NOTE: When the question asks to round off, use the quadratic formula to solve.
Simultaneous linear equations can be solved either by substitution or elimination.
Simultaneous linear and non-linear equations are generally solved by substitution as follows:
Step 1: Obtain an equation in one unknown and solve this equation
Step 2: Substitute the result from step 1 into the equation to find the other unknown.
The points of intersection of two graphs are given by the solution of their simultaneous equations.
You would solve these like you would equations.
Multiplying or dividing by negative would cause a switch of the inequality sign. For example:
y/-3 ≥ 7
Multiply through by -3: y ≤ -7 × -3
∴ y ≤ 21
When two inequalities are present, split into two. For example:
x < 3x – 1 < 2x + 7 ⇒ Split into two groups:
x < 3x – 1; x > 1/2
3x – 1 < 2x + 7; x < 8
For strict inequalities (‘<‘, ‘>’), use broken lines.
For non-strict inequalities (‘≤’, ‘≥’), use solid lines.
Steps to solve:
Draw straight line graphs
Linear Sequence: Find the common, e.g. 3, then multiply it by n and work out what needs to be added.
Quadratic Sequence: Using 2, 6, 12, 20, 30, 42 as an example.
The general formula is un = an2 + bn + c.
To find the nth term of the sequence:
Find the differences between the term sin the sequence
Find the difference between the differences in the above step. Note: The second difference is common (the same throughout).
The coefficient of n2 is gotten by dividing the second difference by 2. Therefore, a = 1.
Then, find the first few terms of the first term (an2); in this case, 1, 4, 9, 16, 25 (we found for the first 5 terms)
Then, create a 3-rowed table; the first column would have the first number of terms from the original sequence (according to the number of terms you found above), the second column would have the terms you found above, and the third term would have the subtractions of the first column and the second column.
Treat dn like a linear sequence and find the formula for its nth term; in this case dn = n (the common difference is 1, and you need not add anything to get the terms)
Therefore, un = n2 + n.
NOTE: If dn = something else, say 4n + 3; un = n2 + 4n + 3.
This is a sequence where the terms have been multiplied by a constant to form the next term.
The nth term of G.P. = ar(n – 1)
a = 1st term
r = common difference
From O to A: Uniform speed.
From B to C: Uniform speed (return journey)
From A to B: Stationary (speed = 0)
On the graph, the gradient is equal to the speed of the journey.
From O to A: Uniform speed.
From A to B: Constant speed (acceleration = 0)
From B to C: Uniform deceleration/retardation
Area under a graph = distance traveled
Gradient = acceleration
If the acceleration is negative, it is called deceleration or retardation. (Moving body is slowing down)
f: x → 2x – 1
Function f such that x maps onto 2x -1
Given two functions f(x) and g(x), the composite function of f and g is the function which maps x onto f(g(x))