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UPCAT Science and Mathematics Core Concepts

Essential formulas, physical laws, biological systems, and algebraic principles for senior high school students taking the UPCAT.

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What is the quadratic formula used to solve ax^2 + bx + c = 0?
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1
What is the quadratic formula used to solve ax^2 + bx + c = 0?
x = (-b +- sqrt(b^2 - 4ac)) / (2a). The discriminant D = b^2 - 4ac determines the nature of the roots.
2
What do Vieta's formulas state for the roots r1 and r2 of ax^2 + bx + c = 0?
Sum of roots r1 + r2 = -b/a; Product of roots r1 * r2 = c/a.
3
What is the Remainder Theorem for a polynomial P(x) divided by (x - c)?
The remainder is equal to P(c). If P(c) = 0, then (x - c) is a factor of P(x) (Factor Theorem).
4
What is the formula for the sum of an infinite geometric series with |r| < 1?
S = a / (1 - r), where a is the first term and r is the common ratio.
5
What are the trigonometric values of sin(30 deg), cos(30 deg), and tan(30 deg)?
sin(30 deg) = 1/2, cos(30 deg) = sqrt(3)/2, and tan(30 deg) = sqrt(3)/3.
6
What are the double-angle formulas for sin(2x) and cos(2x)?
sin(2x) = 2 sin(x) cos(x); cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x).
7
What is the Law of Cosines for any triangle ABC?
c^2 = a^2 + b^2 - 2ab cos(C), relating the three side lengths and one included angle.
8
What is the Law of Sines for any triangle ABC with circumradius R?
a / sin(A) = b / sin(B) = c / sin(C) = 2R.
9
What is the distance formula between points (x1, y1) and (x2, y2) in the Cartesian plane?
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
10
What is the standard form equation of a circle centered at (h, k) with radius r?
(x - h)^2 + (y - k)^2 = r^2.
11
What is the formula for combinations of n items taken r at a time without replacement?
C(n, r) = n! / (r! * (n - r)!), used when order of selection does not matter.
12
What is Newton's Second Law of Motion?
F_net = m a (Net force equals mass times acceleration), with force in Newtons (kgm/s^2).
13
What are the formulas for kinetic energy and gravitational potential energy?
Kinetic Energy = (1/2) m v^2; Potential Energy = m g h.
14
What is Ohm's Law relating voltage (V), current (I), and resistance (R)?
V = I R. Power dissipated in a resistor is P = V I = I^2 * R = V^2 / R.
15
What is Snell's Law of refraction at an optical interface?
n1 sin(theta_1) = n2 sin(theta_2), where n is the refractive index of each medium.
16
What is the Ideal Gas Law equation?
PV = nRT, where P is pressure, V is volume, n is moles, R is gas constant (0.0821 Latm/(molK)), and T is Kelvin temperature.
17
What is the definition of pH in aqueous chemistry?
pH = -log10[H+], where [H+] is the molar hydronium ion concentration. Neutral water has pH 7 at 25°C.
18
What is Le Chatelier's principle in chemical equilibrium?
When a dynamic equilibrium system is subjected to a disturbance (temperature, pressure, concentration), it shifts in the direction that counteracts the change.
19
What are the three main types of plate boundaries in plate tectonics?
Convergent (plates collide, forming trenches/mountains), Divergent (plates separate, forming ridges), and Transform (plates slide horizontally).
20
Why is the Philippines prone to earthquakes and volcanic activity?
It lies along the Pacific Ring of Fire where the Philippine Sea Plate subducts beneath the Eurasian/Sundaland Plate.
21
What is the difference between P-waves and S-waves in seismology?
P-waves are fast longitudinal compressional waves traveling through solids and liquids; S-waves are slower transverse shear waves traveling only through solids.
22
What is the central dogma of molecular biology?
Genetic information flows from DNA to RNA via transcription, and from RNA to protein via translation.
23
What are the phases of mitosis in sequential order?
Prophase, Metaphase (chromosomes align at equator), Anaphase (sister chromatids separate), and Telophase (nuclear envelopes reform).
24
What is the expected phenotypic ratio in a Mendelian monohybrid cross between two heterozygotes (Aa x Aa)?
3 dominant to 1 recessive (3 : 1 ratio).
25
What are the primary vascular tissues in plants, and what do they transport?
Xylem transports water and dissolved inorganic minerals upward; Phloem transports organic sugars and nutrients bidirectionally.
26
What is Lindeman's 10% rule in ecosystem trophic dynamics?
Approximately 10% of chemical energy stored as biomass is transferred from one trophic level to the next higher level; 90% is lost as metabolic heat.

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