Finding number of Factors
Sum of n natural numbers
-->
To find the number of factors of a given number, express the number as a product of powers of prime numbers.
In this case, 48 can be written as 16 * 3 = (24 * 3)
Now, increment the power of each of the prime numbers by 1 and multiply the result.
In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)
Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors.
Sum of n natural numbers
-> The sum of first n natural numbers = n (n+1)/2
-> The sum of squares of first n natural numbers is n (n+1)(2n+1)/6
-> The sum of first n even numbers= n (n+1)
-> The sum of first n odd numbers= n^2
Finding Squares of numbers
To find the squares of numbers near numbers of which squares are known
To find 41^2 , Add 40+41 to 1600 =1681
To find 59^2 , Subtract 60^2-(60+59) =3481
Finding number of Positive Roots
If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then.
Eg: x^4+3x^2+2x+6=0 has no positive roots .
Finding number of Imaginary Roots
For an equation f(x)=0 , the maximum number of positive roots it can
have is the number of sign changes in f(x) ; and the maximum number of
negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the
equation(Since we also know that the index of the maximum power of x is
the number of roots of an equation.)
Reciprocal Roots
The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a
Roots
Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1
Finding Sum of the rootsFor a
cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the
product of the roots taken two at a time = c/a product of the roots =
-d/a
For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = -
b/a sum of the product of the roots taken three at a time = c/a sum of
the product of the roots taken two at a time = -d/a product of the roots
= e/a
Maximum/Minimum
-> If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4
-> If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) .
Inequalties
-> x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some inequations)
-> a+b=a+b if a*b>=0 else a+b >= a+b
->
2<= (1+1/n)^n <=3 -> (1+x)^n ~ (1+nx) if x<<<1>
When you multiply each side of the inequality by -1, you have to reverse
the direction of the inequality.
Product Vs HCF-LCM
Product
of any two numbers = Product of their HCF and LCM . Hence product of
two numbers = LCM of the numbers if they are prime to each other
AM GM HM
For
any 2 numbers a>b a>AM>GM>HM>b (where AM, GM ,HM stand
for arithmetic, geometric , harmonic menasa respectively) (GM)^2 = AM *
HM
Sum of Exterior Angles
For any regular polygon , the sum of the exterior angles is equal to 360
degrees hence measure of any external angle is equal to 360/n. ( where n
is the number of sides)
For any regular polygon , the sum of interior angles =(n-2)180 degrees
So measure of one angle in
Square-----=90
Pentagon--=108
Hexagon---=120
Heptagon--=128.5
Octagon---=135
Nonagon--=140
Decagon--=144
Problems on clocks
Problems on clocks can be tackled as assuming two runners going round a
circle , one 12 times as fast as the other . That is , the minute hand
describes 6 degrees /minute the hour hand describes 1/2 degrees /minute .
Thus the minute hand describes 5(1/2) degrees more than the hour hand
per minute .
The hour and the minute hand meet each other after every 65(5/11)
minutes after being together at midnight. (This can be derived from the
above) .
Co-ordinates
Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the
coordinates of the meeting point of the diagonals can be found out by
solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]
Ratio
If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to
(k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+
k2*b2+k3*b3+..............) , which is also equal to
(a1+a2+a3+............./b1+b2+b3+..........)
Finding multiples
x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very
useful for finding multiples .For example (17-14=3 will be a multiple of
17^3 - 14^3)
Exponents
e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 <>GP
-> In a GP the product of any two terms equidistant from a term is always constant .
-> The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .
Mixtures
If Q be the volume of a vessel q qty of a mixture of water and wine be
removed each time from a mixture n be the number of times this operation
be done and A be the final qty of wine in the mixture then ,
A/Q = (1-q/Q)^n
