Tamil Nadu SSLC Mathematics cheat sheet

For finite sets A and B, A×B={(a,b):a∈A,b∈B}; n(A×B)=n(A)·n(B). A×B=B×A if A=B or one set is empty, but not generally otherwise.

A relation R from A to B is any subset of A×B. Its domain is {a∈A:(a,b)∈R for some b}, and its range is {b∈B:(a,b)∈R for some a}.

A relation on A is reflexive if aRa for all a∈A, symmetric if aRb⇒bRa, and transitive if aRb and bRc⇒aRc; reflexive+symmetric+transitive gives an equivalence relation.

A function f:A→B assigns exactly one element of B to each element of A. Injective means f(x₁)=f(x₂)⇒x₁=x₂; surjective means range=codomain; bijective means both.

For |A|=m and |B|=n, the number of relations from A to B is 2^{mn}; the number of functions is n^m; the number of one-one functions is n(n−1)…(n−m+1) when n≥m.

Composition: (f∘g)(x)=f(g(x)); it is defined when range(g)⊆domain(f). Composition is associative but not generally commutative: f∘g≠g∘f.

A function f has an inverse f⁻¹ exactly when f is bijective; then f⁻¹(y)=x iff f(x)=y, and (f⁻¹∘f)(x)=x.

Vertical line test: a graph represents a function if no vertical line meets it more than once; the horizontal line test detects one-one functions.

Euclid's division lemma: for integers a and b>0, there are unique q,r with a=bq+r and 0≤r<b; the HCF is the last nonzero remainder in the Euclidean algorithm.

If a≡b (mod n), then n divides a−b; congruence is preserved under addition, subtraction, and multiplication, so remainders may be reduced before operating.

Fundamental theorem of arithmetic: every integer >1 has a unique prime factorisation up to order; HCF takes the least power of each common prime, LCM takes the greatest power.

For two positive integers a and b, HCF(a,b)×LCM(a,b)=a×b; this identity does not extend directly to three numbers.

AP: a_n=a+(n−1)d; S_n=(n/2)[2a+(n−1)d]=(n/2)(a+l), where l is the last term.

GP: a_n=ar^{n−1}; S_n=a(r^n−1)/(r−1) for r≠1; S_∞=a/(1−r) when |r|<1.

Sums: 1+2+…+n=n(n+1)/2; 1²+2²+…+n²=n(n+1)(2n+1)/6; 1³+2³+…+n³=[n(n+1)/2]².

For positive numbers, AM≥GM, with equality only when the numbers are equal; the arithmetic mean of a and b is (a+b)/2 and the geometric mean is √(ab).

For a polynomial p(x), a zero is a value c with p(c)=0; its graph meets the x-axis at (c,0). A quadratic has at most two real zeros.

For ax²+bx+c=0, discriminant Δ=b²−4ac: Δ>0 real distinct roots, Δ=0 real equal roots, Δ<0 no real roots; roots x=(−b±√Δ)/(2a).

For roots α,β of ax²+bx+c=0: α+β=−b/a, αβ=c/a; the quadratic with these roots is x²−(α+β)x+αβ=0.

The graph of y=ax²+bx+c is a parabola with axis x=−b/(2a); the vertex is at x=−b/(2a), and it opens upward if a>0, downward if a<0.

For two polynomials, product=GCD×LCM up to a nonzero constant factor; GCD divides both and LCM is divisible by both.

Square root of a polynomial can be found by long division; the polynomial is a perfect square if the remainder is zero and the quotient is the square root.

Matrices: A+B requires equal orders; AB requires columns of A = rows of B; matrix multiplication is associative but not commutative; (AB)^T=B^T A^T.

For A=[[a,b],[c,d]], det A=ad−bc; A is singular if det A=0. Inverse A⁻¹=(1/det A)[[d,−b],[−c,a]] when det A≠0.

Cramer's rule for ax+by=e, cx+dy=f: Δ=ad−bc, Δ_x=ed−bf, Δ_y=af−ec; if Δ≠0 then x=Δ_x/Δ, y=Δ_y/Δ.

Thales (basic proportionality) theorem: a line parallel to one side of a triangle divides the other two sides proportionally; the converse is also true.

Similar triangles: corresponding angles equal and corresponding sides proportional; criteria AA, SSS (all three side ratios equal), SAS (two sides proportional and included angle equal).

Areas of similar triangles are in the ratio of the squares of corresponding sides; volumes of similar solids are in the ratio of cubes.

Pythagoras theorem: in a right triangle, hypotenuse²=sum of squares of the other two sides; its converse: if this holds, the triangle is right-angled.

Tangent–radius theorem: a tangent is perpendicular to the radius at the point of contact; tangents drawn from an external point to a circle are equal in length.

In a circle, a perpendicular from the centre to a chord bisects the chord; equal chords are equidistant from the centre; the angle in a semicircle is a right angle.

Cyclic quadrilateral: opposite angles are supplementary; the exterior angle equals the interior opposite angle. Alternate segment theorem: angle between tangent and chord equals angle in alternate segment.

Concurrency: medians meet at the centroid, which divides each median 2:1 from the vertex; perpendicular bisectors meet at the circumcentre; angle bisectors meet at the incentre; altitudes at the orthocentre.

Constructions: to divide a segment in ratio m:n, draw a ray making an acute angle, mark m+n equal segments, join the last point to the segment end, then draw a parallel through the m-th point.

Distance between (x₁,y₁) and (x₂,y₂): √[(x₂−x₁)²+(y₂−y₁)²]; distance from origin is √(x²+y²).

Midpoint of segment joining (x₁,y₁),(x₂,y₂): ((x₁+x₂)/2,(y₁+y₂)/2); centroid of triangle with vertices (xᵢ,yᵢ): ((Σxᵢ)/3,(Σyᵢ)/3).

Internal section formula: point dividing join of (x₁,y₁),(x₂,y₂) in ratio m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

External section formula: same ratio externally gives ((mx₂−nx₁)/(m−n),(my₂−ny₁)/(m−n)), m≠n.

Slope through (x₁,y₁),(x₂,y₂): m=(y₂−y₁)/(x₂−x₁), x₁≠x₂; horizontal lines slope 0, vertical lines have undefined slope.

Parallel lines have equal slopes; perpendicular nonvertical lines satisfy m₁m₂=−1. Angle θ between lines: tanθ=|(m₁−m₂)/(1+m₁m₂)|.

Line forms: y=mx+c; y−y₁=m(x−x₁); through two points (y−y₁)/(y₂−y₁)=(x−x₁)/(x₂−x₁); intercept form x/a+y/b=1.

General line ax+by+c=0 has slope −a/b (b≠0); distance from (x₁,y₁) to it is |ax₁+by₁+c|/√(a²+b²).

Area of triangle with vertices: ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|; points are collinear exactly when this area is zero.

Right-triangle ratios: sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, tanθ=opposite/adjacent; cosec, sec, cot are reciprocals.

Standard values: sin30°=1/2, sin45°=1/√2, sin60°=√3/2; cos30°=√3/2, cos45°=1/√2, cos60°=1/2; tan30°=1/√3, tan45°=1, tan60°=√3.

Pythagorean identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.

Quotient identities: tanθ=sinθ/cosθ, cotθ=cosθ/sinθ; reciprocal identities: cosecθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ.

Complementary angles: sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, tan(90°−θ)=cotθ, cosec(90°−θ)=secθ, sec(90°−θ)=cosecθ, cot(90°−θ)=tanθ.

In heights-and-distances problems, angle of elevation is measured upward from horizontal to the line of sight; angle of depression is measured downward from horizontal.

For a right triangle in a height problem, tan(angle)=vertical height/horizontal distance; use sin or cos when the line of sight is the hypotenuse.

Signs by quadrant: All positive in I; sin and cosec positive in II; tan and cot positive in III; cos and sec positive in IV.

Cylinder radius r, height h: curved surface 2πrh; total surface 2πr(r+h); volume πr²h.

Right circular cone radius r, height h, slant l=√(r²+h²): curved surface πrl; total surface πr(l+r); volume (1/3)πr²h.

Sphere radius r: surface area 4πr²; volume (4/3)πr³. Hemisphere: curved surface 2πr²; total surface 3πr²; volume (2/3)πr³.

Frustum with base radii R,r and height h: slant l=√[h²+(R−r)²]; curved surface π(R+r)l; total surface π(R+r)l+πR²+πr²; volume (1/3)πh(R²+Rr+r²).

Cuboid length l, breadth b, height h: surface area 2(lb+bh+hl); volume lbh; diagonal √(l²+b²+h²). Cube edge a: surface 6a², volume a³, diagonal a√3.

For combined solids, volumes add; total surface area adds only exposed surfaces, omitting faces where solids join.

If a solid is melted and recast, volume is conserved; equate initial and final volumes to find the unknown dimension.

For similar solids with linear scale factor k, surface areas scale by k² and volumes by k³; hence volume ratio is the cube of the surface-area ratio's square root.

Range=max−min; it uses only two extreme observations and ignores the other values.

Variance of n observations: σ²=(1/n)Σ(xᵢ−x̄)²=(1/n)Σxᵢ²−x̄²; standard deviation σ is its positive square root.

For grouped data with midpoints xᵢ and frequencies fᵢ, use σ²=(1/N)Σfᵢ(xᵢ−x̄)², where N=Σfᵢ and x̄=(1/N)Σfᵢxᵢ.

Adding a constant to every observation leaves variance and standard deviation unchanged; multiplying every observation by k multiplies standard deviation by |k| and variance by k².

Coefficient of variation CV=(σ/x̄)×100%, used to compare variability of data sets with different means or units.

Probability: P(E)=number of favourable outcomes/number of equally likely outcomes in sample space S; 0≤P(E)≤1.

Complement and addition: P(E′)=1−P(E); P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive A,B, P(A∪B)=P(A)+P(B).

Independent events: P(A∩B)=P(A)P(B); for two dice, the sample space has 36 equally likely ordered outcomes.