All flashcard decksNew Zealand Scholarship syllabus →
26 cardsNew Zealand Scholarship

Scholarship Calculus and Physical Sciences

High-level definitions, integration techniques, mechanics laws and thermodynamic principles for New Zealand Scholarship candidates.

Start studyingPlay Match
PDF study sheet

Free account needed to save, copy or download. We bring you back here.

1 / 26
0 known
TermClick or press Space to flip
What is the formula for integration by parts?
GetMyMockTestQuestion side

Full list

Terms in this deck (26)

New Zealand Scholarship mock tests →
1
What is the formula for integration by parts?
integral(u dv) = u v - integral(v * du).
2
What does L'Hopital's rule state for evaluating indeterminate limits of the form 0/0 or infinity/infinity?
lim [f(x) / g(x)] = lim [f'(x) / g'(x)], provided the derivatives exist near the point.
3
What is the general solution of the separable differential equation dy/dx = k * y?
y = y_0 e^(k x), where y_0 is the initial condition value at x = 0.
4
What is de Moivre's theorem for complex numbers in polar form?
[r (cos(theta) + isin(theta))]^n = r^n (cos(ntheta) + isin(ntheta)) = r^n e^(in*theta).
5
How is the volume of revolution about the x-axis calculated between x = a and x = b?
V = pi * integral_a^b [y(x)]^2 dx.
6
What is the definition of the dot product of two vectors u and v, and what does u . v = 0 imply?
u . v = |u| |v| cos(theta). If u . v = 0 for non-zero vectors, they are orthogonal (perpendicular).
7
What is the relationship between torque tau and angular acceleration alpha for a rigid body?
tau_net = I * alpha, where I is the rotational moment of inertia about the axis.
8
How does moment of inertia change with axis shift according to the parallel axis theorem?
I = I_cm + M * d^2, where I_cm is about the centre of mass and d is the displacement distance.
9
What is the total mechanical energy of an ideal simple harmonic oscillator of mass m, amplitude A, and spring constant k?
E = 0.5 k A^2 = 0.5 m (omega * A)^2.
10
What is the orbital speed of a satellite in circular orbit of radius r around a central body of mass M?
v = sqrt(G * M / r).
11
What is the escape velocity from the surface of a spherical planet of mass M and radius R?
v_esc = sqrt(2 G M / R).
12
What is the condition for destructive interference in a double-slit experiment?
Path difference delta x = (m + 0.5) * lambda, where m = 0, 1, 2, ...
13
What is Einstein's photoelectric equation relating photon frequency f to kinetic energy?
E_k(max) = h * f - phi, where phi is the work function of the target metal.
14
What is the de Broglie wavelength of a particle with linear momentum p?
lambda = h / p = h / (m * v).
15
What is the radius of the circular path of a particle of charge q and mass m moving with speed v perpendicular to magnetic field B?
r = m v / (q B).
16
What is Faraday's law of induction representing induced electromotive force?
emf = -d(Phi_B) / dt, where Phi_B is the magnetic flux linked through the circuit.
17
What does Lenz's law state regarding the direction of induced current?
The induced current produces a magnetic field that opposes the change in magnetic flux that generated it.
18
What is the resonant frequency of an ideal LC circuit with inductance L and capacitance C?
f_0 = 1 / (2 pi sqrt(L * C)).
19
What is the radioactive decay law connecting remaining nuclei N(t) to initial amount N_0?
N(t) = N_0 e^(-lambda t), where half-life T_half = ln(2) / lambda.
20
What is the Gibbs free energy equation relating enthalpy Delta H, entropy Delta S, and temperature T?
Delta G = Delta H - T * Delta S. A negative Delta G indicates thermodynamic spontaneity.
21
What is the relationship between standard cell potential E^0_cell and standard Gibbs energy change Delta G^0?
Delta G^0 = -n F E^0_cell, where n is moles of electrons and F is the Faraday constant.
22
How does the equilibrium constant K relate to standard Gibbs free energy change Delta G^0?
Delta G^0 = -R T ln(K).
23
What is the Henderson-Hasselbalch equation for the pH of an acid-conjugate base buffer solution?
pH = pK_a + log10([A^-] / [HA]).
24
What is the solubility product expression K_sp for calcium fluoride, CaF2?
K_sp = [Ca^2+] [F^-]^2 = (s) (2s)^2 = 4*s^3, where s is molar solubility.
25
What does the (n + 1) rule in proton NMR spectroscopy indicate about peak multiplicity?
A proton resonance splits into (n + 1) peaks, where n is the number of chemically non-equivalent protons on adjacent carbons.
26
What is Brewster's angle theta_B and what property does reflected light exhibit at this angle?
tan(theta_B) = n_2 / n_1; light reflected at Brewster's angle is 100% linearly polarized perpendicular to the plane of incidence.

More New Zealand Scholarship practice

New Zealand Scholarship mock tests

All New Zealand Scholarship practice →