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EJU Mathematics Course 1: Core Theorems & Identities

Essential formulas, identities, and analytical techniques for EJU Examination for Japanese University Admission (Mathematics Course 1).

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Vertex Form of a Quadratic Function

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Vertex Form of a Quadratic Function

y = a (x - p)^{2} + q

, where

(p, q)

is the vertex coordinates and

x = p

is the axis of symmetry.

Discriminant Conditions for

a x^{2} + b x + c = 0

D = b^{2} - 4 a c

D > 0

(2 distinct real roots),

D = 0

(1 repeated real root, tangency),

D < 0

(no real roots).

Factorization of Difference of Cubes

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

and

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

Sum and Product of Quadratic Roots (Vieta's Formulas)

For

a x^{2} + b x + c = 0

α + β = - \frac{b}{a}

and

α β = \frac{c}{a}

Distance from Point

(x_{0}, y_{0})

to Line

a x + b y + c = 0

d = \frac{∣ a x _{0} + b y _{0} + c ∣}{a ^{2} + b ^{2}}

Fundamental Pythagorean Trigonometric Identity

sin^{2} θ + cos^{2} θ = 1

1 + tan^{2} θ = \frac{1}{c o s ^{2} θ}

1 + cot^{2} θ = \frac{1}{s i n ^{2} θ}

Trigonometric Addition Theorem for Sine

sin (α \pm β) = sin α cos β \pm cos α sin β

Trigonometric Addition Theorem for Cosine

cos (α \pm β) = cos α cos β \mp sin α sin β

Double-Angle Formulas for

sin (2 θ)

and

cos (2 θ)

sin (2 θ) = 2 sin θ cos θ

;

cos (2 θ) = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ

Law of Sines with Circumradius

R

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

Law of Cosines for Any Triangle

a^{2} = b^{2} + c^{2} - 2 b c cos A

Area of Triangle Using Two Sides and Included Angle

S = \frac{1}{2} ab sin C

Arithmetic Progression

n

-th Term Formula

a_{n} = a_{1} + (n - 1) d

, where

d

is the common difference.

Sum of First

n

Terms of an Arithmetic Progression

S_{n} = \frac{n}{2} (a_{1} + a_{n}) = \frac{n}{2} [2 a_{1} + (n - 1) d]

Geometric Progression

n

-th Term Formula

a_{n} = a_{1} \cdot r^{n - 1}

, where

r

is the common ratio.

Sum of First

n

Terms of a Geometric Progression

S_{n} = \frac{a _{1} ( 1 - r ^{n} )}{1 - r}

(

r \neq = 1

). Infinite sum if

∣ r ∣ < 1

S_{\infty} = \frac{a _{1}}{1 - r}

Formula for

\sum_{k = 1}^{n} k

and

\sum_{k = 1}^{n} k^{2}

\sum k = \frac{n ( n + 1 )}{2}

;

\sum k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

Formula for

\sum_{k = 1}^{n} k^{3}

\sum_{k = 1}^{n} k^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Binomial Theorem Expansion

(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k}

Circular Permutation of

n

Objects

(n - 1)!

arrangements around a fixed circle.

Combination with Repetition

_{n} H_{r}

_{n} H_{r} = (r n + r - 1)

Conditional Probability Definition

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

with

P (B) > 0

Addition Rule for Probabilities

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Standard Deviation and Variance Relationship

Variance σ^{2} = \frac{1}{n} \sum (x_{i} - \overset{x}{ˉ})^{2} = \overline{x^{2}} - (\overset{x}{ˉ})^{2}

;

σ = σ^{2}

Centroid Coordinates of Triangle with Vertices

(x_{i}, y_{i})

G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})

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