MSCE Additional Mathematics flashcards

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

Discriminant

For the quadratic ax² + bx + c = 0, the discriminant is b² − 4ac. Positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Remainder Theorem

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). Dividing by (ax − b) leaves remainder P(b/a).

Factor Theorem

The expression (x − a) is a factor of polynomial P(x) if and only if P(a) = 0. It is used to find unknown constants in polynomial identities.

Logarithm change of base

log_a b = log_c b / log_c a for any valid base c. This converts logarithms to a common base, often 10 or e, for evaluation.

Inverse function

f⁻¹ reverses f, so f⁻¹(f(x)) = x. Its graph is the reflection of y = f(x) in the line y = x. Only one-to-one functions have inverses.

Composite function domain

For fg(x) = f(g(x)), the domain is the set of x in the domain of g for which g(x) lies in the domain of f.

Horizontal stretch

The graph of y = f(ax) is the graph of y = f(x) stretched parallel to the x-axis by scale factor 1/a, where a is positive.

Even and odd functions

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis. Odd functions satisfy f(−x) = −f(x) and have rotational symmetry of order two about the origin.

Perpendicular gradient condition

Two straight lines are perpendicular exactly when the product of their gradients equals −1, so m₁ × m₂ = −1. Parallel lines have equal gradients.

Circle equation

A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². Expanded, x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f).

Tangent to a circle

At any point on a circle, the tangent is perpendicular to the radius drawn to that point. This gives the gradient of the tangent from the centre gradient.

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) in the ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Sum of geometric series

For a geometric progression with first term a and ratio r, the sum of n terms is a(1 − rⁿ)/(1 − r) when r ≠ 1, and na when r = 1.

Sum to infinity

A geometric series has a sum to infinity S∞ = a/(1 − r) only when the common ratio satisfies |r| < 1. Otherwise the series diverges.

General binomial term

In the expansion of (a + b)ⁿ, the term containing b^r is ⁿCᵣ a^(n−r) b^r. Setting the power of x to zero finds the constant term.

Arithmetic series sum

For an arithmetic progression with first term a and common difference d, the sum of n terms is Sₙ = n/2 [2a + (n − 1)d].

Cosine rule

In any triangle, a² = b² + c² − 2bc cos A. It finds a side given two sides and the included angle, or an angle given all three sides.

Double angle identities

sin 2A = 2 sin A cos A. cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A. tan 2A = 2tan A/(1 − tan²A).

Radians and sector area

π radians equal 180°. For angle θ in radians, arc length is s = rθ and sector area is ½r²θ, where r is the radius.

R sin(θ + α) form

The expression a sin θ + b cos θ equals R sin(θ + α) where R = √(a² + b²) and tan α = b/a. This solves equations of the form a sin θ + b cos θ = c.

Second derivative test

At a stationary point where dy/dx = 0, d²y/dx² > 0 indicates a minimum and d²y/dx² < 0 indicates a maximum. If it equals zero the test is inconclusive.

Quotient rule

For y = u/v, dy/dx = (v du/dx − u dv/dx)/v². It applies to ratios of differentiable functions, such as polynomial over polynomial.

Definite integral

The definite integral from a to b of f(x) dx equals F(b) − F(a), where F is any antiderivative of f. It gives the signed area under the curve.

Normal gradient

The normal to a curve at a point is perpendicular to the tangent there. Its gradient is −1 divided by the tangent gradient dy/dx at that point.

Scalar product

For vectors a and b, a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Perpendicular vectors have scalar product zero.

Unit vector

The unit vector in the direction of a nonzero vector a is a/|a|. It has magnitude one and preserves the direction of a.

Vector line equation

A straight line has equation r = a + t b, where a is the position vector of a point on the line, b is a direction vector and t is a scalar.

Inverse of 2×2 matrix

For M = [[a, b], [c, d]], the inverse is (1/(ad − bc)) [[d, −b], [−c, a]], provided ad − bc ≠ 0. A zero determinant means no inverse exists.

Rotation matrix

An anticlockwise rotation by angle θ about the origin has matrix [[cos θ, −sin θ], [sin θ, cos θ]]. A clockwise rotation uses −θ in place of θ.

Matrix of a transformation

The image of the unit vectors (1, 0) and (0, 1) under a linear transformation form the columns of its matrix. This determines the matrix.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. Events A and B are independent when P(A|B) equals P(A).

Binomial distribution

If X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ (1 − p)^(n−r). The mean is np and the variance is np(1 − p).