Assam ASSEB Higher Secondary Mathematics flashcards

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

Equivalence relation

A relation on a set that is reflexive, symmetric and transitive. Congruence modulo n on the integers is the standard example; it partitions the set into disjoint equivalence classes.

Bijective function

A function that is both one-one and onto: distinct inputs give distinct outputs, and every element of the codomain is an image of some input, so an inverse function exists.

Inverse function condition

A function f has an inverse f⁻¹ exactly when f is bijective. Then (f⁻¹∘f)(x)=x, and the graph of f⁻¹ is the mirror image of the graph of f in the line y=x.

Principal value of sin⁻¹x

The value chosen in the interval [−π/2, π/2]. On that interval sine is one-one, so sin⁻¹(1)=π/2 and sin⁻¹(−1)=−π/2.

Number of relations

From a set with m elements to a set with n elements there are 2^(mn) relations, because each of the mn ordered pairs may independently be included or excluded.

Composition of functions

(g∘f)(x)=g(f(x)). It is defined only when the range of f lies inside the domain of g, and in general g∘f is not equal to f∘g.

Modulus and conjugate

For z=a+ib, |z|=√(a²+b²) and z̄=a−ib. The product z·z̄=|z|² is real and non-negative, and |z₁z₂|=|z₁||z₂|.

Cube roots of unity

The three numbers 1, ω and ω², where ω=(−1+i√3)/2. They satisfy ω³=1 and 1+ω+ω²=0, and ω² is the conjugate of ω.

Sum and product of roots

For ax²+bx+c=0 with a≠0, roots α and β satisfy α+β=−b/a and αβ=c/a. The discriminant b²−4ac decides whether the roots are real or complex.

General term of binomial

In the expansion of (a+b)^n, the term containing b^r is T_{r+1}=C(n,r)a^(n−r)b^r. Putting r=0 gives the first term a^n.

Combination formula

C(n,r)=n!/(r!(n−r)!) counts the number of ways to choose r objects from n without regard to order. Also C(n,r)=C(n,n−r) and C(n,0)=C(n,n)=1.

AM-GM inequality

For positive real numbers a and b, the arithmetic mean is at least the geometric mean: (a+b)/2 ≥ √(ab), with equality exactly when a=b.

Limit of sin x over x

As x approaches 0 with x measured in radians, sin x/x approaches 1. This standard limit underlies the derivatives of sine and cosine.

Differentiability implies continuity

If f is differentiable at a point then it is continuous there. The converse fails: f(x)=|x| is continuous at 0 but has no derivative at 0.

Derivative of tan⁻¹x

d/dx(tan⁻¹x)=1/(1+x²) for every real x. Equivalently, the integral of 1/(1+x²) with respect to x is tan⁻¹x plus a constant.

Second derivative test

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

Integration by parts

∫u dv = uv − ∫v du. Choose u using the ILATE order (inverse, logarithmic, algebraic, trigonometric, exponential) so that the remaining integral is simpler.

Area under a curve

The area bounded by y=f(x), the x-axis and the lines x=a and x=b is ∫ from a to b of |f(x)|dx, splitting the interval wherever f changes sign.

Dot product

a·b=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃. The result is a scalar, and for non-zero vectors a·b=0 exactly when a and b are perpendicular.

Cross product magnitude

|a×b|=|a||b|sinθ equals the area of the parallelogram with adjacent sides a and b. Half of this value is the area of the triangle they span.

Scalar triple product

[a b c]=a·(b×c). Its absolute value is the volume of the parallelepiped with edges a, b, c, and it equals zero exactly when the three vectors are coplanar.

Direction cosines identity

If a vector makes angles α, β, γ with the coordinate axes, then l=cosα, m=cosβ, n=cosγ satisfy l²+m²+n²=1, and (l,m,n) is a unit vector.

Distance from point to plane

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

Feasible region

The set of all points satisfying every constraint and the non-negativity restrictions of a linear programming problem. It may be bounded, unbounded or empty.

Corner point method

Evaluate the objective Z at every vertex of the feasible region. For a bounded region, the largest and smallest of these values are the maximum and minimum of Z.

Slack variable

A non-negative variable added to a ≤ constraint to convert it into an equation: 2x+y≤10 becomes 2x+y+s=10 with s≥0, measuring unused capacity.

Unbounded feasible region

If the feasible region is unbounded, a linear objective may increase without limit. Compare corner values with a level line to decide whether an optimum actually exists.

Conditional probability

P(A|B)=P(A∩B)/P(B), defined when P(B)>0. It is the probability of A computed on the reduced sample space consisting of the outcomes in B.

Independent events

Events A and B are independent when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0. The occurrence of one gives no information about the other.

Bayes' theorem

For a partition E₁,…,Eₙ: P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ) divided by ΣP(Eⱼ)P(A|Eⱼ). It reverses a conditional probability using the total probability rule.

Binomial mean and variance

For X~B(n,p), the mean is np and the variance is npq with q=1−p. The probability of exactly r successes is C(n,r)p^r q^(n−r).

Variance of random variable

Var(X)=Σx²P(x)−μ², where μ=ΣxP(x) is the mean. The standard deviation is the non-negative square root of the variance.