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CSAT Mathematics Formulae and Rules

Key formulae and rules for CSAT mathematics: algebra, logarithms, trigonometry, sequences, calculus, probability and geometry.

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Quadratic formula
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1
Quadratic formula
For ax^2 + bx + c = 0, x = (-b plus or minus root(b^2 - 4ac)) / 2a.
2
Discriminant
b^2 - 4ac. Positive: two real roots. Zero: one repeated root. Negative: no real roots.
3
Sum and product of the roots
Sum = -b/a and product = c/a.
4
Remainder theorem
The remainder when f(x) is divided by x - a is f(a).
5
Logarithm of a product
log (xy) = log x + log y.
6
Logarithm of a quotient
log (x/y) = log x - log y.
7
Change of base
log_a b = log b / log a.
8
Sine rule
a / sin A = b / sin B = c / sin C = 2R.
9
Cosine rule
c^2 = a^2 + b^2 - 2ab cos C.
10
Addition formula for sine
sin (A + B) = sin A cos B + cos A sin B.
11
Double angle for cosine
cos 2A = cos^2 A - sin^2 A = 1 - 2 sin^2 A = 2 cos^2 A - 1.
12
Period of y = a sin (bx)
2 pi / b.
13
Arithmetic sequence: nth term
a_n = a + (n - 1)d.
14
Arithmetic sequence: sum
S_n = n/2 [2a + (n - 1)d].
15
Geometric sequence: sum
S_n = a (r^n - 1)/(r - 1). For |r| < 1 the infinite sum is a/(1 - r).
16
Sum of squares
The sum of k^2 for k = 1 to n is n(n + 1)(2n + 1)/6.
17
Sum of cubes
The sum of k^3 for k = 1 to n is [n(n + 1)/2]^2.
18
Derivative of x^n
n x^(n - 1).
19
Product rule
(uv)' = u'v + uv'.
20
Chain rule
d/dx f(g(x)) = f'(g(x)) g'(x).
21
Integral of x^n
x^(n + 1)/(n + 1) + C, for n not equal to -1.
22
Area under a curve
The definite integral of f(x) from a to b, when f(x) is at least 0.
23
Combinations
nCr = n! / (r! (n - r)!).
24
Permutations
nPr = n! / (n - r)!.
25
Binomial probability
P(X = k) = nCk p^k (1 - p)^(n - k).
26
Conditional probability
P(A given B) = P(A and B) / P(B).
27
Mean and variance of aX + b
Mean = a E(X) + b and variance = a^2 Var(X).
28
Standardising a normal variable
Z = (X - mean) / standard deviation.
29
Distance between two points
root[(x2 - x1)^2 + (y2 - y1)^2].
30
Distance from a point to a line
For ax + by + c = 0 the distance from (p, q) is |ap + bq + c| / root(a^2 + b^2).
31
Equation of a circle
(x - h)^2 + (y - k)^2 = r^2 with centre (h, k).
32
Dot product
a . b = |a||b| cos theta = a1 b1 + a2 b2. It is zero for perpendicular vectors.

CSAT Suneung in Korea, Republic of

In Korea, Republic of, CSAT Suneung is part of: National exams in Korea, Republic of.

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