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)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1) | n is a natural number.
13. a^maⁿ = a^m+n 
14. (ab)^m = a^mb^m 
15. (a^m)ⁿ = a^mn
16. a^m/ aⁿ = a^m-n
17. (a/b)ⁿ = aⁿ/ bⁿ
18. a⁰ = 1
19. a^-n = 1/ aⁿ
20. log_amn = log_am + log_an
21. log_a(m/n) = log_am – log_an
22. log_amⁿ = n.log_am
23. log_ba = log_ka / log_kb
24. log_ba = 1/ log_ab
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

• n^th Term = T_n = a + (n-1)d
• Sum of n Terms = S_n = 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

• n^th Term = T_n = ar^n-1
• Sum of n Terms = S_n = a(1-rⁿ) / (1-r)
• Sum of ∞ Terms = S∞ = a / (1-r)

  3-Harmonic Progression(HP)
   a_1, a_2, a_3, a_4, …, …. are said to be in Harmonic Progression(HP) if and only             if 1/a_1, 1/a_12, 1/a_3, 1/a_4, … are in Arithmetic Progession(AP).  

VBODMAS Rule

   V- Vinculumor or Bar
   B- Bracket
   O- Of
   D- Division
   M- Multiplication
   A- Addition
   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

Percentage

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)

ⁿC_r = n! / r!(n-r)!

Permutation

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

ⁿP_r = n! / (n-r)!

Probability

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

Mensuration

Triangle

Equilateral Triangle:

• All three sides of the Equilateral Triangle are equal.
• Each angle of the Equilateral Triangle is 60^o 
• 

• 

• 
• 

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 90^o 

Isosceles Right Angle Triangle:

• One of the angle is 90^o 
• 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

Quadrilateral:

• Figure enclosed by 4 sides.
• Sum of all the 4 angles = 360^o 

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 90^o 
•  Diagonal of a square are equal & bisect each other at 90^o 

Rectangle:

• It is a parallelogram with equal opposite sides and each angle is 90^o 
• 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 90^o .
• It has unequal diagonal but they bisect each other at 90^o .

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:

Volume Surface & Area

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: