25 cardsSIMAK UI
SIMAK UI Core Formulas
Essential formulas and core relationships for mathematics and natural sciences tested in Universitas Indonesia entrance exam.
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Vieta's Formulas (Quadratic)
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1
Vieta's Formulas (Quadratic)
For ax^2 + bx + c = 0 with roots x_1, x_2: sum x_1 + x_2 = -b/a, product x_1 x_2 = c/a.
2
Sum of Roots Cubed
x_1^3 + x_2^3 = (x_1 + x_2)^3 - 3 x_1 x_2 (x_1 + x_2).
3
Arithmetic Progression nth Term and Sum
a_n = a_1 + (n - 1)d, and S_n = (n/2)[2a_1 + (n - 1)d] = (n/2)(a_1 + a_n).
4
Infinite Geometric Series Sum
S_inf = a / (1 - r), valid when |r| < 1.
5
Logarithm Change of Base
log_a(b) = log_c(b) / log_c(a) = 1 / log_b(a).
6
Matrix Inverse (2x2)
If A = [[a, b], [c, d]], then A^(-1) = (1 / (ad - bc)) * [[d, -b], [-c, a]], provided det(A) != 0.
7
Double Angle Formula for Sine
sin(2theta) = 2 sin(theta) cos(theta).
8
Double Angle Formula for Cosine
cos(2theta) = cos^2(theta) - sin^2(theta) = 2 cos^2(theta) - 1 = 1 - 2 sin^2(theta).
9
Derivative of Quotient Function
d/dx [u/v] = (u'v - uv') / v^2.
10
Permutations vs Combinations
P(n, r) = n! / (n - r)! (order matters); C(n, r) = n! / [r! (n - r)!] (order does not matter).
11
Kinematic Equation (Velocity-Displacement)
v^2 = u^2 + 2as, where u is initial velocity, a is acceleration, and s is displacement.
12
Centripetal Acceleration
a_c = v^2 / r = omega^2 * r, directed radially inward toward center.
13
Simple Pendulum Period
T = 2pi * sqrt(L / g), independent of mass and amplitude for small angles.
14
Spring-Mass Oscillator Period
T = 2pi * sqrt(m / k), where m is oscillating mass and k is spring stiffness.
15
Ideal Gas Law
PV = nRT = N k_B T, where R = 8.314 J/(mol K) and k_B = 1.38 x 10^(-23) J/K.
16
First Law of Thermodynamics
Delta U = Q - W, where Q is heat added to system and W is work done by system.
17
Carnot Engine Maximum Efficiency
eta = 1 - T_C / T_H = (T_H - T_C) / T_H, with absolute temperatures in Kelvin.
18
Coulomb's Law
F = k * |q_1 q_2| / r^2, where k = 1 / (4 pi epsilon_0) approx 8.99 x 10^9 N m^2 / C^2.
19
Faraday's Law of Induction
EMF = -N * (dPhi_B / dt), where Phi_B = B A cos(theta) is magnetic flux.
20
Snell's Law of Refraction
n_1 sin(theta_1) = n_2 sin(theta_2); critical angle theta_c = arcsin(n_2 / n_1) for n_1 > n_2.
21
Molarity and Dilution
M = moles of solute / liters of solution; dilution formula: M_1 V_1 = M_2 V_2.
22
pH and pOH Definitions
pH = -log10[H^+], pOH = -log10[OH^-], and at 25 deg C: pH + pOH = 14.
23
First-Order Radioactive Half-Life
t_half = ln(2) / k approx 0.693 / k; remaining amount N(t) = N_0 * (1/2)^(t / t_half).
24
Standard Galvanic Cell Potential
E_cell^0 = E_cathode^0 - E_anode^0; spontaneous if E_cell^0 > 0.
25
Central Dogma of Molecular Biology
DNA is transcribed to mRNA in nucleus; mRNA is translated to polypeptide at ribosome.