Kochi Entrance Mathematics key terms

For a quadratic ax²+bx+c, the quantity b²−4ac. Its sign decides how many real solutions the equation has: positive gives two distinct roots, zero gives one repeated root, and negative gives none.

The vertical line x = −b/(2a) passing through the vertex of the parabola y = ax²+bx+c. Reflecting the graph across this line leaves the curve unchanged.

Rewriting ax²+bx+c in the form a(x−p)²+q by adding and subtracting a constant. This exposes the vertex and makes the maximum or minimum value easy to read off.

For ax²+bx+c=0 with roots α and β, the sum α+β equals −b/a and the product αβ equals c/a. These let you evaluate symmetric expressions in the roots without solving the equation.

Dividing a polynomial P(x) by x−k leaves remainder P(k). As a consequence, x−k is a factor exactly when P(k)=0, which is the factor theorem.

The angle at a circle's centre subtended by an arc whose length equals the radius. One full turn is 2π radians, so degrees convert to radians by multiplying by π/180.

In a triangle with sides a, b, c opposite angles A, B, C: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It applies when a side and its opposite angle are known.

Identities such as sin(α+β) = sin α cos β + cos α sin β, with matching versions for cosine and tangent. They express a trigonometric function of a sum or difference through the separate angles.

The smallest positive p satisfying f(x+p)=f(x) for every x. Sine and cosine have period 2π, tangent has period π, and sin(kx) has period 2π/|k|.

For base a with a>0 and a≠1, log_a x is the exponent that turns a into x. It is defined only for x>0 and undoes exponentiation with the same base.

log_a x = log_b x / log_b a. It rewrites a logarithm in any convenient base, usually 10 or e, for calculation or for comparing logarithmic expressions.

For a positive number N, log₁₀N splits into an integer part plus a decimal part between 0 and 1. The mantissa fixes the digit pattern of N, the characteristic its order of magnitude.

A function of the form y = a^x with a>0 and a≠1. It is defined for every real x, takes only positive values, and increases when a>1 but decreases when 0<a<1.

A sequence in which each term after the first comes from adding a fixed number d, the common difference. Its nth term is a+(n−1)d and its partial sums grow quadratically.

A sequence in which each term is the previous term multiplied by a fixed nonzero number r, the common ratio. Its nth term is a·r^(n−1).

A rule that defines each term of a sequence using earlier terms, together with starting values. For example a_{n+1} = 2a_n + 1 with a₁ = 1.

The symbol Σ writes a sum compactly: Σ from k=1 to n of a_k means a₁+a₂+…+a_n. Standard formulas for sums of k, k² and k³ are used to evaluate such expressions.

A proof technique that establishes a statement for n=1 and shows that its truth for n forces its truth for n+1. Together these prove it for all positive integers n.

The value that f(x) approaches as x approaches a, written lim_{x→a} f(x) = L. The limit may exist even when f(a) itself is undefined.

The limit of the difference quotient {f(x+h)−f(x)}/h as h tends to 0. It gives the instantaneous rate of change, equal to the slope of the tangent to y=f(x) at x.

The rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x))·g′(x). The outer derivative is multiplied by the derivative of the inner function.

The straight line that touches a curve at a point and has the same slope as the curve there. For y=f(x) at x=a its equation is y−f(a) = f′(a)(x−a).

An expression such as 0/0 or ∞/∞ whose value cannot be read off directly from the parts. Limits of this type are resolved by factoring, cancelling, or substituting a new variable.

A function F whose derivative is f, that is F′(x)=f(x). Any two antiderivatives of f differ only by a constant, so the indefinite integral is written F(x)+C.

For a continuous function f on [a,b], the limit of sums of f(x)·Δx over finer and finer partitions. It equals F(b)−F(a) for any antiderivative F and represents signed area.

A technique that changes the variable to simplify an integral, using ∫f(g(x))g′(x)dx = ∫f(u)du with u=g(x). In a definite integral the limits must be converted as well.

The rule ∫u v′ dx = uv − ∫u′ v dx, obtained by reversing the product rule. It suits integrals of products such as x·e^x or x·cos x.

When the region under y=f(x) between a and b is rotated about the x-axis, the volume is π∫ from a to b of f(x)² dx, obtained by adding thin circular slices.

For vectors a=(a₁,a₂) and b=(b₁,b₂), the dot product is a₁b₁+a₂b₂, which also equals |a||b|cos θ. It measures alignment, and it is zero exactly when the vectors are perpendicular.

The vector drawn from the origin to a point P, whose components equal the coordinates of P. It lets geometric points be treated as vectors when studying lines, distances and ratios.

For z = a+bi the conjugate is z̄ = a−bi. The product z·z̄ = a²+b² is a real number, which is used to divide complex numbers and to compute moduli.

For z = a+bi, the modulus |z| = √(a²+b²) is the distance from the origin in the complex plane, and the argument is the angle measured from the positive real axis.

For any integer n, (cos θ + i sin θ)^n = cos nθ + i sin nθ. It computes powers and roots of complex numbers written in polar form.

An ordered arrangement of r objects selected from n distinct objects, counted by n!/(n−r)!. Order matters here, which is what separates a permutation from a combination.

A selection of r objects from n distinct objects in which order does not matter, counted by n!/(r!(n−r)!). It counts subsets, and nCr always equals nC(n−r).

The probability-weighted average of a random variable X, computed as the sum of each value multiplied by its probability. It describes the long-run mean over repeated trials.

The expected value of the squared deviation of X from its mean, measuring how spread out the distribution is. Its square root is the standard deviation, and it equals E(X²)−{E(X)}².

The probability that event A occurs given that event B has occurred, written P(A|B) = P(A∩B)/P(B) and defined when P(B)>0. It effectively shrinks the sample space to B.

The statement a ≡ b (mod m) means m divides a−b, so a and b leave the same remainder when divided by m. It underpins divisibility proofs and remainder problems.