MSCE Mathematics flashcards

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

Quadratic formula

For ax² + bx + c = 0, x = [-b ± √(b² - 4ac)] / (2a). Use when factorising is not straightforward.

Discriminant

D = b² - 4ac. If D > 0, two distinct real roots; D = 0, one repeated root; D < 0, no real roots.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C. Use with two angles and any side, or two sides and a non-included angle.

Cosine rule

a² = b² + c² - 2bc cos A. Use with two sides and the included angle, or all three sides to find an angle.

Area of triangle using sine

Area = ½ab sin C, where C is the angle between sides a and b. Works for any triangle, not just right-angled.

Exact trig values

sin 30°=½, cos 30°=√3/2, tan 30°=1/√3; sin 45°=cos 45°=1/√2, tan 45°=1; sin 60°=√3/2, cos 60°=½, tan 60°=√3.

Circle theorem: angle in semicircle

The angle subtended by a diameter at any point on the circumference is 90°. Also, angle at centre is twice angle at circumference.

Interior angle sum of polygon

For an n-sided polygon, sum of interior angles = (n - 2) × 180°. Each interior angle of a regular n-gon is (n-2)×180°/n.

Pythagoras' theorem

In a right-angled triangle, a² + b² = c², where c is the hypotenuse (side opposite the right angle).

Volume of a cone

V = ⅓πr²h, where r is base radius and h is perpendicular height. Curved surface area = πrl, l = slant height.

Volume of a sphere

V = (4/3)πr³. Surface area = 4πr². Use for hemispheres: halve the volume and add the flat circular area for total surface.

Volume of a pyramid

V = ⅓ × base area × perpendicular height. Works for any pyramid, including cones.

Compound interest formula

A = P(1 + r/100)^n, where P is principal, r is annual interest rate %, and n is number of years. Gives total amount after n years.

Rules of indices

a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m-n); (a^m)^n = a^(mn); a^0 = 1; a^(-n) = 1/a^n; a^(1/n) = n√a.

Rationalising a surd denominator

Multiply numerator and denominator by the surd or conjugate. For 1/√a, result is √a/a; for 1/(a+√b), multiply by (a-√b)/(a-√b).

Upper and lower bounds

For a value x rounded to nearest unit, UB = x + 0.5 and LB = x - 0.5. For division, maximum = UB ÷ LB, minimum = LB ÷ UB.

Standard form

A × 10^n where 1 ≤ A < 10 and n is an integer. Multiply: multiply A-parts, add indices. Divide: divide A-parts, subtract indices.

nth term of arithmetic sequence

nth term = a + (n - 1)d, where a is first term and d is common difference. Sum of first n terms = n/2[2a + (n-1)d].

Difference of two squares

a² - b² = (a + b)(a - b). Useful for factorising and simplifying algebraic fractions.

Completing the square

Write ax² + bx + c as a(x + p)² + q. Vertex of parabola is (-p, q); used to find maximum or minimum value.

Perpendicular gradients

If two lines are perpendicular, product of their gradients is -1 (m₁ × m₂ = -1). Parallel lines have equal gradients.

Gradient of a line

m = (y₂ - y₁)/(x₂ - x₁) for two points on the line. In y = mx + c, m is gradient and c is y-intercept.

Inverse function

f⁻¹ reverses f. To find it, swap x and y in y = f(x) then rearrange for y. Graph of f⁻¹ is reflection of f in y = x.

Graph transformation: y = f(x) + a

Translates the graph of y = f(x) vertically by a units (up if a > 0, down if a < 0). The shape is unchanged.

Mean from grouped data

Estimate mean = Σ(f × midpoint) / Σf, where f is frequency and midpoint is the middle of each class interval.

Interquartile range

IQR = upper quartile (Q3) - lower quartile (Q1). Measures spread of middle 50% of data; found from cumulative frequency graph.

Probability of combined events

For independent events: P(A and B) = P(A) × P(B). For mutually exclusive: P(A or B) = P(A) + P(B).

Tree diagram rule

Multiply probabilities along a branch for successive events; add probabilities of different branches for alternative outcomes.

Congruence tests

Triangles are congruent if SSS, SAS, ASA, or RHS. Similar triangles have equal angles and proportional sides.

Arc length and sector area

Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr², where θ is the central angle in degrees.

Density, speed, pressure

Density = mass/volume; speed = distance/time; pressure = force/area. Use consistent units and rearrange as needed.

Surface area of a cylinder

Total surface area = 2πr² + 2πrh, where r is radius and h is height. Curved surface area alone is 2πrh.