Karnataka II PUC Basic Mathematics (KSEAB) cheat sheet

For finite sets, |A ∪ B| = |A| + |B| − |A ∩ B|; |A × B| = |A||B|; a set with n elements has 2^n subsets.

For three finite sets, |A ∪ B ∪ C| = |A|+|B|+|C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C|.

A relation R ⊆ A × A is reflexive if (a,a) ∈ R for every a ∈ A, symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R, and transitive if (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R.

An equivalence relation is reflexive, symmetric and transitive; it partitions A into disjoint equivalence classes.

f: A → B is injective if f(a1)=f(a2) ⇒ a1=a2; surjective if its range is B; bijective if both, and only then f^{-1}: B → A exists.

For |A|=m and |B|=n, the number of relations from A to B is 2^{mn}; the number of functions from A to B is n^m.

Composition (g∘f)(x)=g(f(x)) is defined when range(f) ⊆ domain(g); generally g∘f ≠ f∘g.

De Morgan: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

For n×n matrices, det(AB)=det(A)det(B), det(A^T)=det(A), det(kA)=k^n det(A), and det(A^{-1})=1/det(A) when A is invertible.

A 2×2 determinant is ad−bc for rows (a,b) and (c,d); a 3×3 determinant is expanded along any row or column using signed cofactors.

A square matrix A is invertible iff det(A) ≠ 0; then A^{-1} = adj(A)/det(A).

adj(A) is the transpose of the cofactor matrix and satisfies A adj(A) = adj(A) A = |A| I.

For AX=B with A square, a unique solution exists iff det(A)≠0; it is X=A^{-1}B.

Cramer's rule: x_i=Δ_i/Δ, where Δ=det(A) and Δ_i replaces column i by B; if Δ=0, no unique Cramer solution exists.

Symmetric means A^T=A; skew-symmetric means A^T=−A, so every diagonal entry is 0; orthogonal means AA^T=I.

Matrix multiplication AB is defined when columns(A)=rows(B); addition requires identical orders; generally AB≠BA.

nPr = n!/(n−r)! counts ordered arrangements of r objects from n distinct objects; nCr = n!/(r!(n−r)!) counts unordered selections.

nCr = nC(n−r); nC0 = nCn = 1; nPr = r! nCr; nCr + nC(r−1) = (n+1)Cr.

Arrangements of n objects with p,q,... identical objects of different types: n!/(p! q! ...).

Circular permutations of n distinct objects: (n−1)!; if clockwise and anticlockwise arrangements are the same, divide by 2.

Binomial theorem: (a+b)^n = Σ_{r=0}^n nCr a^{n−r} b^r.

General term: T_{r+1}=nCr a^{n−r}b^r. For even n the middle term is T_{n/2+1}; for odd n there are two middle terms.

Sum of binomial coefficients Σ nCr = 2^n; alternating sum Σ (−1)^r nCr = 0; sum over even r nCr = 2^{n−1}.

Number of terms in (a+b)^n is n+1; term independent of x is found by setting the exponent of x to 0 in T_{r+1}.

AP: nth term a+(n−1)d; sum S_n = n/2[2a+(n−1)d] = n/2(a+l) when last term is l.

GP: nth term ar^{n−1}; sum S_n = a(r^n−1)/(r−1) for r≠1; infinite sum S∞=a/(1−r) only if |r|<1.

For positive a,b: AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b), and AM ≥ GM ≥ HM.

If three numbers are in AP, write them as a−d, a, a+d; if in GP, write them as a/r, a, ar.

Sum of first n natural numbers = n(n+1)/2; squares = n(n+1)(2n+1)/6; cubes = [n(n+1)/2]^2.

Inserting n arithmetic means between a and b gives common difference d=(b−a)/(n+1).

For a GP, the product of terms equidistant from the beginning and end is constant; in an AP, the sum of such terms is constant.

If a,b,c are in GP then b²=ac; if in AP then 2b=a+c.

Slope of line through (x1,y1),(x2,y2) is m=(y2−y1)/(x2−x1), provided x1≠x2; vertical line has undefined slope.

Point-slope form: y−y1=m(x−x1); slope-intercept: y=mx+c; intercept form: x/a+y/b=1.

General line Ax+By+C=0 has slope −A/B (B≠0); parallel lines have equal slopes; perpendicular lines satisfy m1m2=−1.

Distance from (x1,y1) to Ax+By+C=0 is |Ax1+By1+C|/√(A²+B²).

Distance between parallel lines Ax+By+C1=0 and Ax+By+C2=0 is |C1−C2|/√(A²+B²).

Angle θ between lines with slopes m1,m2 satisfies tanθ=|(m1−m2)/(1+m1m2)|.

Two-point form: (y−y1)/(y2−y1)=(x−x1)/(x2−x1).

Every line through the intersection of L1=0 and L2=0 has equation L1+λL2=0 for some λ.

Circle center (h,k), radius r: (x−h)²+(y−k)²=r². General x²+y²+2gx+2fy+c=0 has center (−g,−f), radius √(g²+f²−c).

Circle with diameter endpoints (x1,y1),(x2,y2): (x−x1)(x−x2)+(y−y1)(y−y2)=0.

Tangent to x²+y²=r² at (x1,y1) is xx1+yy1=r²; line y=mx+c is tangent iff c²=r²(1+m²).

Parabola y²=4ax: vertex (0,0), focus (a,0), directrix x=−a, latus rectum 4a; x²=4ay has focus (0,a).

Ellipse x²/a²+y²/b²=1 (a>b): e=√(1−b²/a²), foci (±ae,0), latus rectum 2b²/a.

Hyperbola x²/a²−y²/b²=1: e=√(1+b²/a²), foci (±ae,0), asymptotes y=±(b/a)x.

Eccentricity: circle e=0, parabola e=1, ellipse 0<e<1, hyperbola e>1.

Tangency condition: y=mx+c tangent to ellipse x²/a²+y²/b²=1 iff c²=a²m²+b²; to hyperbola iff c²=a²m²−b².

Limit exists at a iff left-hand and right-hand limits both exist and are equal; continuity at a also requires equality to f(a).

Standard limits: lim_{x→0} sinx/x=1; lim_{x→0}(1−cosx)/x²=1/2; lim_{x→0}(e^x−1)/x=1; lim_{x→0}(a^x−1)/x=ln a; lim_{x→0}(1+x)^{1/x}=e.

Differentiability at a implies continuity at a; continuity does not imply differentiability.

Derivative from first principles: f′(x)=lim_{h→0}[f(x+h)−f(x)]/h.

Rules: (uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²; chain rule d/dx f(g(x))=f′(g(x))g′(x).

Standard derivatives: d/dx x^n=nx^{n−1}; sinx→cosx; cosx→−sinx; tanx→sec²x; lnx→1/x; e^x→e^x.

For y=f(x)^{g(x)}, take ln y=g(x)ln f(x) before differentiating; for implicit F(x,y)=0, differentiate both sides and solve for dy/dx.

L'Hôpital's rule applies to 0/0 or ∞/∞ indeterminate forms when the derivatives have a finite or infinite limit.

f is increasing where f′>0 and decreasing where f′<0; stationary points satisfy f′=0.

At a stationary point, f″>0 gives a local minimum, f″<0 gives a local maximum; f″=0 is inconclusive.

Absolute extrema on [a,b] occur at critical points or endpoints; evaluate f at all such candidates.

Tangent at x=x1 has slope f′(x1); normal slope is −1/f′(x1) when f′(x1)≠0.

Indefinite integrals: ∫x^n dx=x^{n+1}/(n+1)+C (n≠−1); ∫1/x dx=ln|x|+C; ∫e^x dx=e^x+C; ∫sinx dx=−cosx+C; ∫cosx dx=sinx+C.

Definite integral: ∫_a^b f(x)dx=F(b)−F(a), where F′=f; areas are taken as positive.

Integration by substitution reverses the chain rule; integration by parts uses ∫u dv=uv−∫v du.

Area between y=f(x) and y=g(x) from a to b is ∫_a^b |f(x)−g(x)| dx; split where curves cross.

Grouped mean = Σfx/Σf; variance = Σfx²/Σf − (mean)²; standard deviation is its square root.

Probability P(A)=number of favourable outcomes/number of equally likely outcomes; 0≤P(A)≤1 and P(A′)=1−P(A).

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

Conditional probability: P(A|B)=P(A∩B)/P(B), P(B)>0.

A and B are independent iff P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A).

Bayes' theorem: P(A_i|B)=P(A_i)P(B|A_i)/Σ_j P(A_j)P(B|A_j).

Random variable mean μ=ΣxP(x); variance σ²=Σx²P(x)−μ².

For binomial distribution with n trials and success probability p: mean np, variance np(1−p), P(X=r)=nCr p^r(1−p)^{n−r}.

A linear programming problem maximizes or minimizes Z=cx+dy subject to linear inequalities and nonnegativity x,y≥0.

The feasible region is the intersection of all constraint half-planes; it can be bounded, unbounded, or empty.

Corner point theorem: for a nonempty bounded feasible region, an optimal value occurs at a corner point.

Graphical method: draw each constraint as a line, test a point to shade the feasible side, then evaluate Z at every corner.

If the feasible region is unbounded, Z may be unbounded; check whether the objective can increase or decrease indefinitely.

A ≤ constraint introduces a slack variable; a ≥ constraint introduces a surplus variable; equations use artificial variables in simplex.

Parallel objective line to a constraint edge can give infinitely many optimal solutions along that edge.

No feasible region means the problem is infeasible; an unbounded feasible region with no finite optimum means unbounded solution.

Simple interest SI=PRT/100; amount A=P+SI=P(1+RT/100).

Compound interest: A=P(1+r/100)^n for annual compounding; CI=A−P. For n compounds per year use rate r/n per period.

Effective annual rate for nominal r compounded n times yearly is (1+r/n)^n−1.

Ordinary annuity present value PV=R[1−(1+i)^{−n}]/i; future value FV=R[(1+i)^n−1]/i.

Sinking fund payment to accumulate FV: R=FV·i/[(1+i)^n−1].

Perpetuity present value PV=R/i for constant payment R at interest i per period.

Banker's discount BD=Frt; proceeds=F−BD. True discount TD=Frt/(1+rt); banker's gain=BD−TD.

Reducing-balance depreciation: value after n years = P(1−r/100)^n.