Tamil Nadu HSE First Year Mathematics flashcards

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

Power set cardinality

The power set P(A) is the set of all subsets of A; if A has n elements, then P(A) has 2^n elements and A has 2^n−1 proper subsets.

One-one function

A function f from A to B is one-one, or injective, when f(a1)=f(a2) always implies a1=a2; distinct inputs then have distinct images.

Bijective function

A function is bijective when it is both one-one and onto; its range equals the codomain, and only bijective functions possess an inverse function.

Quadratic discriminant

For ax²+bx+c=0 with a≠0, the discriminant is Δ=b²−4ac; Δ>0 gives distinct real roots, Δ=0 equal real roots, and Δ<0 non-real complex roots.

Logarithm quotient rule

For positive m, n and base a>0, a≠1, log_a(m/n)=log_a m−log_a n; the logarithm of a quotient equals the difference of the logarithms.

Absolute value inequality

For a>0, |x|<a is equivalent to −a<x<a, while |x|>a is equivalent to x<−a or x>a; these rules solve simple modulus inequalities.

Pythagorean identity

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

Double angle formulas

sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A). These express ratios of twice an angle using ratios of the original angle.

General solution of sine

If sinθ=sinα, then θ=nπ+(−1)^n α for every integer n; this gives all angles with the same sine value as α.

Combination formula

The number of r-element subsets chosen from n distinct objects is nCr=n!/[r!(n−r)!]; order does not matter, and nCr=nC(n−r).

Circular permutation count

The number of ways to arrange n distinct objects around a circle is (n−1)!; if clockwise and anticlockwise arrangements are considered the same, divide by 2.

Induction proof steps

To prove P(n) for all positive integers n: verify P(1); assume P(k) true; use that assumption to prove P(k+1). This is mathematical induction.

Binomial general term

In the expansion of (a+b)^n, the general term is T_{r+1}=nCr a^{n−r}b^r for r=0,1,…,n; setting an exponent to zero finds a required term.

Geometric mean of two numbers

For positive numbers a and b, their geometric mean is GM=√(ab); it satisfies GM²=AM×HM, where AM=(a+b)/2 and HM=2ab/(a+b).

Infinite geometric series sum

For a geometric series with first term a and common ratio r, the infinite sum is a/(1−r) when |r|<1; otherwise the series does not converge.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1; if one slope is m, the perpendicular slope is −1/m.

Circle center and radius

The circle x²+y²+2gx+2fy+c=0 has center (−g,−f) and radius √(g²+f²−c); it is real only when g²+f²−c>0.

Parabola y²=4ax

For the standard parabola y²=4ax, the focus is (a,0), the directrix is x=−a, the vertex is (0,0), and the latus rectum length is 4a.

Determinant of 2x2 matrix

For A=[[a,b],[c,d]], the determinant is |A|=ad−bc; A is singular when ad−bc=0 and invertible when ad−bc≠0.

Inverse by adjoint

If A is a square matrix with |A|≠0, then A⁻¹=(adj A)/|A|; the adjoint is the transpose of the cofactor matrix, and A(adj A)=|A|I.

Skew-symmetric matrix

A square matrix A is skew-symmetric when Aᵀ=−A; all its diagonal entries must be zero, and every square matrix decomposes into symmetric plus skew-symmetric parts.

Dot product

For vectors a and b, a·b=|a||b|cosθ, where θ is the angle between them; a·b=0 exactly when nonzero a and b are perpendicular.

Scalar triple product

The scalar triple product [a b c]=a·(b×c) gives the signed volume of the parallelepiped formed by a, b, c; it is zero exactly when the three vectors are coplanar.

Limit exists criterion

For lim_{x→a} f(x) to exist, the left-hand limit and right-hand limit must both exist and be equal; the common value is then the limit.

Continuity at a point

A function f is continuous at x=a when f(a) is defined, lim_{x→a}f(x) exists, and the limit equals f(a); all three conditions are required.

Differentiability implies continuity

If f is differentiable at x=a, then f is continuous at a; the converse fails, because a continuous function such as |x| need not have a derivative at zero.

Chain rule

If y=f(u) and u=g(x), then dy/dx=(dy/du)(du/dx); this rule differentiates composite functions by multiplying the outer and inner derivatives.

Derivative of inverse sine

For −1<x<1, d/dx(sin⁻¹x)=1/√(1−x²); the derivative of cos⁻¹x is −1/√(1−x²). Use these with the chain rule when the argument is a function of x.

Integration by parts

Integration by parts uses ∫u dv=uv−∫v du; choose u according to LIATE-logarithmic, inverse trigonometric, algebraic, trigonometric, exponential-to simplify the remaining integral.

Fundamental theorem of calculus

If F is an antiderivative of continuous f on [a,b], then ∫_a^b f(x)dx=F(b)−F(a); the definite integral equals the change in F.

Conditional probability

For P(B)>0, the conditional probability of A given B is P(A|B)=P(A∩B)/P(B); it rescales the probability of A to the reduced sample space B.

Independent events

Events A and B are independent when P(A∩B)=P(A)P(B); equivalently, P(A|B)=P(A) and P(B|A)=P(B) whenever the conditional probabilities are defined.