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NCEA Mathematics and Science Concepts

Essential formulas, theorems and definitions across algebra, calculus, physics, chemistry and genetics for secondary study in New Zealand.

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What does the discriminant Delta = b^2 - 4ac indicate about quadratic roots?
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1
What does the discriminant Delta = b^2 - 4ac indicate about quadratic roots?
Delta > 0 gives two real distinct roots; Delta = 0 gives one repeated real root; Delta < 0 gives no real roots.
2
What is the vertex form of a quadratic parabola, and what are the vertex coordinates?
y = a(x - h)^2 + k, with vertex coordinates located at (h, k).
3
What is the power rule for finding the derivative of f(x) = a*x^n?
f'(x) = n a x^(n - 1).
4
How do you determine whether a stationary point is a local maximum or minimum using the second derivative?
If f''(x) > 0 the curve is concave up (local minimum); if f''(x) < 0 the curve is concave down (local maximum).
5
What is the kinematic relationship connecting displacement s(t), velocity v(t), and acceleration a(t)?
Velocity is the first derivative of displacement: v(t) = s'(t); acceleration is the derivative of velocity: a(t) = v'(t) = s''(t).
6
What is the gradient relationship between two perpendicular straight lines with slopes m1 and m2?
m1 * m2 = -1, meaning m2 is the negative reciprocal of m1: m2 = -1 / m1.
7
What is the addition rule for finding the probability of the union of two events A and B?
P(A union B) = P(A) + P(B) - P(A intersect B).
8
What condition proves that two events A and B are statistically independent?
P(A intersect B) = P(A) * P(B), or equivalently P(A | B) = P(A).
9
What percentage of data falls within 1, 2, and 3 standard deviations in a normal distribution?
Approximately 68% within +/- 1 sigma, 95% within +/- 2 sigma, and 99.7% within +/- 3 sigma.
10
How is the z-score of an observation x calculated from mean mu and standard deviation sigma?
z = (x - mu) / sigma, measuring how many standard deviations x lies from the mean.
11
How is relative risk (RR) calculated from two group risks?
RR = Risk in exposed group / Risk in unexposed group.
12
What is the 1.5 * IQR rule for identifying statistical outliers?
An outlier lies below Q1 - 1.5 IQR or above Q3 + 1.5 IQR, where IQR = Q3 - Q1.
13
What is Newton's second law of motion in algebraic form?
F_net = m * a, where net force equals mass multiplied by acceleration.
14
What is the formula for gravitational potential energy near Earth's surface?
E_p = m g h, where m is mass in kg, g is gravitational acceleration (approx 9.8 or 10 N/kg), and h is height in metres.
15
What is the formula for kinetic energy of an object of mass m moving at velocity v?
E_k = 0.5 m v^2.
16
What is the wave equation relating wave speed v, frequency f, and wavelength lambda?
v = f * lambda.
17
What is Ohm's law connecting voltage V, current I, and electrical resistance R?
V = I * R.
18
Why do solid ionic compounds not conduct electricity while molten ionic compounds do?
Ions are locked in fixed positions in the solid lattice; melting frees ions to move and carry electric charge.
19
Why does graphite conduct electricity whereas diamond does not?
In graphite, each carbon atom forms 3 covalent bonds in layers, leaving one delocalised electron per atom free to conduct current.
20
What is an exothermic reaction in terms of enthalpy change delta H?
A reaction that releases thermal energy to the surroundings, characterised by negative enthalpy change (delta H < 0).
21
What is the general molecular formula for saturated alkanes?
C_n H_(2n + 2).
22
What chemical test distinguishes an alkene from an alkane in the laboratory?
Alkenes rapidly decolourise orange bromine water in the dark; alkanes require ultraviolet light to react.
23
What are the four nitrogenous bases in DNA and how do they pair complementarily?
Adenine pairs with Thymine (A-T via 2 hydrogen bonds); Cytosine pairs with Guanine (C-G via 3 hydrogen bonds).
24
What is the sole source of completely new genetic alleles in a biological population?
Mutation, which alters the nucleotide base sequence of DNA.
25
What are the two major mechanisms during meiosis that generate genetic variation?
Crossing over (recombination) in prophase I and independent assortment of homologous chromosomes in metaphase I.
26
What is the 10 percent rule of ecological energy transfer between trophic levels?
Only about 10% of energy at one trophic level is transferred to the next; 90% is lost as heat, respiration, and metabolic waste.

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