Kuwait University Mathematics Aptitude Test flashcards

33 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 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac determines whether the roots are real, repeated, or complex.

Discriminant test

In ax² + bx + c = 0, the value b² − 4ac is the discriminant. If it is positive there are two distinct real roots, if zero one repeated real root, and if negative no real roots.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, the sum α + β = −b/a and the product αβ = c/a. These follow from expanding a(x − α)(x − β).

Completing the square

Rewrite x² + bx as (x + b/2)² − b²/4. This converts a quadratic into vertex form, revealing the turning point and enabling solution without factoring.

Zero product rule

If the product of two real numbers equals zero, at least one factor is zero. Factoring a polynomial and setting each factor to zero therefore yields all its roots.

Exponent laws

a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n. These hold for real exponents where defined.

Logarithm definition

log_b x = y means b^y = x, requiring b > 0, b ≠ 1, and x > 0. The natural logarithm ln x uses base e, and log x often means base 10.

Logarithm product and power rules

log_b(xy) = log_b x + log_b y and log_b(x^k) = k log_b x. The quotient rule gives log_b(x/y) = log_b x − log_b y. These convert multiplication into addition.

Change of base formula

log_b x = log_c x / log_c b for any valid base c. Choosing c = e or c = 10 allows evaluation on a standard calculator.

Arithmetic progression nth term

A sequence with constant difference d has nth term aₙ = a + (n − 1)d, where a is the first term. The sum of n terms is Sₙ = (n/2)(2a + (n − 1)d).

Geometric progression sum

With first term a and ratio r ≠ 1, the nth term is ar^(n−1) and Sₙ = a(1 − rⁿ)/(1 − r). If |r| < 1 the infinite sum converges to a/(1 − r).

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

Double angle formulas

sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1. The tangent form is tan 2θ = 2 tan θ / (1 − tan²θ).

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its named angle. It applies to acute and obtuse triangles alike, unlike right-triangle ratios.

Cosine rule

a² = b² + c² − 2bc cos A relates three sides to one angle. It reduces to Pythagoras when A = 90°, since cos 90° = 0.

Triangle area formula

Area = ½ · base · height, or Area = ½ ab sin C using two sides and the included angle. Heron's formula uses the semiperimeter s: Area = √(s(s−a)(s−b)(s−c)).

Slope of a line

For points (x₁, y₁) and (x₂, y₂) with x₁ ≠ x₂, slope m = (y₂ − y₁)/(x₂ − x₁). A vertical line has undefined slope; a horizontal line has slope zero.

Point-slope form

A line through (x₁, y₁) with slope m satisfies y − y₁ = m(x − x₁). Rearranging gives slope-intercept form y = mx + b, where b is the y-intercept.

Distance and midpoint formulas

Distance between two points is √((x₂−x₁)² + (y₂−y₁)²). Their midpoint is ((x₁+x₂)/2, (y₁+y₂)/2), the average of the coordinates.

Perpendicular slope rule

Two non-vertical lines are perpendicular exactly when their slopes multiply to −1, so m₂ = −1/m₁. Parallel lines, by contrast, have equal slopes.

Circle equation

A circle with centre (h, k) and radius r satisfies (x − h)² + (y − k)² = r². In expanded form x² + y² + Dx + Ey + F = 0, the centre is (−D/2, −E/2).

Power rule for derivatives

d/dx(xⁿ) = nx^(n−1) for any real n. Combined with linearity, it differentiates any polynomial term by term.

Product rule

If y = uv, then dy/dx = u′v + uv′. Differentiate the first factor times the second, plus the first times the derivative of the second.

Quotient rule

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

Chain rule

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

Derivative of sine and cosine

d/dx(sin x) = cos x and d/dx(cos x) = −sin x when x is measured in radians. Degree measure would introduce an extra constant factor.

Permutations versus combinations

nPr = n!/(n−r)! counts ordered selections of r items from n. nCr = n!/(r!(n−r)!) counts unordered selections, so nPr = nCr · r!.

Addition rule for probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the intersection because outcomes in both events are otherwise counted twice.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. It gives the probability of A given that B has occurred, restricting the sample space to B.

Independent events

Events A and B are independent when P(A ∩ B) = P(A) · P(B), equivalently P(A|B) = P(A). Occurrence of one then gives no information about the other.

Mean, median, mode

The mean is the sum divided by the count; the median is the middle value of ordered data; the mode is the most frequent value. The median resists outliers better than the mean.

Standard deviation

Standard deviation is the square root of variance, the average squared deviation from the mean. A small value means data cluster tightly around the mean.

Limit definition of derivative

The derivative at a point is the limit of (f(x + h) − f(x))/h as h approaches zero. It equals the slope of the tangent line at that point.