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UPCAT & Civil Service: Math & Science Essentials

Core formulas, biochemical principles, and physical laws required for the University of the Philippines College Admission Test and Civil Service exam.

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Quadratic Formula and Discriminant

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Quadratic Formula and Discriminant

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

;

D = b^{2} - 4 a c

determines root nature (

D > 0

two reals,

D = 0

one repeated,

D < 0

non-real).

Distance Formula Between Coordinates

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Equation of Circle in Standard Form

(x - h)^{2} + (y - k)^{2} = r^{2}

, with center

(h, k)

and radius

r

Logarithm Product and Quotient Properties

lo g_{b} (x y) = lo g_{b} (x) + lo g_{b} (y)

and

lo g_{b} (\frac{x}{y}) = lo g_{b} (x) - lo g_{b} (y)

Law of Sines and Law of Cosines

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

;

c^{2} = a^{2} + b^{2} - 2 ab cos (C)

Double Angle Trigonometric Identities

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

Sum of Infinite Geometric Series

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

, provided

∣ r ∣ < 1

Work-Rate Combined Equation

\frac{1}{T} = \frac{1}{t _{1}} + \frac{1}{t _{2}} ⟹ T = \frac{t _{1} t _{2}}{t _{1} + t _{2}}

Kinematic Equation for Uniform Acceleration

v = u + a t

d = u t + \frac{1}{2} a t^{2}

v^{2} = u^{2} + 2 a d

Newton's Second Law of Motion

F_{net} = ma

; unit is Newton (

N = kg \cdot m/s^{2}

Mechanical Work and Gravitational Potential Energy

W = F d cos θ

;

P E = m g h

; Kinetic Energy

K E = \frac{1}{2} m v^{2}

Ohm's Law and Electrical Power

V = I R

; Power

P = I V = I^{2} R = \frac{V ^{2}}{R}

Resistors in Series vs Parallel

Series:

R_{total} = R_{1} + R_{2}

. Parallel:

\frac{1}{R _{total}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}

Le Chatelier's Principle: Exothermic Reactions

Decreasing temperature shifts equilibrium forward (towards products); increasing pressure shifts toward fewer moles of gas.

pH Scale Definition

pH = - lo g_{10} [H^{+}]

;

pOH = - lo g_{10} [OH^{-}]

;

pH + pOH = 14

2 5^{\circ} C

Ideal Gas Law Equation

P V = n R T

, where

R = 0.0821 L \cdot atm / (mol \cdot K)

8.314 J / (mol \cdot K)

Mole-Mass Conversion

Moles (n) = \frac{mass ( g )}{molar mass ( g / mol )}

Central Dogma of Molecular Biology

DNA transcription mRNA translation Protein

Mendelian Monohybrid Cross Ratio

Heterozygous cross (

A a \times A a

): Phenotypic ratio

3 : 1

; Genotypic ratio

1 : 2 : 1

(

AA : A a : aa

Cellular Respiration ATP Yield

Complete aerobic oxidation yields

\approx 30 - 32 ATP

per glucose molecule.

Photosynthesis Overall Equation

6 CO_{2} + 6 H_{2} O + light \to C_{6} H_{12} O_{6} + 6 O_{2}

Mitosis vs Meiosis Differences

Mitosis: 2 identical diploid daughter cells (

2 n

). Meiosis: 4 genetically diverse haploid gametes (

n

) via crossing over.

Lindeman's 10% Trophic Rule

Only about 10% of chemical energy transfers to the next trophic level; 90% is dissipated as metabolic heat.

Enzyme Active Site and Denaturation

Active sites catalyze substrate binding; extreme heat or non-optimal pH denatures tertiary protein structure irreversibly.

Archimedes' Principle (Buoyant Force)

F_{b} = ρ_{fluid} \cdot V_{submerged} \cdot g

, equal to the weight of fluid displaced.

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