs

What is true for angles in a semi circle?

They form a right angle

What is the angle where a tangent and radius meet?

Right angle/ 90 degrees

How to complete the square?

(when y = ax² +bx +c)

1. ½ the b (= d)

2. rewrite to y = (x + d)² + c - d²

3. simplify c - d² to one number

How to find the point of intersection of 2 lines

Equate them

Where do 3 figure bearings begin (aka what direction is 000 degrees)?

North

Percentage increase formula

change / original x 100

Compound interest formula

O x (1 + r)ⁿ, where O is the original value, r is the rate, and n is the number of time periods (usually years). (e.g. 1000 x (1.02)³

How to solve a simultaneous equation?

1. Eliminate x or y

2. Find x on its own

3. Sub into 1 to find y

4. Check in 2

How to do standard deviation?

1. Find the mean

2. Create a table with | mean | mean - values | square column 2 | (ALWAYS remember (-x)² = +x²

3. Enter into formula

How to find the equation of a straight line?

1. Find the gradient (m)

2. Find the y-intercept (sub into y = mx + c)

3. Write in the form y = mx + c

How to find y-intercept(c) in straight line?

Sub m, x and y into y = mx + c and solve for c (m = gradient, x = x-coord, y = y-coord)

How to rationalise the denominator?

Multiply top and bottom by the surd on the bottom then simplify