Maths Formulas and Solved Examples

Math formulae are equations that explain to us how to solve a problem. We utilise them to make problem-solving easier and faster. We apply them by first carefully reading our problem to see what it is asking for, and then looking for a formula that offers us the answer to our difficulty. Learning formulas help the students to better tackle all mathematical problems and prepare well for their exams. Maths formulas help students to revise thoroughly and in turn, also prepare for the competitive entrance exams. In this article, we will look at the different formulas that will help to build a stronger grip on the topic.

Read more: Circumference of Circle

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Maths Formulas

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Math is a mental topic that needs a strong grasp of all formulae. Students can use numerous methods and tricks to solve complicated problems by learning arithmetic formulae. When we look at chapters like fractions and percentages, we can see how they are connected. Similarly, there are linkages between chapters like percentages and profit and loss, then complex numbers and the notions of real numbers and exponents. It indicates that if we grasp the formulas of one chapter, we will be able to comprehend the formulas of subsequent chapters with ease; this is how Maths formulae become important to study.

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Algebra

The formulas pertaining to the Algebra branch of mathematics are as given below:

1. (a+b)²=a²+2ab+b²; a²+b²=(a+b)²-2ab
2. (a-b)²=a²-2ab+b²; a²+b²=(a-b)²+2ab
3. (a+b+c)²=a²+b²+c²+2(ab+bc+ca)
4. (a+b)³=a³+b³+3ab(a+b); a³+b³=(a+b)³-3ab(a+b)
5. (a-b)³=a³-b³-3ab(a-b); a³-b³=(a-b)³+3ab(a-b)
6. a²-b²=(a+b)(a-b)
7. a³-b³=(a-b)a²+ab+b²
8. a³+b³=(a+b)a²-ab+b²
9. aⁿ-bⁿ=(a-b)a^n-1+a^n-2b+a^n-3b²+….+b^n-1
10. aⁿ= a.a.a... n times
11. a^m . aⁿ = a^m+n
12. \(\frac{a^m}{a^n}\)=a^m-n if m>n

=1  if m=n

=\(\frac{1}{a^{n-m}}\) if m<n;a ∈ R,a≠0

Also Read: Three Dimensional Geometry

13. (a^m)ⁿ = a^mn=(aⁿ)^m
14. (ab)ⁿ=aⁿ . bⁿ
15. \((\frac{a}{b})^n\)=\(\frac{a^n}{b^n}\)
16. a⁰=1 where a∈R,a≠0
17. a^-n=\(\frac{1}{a^n}\),aⁿ=\(\frac{1}{a^{-n}}\)
18. a^p/q=^a√a^p
19. If a^m=aⁿ and a ≠ ±1,a ≠ 0 then m=n
20. If aⁿ=bⁿ where n ≠ 0, then a=±b
21. If √x,√y are quadratic surds and if a+√x=√y, then a=0 and x=y
22. If √x,√y are quadratic surds and if a+√x=b+√y then a=b and x=y
23. If a,m,n are positive real numbers and a ≠ 1, then log_amn=log_am+log_an
24. If a,m,n are positive real numbers, a ≠ 1, then log_a(\(\frac{m}{n}\))=log_am-log_an

25. If a and m are positive real numbers, a ≠ 1 then log_amⁿ=nlog_am

26. If a,b and k are positive real numbers, b ≠ 1,k ≠ 1, then log_ba=\(\frac{log_ka}{log_kb}\)
27. a =\(\frac{1}{b}\) where a,b are positive real numbers, a ≠ 1,b ≠ 1,
28. if a,m,n are positive real numbers, a ≠ 1 and if log_am=log_an, then m=n
29. if a+ib=0 where i=√-1, then a=b=0
30. if a+ib=x+iy, where i=√-1, then a=x and b=y
31. The roots of the quadratic equation ax²+bx+c=0; a≠0 are \(\frac{-b\pm \sqrt{ b^2-4ac}}{2a}\)

The solution set of the equation is{ \(\frac{-b+\sqrt{\bigtriangleup}}{{2a}}, \frac{-b- \sqrt{\bigtriangleup}}{2a}\)}

where Δ= discriminant =b²-4ac

i) The roots are real and distinct if Δ>0.

ii) The roots are real and coincident if Δ=0.

iii) The roots are non-real if Δ<0.

32. If and are the roots of the equation ax²+bx+c=0,a≠0 then
    i) α+β =\(\frac{-b}{a}\)=- \(\frac{coeff. of x}{ coeff. of x^2}\)
    ii)α.β =\(\frac{c}{a}\)= \(\frac{constant term}{coeff. of x^2}\)

33. The quadratic equation whose roots are and is (x-α)(x-β)=0
    i.e. x²-(α+β)x+αβ=0
    i.e. x²-Sx+P=0 where S= Sum of the roots and P= Product of the roots.

34. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d).
    i) n^th term =t_n=a+(n-1)d
    ii) The sum of the first (n) terms =S_n=\(\frac{n}{2}\)(a+l)=\(\frac{n}{2}\){2a+(n-1)d} where l= last term =a+(n-1)d.

35. For a geometric progression (G.P.) whose first term is (a) and the common ratio is (),
    i) n^th  term =t_n=aγ^n-1.
    ii) The sum of the first (n) terms:

S_n  =\(\frac{a(1-\Upsilon ^n)}{1- \Upsilon}\) if <1  =\(\frac{a(\Upsilon -1^n)}{\Upsilon - 1}\)  if >1  =na  if =1  

36. For any sequence {t_n},S_n-S_n-1=t_n where S_n=Sum of the first (n) terms.
37. ∑ⁿ _γ=1γ=1+2+3+...+n=n²(n+1).
38. ∑ⁿ _γ=1γ²=1²+2²+3²+...+n²=n6(n+1)(2n+1).
39. ∑ⁿ _γ=1γ³=1³+2³+3³+4³+...+n³=n²4(n+1)².
40. n!=(1)⋅(2)⋅(3)…(n-1)⋅n.
41. n!=n(n-1)!=n(n-1)(n-2)!=….
42. 0!=1.
43. (a+b)ⁿ=aⁿ+na^n-1b+\(\frac{n(n-1)}{2!}\)a^n-2b²+\(\frac{n(n-1)(n-2)}{3!}\)a^n-3b³+...+ bⁿ,n>1

Also Read:

Straight Line Formula

The formulas for straight lines are as given below:

1. Distance Formula:

d=\(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\)

2. Section Formula:

x=\(\frac{mx_2 \pm nx_1}{m \pm n}\);y=\(\frac{my_2 \pm ny_1}{m \pm n}\).

3. Centroid, Incentre & Excentre:
   Centroid G (\(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\)),
   Incentre I \(\frac{ax_1+bx_2+cx_3}{a+b+c},\frac {ay_1+by_2+cy_3}{a+b+c}\)
   Excentre I₁ \(\frac{-ax_1+bx_2+cx_3}{-a+b+c},\frac {-ay_1+by_2+cy_3}{-a+b+c}\)
4. Area of a Triangle:

ΔABC= \(\frac{1}{2}\) \(\mid\)x₁ y₁ 1 x₂ y₂ 1 x₃ y₃ 1\(\mid\)

5. Slope Formula:
   Line Joining two points x₁y₁ & x₂y₂, m=\(\frac{y_1-y_2}{x_1-x_2}\)
6. Condition of collinearity of three points:

Ix₁ y₁ 1 x₂ y₂ 1 x₃ y₃ 1I =0

7. Two Lines:
   ax+by+c=0 and a'x+b'y+c'=0 two lines
8. Parallel if \(\frac{a}{a'}=\frac{b}{b'} \neq \frac{c}{c'}\).
9. Distance between two parallel lines = \(\mid \frac {c_1-c_2}{\sqrt{a^2+b^2}} \mid\).

Perpendicular: If aa'+bb'=0.

10. A point and line:
11. Distance between point and line =\(\mid \frac{ax_1+by_1+c}{\sqrt {a^2+b^2}} \mid\).
12. Reflection of a point about a line:

\(\frac{x-x_1}{a} = \frac{y-y_1}{b} =-2 \frac{ax_1+by_1+c}{a^2+b^2}\)

13. The foot of the perpendicular from any point on the line

\(\frac{x-x_1}{a} = \frac{y-y_1}{b} =- \frac{ax_1+by_1+c}{a^2+b^2}\)

14. Bisectors of the angles between two lines:

\(\frac{ax+by+c}{\sqrt {a^2+b^2}}\)=± \(\frac{ax+by+c}{\sqrt {a'^2+b'^2}}\)

15. Condition of Concurrency:
    of three straight lines a₁x+b_yy+c₁=0, i=1,2,3 is \(\mid\)a₁ b₁ c₁ a₂ b₂ c₂ a₃ b₃ c₃\(\mid\) =0.

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Trigonometry

Trignometry formulas are as given below:

Factorization of the Sum or the Difference of Two Sines or Cosines:
(a) sinC+sinD=2sin\(\frac{C+D}{2}\)cos\(\frac{C-D}{2}\)
(b) sinC-sinD=2cos\(\frac{C+D}{2}\)sin\(\frac{C-D}{2}\)
(c) cosC+cosD=2cos\(\frac{C+D}{2}\)cos\(\frac{C-D}{2}\)
(d) cosC-cosD= -2sin\(\frac{C+D}{2}\)sin\(\frac{C-D}{2}\)
Multiple and Sub-multiple Angles :
(a) cos²A=cos²A-sin²A=2cos²A-1=1-2sin²A;2cos²\(\frac{\theta}{2}\) =1+cos\(\theta\),2sin²\(\frac{\theta}{2}\)=1-cos\(\theta\).
(b) sin²A=\(\frac{2tanA}{1+tan^2A}\),cos²A=\(\frac{1-tan^2A}{1+tan^2A}\)
(c) sin³A=3sinA-4sin³A
(d) cos³A=4cos³A-3cosA
(e) tan³A=\(\frac{3tanA-tan^3A}{1-3tan^2A}\)

Also Read: Vectors
Important Trigonometric Ratios:

Important Trigonometric Ratios

Limit Of Function

The formulas for limits of functions are:

1. Limit of a function f(x) is said to exist as x a when, Limit_h→0+f(a-h)=Limit_h→0f(a+h) = some finite value M.
   (Left-hand limit) and (Right-hand limit)
2. Indeterminant Forms:

\(\frac{0}{0}\),\(\frac{\infty}{\infty}\),0×∞,∞-∞,∞^o,0^o, and 1.

3. Standard Limits:

                           Standard Limits

4. Limits Using Expansion
   

                   Limits Using Expansion

Differentiation of Basic Functions

The formulas for differentiation of basic functions are:

1. \(\frac{d}{dx}\)(xⁿ)=nx^n-1
2. \(\frac{d}{dx}\)a^x=a^xlna
3. \(\frac{d}{dx}\)(ln|x|)=\(\frac{1}{x}\)
4. \(\frac{d}{dx}\)log_ax=\(\frac{1}{xlna}\)
5. \(\frac{d}{dx}\)(sinx)=cosx
6. \(\frac{d}{dx}\)(cosx)=-sinx
7. \(\frac{d}{dx}\)(secx)=secxtanx
8. \(\frac{d}{dx}\)(cosecx)=-cosecxcotx
9. \(\frac{d}{dx}\)(tanx)=sec²x
10. \(\frac{d}{dx}\)(cotx)=-cosec²x

Also Read:

Basic Theorems

• \(\frac{d}{dx}\)f±g=f'(x)±g'x
• \(\frac{d}{dx}\)(kf(x))=k\(\frac{d}{dx}\)f(x)
• \(\frac{d}{dx}\)(f(x)⋅g(x))=f(x)g'(x)+g(x)f'(x)
• \(\frac{d}{dx}\)\((\frac{f(x)}{g(x)})\)=\(\frac{g(x)f'(x)-f(x)g'(x)}{g^2(x)}\)
• \(\frac{d}{dx}\)(f(g(x)))=f'(g(x))g'(x)

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Derivative Of Inverse Trigonometric Functions

The derivatives of inverse trigonometric functions are:

Derivative Of Inverse Trigonometric Functions

Differentiation using substitution

Differentiation using substitution

Indefinite Integration

Indefinite Integration

Definite Integration

The formulas for definite integration are:

Definite Integration

Mensuration Formulas

The formulas for mensuration are:

Cylinder

Curved surface area = 2Πrh

Total surface area = 2Πr(h + r)

Volume = Πr²h

Cone

Curved surface area = Πrl

Total surface area = 2Πr(l + r)

Volume = Πr2h/3

Sphere

Surface area = 4Πr²

Volume = 4Πr³/3

Cube

Surface area = 4a²

Total surface area = 6a²

Cuboid

Surface area = 2(hw + lh)

Total surface area = 2(lw + hw + lh)

Volume = l x w x h

Also Read: Maxima and Minima

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Things to Remember

• For students studying for competitive exams and board exams, Math Formulas are essential. 
• Students can use math formulas to get hands-on practice and improve their grades on in-class exams and standardized tests. On-time completion of the curriculum is required. Prepare for various types of entrance tests, such as the JEE Main.
• Mathematical equations are also used in traffic control, airplanes, the space program, and medicine, among other applications. 
• As a result, we must always keep in mind that the outcome of each arithmetic equation has the ability to change the world. That is why all mathematical equations are crucial in our daily lives.
• Mathematics requires operations with numbers, and it also aids you in calculating the product price, how many discounted rewards are available, and how quickly you can compute if you are skilled at arithmetic.

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