Tips Flashcards
For a given annual % of interest & initial principal
Increasing the frequency of compounding, increases the interest amount accumulated
If multiply/divide both sides of an inequality with a -ve number
DIRECTION of the inequality changes (reverses)
Listing down factors of a number
Use the T method (stop when pairs start repeating)
Multiple of every +ve integer
0
Standard form for a Quadratic equation
y=ax²+bx+c; if |a| > 1 skinny, if |a| < 1 wide; if a>0, graph opens upwards, if a<0, graph opens downwards
Impact of outliers on Median
Changing highest/lowest no on a list, DOESN’T change median
When there is a distinct outlier or a set of outliers in one direction
That pulls the mean away from the median
Exponents of prime factors of a square
Must all be EVEN
Numbers with exactly 3 divisors
Squares of prime numbers (1, p, p²)
Numbers with 5 factors can be
Square of a prime’s square (1, p, p², p³, p⁴)
Some mathematical exceptions (x, x²)
For most +ve nos, x² > x; For x=0 & x=1, x²=x; For fractions b/w 0 & 1, x²<x
If divisor > dividend, q=?, r=?
q=0, r=dividend
Squaring both sides of an inequality
If both sides are +ve, then can square; If opp. signs, then can’t square
Adding mean term to a list, shifts standard deviation
to LOWER value
LCM & HCF
LCM=product of HIGHEST powers; HCF=product of LOWEST powers
Perfect squares
The only integers with an ODD number of factors
Numbers with 4 factors can be
Product of 2 prime numbers (1, p, q, pxq) / Cube of a prime number (1, p, p². p³)
Always +ve
Factors of a +ve integer/ Remainders/ Prime numbers
Always integers
Multiples of a +ve integer / Odd & even numbers
[Exception] Quadratics have 2 solutions
(a-b)² = 0 & (a+b)² = 0 Yield only 1 solution
A set of ‘n’ consecutive inegers will always contain one no.
Divisible by n (if n is odd, sum of n consecutive integers will be divisible by n)
a⁰ = ?
1 (for all a except 0, i.e., 0⁰ = undefined)
Addition with mod
|x+y| <= |x| + |y| (equal when x,y have same sign)
Subraction with mod
|x-y| => |x| - |y| (equal when x,y have same sign & |x| >= |y|
Impact of outliers on Mean
Changing highest/lowest no on a list, CHANGES mean
Mutually exclusive events
2 events canNOT occur at the same time
Independent events
One event remains UNAFFECTED by the occurence of the other event
Data set values x -> ax+b, μ=? σ=?
a -> aμ + b; σ -> aσ
If xᵃ = 1
then a=0 & x = any number (or) a = any number & x=1
Data set that is symmetric about its average
Average = Average(least, greatest)
Terminating decimal depends on
Factors other than 2 & 5 (i.e., fractions with 2s and 5s terminate, while others recur)
Consecutive multiples of same number
μ = Median
Consecutive integers
μ = Median
Evenly spaced numbers / Any arithmetic sequence
μ = Median
Symmetrical list
μ = Median
If 2 series of consecutive multiples of ‘p’ have the same middle no
then the 2 series have the same μ
If 2 series of consecutive multiples of ‘p’ have the same no of terms
then the 2 series have the same σ
If |a| = b then
a = b (or) a = -b
Even roots of
positive numbers -> +ve output; -ve numbers -> doesn’t apply
If (x+y) is divisible by d & x is divisible by d
y is divisible by d
aˣ = ?
Every prime factor within the prime factorisation of that integer MUST have a power of x/its multiples
Atleast 1 even factor in a product => product will be…
EVEN
Fractions s ⁄ p & s ⁄ q where p>q [Bigger Dr make…]
s ⁄ p < s ⁄ q […Smaller fractions]
If x & y are each divisible by d
then (x+y) is divisible by d
Fraction a/b; Adding p&q (a+p)/(b+q); result?
(a+p)/(b+q) will be closer to p/q than a/b; If a/b < p/q => a/b < resultant; If a/b > p/q => a/b > resultant
[Exception] We need N equations to solve for N variables
Equations must be independent to count as separate / Cancellation of several variables at once helps
Odd roots of
positive numbers -> +ve output; negative numbers -> -ve output
x is what % of y
x/y * 100
x is what % > or < y
(x/y - 1)100 or (y/x - 1)100
If more than half the numbers in a list have the same value (x), median=?
x (regardless of values of remaining numbers)
Product can be ODD if and only if
Every single factor is ODD
Units digit of any product will be influenced by
ONLY the units digits of the 2 factors
If Nr increases & Dr decreases, fraction?
Fraction increases
p is a factor of q
q = kp
q is a multiple of p
q = kp
Inequality/Equality of type x-y & y-x on two sides
Unless x & y are equal +ve on one side & -ve on the other (opp signs)
When a perfect square ends with an even number of zeros, the square root of such a perfect square will have exactly ___ of the number of zeros to the right of the final
nonzero digit as the perfect square.
half
If a decimal with a finite number of decimal places is a perfect square, its square root will have exactly ___ of the number of
decimal places. Thus, a perfect square decimal must have an ___ number of decimal places.
Half, Even
If two absolute values are equal, it must be true that the expressions within the absolute value bars are either
equals or opposites
If x² < |x| and x≠0, the range of x is
-1 < x < 1
