NEB Class 12 Mathematics cheat sheet

Quadratic equation: ax²+bx+c=0 has roots x=[−b±√(b²−4ac)]/(2a); sum of roots = −b/a; product = c/a; discriminant Δ=b²−4ac: Δ>0 real distinct, Δ=0 real equal, Δ<0 complex conjugate.

Complex numbers: i²=−1; for z=a+bi, |z|=√(a²+b²), z̄=a−bi, z·z̄=|z|²; division: (a+bi)/(c+di)=[(a+bi)(c−di)]/(c²+d²).

AP: tn=a+(n−1)d; Sn=n/2[2a+(n−1)d]; if three terms are in AP, the middle term is the arithmetic mean (a+c)/2.

GP: tn=ar^(n−1); Sn=a(r^n−1)/(r−1) for r≠1; S∞=a/(1−r) for |r|<1; geometric mean of a and b is √(ab).

Permutations and combinations: nPr=n!/(n−r)!; nCr=n!/[r!(n−r)!]; nCr=nC(n−r); nPr=r!·nCr; number of subsets of an n-set is 2^n.

Binomial theorem: (a+b)^n=Σ_{r=0}^n nCr a^(n−r)b^r; general term Tr+1=nCr a^(n−r)b^r; number of terms n+1; middle term exists for even n at r=n/2.

Matrices: for A=[[a,b],[c,d]], det A=ad−bc; A⁻¹=(1/det A)[[d,−b],[−c,a]] if det A≠0; singular means det A=0.

Logarithms: log_a(mn)=log_a m+log_a n; log_a(m/n)=log_a m−log_a n; log_a(m^k)=k log_a m; log_a b=log_c b/log_c a; a^0=1 and log_a 1=0.

Mathematical induction: prove the base case P(1); assume P(k) is true; prove P(k+1) from P(k); conclude P(n) is true for all n≥1.

Pythagorean identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=csc²θ; quotient tanθ=sinθ/cosθ, cotθ=cosθ/sinθ.

Addition/subtraction: sin(A±B)=sinA cosB±cosA sinB; cos(A±B)=cosA cosB∓sinA sinB; tan(A±B)=(tanA±tanB)/(1∓tanA tanB).

Double-angle: sin2θ=2sinθcosθ; cos2θ=cos²θ−sin²θ=2cos²θ−1=1−2sin²θ; tan2θ=2tanθ/(1−tan²θ).

Half-angle: sin²(θ/2)=(1−cosθ)/2; cos²(θ/2)=(1+cosθ)/2; tan(θ/2)=sinθ/(1+cosθ)=(1−cosθ)/sinθ.

General solutions: sinθ=sinα ⇒ θ=nπ+(−1)^n α; cosθ=cosα ⇒ θ=2nπ±α; tanθ=tanα ⇒ θ=nπ+α, n∈Z.

Inverse trig principal ranges: sin⁻¹x ∈ [−π/2,π/2]; cos⁻¹x ∈ [0,π]; tan⁻¹x ∈ (−π/2,π/2); domains: sin⁻¹ and cos⁻¹ x∈[−1,1], tan⁻¹ x∈R.

Triangle: sine rule a/sinA=b/sinB=c/sinC; cosine rule a²=b²+c²−2bc cosA; area=½bc sinA=½ab sinC.

Product-to-sum: 2sinA cosB=sin(A+B)+sin(A−B); 2cosA cosB=cos(A+B)+cos(A−B); 2sinA sinB=cos(A−B)−cos(A+B).

Distance d=√[(x2−x1)²+(y2−y1)²]; midpoint ((x1+x2)/2,(y1+y2)/2); section formula for ratio m:n internally ((mx2+nx1)/(m+n),(my2+ny1)/(m+n)).

Slope m=(y2−y1)/(x2−x1); line forms: y−y1=m(x−x1); y=mx+c; x/a+y/b=1; general ax+by+c=0 has slope −a/b.

Parallel: m1=m2; perpendicular: m1m2=−1; angle between lines tanθ=|(m1−m2)/(1+m1m2)|; distance from (x1,y1) to ax+by+c=0 is |ax1+by1+c|/√(a²+b²).

Circle: centre (h,k), radius r ⇒ (x−h)²+(y−k)²=r²; general x²+y²+2gx+2fy+c=0 centre (−g,−f), radius √(g²+f²−c); tangent at (x1,y1) to x²+y²=r² is xx1+yy1=r².

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

Ellipse x²/a²+y²/b²=1 (a>b): centre (0,0), foci (±c,0), c²=a²−b², eccentricity e=c/a<1, directrices x=±a/e, latus rectum 2b²/a.

Hyperbola x²/a²−y²/b²=1: foci (±c,0), c²=a²+b², e=c/a>1, directrices x=±a/e, asymptotes y=±(b/a)x; conjugate hyperbola swaps signs.

General second-degree conic Ax²+Bxy+Cy²+Dx+Ey+F=0: discriminant B²−4AC <0 ellipse (or circle), =0 parabola, >0 hyperbola; degenerate cases possible.

Tangent to circle x²+y²+2gx+2fy+c=0 at (x1,y1): xx1+yy1+g(x+x1)+f(y+y1)+c=0.

Vector a=a1i+a2j+a3k; magnitude |a|=√(a1²+a2²+a3²); unit vector â=a/|a|; direction cosines l=a1/|a|, m=a2/|a|, n=a3/|a| with l²+m²+n²=1.

Dot product: a·b=a1b1+a2b2+a3b3=|a||b|cosθ; perpendicular ⇔ a·b=0; projection of a on b is (a·b)/|b|; component vector is [(a·b)/|b|²]b.

Cross product: a×b=(a2b3−a3b2)i−(a1b3−a3b1)j+(a1b2−a2b1)k; |a×b|=|a||b|sinθ; parallel ⇔ a×b=0.

Area: parallelogram spanned by a,b has area |a×b|; triangle has area ½|a×b|; if vertices A,B,C, area=½|AB×AC|.

Scalar triple product: [a b c]=a·(b×c); volume of parallelepiped=| [a b c] |; a,b,c coplanar ⇔ [a b c]=0.

Line: r=a+λb passes through a and is parallel to b; Cartesian (x−x1)/l=(y−y1)/m=(z−z1)/n; angle between lines uses direction vectors.

Plane: r·n=d has normal n; Cartesian ax+by+cz=d; distance from point p to plane r·n=d is |p·n−d|/|n|.

Angle between vectors: cosθ=(a·b)/(|a||b|); angle between line and plane: sinφ=|b·n|/(|b||n|), where b is line direction and n plane normal.

Limits: lim x→0 (sin x)/x=1; lim x→0 (tan 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; lim x→0 (a^x−1)/x=ln a.

Continuity at x=a requires f(a) defined, lim x→a f(x) exists, and limit equals f(a); differentiability implies continuity, converse false.

Derivative rules: (u±v)′=u′±v′; (uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²; chain rule dy/dx=dy/du·du/dx.

Standard derivatives: d/dx(x^n)=nx^(n−1); d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(tan x)=sec²x; d/dx(ln x)=1/x; d/dx(e^x)=e^x; d/dx(a^x)=a^x ln a.

Tangent and normal: slope of tangent at x=a is f′(a); tangent y−f(a)=f′(a)(x−a); normal slope −1/f′(a) if f′(a)≠0.

Monotonicity: f′(x)>0 on an interval ⇒ f increasing; f′(x)<0 ⇒ decreasing; critical points are where f′(x)=0 or undefined; f″<0 local max, f″>0 local min.

Integration: ∫x^n dx=x^(n+1)/(n+1)+C (n≠−1); ∫1/x dx=ln|x|+C; ∫e^x dx=e^x+C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C; ∫sec²x dx=tan x+C.

Methods: substitution reverses chain rule; integration by parts ∫u dv=uv−∫v du; partial fractions for rational functions; definite integral ∫_a^b f(x)dx=F(b)−F(a).

Applications: area under curve y=f(x) from a to b is ∫_a^b |f(x)|dx; area between curves is ∫[upper−lower]dx; separable differential equation dy/dx=f(x)g(y) solved by separating variables.

Grouped data: mean x̄=Σfx/Σf with class midpoints x; median class where cumulative frequency reaches N/2; mode class has highest frequency.

Dispersion: variance σ²=Σf(x−x̄)²/Σf; SD σ=√σ²; coefficient of variation CV=(σ/x̄)×100%; range=max−min.

Quartiles: Q1 at N/4, Q2 at N/2, Q3 at 3N/4 in cumulative frequency; interquartile range IQR=Q3−Q1; quartile deviation=(Q3−Q1)/2.

Correlation: Pearson r=Σ(x−x̄)(y−ȳ)/√[Σ(x−x̄)²Σ(y−ȳ)²]; −1≤r≤1; r>0 positive linear association, r<0 negative; r=0 no linear correlation.

Regression: line of y on x is y−ȳ=byx(x−x̄) with byx=Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²; line of x on y has slope bxy; both pass through (x̄,ȳ).

Probability: 0≤P(A)≤1; P(A∪B)=P(A)+P(B)−P(A∩B); mutually exclusive ⇒ P(A∩B)=0; complement P(A′)=1−P(A).

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

Binomial distribution: P(X=r)=nCr p^r q^(n−r), q=1−p; mean=np; variance=npq; SD=√(npq); applies to n independent Bernoulli trials.

Expected value: E(X)=ΣxP(x); Var(X)=E(X²)−[E(X)]².