SEE Optional Mathematics Nepal cheat sheet

A function f: A→B assigns each x in the domain A exactly one y in the codomain B; f is one-to-one if f(x₁)=f(x₂) implies x₁=x₂, and onto if every element of B is an image.

Composite (f∘g)(x)=f(g(x)) is defined only when the range of g lies within the domain of f; the inverse f⁻¹ exists only for one-to-one onto functions and satisfies f(f⁻¹(x))=x.

Remainder theorem: dividing a polynomial f(x) by x−a leaves remainder f(a); factor theorem: x−a is a factor exactly when f(a)=0.

Arithmetic sequence: tn=a+(n−1)d and Sn=n/2[2a+(n−1)d]=n/2(a+tn); the arithmetic mean between a and b is (a+b)/2.

Geometric sequence: tn=ar^(n−1); Sn=a(r^n−1)/(r−1) for r≠1; S∞=a/(1−r) when |r|<1; the geometric mean between positive a and b is √(ab).

For ax²+bx+c=0 with a≠0, x=[−b±√(b²−4ac)]/(2a); the discriminant D=b²−4ac gives two distinct real roots if D>0, equal real roots if D=0, and no real roots if D<0.

If α and β are roots of ax²+bx+c=0, then α+β=−b/a and αβ=c/a; the equation with these roots is x²−(α+β)x+αβ=0, and the vertex is at x=−b/(2a).

Linear programming: graph each constraint as an inequality, identify the feasible region, and evaluate the objective P=ax+by at every corner point; by the corner-point theorem an optimum occurs at a vertex when it exists.

Pythagorean identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and 1+cot²θ=csc²θ, valid wherever the functions involved are defined.

Compound angles: sin(A±B)=sinAcosB±cosAsinB; cos(A±B)=cosAcosB∓sinAsinB; tan(A±B)=(tanA±tanB)/(1∓tanAtanB).

Double angles: sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A).

Product-to-sum: 2sinAcosB=sin(A+B)+sin(A−B); 2cosAcosB=cos(A+B)+cos(A−B); 2sinAsinB=cos(A−B)−cos(A+B); sinC+sinD=2sin((C+D)/2)cos((C−D)/2).

General solutions: sinθ=sinα gives θ=nπ+(−1)^n α; cosθ=cosα gives θ=2nπ±α; tanθ=tanα gives θ=nπ+α, where n is an integer.

Standard values: sin30°=1/2, cos30°=√3/2, tan30°=1/√3; sin45°=cos45°=1/√2; sin60°=√3/2, cos60°=1/2, tan60°=√3; sin90°=1 and cos90°=0.

Angle of elevation is measured upward from the horizontal and angle of depression downward; in a right triangle tanθ=opposite/adjacent, and two observation points give two tangent equations for height and distance.

Sine rule: a/sinA=b/sinB=c/sinC=2R; cosine rule: a²=b²+c²−2bc cosA; area of triangle ABC=½ab sinC.

The angles of a triangle sum to 180°; an exterior angle equals the sum of the two opposite interior angles; congruence tests are SSS, SAS, ASA, AAS, and RHS.

Similar triangles have equal corresponding angles and proportional corresponding sides; their areas are in the square of the side ratio, and a line parallel to one side divides the other two sides proportionally.

In a right triangle, hypotenuse²=sum of squares of the legs (Pythagoras); area of any triangle is ½base×height=½ab sinC=√[s(s−a)(s−b)(s−c)], where s=(a+b+c)/2.

Parallelogram: opposite sides and opposite angles are equal, and diagonals bisect each other; midpoint theorem: the segment joining the midpoints of two sides is parallel to the third side and half its length.

Circle: the angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal; the angle in a semicircle is 90°.

Cyclic quadrilateral: opposite angles sum to 180° and an exterior angle equals the interior opposite angle; equal chords subtend equal angles at the centre and are equidistant from the centre.

Tangent: it is perpendicular to the radius at the point of contact; tangents from an external point are equal; the angle between a tangent and a chord equals the angle in the alternate segment.

Construction: the perpendicular bisector of a segment is the locus of points equidistant from its endpoints; the angle bisector is the locus of points equidistant from the two sides; triangles are constructed from SSS, SAS, or ASA data.

Transformation: translation by (a,b) sends (x,y) to (x+a,y+b); reflection in the x-axis gives (x,−y), in the y-axis (−x,y), in y=x (y,x); 90° anticlockwise rotation gives (x,y)→(−y,x); enlargement centre origin factor k gives (kx,ky).

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

Section formula: the point dividing AB internally in ratio m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)); the midpoint is the case m:n=1:1.

Slope m=(y₂−y₁)/(x₂−x₁); parallel lines satisfy m₁=m₂; perpendicular lines satisfy m₁m₂=−1.

Line forms: y=mx+c, y−y₁=m(x−x₁), x/a+y/b=1, and ax+by+c=0 with slope −a/b when b≠0.

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

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

Vector ai+bj has magnitude √(a²+b²); its unit vector is v/|v|; addition and subtraction are performed componentwise.

Dot product: a·b=|a||b|cosθ=a₁b₁+a₂b₂; a·b=0 exactly when the vectors are perpendicular; cosθ=(a·b)/(|a||b|), and vectors are collinear when one is a scalar multiple of the other.

Grouped mean: x̄=Σfx/Σf, where x is each class midpoint and f its frequency; the assumed-mean method uses x̄=A+Σfd/Σf with d=x−A.

Grouped median: locate the class where cumulative frequency reaches N/2, then use median=L+[(N/2−cf)/f]×h, with L the lower boundary, cf the preceding cumulative frequency, f the class frequency, and h the class width.

Grouped mode: the modal class has the highest frequency; mode=L+[(f₁−f₀)/(2f₁−f₀−f₂)]×h, where f₁ is the modal frequency and f₀, f₂ are the adjacent frequencies.

Quartiles: Q₁ is at cumulative frequency N/4 and Q₃ at 3N/4; interquartile range=Q₃−Q₁; range=maximum−minimum.

Standard deviation: σ=√[Σf(x−x̄)²/Σf]=√[Σfx²/Σf−x̄²]; variance=σ².

Probability of an event E is P(E)=favourable outcomes/total equally likely outcomes, with 0≤P(E)≤1; complement rule: 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).

Multiplication rule: P(A∩B)=P(A)P(B|A); for independent events P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A), and conditional probability is P(A|B)=P(A∩B)/P(B) when P(B)>0.