NSSCO Mathematics (6131) flashcards

35 cards. Read the term and the definition here. The deck at the top of this page is optional.

Standard form

A number written as a × 10^n with 1 ≤ a < 10 and n an integer. Used for very large or small values, e.g. 3.6 × 10^5 = 360 000.

Compound interest formula

A = P(1 + r/100)^n, where P is the principal, r the rate per period as a percentage and n the number of periods. Interest is added to the principal each period.

Rationalising a denominator

Removing a surd from the bottom of a fraction by multiplying numerator and denominator by that surd, so 1/√5 becomes √5/5.

Limits of accuracy

The range in which a rounded value truly lies. A value rounded to the nearest d has maximum error d/2, so 6.4 to 1 decimal place lies between 6.35 and 6.45.

Dividing in a given ratio

Split the total into the sum of the ratio parts, find the value of one part, then multiply by each part. £60 in ratio 2:3 gives £24 and £36.

Difference of two squares

The identity a² − b² = (a − b)(a + b). It factorises expressions such as x² − 49 into (x − 7)(x + 7).

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)] / 2a. It solves any quadratic, including ones that do not factorise.

Discriminant

The value b² − 4ac for the quadratic ax² + bx + c. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Solving simultaneous equations

Find values satisfying both equations at once. Multiply equations to match coefficients, then add or subtract to eliminate one variable and substitute back.

Inequality sign reversal

When both sides of an inequality are multiplied or divided by a negative number, the inequality sign reverses, so −2x > 6 gives x < −3.

Laws of logarithms

log(xy) = log x + log y, log(x/y) = log x − log y and log x^n = n log x. These convert products, quotients and powers into sums, differences and multiples.

Changing the subject

Rearranging a formula to isolate one variable. Apply inverse operations in reverse order, and square both sides to remove a square root.

Interior angle sum of polygon

For a polygon with n sides, the interior angles add to (n − 2) × 180°. A pentagon therefore has an interior angle sum of 540°.

Sector area and arc length

For central angle θ, sector area = (θ/360) × πr² and arc length = (θ/360) × 2πr. Both are fractions of the whole circle.

Volume of a cone

V = ⅓πr²h, one third of the volume of a cylinder with the same base radius and height. The slant height is not used in the volume.

Similar shapes ratios

If corresponding lengths are in ratio k, areas are in ratio k² and volumes in ratio k³. Doubling lengths multiplies area by 4 and volume by 8.

Triangle congruence tests

Two triangles are congruent if SSS, SAS, ASA (or AAS) or RHS holds. AAA and SSA do not guarantee congruence.

Pythagoras' theorem

In a right-angled triangle, a² + b² = c², where c is the hypotenuse opposite the right angle. It finds a missing side and tests for right angles.

Trigonometric ratios

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, remembered as SOH-CAH-TOA.

Cosine rule

a² = b² + c² − 2bc cos A. Use it when two sides and the included angle are known, or when all three sides are known.

Sine rule

a/sin A = b/sin B = c/sin C in any triangle. Use it when a side and its opposite angle are both known.

Area of any triangle

Area = ½ab sin C, where a and b are two sides and C is the angle between them. It needs no height measurement.

Three-figure bearings

Bearings are measured clockwise from north and written with three digits, such as 072°. A back bearing differs from the forward bearing by 180°.

Angle of elevation

The angle measured upwards from the horizontal to an object above the observer. It forms part of a right-angled triangle used with trigonometry.

Mean of grouped data

Estimate by mean = Σfx ÷ Σf, where x is each class midpoint and f its frequency. The result is an estimate because exact values are unknown.

Interquartile range

Upper quartile minus lower quartile. It measures the spread of the middle half of the data and is not affected by extreme values.

Independent events

Events where one outcome does not affect the other. For independent events, P(A and B) = P(A) × P(B).

Mutually exclusive events

Events that cannot happen at the same time. For these, P(A or B) = P(A) + P(B) and P(A and B) = 0.

Frequency density

Frequency ÷ class width, used for histograms with unequal class widths. The area of each bar equals the frequency of that class.

Inverse function

A function that reverses f, written f⁻¹. Find it by writing y = f(x), rearranging to make x the subject, then swapping x and y.

Composite function

fg(x) means apply g first, then f to the result. Order matters, so fg(x) is usually not the same as gf(x).

Gradient of a straight line

m = (y₂ − y₁)/(x₂ − x₁) for two points on the line. It equals the tangent of the angle the line makes with the horizontal.

Perpendicular gradients

Two lines are perpendicular when the product of their gradients is −1, so a line of gradient 3 is perpendicular to one of gradient −⅓.

Position vector

The vector from the origin to a point, written OA. Then AB = OB − OA, and the midpoint of AB has position vector ½(OA + OB).

Magnitude of a vector

For column vector (x, y), magnitude = √(x² + y²). A vector multiplied by scalar k has magnitude multiplied by |k|.