SEE Optional Mathematics Nepal flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Function

A rule assigning each element of a domain to exactly one element of a codomain; the set of actual outputs is the range.

Remainder theorem

When a polynomial f(x) is divided by x−a, the remainder equals f(a); substituting a into the polynomial gives the remainder directly.

Factor theorem

A polynomial f(x) has factor x−a exactly when f(a)=0; this follows from the remainder theorem and is used to find polynomial factors.

Arithmetic sequence nth term

For first term a and common difference d, the nth term is tn=a+(n−1)d; consecutive terms differ by the same constant d.

Geometric series sum to infinity

When |r|<1, the sum of infinitely many terms of a geometric series is S∞=a/(1−r), where a is the first term and r the common ratio.

Quadratic discriminant

For ax²+bx+c=0, D=b²−4ac; D>0 gives two distinct real roots, D=0 gives equal real roots, and D<0 gives no real roots.

Sum and product of roots

For ax²+bx+c=0 with roots α and β, α+β=−b/a and αβ=c/a; these build the equation x²−(α+β)x+αβ=0.

Pythagorean identity

For any angle θ where the functions are defined, sin²θ+cos²θ=1; dividing by cos²θ or sin²θ gives 1+tan²θ=sec²θ and 1+cot²θ=csc²θ.

Compound angle formula for sine

sin(A+B)=sinAcosB+cosAsinB and sin(A−B)=sinAcosB−cosAsinB; these express the sine of a sum or difference through sines and cosines of the separate angles.

Double angle formula for cosine

cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; the three forms are equivalent, and the choice depends on which trigonometric ratio is known.

General solution of sinθ=sinα

The complete solution is θ=nπ+(−1)^n α for integer n; this captures both the direct angle α and the supplementary angle π−α in every revolution.

Angle of elevation

The angle measured upward from a horizontal line to the line of sight of an object above the observer; in a right triangle it is used with tanθ=opposite/adjacent.

Sine rule

In any triangle ABC, a/sinA=b/sinB=c/sinC=2R, where R is the circumradius; it is used when two angles and one side or two sides and a non-included angle are known.

Cyclic quadrilateral opposite angles

A quadrilateral whose four vertices lie on a circle has opposite angles supplementary, so each pair sums to 180°; conversely, supplementary opposite angles imply the vertices are concyclic.

Alternate segment theorem

The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment subtended by that chord.

Tangent-radius theorem

A tangent to a circle is perpendicular to the radius drawn to the point of contact; also, two tangents drawn from an external point to the same circle are equal in length.

Midpoint theorem

In a triangle, the segment joining the midpoints of two sides is parallel to the third side and half its length; conversely, a line through one midpoint parallel to a side bisects the other side.

Similar triangle area ratio

If two triangles are similar with corresponding sides in ratio k, their areas are in ratio k²; all corresponding angles are equal and corresponding sides are proportional.

Exterior angle of triangle

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles; this follows because it is supplementary to the adjacent interior angle, which completes 180° with the other two.

Rotation by 90 degrees anticlockwise

About the origin, the mapping is (x,y)→(−y,x); a 90° clockwise rotation gives (x,y)→(y,−x), and rotations preserve distances and angles.

Distance formula

The distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]; it follows from Pythagoras applied to the horizontal and vertical differences.

Section formula

The point dividing the segment from A(x₁,y₁) to B(x₂,y₂) internally in ratio m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

Perpendicular lines condition

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, so m₁m₂=−1; a horizontal line is perpendicular to a vertical line.

Dot product

For vectors a and b, a·b=|a||b|cosθ=a₁b₁+a₂b₂, where θ is the angle between them; the dot product is zero exactly when the vectors are perpendicular.

Unit vector

A vector of length 1 in the direction of a given nonzero vector v; it is found by dividing v by its magnitude, v/|v|.

Area of triangle by coordinates

For vertices (x₁,y₁),(x₂,y₂),(x₃,y₃), area=½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|; the result is zero exactly when the three points are collinear.

Mean of grouped data

For grouped data, x̄=Σfx/Σf, where x is each class midpoint and f its frequency; the mean estimates the balance point of the distribution.

Median of grouped data

Median=L+[(N/2−cf)/f]×h, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency, h the class width, and N=Σf.

Standard deviation

A measure of spread defined by σ=√[Σf(x−x̄)²/Σf]; equivalently σ=√[Σfx²/Σf−x̄²], with variance equal to σ².

Complement rule

For any event E, P(E′)=1−P(E), where E′ is the complement of E; this is useful when the probability of the complement is easier to calculate.

Mutually exclusive events

Events that cannot occur together, so P(A∩B)=0 and P(A∪B)=P(A)+P(B); the occurrence of one rules out the occurrence of the other.

Independent events

Events A and B are independent when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A); knowing one occurred does not change the probability of the other.