**Maths Formulas**

Every chapter as well as subject plays a vital role in class 10 and 12 and so does Maths, which is incomplete without **Maths Formulas**…! For those visitors who are in search of complete Class 10 And 12 Basic/Vedic Topics Wise *Important Maths Formulas*, we have gathered on this single page. Speaking from the Knowledge point of view, Math Formulas is quite helpful in scoring the best grade in subject Math which will automatically affect your percentage. So Candidates, start your preparation according to the latest Maths Formulas and along with that check the best Sample Questions from here, you also have option of downloading the same in the form of PDF.

At the time of preparation of Math subject, students are advised to make proper chart of all formulas that are stated below and stick it on your wall, revise them regularly for the best results. Practice is what you need in Math subject, so start preparation with the collection of formulas given below. Apart from this, Candidates should know the accuracy and speed are other components that students have to make level of both in Math exam. You must remember that the marks of these classes are very vital for further life. Go through this page of www.recruitmentinboxx.com and save your time in searching it.

**Maths Formulas**

**Maths Formula for 10th Class**

Below is the list of important formulas of class 10, it will be definitely easy for the candidates to collect all according to their chapters such as Algebra, Geometry, Circles Constructions, Trigonometry, Probability, Coordinate Geometry and Menstruation. Check the marking scheme as well topics from below stipulated table and list of formulas.

Chapter | Topics | Marks |

Algebra | Quadratic Equations, Arithmetic Progressions | 26 |

Geometry | 12 | |

Circles Constructions | 22 | |

Trigonometry | 10 | |

Probability | 12 | |

Coordinate Geometry | 08 | |

Mensuration | 10 | |

Total | 100 |

__Maths Algebraic Identities Formula__

- (a+b)
^{2}=a^{2}+2ab+b^{2} - (a−b)
^{2}=a^{2}−2ab+b^{2} - (a+b)(a–b)=a
^{2}–b^{2} - (x+a)(x+b)=x
^{2}+(a+b)^{x}+ab - (x+a)(x–b)=x
^{2}+(a–b)^{x}–ab - (x–a)(x+b)=x
^{2}+(b–a)^{x}–ab - (x–a)(x–b)=x
^{2}–(a+b)^{x}+ab - (a+b)
^{3}=a^{3}+b^{3}+3ab(a+b) - (a–b)
^{3}=a^{3}–b^{3}–3ab(a–b) - (a – b)
^{4}= a^{4}– 4a^{3b}+ 6a2^{b2}– 4ab^{3}+ b^{4}) - a
^{4 }– b^{4}= (a – b)(a + b)(a^{2}+ b^{2}) - a5 – b5 = (a – b)(a4 + a
^{3b}+ a2^{b2}+ ab^{3}+ b^{4}) - If n is a natural number, a
^{n }– b^{n}= (a – b)(a^{n}-1 + a^{n}-2^{b}+…+ bn-2^{a}+ b^{n}-1) - If n is even (n = 2
^{k}), a^{n}+ b^{n}= (a + b)(a^{n}-1 – a^{n}-2^{b}+…+ b^{n}-2^{a}– b^{n}-1) - If n is odd (n = 2
^{k}+ 1), a^{n}+ b^{n}= (a + b)(a^{n}-1 – a^{n}-2^{b}+…- b^{n}-2^{a}+ b^{n}-1) - (a + b + c + …)2 = a
^{2}+ b^{2}+ c^{2}+ … + 2(a^{b}+ a^{c }+ b^{c}+ ….

Example:

Question: Which of the following expressions is in the sum-of-products (SOP) form?

- (A + B)(C + D)
- (A)B(CD)
- AB(CD)
- AB + CD

Answer: AB + CD

__Combination__:

Number of combinations of *n* different things taken *r* at a time = ⁿCr = n!r!(n−r)!n!r!(n−r)!

ⁿP₀ = r!∙ ⁿC₀.

ⁿC₀ = ⁿCn = 1

ⁿCr = ⁿCn – r

ⁿCr + ⁿCn – 1 = n+1n+1C_{r}

If p ≠ q and ⁿCp = ⁿCq then p + q = n.

ⁿCr/ⁿCr – 1 = (n – r + 1)/r.

The total number of combinations of n different things taken any number at a time = ⁿC₁ + ⁿC₂ + ⁿC₃ + …………. + ⁿC₀ = 2ⁿ – 1.

The total number of combinations of (p + q + r + . . . .) things of which p things are alike of a first kind, q things are alike of a second kind r things are alike of a third kind and so on, taken any number at a time is [(p + 1) (q + 1) (r + 1) . . . . ] – 1.

__Maths Logarithm Formula__

If a^{x} = M then log_{a }M = x and conversely.

- log
_{a }1 = 0. - log
_{a}a = 1. _{a}log_{a}^{m}= M.- log
_{a}MN = log_{a}M + log_{a} - log
_{a}(M/N) = log_{a}M – log_{a} - log
_{a}M^{n}= n log_{a} - log
_{a}M = log_{b}M x log_{a} - log
_{b}a x 1og_{a}b = 1. - log
_{b}a = 1/log_{b} - log
_{b }M = log_{b}M/log_{a}

__Maths Exponential Series__

- For all x, e
^{x}= 1 + x/1! + x^{2}/2! + x^{3}/3! + …………… + x^{r}/r! + ………….. ∞. - e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
- 2 < e < 3; e = 2.718282 (correct to six decimal places).
- a
^{x }= 1 + (log_{e}a) x + [(log_{e}a)^{2}/2!] ∙ x^{2 }+ [(log_{e}a)^{3}/3!] ∙ x^{3}+ …………….. ∞.

__Maths Logarithmic Series__

- log
_{e}(1 + x) = x – x^{2}/2 + x^{3}/3 – ……………… ∞ (-1 < x ≤ 1). - log
_{e}(1 – x) = – x – x^{2}/ 2 – x3/3 – ………….. ∞ (- 1 ≤ x < 1). - ½ log
_{e}[(1 + x)/(1 – x)] = x + x^{3}/3 + x^{5}/5 + ……………… ∞ (-1 < x < 1). - log
_{e}2 = 1 – 1/2 + 1/3 – 1/4 + ………………… ∞. - log
_{10}m = µ log_{e}m where µ = 1/log_{e}10 = 0.4342945 and*m*is a positive number.

__Maths Laws of Exponents__

- (a
^{m})(a^{n}) = a^{m+n} - (ab)
^{m}= a^{m}b^{m} - (am)
^{n}= a^{mn}

__Maths Fractional Exponents__

- a
^{0 }= 1 - a
^{m}/a^{n}=a^{m−n} - a
^{m}=1/ a^{−m} - a
^{−m}=1/a^{m}

__All Mensuration Formulas In Maths__

__Basic Formula__:

- sin(−θ)=−sinθ
- cos(−θ)=cosθ
- tan(−θ)=−tanθ
- cosec(−θ)=−cosecθ
- sec(−θ)=secθ
- cot(−θ)=−cotθ

__Maths Trigonometry Formulas__**:**

Check Out: __Compound Interest Formula – Monthly/Quarterly/Half Yearly With Example__

**Math Formula for 12th Class**

Candidates who are going to attempt class 12th board exams must collect all vital formulas of math subject from below

__Laws of Indices__:

aᵐ ∙ aⁿ = aᵐ + ⁿ

aᵐ/aⁿ = aᵐ – ⁿ

(iii) (aᵐ)ⁿ = aᵐⁿ

(iv) a = 1 (a ≠ 0).

(v) a-ⁿ = 1/aⁿ

(vi) ⁿ√aᵐ = aᵐ/ⁿ

(vii) (ab)ᵐ = aᵐ ∙ bⁿ.

(viii) (a/b)ᵐ = aᵐ/bⁿ

(ix) If aᵐ = bᵐ (m ≠ 0), then a = b.

(x) If aᵐ = aⁿ then m = n.

__Surds__:

- The surd conjugate of √a + √b (or a + √b) is √a – √b (or a – √b) and conversely.
- If a is rational, √b is a surd and a + √b (or, a – √b) = 0 then a = 0 and b = 0.
- If a and x are rational, √b and √y are surds and a + √b = x + √y then a = x and b = y.

__Complex Numbers__:

(i) The symbol z = (x, y) = x + iy where x, y are real and i = √-1, is called a complex (or, imaginary) quantity;x is called the real part and y, the imaginary part of the complex number z = x + iy.

(ii) If z = x + iy then z = x – iy and conversely; here, z is the complex conjugate of z.

(iii) If z = x+ iy then

(a) mod. z (or, | z | or, | x + iy | ) = + √(x² + y²) and

(b) amp. z (or, arg. z) = Ф = tan−1−1 y/x (-π < Ф ≤ π).

(iv) The modulus – amplitude form of a complex quantity z is

z = r (cosф + i sinф); here, r = | z | and ф = arg. z (-π < Ф <= π).

(v) | z | = | -z | = z ∙ z = √ (x² + y²).

(vi) If x + iy= 0 then x = 0 and y = 0(x,y are real).

(vii) If x + iy = p + iq then x = p and y = q(x, y, p and q all are real).

(viii) i = √-1, i² = -1, i³ = -i, and i⁴ = 1.

(ix) | z₁ + z₂| ≤ | z₁ | + | z₂ |.

(x) | z₁ z₂ | = | z₁ | ∙ | z₂ |.

(xi) | z₁/z₂| = | z₁ |/| z₂ |.

(xii) (a) arg. (z₁ z₂) = arg. z₁ + arg. z₂ + m

(b) arg. (z₁/z₂) = arg. z₁ – arg. z₂ + m where m = 0 or, 2π or, (- 2π).

(xiii) If ω be the imaginary cube root of unity then ω = ½ (- 1 + √3i) or, ω = ½ (-1 – √3i)

(xiv) ω³ = 1 and 1 + ω + ω² = 0

Do You Know: __How to Prepare For Maths Exam__

__Variation__:

(i) If x varies directly as y, we write x ∝ y or, x = ky where k is a constant of variation.

(ii) If x varies inversely as y, we write x ∝ 1/y or, x = m ∙ (1/y) where m is a constant of variation.

(iii) If x ∝ y when z is constant and x ∝ z when y is constant then x ∝ yz when both y and z vary.

__Arithmetical Progression (A.P.)__:

(i) The general form of an A. P. is a, a + d, a + 2d, a + 3d,…..

where a is the first term and d, the common difference of the A.P.

(ii) The nth term of the above A.P. is t₀ = a + (n – 1)d.

(iii) The sum of first n terns of the above A.P. is s = n/2 (a + l) = (No. of terms/2)[1st term + last term] or, S = ⁿ/₂ [2a + (n – 1) d]

(iv) The arithmetic mean between two given numbers a and b is (a + b)/2.

(v) 1 + 2 + 3 + …… + n = [n(n + 1)]/2.

(vi) 1² + 2² + 3² +……………. + n² = [n(n+ 1)(2n+ 1)]/6.

(vii) 1³ + 2³ + 3³ + . . . . + n³ = [{n(n + 1)}/2 ]².

__Geometrical Progression (G.P.)__:

(i) The general form of a G.P. is a, ar, ar², ar³, . . . . . where a is the first term and r, the common ratio of the G.P.

(ii) The n th term of the above G.P. is t₀ = a.rn−1n−1 .

(iii) The sum of first n terms of the above G.P. is S = a ∙ [(1 – rⁿ)/(1 – r)] when -1 < r < 1

or, S = a ∙ [(rⁿ – 1)/(r – 1) ]when r > 1 or r < -1.

(iv) The geometric mean of two positive numbers a and b is √(ab) or, -√(ab).

(v) a + ar + ar² + ……………. ∞ = a/(1 – r) where (-1 < r < 1).

__Theory of Quadratic Equation__:

ax² + bx + c = 0 … (1)

(i) Roots of the equation (1) are x = {-b ± √(b² – 4ac)}/2a.

(ii) If α and β be the roots of the equation (1) then,

sum of its roots = α + β = – b/a = – (coefficient of x)/(coefficient of x² );

and product of its roots = αβ = c/a = (Constant term /(Coefficient of x²).

(iii) The quadratic equation whose roots are α and β is

x² – (α + β)x + αβ = 0

i.e. , x² – (sum of the roots) x + product of the roots = 0.

(iv) The expression (b² – 4ac) is called the discriminant of equation (1).

(v) If a, b, c are real and rational then the roots of equation (1) are

(a) real and distinct when b² – 4ac > 0;

(b) real and equal when b² – 4ac = 0;

(c) imaginary when b² – 4ac < 0;

(d) rational when b²- 4ac is a perfect square and

(e) irrational when b² – 4ac is not a perfect square.

(vi) If α + iβ be one root of equation (1) then its other root will be conjugate complex quantity α – iβ and conversely (a, b, c are real).

(vii) If α + √β be one root of equation (1) then its other root will be conjugate irrational quantity α – √β (a, b, c are rational).

START NOW: __Logical Reasoning Test__

__Permutation__:

(i) ⌊n (or, n!) = n (n – 1) (n – 2) ∙∙∙∙∙∙∙∙∙ 3∙2∙1.

(ii) 0! = 1.

(iii) Number of permutations of n different things taken r ( ≤ n) at a time ⁿP₀ = n!/(n – 1)! = n (n – 1)(n – 2) ∙∙∙∙∙∙∙∙ (n – r + 1).

(iv) Number of permutations of n different things taken all at a time = ⁿP₀ = n!.

(v) Number of permutations of n things taken all at a time in which p things are alike of a first kind, q things are alike of a second kind, r things are alike of a third kind and the rest are all different, is ⁿ<span style=’font-size: 50%’>!/₀

(vi) Number of permutations of n different things taken r at a time when each thing may be repeated upto r times in any permutation, is nʳ .

__Combination__:

(i) Number of combinations of n different things taken r at a time = ⁿCr = n!r!(n−r)!n!r!(n−r)!

(ii) ⁿP₀ = r!∙ ⁿC₀.

(iii) ⁿC₀ = ⁿCn = 1.

(iv) ⁿCr = ⁿCn – r.

(v) ⁿCr + ⁿCn – 1 = n+1n+1Crr

(vi) If p ≠ q and ⁿCp = ⁿCq then p + q = n.

(vii) ⁿCr/ⁿCr – 1 = (n – r + 1)/r.

(viii) The total number of combinations of n different things taken any number at a time = ⁿC₁ + ⁿC₂ + ⁿC₃ + …………. + ⁿC₀ = 2ⁿ – 1.

(ix) The total number of combinations of (p + q + r + . . . .) things of which p things are alike of a first kind, q things are alike of a second kind r things are alike of a third kind and so on, taken any number at a time is [(p + 1) (q + 1) (r + 1) . . . . ] – 1.

__Also Know About__:

__Permutation and Combination Formula, Aptitude Questions, Problems and Solutions__

__Binomial Theorem__:

(i) Statement of Binomial Theorem : If n is a positive integer then

(a + x)n = an + nC1 an – 1 x + nC2 an – 2 x2 + …………….. + nCr an – r xr + ………….. + xn …….. (1)

(ii) If n is not a positive integer then

(1 + x)n = 1 + nx + [n(n – 1)/2!] x2 + [n(n – 1)(n – 2)/3!] x3 + ………… + [{n(n-1)(n-2)………..(n-r+1)}/r!] xr+ ……………. ∞ (-1 < x < 1) ………….(2)

(iii) The general term of the expansion (1) is (r+ 1)th term

= tr + 1 = nCr an – r xr

(iv) The general term of the expansion (2) is (r + 1) th term

= tr + 1 = [{n(n – 1)(n – 2)….(n – r + l)}/r!] ∙ xr.

(v) There is one middle term is the expansion ( 1 ) when n is even and it is (n/2 + 1)th term ; the expansion ( I ) will have two middle terms when n is odd and they are the {(n – 1)/2 + 1} th and {(n – 1)/2 + 1} th terms.

(vi) (1 – x)-1 = 1 + x + x2 + x3 + ………………….∞.

(vii) (1 + x)-1 = I – x + x2 – x3 + ……………∞.

(viii) (1 – x)-2 = 1 + 2x + 3×2 + 4×3 + . . . . ∞ .

(ix) (1 + x)-2 = 1 – 2x + 3×2 – 4×3 + . . . . ∞ .

__Logarithm__:

(i) If ax = M then loga M = x and conversely.

(ii) loga 1 = 0.

(iii) loga a = 1.

(iv) a logam = M.

(v) loga MN = loga M + loga N.

(vi) loga (M/N) = loga M – loga N.

(vii) loga Mn = n loga M.

(viii) loga M = logb M x loga b.

(ix) logb a x 1oga b = 1.

(x) logb a = 1/logb a.

(xi) logb M = logb M/loga b.

Must Read This: __How to Reduce Negative Marking in Exam__

__Exponential Series__:

(i) For all x, ex = 1 + x/1! + x2/2! + x3/3! + …………… + xr/r! + ………….. ∞.

(ii) e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.

(iii) 2 < e < 3; e = 2.718282 (correct to six decimal places).

(iv) ax = 1 + (loge a) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.

__Logarithmic Series__:

(i) loge (1 + x) = x – x2/2 + x3/3 – ……………… ∞ (-1 < x ≤ 1).

(ii) loge (1 – x) = – x – x2/ 2 – x3/3 – ………….. ∞ (- 1 ≤ x < 1).

(iii) ½ loge [(1 + x)/(1 – x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).

(iv) loge 2 = 1 – 1/2 + 1/3 – 1/4 + ………………… ∞.

(v) log10 m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.

**Maths Sample Questions**

Well, we all know that math subject needs a lot of knowledge which involve lot of hard work but if you want to do it easier way the Math previous year paper, Math sample paper is the best option. Visitors are advised to do practice for this vital subject from the below settled sample paper, you just have to press the links given below.

**Press Here: Maths Sample Paper for class 10 **

**Press Here: Maths Sample Paper for class 12 **

We hope the above provided Maths Formulas will be beneficial for your in preparation of your exams. Stay connected with this web portal of www.recruitmentinboxx.com as will update you time to time about the same and if you have any query then must comment in the below framed comment box, our experts will solve it as soon as possible.

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