SURDS-irrational roots




INDICES
a⁰ = 1
a^m × aⁿ = a^m+n
aⁿ × bⁿ = (ab)ⁿ
(a^m)ⁿ = a^mn






LOGARITHMS
a^b = C (index form) log_aC = b (logarithmic form) C>0,a>0,a1

log_a xy = log_ax + log_ay
log_a (x/y) = log_ax - log_ay
log_a xⁿ = n log_ax
log_a a = 1
log_a 1 = 0
lg = log₁₀
lg C = b 10^b = C

log_e x = ln x
e^b = C ln C = b
ln xy = ln x + ln y
ln (x/y) = ln x - ln y
ln xⁿ = n ln x
ln e = 1




a^x = b, (b = aⁿ) a^x = aⁿ > x = n (a -1,0,1)

QUADRATIC EQUATIONS: ax² + bx + c = 0

Discriminant = b² - 4ac
b² - 4ac > 0 real and distinct roots
b² - 4ac = 0 real and equal roots
b² - 4ac < 0 no real roots

y = ax² + bx + c
a(x - a)(x - b) [turning point.: x = (a+b)÷2]
a(x - h)² + k [(h,k) = turning pt.]
a < 0 max.pt., a > 0 min. pt.



(x - a)(x - b) < 0 a < x < b
(x - a)(x - b) > 0 x < a or x > b


REMAINDER THEOREM
f(x) divided by (x - a) > remainder = f(a)

FACTOR THEOREM
(x - a): factor of f(x) > f(a) = 0

CUBIC EQUATIONS
px³ + qx² + rx + s = 0 (x - a)(ax² + bx + c) = 0   [s 0, (x - a) = a factor)]

COORDINATE GEOMETRY




= ½(x₁ y₂ + x₂ y₃ + x₃ y₁ - y₁ x₂ - y₂ x₃ - y₃ x₁)
[A = (x₁, y₁), B = (x₂, y₂), C = (x₃, y₃), pts. taken in anti-clockwise direction]

Equation of a straight line
y - y₁ = m(x - x₁) [(x₁, y₁) lines on line]

Parallel lines
y = m₁x + c₁ & y = m₂x + c₂ are parallel m₁ = m₂

Perpendicular lines
y = m₁x + c₁ & y = m₂x + c₂ are perpendicular, m₁m₂ = -1

LINEAR LAW
Y = mX + c [m & c = constant, Y & X = variables]

FUNCTIONS
f : x x + 1 f(x) = x+1

Finding the Inverse Function
f : x x + 1: Let f(x) = y y = x + 1 Make x the subject x = y-1 f ^-1 : x y-1

gf(x)= g(f(x))
(gf)^-1 = f^-1g^-1 f^-1f(x) = ff^-1 (x) = 1
In graphs, f^-1(x) = reflection of f(x) in the line y = x


ABSOLUTE VALUES
|x| = absolute value of x (numerical value)
| ±x |= x


Amaths2
Amaths3
Amaths4
Amaths5

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