Assam ASSEB Higher Secondary Mathematics cheat sheet

A relation R from A to B is a subset of A×B; its domain is the set of first coordinates and its range is the set of second coordinates actually appearing.

A function f:A→B assigns exactly one element f(x) of B to each x in A; on a graph this is checked by the vertical line test.

One-one (injective): f(x₁)=f(x₂) implies x₁=x₂. Onto (surjective): range of f equals the codomain B. Bijective means both.

Composition: (g∘f)(x)=g(f(x)) is defined only when range(f) ⊆ domain(g); in general g∘f ≠ f∘g.

f⁻¹ exists exactly when f is bijective; then (f⁻¹∘f)(x)=x and the graph of f⁻¹ is the reflection of the graph of f in the line y=x.

An equivalence relation is reflexive, symmetric and transitive; congruence modulo n on the integers is the standard example and partitions the set into disjoint classes.

For finite sets with m and n elements: number of relations from A to B is 2^(mn), number of functions from A to B is n^m.

Principal value ranges: sin⁻¹x ∈ [−π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (−π/2, π/2), cot⁻¹x ∈ (0, π).

Identities: sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1,1]; tan⁻¹x + cot⁻¹x = π/2 for all real x; tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy<1.

Complex numbers: i²=−1; for z=a+ib, |z|=√(a²+b²), z̄=a−ib and z·z̄=|z|²; division is done by multiplying numerator and denominator by the conjugate.

Polar form z=r(cosθ+i sinθ) with r=|z| and θ=arg z; De Moivre: (cosθ+i sinθ)^n = cos nθ + i sin nθ.

Cube roots of unity are 1, ω, ω² where ω=(−1+i√3)/2; they satisfy ω³=1 and 1+ω+ω²=0.

For ax²+bx+c=0 (a≠0): roots = (−b±√(b²−4ac))/(2a); sum = −b/a, product = c/a; discriminant >0, =0, <0 gives real distinct, real equal, complex conjugate roots.

Binomial theorem: (a+b)^n = Σ_{r=0}^{n} C(n,r)a^{n−r}b^r; general term T_{r+1}=C(n,r)a^{n−r}b^r.

Middle term: if n is even there is one middle term T_{n/2+1}; if n is odd there are two, T_{(n+1)/2} and T_{(n+3)/2}.

Permutations P(n,r)=n!/(n−r)!; combinations C(n,r)=n!/(r!(n−r)!); C(n,r)=C(n,n−r), and C(n,r)+C(n,r−1)=C(n+1,r).

AM ≥ GM for positive reals: (a+b)/2 ≥ √(ab), with equality only when a=b.

AP sum S_n = n/2[2a+(n−1)d]; GP sum S_n = a(rⁿ−1)/(r−1) for r≠1, and S_∞=a/(1−r) for |r|<1.

lim_{x→a} f(x) exists iff both one-sided limits exist and are equal; standard limits: lim_{x→0} sin x/x = 1, lim_{x→0}(1−cos x)/x² = 1/2, lim_{x→0}(e^x−1)/x = 1, lim_{x→0} ln(1+x)/x = 1.

Continuity at a requires lim_{x→a} f(x)=f(a). Differentiability at a implies continuity at a, but continuity does not imply differentiability (f(x)=|x| at 0).

Derivatives: (sin x)'=cos x, (cos x)'=−sin x, (tan x)'=sec²x, (sec x)'=sec x tan x, (cot x)'=−csc²x, (csc x)'=−csc x cot x.

Derivatives: (e^x)'=e^x, (a^x)'=a^x ln a, (ln x)'=1/x, (log_a x)'=1/(x ln a), (sin⁻¹x)'=1/√(1−x²), (tan⁻¹x)'=1/(1+x²).

Rules: (uv)'=u'v+uv'; (u/v)'=(u'v−uv')/v²; chain rule dy/dx=(dy/du)(du/dx).

Monotonicity: f is increasing where f'(x)>0 and decreasing where f'(x)<0; stationary points satisfy f'(x)=0.

Second derivative test: at f'(a)=0, f''(a)>0 gives a local minimum, f''(a)<0 gives a local maximum, f''(a)=0 is inconclusive.

Standard integrals: ∫xⁿdx=x^{n+1}/(n+1)+C (n≠−1), ∫(1/x)dx=ln|x|+C, ∫e^xdx=e^x+C, ∫sin x dx=−cos x+C, ∫cos x dx=sin x+C, ∫dx/(1+x²)=tan⁻¹x+C.

Definite integrals: ∫_a^b f = −∫_b^a f; ∫_a^b f = ∫_a^c f + ∫_c^b f; integration by parts ∫u dv = uv − ∫v du, with u chosen by ILATE order.

Position vector of the point dividing the join of points with position vectors a and b in the ratio m:n internally is (m b + n a)/(m+n); externally it is (m b − n a)/(m−n).

Dot product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃; a and b are perpendicular iff a·b=0, and a·a=|a|².

Cross product: |a×b|=|a||b|sinθ, direction given by the right-hand rule; area of a triangle with sides a, b is ½|a×b|, of the parallelogram |a×b|.

Scalar triple product [a b c]=a·(b×c) gives the signed volume of the parallelepiped; |[a b c]| is the volume and [a b c]=0 iff the three vectors are coplanar.

If a vector makes angles α, β, γ with the coordinate axes, its direction cosines l=cosα, m=cosβ, n=cosγ satisfy l²+m²+n²=1.

Line through point with position vector a and direction b: r = a + λb; cartesian form (x−x₁)/p = (y−y₁)/q = (z−z₁)/r.

Plane: r·n = d, or ax+by+cz=d in cartesian form; the vector n=(a,b,c) is normal to the plane.

Perpendicular distance from (x₁,y₁,z₁) to ax+by+cz=d is |ax₁+by₁+cz₁−d|/√(a²+b²+c²).

Angle between line direction b and plane normal n: sinθ=|b·n|/(|b||n|); angle between planes: cosθ=|n₁·n₂|/(|n₁||n₂|); shortest distance between skew lines r=a₁+λb₁ and r=a₂+μb₂ is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

A linear programming problem maximises or minimises a linear objective Z=c₁x₁+c₂x₂+… subject to linear constraints and non-negativity restrictions xᵢ≥0.

The feasible region is the intersection of the half-planes defined by the constraints; it is always a convex set (possibly empty, bounded or unbounded).

Each constraint written as an equation gives a boundary line; the half-plane satisfying the inequality is found by testing a point not on the line, such as the origin.

Corner point method: for a bounded feasible region, the maximum and minimum of a linear objective both occur at vertices of the feasible region.

If the feasible region is unbounded, an optimum may fail to exist; compare the values at corner points with the behaviour of the objective along the unbounded directions.

Slack variable: a ≤ constraint gains a non-negative slack variable to become an equation, e.g. 2x+y≤10 becomes 2x+y+s=10, s≥0.

Surplus variable: a ≥ constraint loses a non-negative surplus variable to become an equation, e.g. x+y≥4 becomes x+y−s=4, s≥0.

Level lines of Z=ax+by are parallel lines ax+by=k; changing k shifts the line, and the optimum vertex is where the line last touches the feasible region.

A redundant constraint does not alter the feasible region, so removing it leaves the optimal value of the objective unchanged.

For a finite sample space with equally likely outcomes, P(E)=n(E)/n(S); 0≤P(E)≤1 and P(E′)=1−P(E).

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0 so P(A∪B)=P(A)+P(B).

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0; multiplication rule P(A∩B)=P(A)P(B|A).

Events A and B are independent iff P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0.

Total probability: for a partition E₁,…,Eₙ, P(A)=ΣP(Eᵢ)P(A|Eᵢ). Bayes: P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ).

Random variable: mean μ=ΣxP(x); variance σ²=Σx²P(x)−μ²; standard deviation σ=√σ².

Binomial distribution for X~B(n,p): P(X=r)=C(n,r)p^r q^{n−r} with q=1−p; mean=np, variance=npq.

Inclusion–exclusion: n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C).

Two fair dice give 36 equally likely outcomes; the sum 7 occurs in 6 ways (probability 1/6), while sums 2 and 12 each occur once (probability 1/36).