Mathes Important Formulas

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.

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.

Maths Algebraic Identities Formula
(a+b)2=a2+2ab+b2
(a−b)2=a2−2ab+b2
(a+b)(a–b)=a2–b2
(x+a)(x+b)=x2+(a+b)x+ab
(x+a)(x–b)=x2+(a–b)x–ab
(x–a)(x+b)=x2+(b–a)x–ab
(x–a)(x–b)=x2–(a+b)x+ab
(a+b)3=a3+b3+3ab(a+b)
(a–b)3=a3–b3–3ab(a–b)
(a – b)4= a4 – 4a3b + 6a2b2 – 4ab3 + b4)
a4 – b4= (a – b)(a + b)(a2 + b2)
a5 – b5 = (a – b)(a4 + a3b + a2b2+ ab3 + b4)
If n is a natural number, an – bn= (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
If n is even (n = 2k), an+ bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
If n is odd (n = 2k + 1), an+ bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
(a + b + c + …)2 = a2+ b2 + c2 + … + 2(ab + ac + bc + ….
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+1Cr
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 ax = M then loga M = x and conversely.
loga 1 = 0.
logaa = 1.
alogam = M.
logaMN = loga M + loga
loga(M/N) = loga M – loga
logaMn = n loga
logaM = logb M x loga
logba x 1oga b = 1.
logba = 1/logb
logb M = logbM/loga

Maths Exponential Series
For all x, ex= 1 + x/1! + x2/2! + x3/3! + …………… + xr/r! + ………….. ∞.
e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
2 < e < 3; e = 2.718282 (correct to six decimal places).
ax = 1 + (logea) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.
Maths Logarithmic Series
loge(1 + x) = x – x2/2 + x3/3 – ……………… ∞ (-1 < x ≤ 1).
loge(1 – x) = – x – x2/ 2 – x3/3 – ………….. ∞ (- 1 ≤ x < 1).
½ loge[(1 + x)/(1 – x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).
loge2 = 1 – 1/2 + 1/3 – 1/4 + ………………… ∞.
log10m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.

Maths Laws of Exponents
(am)(an) = am+n
(ab)m= ambm
(am)n= amn

Maths Fractional Exponentsb
a0 = 1
am/an=am−n
am=1/ a−m
a−m=1/am

Maths Algebraic Identities Formula
(a+b)2=a2+2ab+b2
(a−b)2=a2−2ab+b2
(a+b)(a–b)=a2–b2
(x+a)(x+b)=x2+(a+b)x+ab
(x+a)(x–b)=x2+(a–b)x–ab
(x–a)(x+b)=x2+(b–a)x–ab
(x–a)(x–b)=x2–(a+b)x+ab
(a+b)3=a3+b3+3ab(a+b)
(a–b)3=a3–b3–3ab(a–b)
(a – b)4= a4 – 4a3b + 6a2b2 – 4ab3 + b4)
a4 – b4= (a – b)(a + b)(a2 + b2)
a5 – b5 = (a – b)(a4 + a3b + a2b2+ ab3 + b4)
If n is a natural number, an – bn= (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
If n is even (n = 2k), an+ bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
If n is odd (n = 2k + 1), an+ bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
(a + b + c + …)2 = a2+ b2 + c2 + … + 2(ab + ac + bc + ….
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

Combinations:
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+1Cr
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 ax = M then loga M = x and conversely.
loga 1 = 0.
logaa = 1.
alogam = M.
logaMN = loga M + loga
loga(M/N) = loga M – loga
logaMn = n loga
logaM = logb M x loga
logba x 1oga b = 1.
logba = 1/logb
logb M = logbM/loga

Maths Exponential Series
For all x, ex= 1 + x/1! + x2/2! + x3/3! + …………… + xr/r! + ………….. ∞.
e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
2 < e < 3; e = 2.718282 (correct to six decimal places).
ax = 1 + (logea) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.

Maths Logarithmic Series
loge(1 + x) = x – x2/2 + x3/3 – ……………… ∞ (-1 < x ≤ 1).
loge(1 – x) = – x – x2/ 2 – x3/3 – ………….. ∞ (- 1 ≤ x < 1).
½ loge[(1 + x)/(1 – x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).
loge2 = 1 – 1/2 + 1/3 – 1/4 + ………………… ∞.
log10m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.

Maths Laws of Exponents
(am)(an) = am+n
(ab)m= ambm
(am)n= amn

Maths Fractional Exponents
a0 = 1
am/an=am−n
am=1/ a−m
a−m=1/am

Maths Algebraic Identities Formula
(a+b)2=a2+2ab+b2
(a−b)2=a2−2ab+b2
(a+b)(a–b)=a2–b2
(x+a)(x+b)=x2+(a+b)x+ab
(x+a)(x–b)=x2+(a–b)x–ab
(x–a)(x+b)=x2+(b–a)x–ab
(x–a)(x–b)=x2–(a+b)x+ab
(a+b)3=a3+b3+3ab(a+b)
(a–b)3=a3–b3–3ab(a–b)
(a – b)4= a4 – 4a3b + 6a2b2 – 4ab3 + b4)
a4 – b4= (a – b)(a + b)(a2 + b2)
a5 – b5 = (a – b)(a4 + a3b + a2b2+ ab3 + b4)
If n is a natural number, an – bn= (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
If n is even (n = 2k), an+ bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
If n is odd (n = 2k + 1), an+ bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
(a + b + c + …)2 = a2+ b2 + c2 + … + 2(ab + ac + bc + ….

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

Combinations:
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+1Cr
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 ax = M then loga M = x and conversely.
loga 1 = 0.
logaa = 1.
alogam = M.
logaMN = loga M + loga
loga(M/N) = loga M – loga
logaMn = n loga
logaM = logb M x loga
logba x 1oga b = 1.
logba = 1/logb
logb M = logbM/loga

Maths Exponential Series
For all x, ex= 1 + x/1! + x2/2! + x3/3! + …………… + xr/r! + ………….. ∞.
e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
2 < e < 3; e = 2.718282 (correct to six decimal places).
ax = 1 + (logea) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.

Maths Logarithmic Series
loge(1 + x) = x – x2/2 + x3/3 – ……………… ∞ (-1 < x ≤ 1).
loge(1 – x) = – x – x2/ 2 – x3/3 – ………….. ∞ (- 1 ≤ x < 1).
½ loge[(1 + x)/(1 – x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).
loge2 = 1 – 1/2 + 1/3 – 1/4 + ………………… ∞.
log10m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.

Maths Laws of Exponents
(am)(an) = am+n
(ab)m= ambm
(am)n= amn

Maths Fractional Exponents
a0 = 1
am/an=am−n
am=1/ a−m
a−m=1/am