Kerala Plus Two Mathematics (DHSE) flashcards

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

Bijective function

A function that is both one-one and onto. Only bijections have inverses, and the inverse swaps domain and range.

Equivalence relation

A relation that is reflexive, symmetric and transitive. Its equivalence classes partition the underlying set into disjoint non-empty subsets.

Principal value of sin inverse

sin⁻¹x lies in [−π/2, π/2] for x in [−1,1]. This restricted branch makes sine invertible; values outside this interval are not principal.

tan inverse addition formula

tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)) when xy<1. If xy>1, add π for positive x,y and subtract π for negative x,y.

Skew-symmetric matrix

A square matrix with Aᵀ=−A, so all diagonal entries are zero. Every square matrix splits into symmetric plus skew-symmetric parts.

Inverse of a 2x2 matrix

For A=[[a,b],[c,d]] with determinant ad−bc nonzero, A⁻¹=(1/(ad−bc))[[d,−b],[−c,a]]. A zero determinant means no inverse exists.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c in (a,b) satisfies f′(c)=0.

Chain rule

If y=f(u) and u=g(x), then dy/dx=f′(u)·g′(x). It handles composite functions such as sin(x²) or e^(3x).

Second derivative test

At a stationary point c, f″(c)>0 gives a local minimum and f″(c)<0 gives a local maximum. If f″(c)=0 the test is inconclusive.

Absolute extrema on closed interval

For continuous f on [a,b], compare values at all critical points inside (a,b) and at endpoints a and b; the largest is the maximum.

Integration by parts

∫u v′dx=uv−∫u′v dx. Choose u using ILATE order: inverse trig, logarithmic, algebraic, trigonometric, exponential.

Even and odd integral rule

For f on [−a,a]: if f is odd, the integral is 0; if f is even, it equals 2∫₀ᵃf(x)dx.

Integrating factor

For dy/dx+Py=Q, the integrating factor is e^(∫P dx). Multiplying through makes the left side the derivative of y times that factor.

Order and degree

Order is the highest derivative in the equation. Degree is the power of that highest derivative once the equation is polynomial in derivatives.

Scalar triple product

a⃗·(b⃗×c⃗) equals the volume of the parallelepiped formed by the three vectors. It is zero exactly when the vectors are coplanar.

Distance from point to plane

For plane ax+by+cz+d=0 and point (x₁,y₁,z₁), distance equals |ax₁+by₁+cz₁+d| divided by √(a²+b²+c²).

Corner point theorem

For a bounded feasible region in a linear program, the optimal value of the objective function occurs at a vertex of the region.

Multiple optimal solutions

If two vertices of the feasible region give the same optimal objective value, every point on the segment joining them is also optimal.

Conditional probability

P(A|B)=P(A∩B)/P(B) for P(B)>0. It measures the chance of A given that B has already occurred.

Binomial distribution mean

For X following B(n,p), the mean is np and the variance is np(1−p). The probability of r successes is ⁿCᵣpʳ(1−p)ⁿ⁻ʳ.

Contrapositive equivalence

The statement p⇒q is logically equivalent to its contrapositive ~q⇒~p. Converse q⇒p and inverse ~p⇒~q are not equivalent to it.

Coefficient of variation

CV=(standard deviation/mean)×100. It compares relative variability of two data series; a larger CV indicates greater spread.

Combination formula

ⁿCᵣ=n!/(r!(n−r)!) counts selections without order. Useful identities: ⁿCᵣ=ⁿCₙ₋ᵣ and ⁿCᵣ+ⁿCᵣ₋₁=ⁿ⁺¹Cᵣ.

General term of binomial expansion

In (a+b)ⁿ, the term Tᵣ₊₁=ⁿCᵣaⁿ⁻ʳbʳ. Setting the exponent of a variable to a target value locates a specific term.

Onto function

A function whose range equals its codomain, so every element of the codomain has at least one preimage. Onto plus one-one gives bijection.

cos inverse reflection identity

cos⁻¹(−x)=π−cos⁻¹x for x in [−1,1]. Unlike sine inverse, cosine inverse is not an odd function on its principal branch.

Determinant of product

For square matrices of the same order, |AB|=|A||B|. Also |kA|=kⁿ|A| for order n and |adj A|=|A|^(n−1).

Differentiability implies continuity

If f is differentiable at a point then it is continuous there. The converse fails: |x| is continuous but not differentiable at 0.

Related rates

If y depends on x and x changes with time, then dy/dt=f′(x)·dx/dt. Differentiate the relating equation with respect to time t.

Area between two curves

If f(x)≥g(x) on [a,b], the enclosed area is ∫ₐᵇ(f(x)−g(x))dx. Locate intersection points to determine the limits a and b.

Variable separable equation

An equation dy/dx=f(x)g(y) is solved by separating: ∫dy/g(y)=∫f(x)dx, then integrating both sides and adding one constant.

Angle between line and plane

If b⃗ is the line direction and n̂ the plane unit normal, then sin θ=|b⃗·n̂|/(|b⃗|), giving the acute angle between line and plane.

Bayes theorem

P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ), where events Eⱼ partition the sample space and P(A)>0.