UEE Mathematics flashcards

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

Quadratic discriminant

For ax² + bx + c = 0, Δ = b² − 4ac. Positive Δ gives two distinct real roots, zero gives one repeated root, negative gives no real roots but two complex conjugates.

Remainder theorem

When polynomial P(x) is divided by (x − a), the remainder equals P(a). Consequently (x − a) is a factor of P exactly when P(a) = 0.

Change of base formula

logₐx = log_b x / log_b a for any valid base b. In particular logₐx = ln x / ln a, which lets any calculator evaluate logarithms in an arbitrary base.

Sum of geometric series

For first term a and ratio r ≠ 1, Sₙ = a(1 − rⁿ)/(1 − r). If |r| < 1 the infinite sum converges to a/(1 − r); if |r| ≥ 1 it diverges.

Binomial general term

In the expansion of (a + b)ⁿ, the term containing b^k is C(n,k)·a^(n−k)·b^k, where C(n,k) = n!/(k!(n−k)!). This locates any single coefficient without full expansion.

Completing the square vertex

Writing y = ax² + bx + c as y = a(x − h)² + k gives vertex (h, k) with h = −b/(2a). The axis of symmetry is the vertical line x = h.

Inverse function condition

A function has an inverse only if it is one-to-one, meaning each output comes from exactly one input. The graphs of f and f⁻¹ are mirror images across the line y = x.

Horizontal asymptote rule

For a rational function, compare degrees of numerator and denominator: lower numerator degree gives y = 0, equal degrees give the ratio of leading coefficients, higher numerator degree gives none.

Odd and even functions

A function is even when f(−x) = f(x), producing symmetry about the y-axis. It is odd when f(−x) = −f(x), producing rotational symmetry about the origin.

Exponential graph properties

The graph of y = aˣ with a > 1 rises steadily, passes through (0, 1), stays positive, and approaches the x-axis as a horizontal asymptote as x tends to negative infinity.

Perpendicular gradient condition

Two non-vertical lines are perpendicular exactly when the product of their gradients equals −1, so one gradient is the negative reciprocal of the other, such as 3 and −1/3.

Heron's formula

The area of a triangle with sides a, b, c is √(s(s − a)(s − b)(s − c)), where s is the semiperimeter (a + b + c)/2. It needs no height or angle.

Similar solids scaling

If two similar solids have linear scale factor k, their surface areas scale by k² and their volumes by k³. Doubling every length multiplies volume by eight.

Angle in a semicircle

Any angle subtended by a diameter at a point on the circumference is a right angle, 90°. This is a special case of the angle at the centre being twice the angle at the circumference.

Sector area in radians

For a circle of radius r and central angle θ measured in radians, the sector area is ½r²θ and the corresponding arc length is rθ. Degrees must be converted first.

Interior angle sum of polygon

An n-sided polygon has interior angles summing to (n − 2) × 180°. In a regular polygon each interior angle equals that sum divided by n, and each exterior angle is 360°/n.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Cosine rule

In any triangle, a² = b² + c² − 2bc·cos A, where A is the angle opposite side a. It applies when two sides and the included angle, or all three sides, are known.

Double angle for cosine

cos 2A has three equivalent forms: cos²A − sin²A, 1 − 2sin²A, and 2cos²A − 1. Choose the form that matches the other expressions in the problem.

Amplitude and period

For y = a sin(bx) + d, the amplitude is |a| and the period is 360°/|b| in degrees, or 2π/|b| in radians. The constant d shifts the graph vertically.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its matching angle. It is used with a known side–angle pair plus one more piece of data.

CAST sign rule

In the four quadrants, All ratios are positive in the first, only Sine in the second, only Tangent in the third, and only Cosine in the fourth. This determines signs of trigonometric ratios.

Independent events multiplication

Events A and B are independent when P(A ∩ B) = P(A) × P(B), meaning one occurring does not change the probability of the other. This fails for mutually exclusive events with nonzero probabilities.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0. It gives the probability of A given that B has already occurred, and rearranges to the multiplication rule.

Binomial distribution mean and variance

For X ~ B(n, p) counting successes in n independent trials, the mean is np and the variance is np(1 − p). The standard deviation is the square root of that variance.

Standard deviation definition

The standard deviation is the square root of the variance, so σ = √(Σ(x − x̄)²/n) for a population. It measures typical spread in the same units as the data.

Correlation coefficient range

The Pearson correlation coefficient r always lies between −1 and 1. Values near ±1 indicate strong linear association, values near 0 weak linear association, and r² gives the proportion of variance explained.

Chain rule

For a composite function y = f(g(x)), the derivative is dy/dx = f′(g(x)) · g′(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Quotient rule

For y = u/v with v ≠ 0, the derivative is (u′v − uv′)/v². The numerator order matters: it is the derivative of the top times the bottom minus the top times the derivative of the bottom.

Second derivative test

At a stationary point where f′(x) = 0, f″(x) > 0 indicates a local minimum and f″(x) < 0 a local maximum. If f″(x) = 0 the test is inconclusive.

Definite integral evaluation

If F is an antiderivative of f, then the integral of f from a to b equals F(b) − F(a). This fundamental theorem converts area accumulation into evaluation at the endpoints.

Normal line gradient

The normal to a curve at a point is perpendicular to the tangent there, so its gradient is −1/f′(a) provided f′(a) is nonzero. A horizontal tangent gives a vertical normal.