Must know formulas for CSAT

Must know formulas for CSAT

Algebraic Identities

Must know formulas for CSAT

Algebraic Identities

n(n + l)(2n + 1) is always divisible by 6.
32n leaves remainder = 1 when divided by 8
n3 + (n + 1 )3 + (n + 2 )3 is always divisible by 9
102n + 1 + 1 is always divisible by 11
n(n2- 1) is always divisible by 6
n2+ n is always even
23n-1 is always divisible by 7
152n-1 +l is always divisible by 16
n3 + 2n is always divisible by 3
34n – 4 3n is always divisible by 17
n! + 1 is not divisible by any number between 2 and n(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800
The product of n consecutive numbers is always divisible by n!.
If n is a positive integer and p is a prime, then np – n is divisible by p.
|x| = x if x ≥ 0 and |x| = – x if x ≤ 0.
Minimum value of a2.sec2Ɵ + b2.cosec2Ɵ is (a + b)2; (0° < Ɵ < 90°)for eg. minimum value of 49 sec2Ɵ + 64.cosec2Ɵ is (7 +2 = 225.
Among all shapes with the same perimeter a circle has the largest area.
If one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.
sum of all the angles of a convex quadrilateral = (n – 2)180°
Number of diagonals in a convex quadrilateral = 0.5n(n – 3)
let P, and Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD. Then,

ΔAPD = ΔCQB.

a2 – b2 = (a + b)(a – b)
a2 + b2 + c2 + 2(ab + bc + ca) = (a + b + c)2
(a ± b)2 = a2 + b2± 2ab
(a + b + c + d)2 = a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd)
(a ± b)3 = a3 ± b3 ± 3ab(a ± b)
(a ± b)(a2 + b2 m ab) = a3 ± b3
(a + b + c)(a2 + b2 + c2 -ab – bc – ca) = a3 + b3 + c3 – 3abc =
1/2 (a + b + c)[(a - b)2 + (b - c)2 + (c - a)2]

when a + b + c = 0, a3 + b3 + c3 = 3abc

(x + a)(x + b) (x + c) = x3 + (a + b + c) x2 + (ab + bc + ac)x + abc
(x – a)(x – b) (x – c) = x3 – (a + b + c) x2 + (ab + bc + ac)x – abc
a4 + a2b2 + b4 = (a2 + ab + b2)( a2 – ab + b2)
a4 + b4 = (a2 – √2ab + b2)( a2 + √2ab + b2)
an + bn = (a + b) (a n-1 – a n-2 b + a n-3 b2 – a n-4 b3 +…….. + b n-1)(valid only if n is odd)
an – bn = (a – b) (a n-1 + a n-2 b + a n-3 b2 + a n-4 b3 +……… + b n-1){were n ϵ N)
(a ± b)2n is always positive while -(a ± b)2n is always negative, for any real values of a and b
(a – b)2n = (b – a)2” and (a – b)2n+1 = – (b – a)2n+1
if α and β are the roots of equation ax2 + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β. if α and β are the roots of equation ax2 + bx + c = 0, roots of ax2 – bx + c = 0 are -α and -β.

Divisibility Rule

2 - An even number or a number with an even last digit, such as 0, 2, 4, 6, and 8.

3 - The number should be divisible by 3 when the sum of all its digits is considered.

4 - The result of the number's last two digits must be either 00 or a multiple of 4.

5 - Numbers with the digits 0 or 5 in the ones place.

6 - A number that can be divided by both 2 and 3

7 - Multiple of 7 is obtained by deducting twice the number's last digit from the other digits.

8 - The sum of a number's last three digits must be divisible by 8 or equal to 000.

9 - The number should be divisible by 9 when the sum of all its digits is considered.

10 - Any number with the digit 0 in the ones position.

11 - A number is divisible by 11 when the sums of its alternative digits differ from each other.

12 - A number that can be divided by 3 and 4.

Profit Loss: Formulas

Cost Price = Selling Price ( No profit No loss)
Gain Percentage = (Gain × 100)/(C.P.)
Loss Percentage = (Loss × 100)/(C.P.)
Profit, P = SP – CP; SP>CP
Loss, L = CP – SP; CP>SP
P% = (P/CP) x 100
L% = (L/CP) x 100
SP = {(100 + P%)/100} x CP
SP = {(100 – L%)/100} x CP
CP = {100/(100 + P%)} x SP
CP = {100/(100 – L%)} x SP
Discount = MP – SP
SP = MP -Discount
For false weight, profit percentage will be P% = [(True weight – false weight)/ false weight] x 100.
When there are two successful profits, say m% and n%, then the net percentage profit equals to [m+n+(mn/100)]
When the profit is m%, and loss is n%, then the net % profit or loss will be: [m-n-(mn/100)]
If a product is sold at m% profit and then again sold at n% profit then the actual cost price of the product will be: CP = [100 x 100 x P/(100+m)(100+n)]. In case of loss, CP = [100 x 100 x L/(100-m)(100-n)]
If P% and L% are equal then, P = L and %loss = P2/100

Interest Formulas for SI and CI

(Simple and Compound)	Formula
S.I.	Principal × Rate × Time
C.I.	Principal (1 + Rate)^Time − Principal

Arithmetic Progression

There are a few key terms we'll encounter in AP that are defined as:

First term (a)
Common difference (d)
nth Term (an)
Sum of the first n terms (Sn)

All three terms stand for the arithmetic progression property.

First Term of AP

The AP can also be written in terms of common differences, as follows;

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

where “a” is the first term of the progression.

Common Difference in Arithmetic Progression

In this progression, for a given series, the terms used are the first term, the common difference and nth term. Suppose, a1, a2, a3, ……………., an is an AP, then; the common difference “ d ” can be obtained as;

d = a2 – a1 = a3 – a2 = ……. = an – an – 1

Where “d” is a common difference. It can be positive, negative or zero.

General Form of an AP

Consider an AP to be: a1, a2, a3, ……………., an

Position of Terms	Representation of Terms	Values of Term
1	a1	a = a + (1-1) d
2	a2	a + d = a + (2-1) d
3	a3	a + 2d = a + (3-1) d
4	a4	a + 3d = a + (4-1) d
.	.	.
.	.	.
.	.	.
.	.	.
n	an	a + (n-1)d

General Form of AP	a, a + d, a + 2d, a + 3d, . . .
The nth term of AP	an = a + (n – 1) × d
Sum of n terms in AP	S = n/2[2a + (n − 1) × d]
Sum of all terms in a finite AP with the last term as ‘l’	n/2(a + l)

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