Nepal SEE Mathematics cheat sheet

Roster form lists elements; set-builder form states a property; n(A) counts distinct elements.

A ⊆ B means every element of A is in B; A ⊂ B means proper subset; the empty set is a subset of every set.

A set with n elements has 2^n subsets and 2^n − 1 proper subsets.

A ∪ B = elements in A or B or both; A ∩ B = elements in both; A − B = elements in A not in B.

Complement A' = U − A; (A')' = A; A ∪ A' = U and A ∩ A' = ∅.

Two-set formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); n(A only) = n(A) − n(A ∩ B).

De Morgan's laws: (A ∪ B)' = A' ∩ B'; (A ∩ B)' = A' ∪ B'.

Disjoint sets have A ∩ B = ∅, so n(A ∪ B) = n(A) + n(B).

Profit = SP − CP; profit% = (SP−CP)/CP ×100; SP = CP(1+profit%/100); loss uses negative profit.

Discount = MP − SP; discount% = (MP−SP)/MP ×100; SP = MP(1−discount%/100).

Simple interest I = PTR/100; amount A = P + I; rate and time in years.

Compound amount A = P(1+R/100)^T; CI = A−P; for half-yearly use rate R/2 and periods 2T.

Depreciation/population: final = initial(1 ± R/100)^T; use minus for depreciation, plus for growth.

VAT amount = taxable value × VAT%/100; SP with VAT = SP(1+VAT%/100); VAT is not charged on VAT itself.

Income tax: taxable income = total income − allowable deductions; apply slab rates to successive portions.

Currency conversion: foreign→NPR multiply by bank buying rate; NPR→foreign divide by bank selling rate.

Bank selling rate is used when customer buys foreign currency; buying rate when customer sells foreign currency.

Triangle area = ½ × base × height; Heron: √[s(s−a)(s−b)(s−c)], s=(a+b+c)/2; equilateral area = (√3/4)a².

Parallelogram area = base × height; trapezium area = ½(a+b)h; rhombus area = ½d₁d₂.

Circle area = πr²; circumference = 2πr; sector area = (θ/360)πr²; arc length = (θ/360)2πr.

Cuboid: LSA = 2h(l+b); TSA = 2(lb+bh+hl); volume = lbh; space diagonal = √(l²+b²+h²).

Cube: TSA = 6a²; volume = a³; face diagonal = a√2; space diagonal = a√3.

Cylinder: CSA = 2πrh; TSA = 2πr(r+h); volume = πr²h.

Cone: l = √(r²+h²); CSA = πrl; TSA = πr(l+r); volume = (1/3)πr²h.

Sphere: SA = 4πr²; volume = (4/3)πr³; hemisphere TSA = 3πr²; hemisphere volume = (2/3)πr³.

Prism volume = base area × height; pyramid volume = (1/3) × base area × height.

(a±b)² = a² ± 2ab + b²; a²−b²=(a−b)(a+b); (a±b)³ = a³ ± 3a²b + 3ab² ± b³.

Quadratic formula: x = [−b ± √(b²−4ac)]/(2a) for ax²+bx+c=0, a≠0.

Discriminant D=b²−4ac: D>0 real distinct; D=0 real equal; D<0 no real roots.

Sum of roots = −b/a; product = c/a; equation with roots α, β is x²−(α+β)x+αβ=0.

Index laws: a^m a^n=a^{m+n}; a^m/a^n=a^{m−n}; (a^m)^n=a^{mn}; a^0=1; a^{−n}=1/a^n.

Fractional index: a^{m/n} = (ⁿ√a)^m; √a × √b = √(ab); √a/√b = √(a/b).

Rationalise denominator: multiply by conjugate; 1/(√a+√b) = (√a−√b)/(a−b).

Inequality: same number added/subtracted both sides preserves sign; multiply/divide by negative reverses sign.

Linear simultaneous equations: elimination aligns coefficients; substitution expresses one variable in terms of the other.

Triangle interior angles sum to 180°; exterior angle equals sum of two opposite interior angles.

Congruence tests: SSS, SAS, ASA, AAS, RHS; corresponding parts of congruent triangles are equal.

Similar triangles: AA, SSS (proportional sides), SAS (two sides proportional and included angle equal); corresponding sides proportional.

Area ratio of similar triangles = square of corresponding side ratio; Pythagoras: a²+b²=c² in right triangle.

Parallelogram: opposite sides and angles equal; diagonals bisect each other; midpoint theorem joins midpoints of two sides parallel to third and half its length.

Circle: angle at centre = 2 × angle at circumference on same arc; angles in same segment equal; angle in semicircle = 90°.

Cyclic quadrilateral: opposite angles sum to 180°; exterior angle equals interior opposite angle.

Tangent: perpendicular to radius at contact; two tangents from external point are equal; tangent-chord angle equals angle in alternate segment.

Construction: perpendicular bisector uses equal arcs from endpoints; angle bisector uses equal arcs from sides; triangle construction uses given sides/angles and compass arcs.

Right triangle: sinθ=opp/hyp; cosθ=adj/hyp; tanθ=opp/adj; cosec, sec, cot are reciprocals.

Standard values: sin 0°=0, 30°=1/2, 45°=√2/2, 60°=√3/2, 90°=1; cos reverses; tan 30°=1/√3, 45°=1, 60°=√3.

Identity sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.

Complementary angles: sin(90°−θ)=cosθ; cos(90°−θ)=sinθ; tan(90°−θ)=cotθ.

45-45-90 triangle sides ratio 1:1:√2; 30-60-90 triangle sides ratio 1:√3:2.

Angle of elevation is measured upward from horizontal; angle of depression is measured downward from horizontal.

Angle of depression from A to B equals angle of elevation from B to A when horizontal lines are parallel.

Heights and distances: sketch right triangle, mark known angle and side, use tan for opposite/adjacent, sin or cos when hypotenuse involved.

Mean of raw data = sum of values ÷ number of values; grouped mean = Σfx/Σf where x is class midpoint.

Median of ordered raw data: middle value if n odd; average of two middle values if n even.

Grouped median = L + [(N/2 − cf)/f] × h; L lower boundary of median class, cf cumulative frequency before, f frequency, h class width.

Mode = most frequent value; grouped mode uses highest-frequency class and neighbouring frequencies.

Range = largest value − smallest value; cumulative frequency is running total of frequencies.

Probability P(E)=number of favourable outcomes ÷ total equally likely outcomes; 0≤P(E)≤1.

P(not E)=1−P(E); for mutually exclusive events P(A or B)=P(A)+P(B).

For independent events P(A and B)=P(A)×P(B); a standard deck has 52 cards, 4 suits of 13, 26 red, 12 face cards.

Translation by vector (a,b): (x,y)→(x+a,y+b); length, angle, shape and orientation unchanged.

Reflection in x-axis: (x,y)→(x,−y); in y-axis: (−x,y); in y=x: (y,x); in y=−x: (−y,−x).

Reflection in line x=a: (x,y)→(2a−x,y); in line y=b: (x,y)→(x,2b−y).

Rotation about origin 90° clockwise: (x,y)→(y,−x); 90° anticlockwise: (−y,x); 180°: (−x,−y).

Enlargement centre origin scale factor k: (x,y)→(kx,ky); area scale factor k²; angles unchanged.

Isometry preserves length and angle: translation, reflection and rotation are isometries; enlargement is not.

Combined transformation applies the second transformation to the image of the first; order generally matters.

Distance between (x₁,y₁) and (x₂,y₂) = √[(x₂−x₁)²+(y₂−y₁)²]; midpoint = ((x₁+x₂)/2,(y₁+y₂)/2).

Section formula internal division in ratio m:n: ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

Slope m=(y₂−y₁)/(x₂−x₁); horizontal line slope 0; vertical line slope undefined.

Parallel lines have equal slopes; perpendicular lines have m₁m₂=−1.

Line equation: y=mx+c; point-slope y−y₁=m(x−x₁); two-point (y−y₁)/(y₂−y₁)=(x−x₁)/(x₂−x₁).

Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.

Vector a=xi+yj has magnitude √(x²+y²); unit vector = a/|a|; position vector of point P is OP.

Collinear points have equal slopes or one vector is a scalar multiple of another; vector addition adds corresponding components.