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Bagrut Core Mathematics and Physics Principles

25 flashcards reviewing essential formulas and laws across 5-unit Mathematics and Physics for Bagrut study.

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What is De Moivre's theorem for raising a complex number in polar form to an integer power n?
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1
What is De Moivre's theorem for raising a complex number in polar form to an integer power n?
[r (cos(theta) + isin(theta))]^n = r^n [cos(ntheta) + isin(ntheta)].
2
What is the sum of all n roots of unity for the equation z^n = 1 (where n > 1)?
The sum is strictly 0, because the roots are symmetrically spaced on the unit circle in the complex plane.
3
How is the cosine of the angle theta between two 3D vectors u and v calculated?
cos(theta) = (u . v) / (|u| * |v|), where u . v is the scalar dot product.
4
What does the magnitude of the cross product |u x v| represent geometrically?
The area of the parallelogram spanned by vectors u and v. Half of this magnitude gives the triangle area.
5
What is the perpendicular distance formula from point (x_0, y_0, z_0) to the plane Ax + By + C*z + D = 0?
d = |Ax_0 + By_0 + C*z_0 + D| / sqrt(A^2 + B^2 + C^2).
6
What is the quotient rule for differentiating u(x) / v(x)?
(u / v)' = (u' v - u v') / v^2.
7
What is the formula for integration by parts derived from the product rule?
int u v' dx = u v - int u' * v dx.
8
By the Fundamental Theorem of Calculus, what is d/dx [int_a^(u(x)) f(t) dt]?
f(u(x)) * u'(x), applying the chain rule to the variable upper boundary.
9
What is the volume formula for rotating a region under y = f(x) from x = a to x = b about the y-axis (cylindrical shells)?
V = 2 pi int_a^b x * f(x) dx.
10
What is the standard equation of an ellipse centered at the origin with foci on the x-axis at (+-c, 0)?
x^2 / a^2 + y^2 / b^2 = 1, where b^2 = a^2 - c^2 and 2*a is the major axis length.
11
What is the maximum height reached by a projectile launched at speed v_0 and elevation angle theta above horizontal?
h_{max} = (v_0 sin(theta))^2 / (2 g).
12
In an Atwood machine with masses m_2 > m_1 over an ideal pulley, what is the acceleration of the system?
a = (m_2 - m_1) * g / (m_1 + m_2).
13
What is the ideal banking angle theta for a road curve of radius r at speed v without friction?
tan(theta) = v^2 / (r * g).
14
What is the total mechanical energy of a satellite of mass m in a circular orbit of radius r around a planet of mass M?
E = -G M m / (2 * r), which equals half its gravitational potential energy.
15
What is the escape velocity formula from the surface of a spherical body of mass M and radius R?
v_{esc} = sqrt(2 G M / R) = sqrt(2) * v_{orbital}.
16
What is the minimum speed at the apex of a vertical circular loop of radius R to maintain contact?
v_{min} = sqrt(g * R), where normal force drops to zero.
17
In a 2D elastic collision between a moving particle and an identical stationary particle, what is the angle between their post-collision velocities?
Exactly 90 degrees, derived from simultaneous conservation of momentum and kinetic energy.
18
What is the period of oscillation for a mass m attached to an ideal spring of spring constant k?
T = 2 pi sqrt(m / k).
19
What is the capacitance of a parallel-plate capacitor with plate area A and separation d filled with dielectric constant kappa?
C = kappa epsilon_0 A / d.
20
What is the terminal voltage V across a real battery with EMF E and internal resistance r supplying current I?
V = E - I * r.
21
What is the radius of the circular path of a particle of mass m and charge q moving at speed v perpendicular to a magnetic field B?
r = m v / (q B).
22
What is the magnitude of the motional EMF induced in a straight conductor of length L moving at speed v perpendicular to field B?
EMF = B L v.
23
What is Einstein's photoelectric equation relating incident photon frequency f, metal work function Phi, and maximum kinetic energy K_{max}?
K_{max} = h f - Phi = e V_s, where V_s is the stopping potential.
24
What is the de Broglie wavelength of a particle with linear momentum p?
lambda = h / p = h / (m * v).
25
What is the critical angle theta_c for total internal reflection at an interface between medium 1 and medium 2 (n_1 > n_2)?
sin(theta_c) = n_2 / n_1.

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