# Mathematics Formula Collection- Algebra & Mensuration

#### Algebra

1. (a+b)2 = a2 + 2ab + b2
2. (a – b)2 = a2 – 2ab + b2
3. a2 + b2 = (a+b)2 – 2ab = (a – b)2 + 2ab
4. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
5. (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
6. (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
7. a2 – b2 = (a – b)(a + b)
8. (a + b)3 = a3 + b3 + 3ab(a + b)
9. (a – b)3 = a3 – b3 – 3ab(a – b)
10. a3 – b3 = (a – b)(a2 + ab + b2)
11. a3 + b3 = (a + b)(a2 – ab + b2)
12. an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1) | n is a natural number.
13. aman = am+n
14. (ab)m = amb
15. (am)n = amn
16. am/ an = am-n
17. (a/b)n = an/ bn
18. a0 = 1
19. a-n = 1/ an
20. logamn = logam + logan
21. loga(m/n) = logam – logan
22. logamn = n.logam
23. logba = logka / logkb
24. logba = 1/ logab
25. (x+a)(x+b) = x2 + (a+b)x + ab

#### Progression(Series)

1-Arithmetic Progression(AP)
a, a+d, a+2d, a+3d, …, …, …,l(last term)
a-First Term, d-Common Difference, l-Last Term

• nth Term = Tn = a + (n-1)d
• Sum of n Terms = Sn = n/2[2a + (n-1)d] = n/2[first term + last term]

2-Geometric Progression(GP)
a, ar, ar2, ar3, ar4, ….
a-First Term, r-Common Ratio

• nth Term = Tn = arn-1
• Sum of n Terms = Sn = a(1-rn) / (1-r)
• Sum of ∞ Terms = S∞ = a / (1-r)

3-Harmonic Progression(HP)
a1,
a2, a3, a4, …, …. are said to be in Harmonic Progression(HP) if and only             if 1/a1, 1/a12, 1/a3, 1/a4, … are in Arithmetic Progession(AP).

#### VBODMAS Rule

V- Vinculumor or Bar
B- Bracket
O- Of
D- Division
M- Multiplication
S- Subtraction

#### Average

• Average of First n Natural Numbers = (n+1)/2
• Average of First n Even Numbers = n+1
• Average of First n odd Numbers = n
• Average of Consecutive Numbers = (First No. + Last No.) / 2
• Average of squares of First n Natural Numbers = (n+1)(2n+1) / 6
• Average of Cubes of First n Natural Numbers = n(n+1)2 / 4

#### y% of x = x * y/100

• 5% of a number = number/20
• 10% of a number = number/10
• 20% of a number = number/5
• 25% of a number = number/4
• 50% of a number = number/2
•

#### Profit and Loss

• Profit = Selling Price – Cost Price
• Loss = Cost Price – Selling Price
• Profit% = (Profit/Cost Price) * 100
• Loss(Loss/Cost Price) * 100
• Selleng Price = (100 + Profit%) * Cost Price / 100
• Selleng Price = (100 – Loss%) * Cost Price / 100
• Cost Price = 100 * Selling Price / (100 + Profit%)
• Cost Price = 100 * Selling Price / (100 – Loss%)

#### Discount

• Discount = Marked Price – Selling Price
• Discount% = (Discount/Marked Price) * 100
• Selling Price = Marked Price – Discount
• Selling Price = Marked Price * (1 – r/100)

#### Simple Interest

• Simple Interest = (Principal * Rate * Time)/100
• Total Amount = Principal + Simple Interest = P + SI = P + PRT/100
= P(1 + RT/100)
• Principal = (Amount * 100) / (100 + RT)
• Simple Interest = (Amount * Rate * Time) / (100 + RT)
P = Principal Amount, A = Total Amount, R = Rate of Interest (Annual)
T = Time

#### Compound Interest

• In Compound Interest the amount at the end of each year becomes the principal for the next year.
• Total Amount = P(1 + R/100)T
• P = Principal Amount, R = Rate of Interest(Annual), T = Time

#### Factorial

• n! = n(n-1)(n-2)(n-3)……*1

#### Combination

• Number of Combination of n different things when r things are taken into consideration. (Order is not important)

nCr = n! / r!(n-r)!

#### Permutation

• The number of Permutation of n different things taken r at a time

nPr = n! / (n-r)!

#### Probability

Probability of happening of an event = (Number of Favourable events) / (Total Number of possible outcomes)

#### Triangle

##### Equilateral Triangle:
• All three sides of the Equilateral Triangle are equal.
• Each angle of the Equilateral Triangle is 60o
• • ##### • • ##### Isosceles Triangle:
• 2 Sides and 2 Angles are equal.
• Altitude drawn on non equal side bisect it. ##### Scalene Triangle:
• All the 3 sides are unequal. ##### Right Angle Triangle:
• One of the angle is 90o ##### Isosceles Right Angle Triangle:
• One of the angle is 90o
• 2 Sides are equal. ##### Properties of Triangle:
• Sum of any 2 sides of any triangle is greater than the third side.
• For a constant Perimeter, Equilateral Triangle will have the maximum area.
• Median: The line joining the midpoint of any side of triangle to the opposite vertex.
• The median of a triangle divides the triangle into triangle of two equal area. ##### Inner Circle and Circumcircle of Equilateral Triangle:
• 2 Let the side of equilateral triangle is a • Figure enclosed by 4 sides.
• Sum of all the 4 angles = 360o
##### Parallelogram:
• Opposite sides are parallel.
• Diagonals of a parallelogram bisects each other.
• Diagonal of parallelogram divides it into 2 triangles of equal area.
• Opposite angles of a parallelogram are equal. ##### Square:
• It is a parallelogram which has all 4 sides equal and each angle is 90o
•  Diagonal of a square are equal & bisect each other at 90o ##### Rectangle:
• It is a parallelogram with equal opposite sides and each angle is 90o
• Diagonals of a rectangle are equal.
• Diagonal of a rectangle bisects each other. ##### Trapezium:
• It is a quadrilateral with one pair of opposite sides parallel. ##### Rhombus:
• It is a parallelogram with all 4 sides equal.
• Opposite angles of rhombus are equal but not 90o .
• It has unequal diagonal but they bisect each other at 90o . ##### Circle:
• Let r be the radius of circle.
• l is the length of Arc. ##### Semi Circle:
• A circle when separated into 2 parts along its diameter. ##### Circular Ring: ##### Cube
• A solid body having 6 equal faces with equal length, breadth and height is called cube.
• Each face of cube is a square.
• Let a be the side of cube. ##### Cuboid: ##### Cylinder: ##### Hollow Cylinder: ##### Cone: ##### Sphere: ##### Hollow Sphere or Spherical Shell: ##### Hemisphere: error: Content is protected !!