All Important Formula — Class 11 Maths Notes

All Important Formula · Class 11 Maths · 6 topics.

These notes are free to read without an account. Work through them in order, or use the chapter list to revise selectively before a test.

Topics covered in All Important Formula

  1. 1.Algebraic Formulas:

    Algebraic Formulas:

    1. (a+b)2=a2+2ab+b2

      • Expansion of a binomial square (sum of squares).

    2. (a−b)2=a2−2ab+b2

      • Expansion of a binomial square (difference of squares).

    3. (a+b)(a−b)=a2−b2

      • Difference of squares formula.

    4. a3+b3=(a+b)(a2−ab+b2)

      • Sum of cubes.

    5. a3−b3=(a−b)(a2+ab+b2)

      • Difference of cubes.

    6. (a−b2)2=a2−2ab+b24

      • Square of a fraction involving binomial subtraction.

    7. a3−b3=(a−b)(a2+ab+b2)+3ab(a−b)

      • Another variation for the difference of cubes expansion.

    8. (a−b)4=a4−4a3b+6a2b2−4ab3+b4

      • Fourth power binomial expansion (difference).

    9. a3+b3=(a+b)3−3ab(a+b)

      • Sum of cubes in terms of a binomial cube.

    10. (a+b)3=a3+3a2b+3ab2+b3

      • Cube of a binomial (sum).

    11. (a−b)3=a3−3a2b+3ab2−b3

      • Cube of a binomial (difference).

    12. (a−b)3=a3−3a2b+3ab2−b3

      • Repeated cube of a binomial (difference).

    Explanation and Example of Key Formula


    Formula: (a+b)2=a2+2ab+b2


    • Components:
      • a and b are variables or numbers.
      • The square of the sum of a and b expands to a2+2ab+b2.

    Example:

    Let's expand (3+2)2:

    1. Given: a=3, b=2

    2. Plug into the formula:(3+2)2=32+2×3×2+22
    3. Simplify:=9+12+4=25

  2. 2.Factors Formula (Logarithmic Formulas)

    1. ax=N ⟹ x=log⁡aN


    • Explanation: If a raised to the power x equals N, then x is the logarithm of N with base a.

    • Example:
      If 23=8, then log⁡28=3.

    2. log⁡aa=1


    • Explanation: The logarithm of a number to its own base is always 1.

    • Example:
      log⁡55=1.

    3. log⁡1010=1


    • Explanation: The logarithm of 10 to the base 10 is always 1.

    • Example:
      log⁡1010=1.

    4. log⁡a(m×n)=log⁡am+log⁡an


    • Explanation: The logarithm of the product of two numbers is the sum of their logarithms to the same base.

    • Example:

      Find log⁡2(8×4):log⁡2(8×4)=log⁡28+log⁡24Since log⁡28=3 and log⁡24=2:log⁡2(8×4)=3+2=5

    5. log⁡a(mn)=log⁡am−log⁡an


    • Explanation: The logarithm of the division of two numbers is the difference of their logarithms.

    • Example:

      Find log⁡2(84):log⁡2(84)=log⁡28−log⁡24Since log⁡28=3 and log⁡24=2:log⁡2(84)=3−2=1

    6. log⁡amn=nlog⁡am


    • Explanation: The logarithm of a power is the exponent times the logarithm of the base number.

    • Example:
      Find log⁡282:log⁡282=2log⁡28Since log⁡28=3:log⁡282=2×3=6

    7. log⁡ax2=2log⁡a∣x∣


    • Explanation: The logarithm of x2 is twice the logarithm of the absolute value of x.

    • Example:

      Find log⁡2(−4)2:log⁡2(−4)2=2log⁡2∣4∣Since log⁡24=2:log⁡2(−4)2=2×2=4

    8. log⁡ba×log⁡ab=1


    • Explanation: The product of log⁡ba and log⁡ab is always equal to 1.

    • Example:

      If log⁡39=2, then log⁡93=12, and their product:log⁡39×log⁡93=2×12=1

    9. log⁡ab=1log⁡ba


    • Explanation: The logarithm of b with base a is the reciprocal of the logarithm of a with base b.

    • Example:
      If log⁡28=3, then log⁡82=13.

    10. log⁡bx=log⁡axlog⁡ab


    • Explanation: This is the change of base formula, which allows you to convert logarithms from one base to another.

    • Example:
      Find log⁡28 using base 10:log⁡28=log⁡108log⁡102Approximate values:log⁡108≈0.903,log⁡102≈0.301So,log⁡28≈0.9030.301=3

    11. log⁡ba=1log⁡ab

    • Explanation: This is a repetition of Formula 9, stating the reciprocal relationship of logs.

    • Example:

      If log⁡525=2, then log⁡255=12.

  3. 3.Trigonometric Ratios and Formulas

    📏 Trigonometric Ratios and Formulas

    1. Trigonometric Ratios:

      • tan⁡θ=sin⁡θcos⁡θ

      • cot⁡θ=cos⁡θsin⁡θ

      • sec⁡θ=1cos⁡θ

      • csc⁡θ=1sin⁡θ

    2. Pythagorean Identities:

      • sin⁡2θ+cos⁡2θ=1
      • sec⁡2θ−tan⁡2θ=1
      • csc⁡2θ−cot⁡2θ=1

    3. Co-function Identities:

      • sin⁡θ=cos⁡(90∘−θ)
      • cos⁡θ=sin⁡(90∘−θ)
      • tan⁡θ=cot⁡(90∘−θ)
      • cot⁡θ=tan⁡(90∘−θ)
      • sec⁡θ=csc⁡(90∘−θ)
      • csc⁡θ=sec⁡(90∘−θ)

    4. Additional Identities:

      • tan⁡θ×cot⁡θ=1

    📊 Trigonometric Table for Common Angles



    These formulas and table help solve problems related to trigonometry, geometry, and real-world applications like physics, architecture, and engineering.

  4. 4.Differentiation Formulas

    Basic Differentiation Formulas

    1. Power Rule
      ddx(xn)=nxn−1

    2. Exponential Function
      ddx(ex)=ex

    3. Logarithmic Differentiation
      ddx(log⁡ax)=1xlog⁡a

    4. Natural Logarithm
      ddx(log⁡ex)=1x

    5. Exponential with Base a
      ddx(ax)=axlog⁡a

    Trigonometric Functions

    1. Derivative of sin⁡x
      ddx(sin⁡x)=cos⁡x

    2. Derivative of cos⁡x
      ddx(cos⁡x)=−sin⁡x

    3. Derivative of tan⁡x
      ddx(tan⁡x)=sec⁡2x

    4. Derivative of cot⁡x
      ddx(cot⁡x)=−csc⁡2x

    5. Derivative of sec⁡x
      ddx(sec⁡x)=sec⁡xtan⁡x

    6. Derivative of csc⁡x
      ddx(csc⁡x)=−csc⁡xcot⁡x

    Inverse Trigonometric Functions

    1. Derivative of sin⁡−1x
      ddx(sin⁡−1x)=11−x2,for −1<x<1

    2. Derivative of cos⁡−1x
      ddx(cos⁡−1x)=−11−x2,for −1<x<1

    3. Derivative of tan⁡−1x
      ddx(tan⁡−1x)=11+x2

    4. Derivative of cot⁡−1x
      ddx(cot⁡−1x)=−11+x2

    5. Derivative of sec⁡−1x
      ddx(sec⁡−1x)=1∣x∣x2−1,∣x∣>1

    6. Derivative of csc⁡−1x
      ddx(csc⁡−1x)=−1∣x∣x2−1,∣x∣>1

    Hyperbolic Trigonometric Functions

    1. Derivative of sinh⁡x
      ddx(sinh⁡x)=cosh⁡x

    2. Derivative of cosh⁡x
      ddx(cosh⁡x)=sinh⁡x

    3. Derivative of tanh⁡x
      ddx(tanh⁡x)=sech⁡2x

    4. Derivative of coth⁡x
      ddx(coth⁡x)=−csch⁡2x

    5. Derivative of sech⁡x
      ddx(sech⁡x)=−sech⁡xtanh⁡x

    6. Derivative of csch⁡x
      ddx(csch⁡x)=−csch⁡xcoth⁡x

  5. 5.Integration Formulas

    1. Basic Formulas

    1. ∫1 dx=x+C

    2. ∫xn dx=xn+1n+1+C (for n≠−1)

    3. ∫ex dx=ex+C

    4. ∫ax dx=axln⁡a+C (for a>0,a≠1)

    5. ∫1x dx=ln⁡∣x∣+C

    2. Trigonometric Formulas

    1. ∫sin⁡x dx=−cos⁡x+C

    2. ∫cos⁡x dx=sin⁡x+C

    3. ∫sec⁡2x dx=tan⁡x+C

    4. ∫csc⁡2x dx=−cot⁡x+C

    5. ∫sec⁡xtan⁡x dx=sec⁡x+C

    6. ∫csc⁡xcot⁡x dx=−csc⁡x+C

    3. Inverse Trigonometric Formulas

    1. ∫11−x2 dx=sin⁡−1x+C

    2. ∫−11−x2 dx=cos⁡−1x+C

    3. ∫11+x2 dx=tan⁡−1x+C

    4. ∫−11+x2 dx=cot⁡−1x+C

    5. ∫1xx2−1 dx=sec⁡−1∣x∣+C

    6. ∫−1xx2−1 dx=csc⁡−1∣x∣+C

    4. Exponential and Logarithmic Functions

    1. ∫ex dx=ex+C

    2. ∫ln⁡x dx=xln⁡x−x+C

    3. ∫ax dx=axln⁡a+C

    5. Special Forms

    1. ∫1a2+x2 dx=1atan⁡−1xa+C

    2. ∫1a2−x2 dx=sin⁡−1xa+C

    3. ∫1xx2−a2 dx=1asec⁡−1∣x∣a+C

    6. Integration by Substitution

    If u=f(x), then:

    ∫f′(x)g(f(x)) dx=∫g(u) du

    7. Integration by Parts

    ∫uv dx=u∫v dx−∫(dudx∫v dx)dx

    Where u and v are functions of x.

    8. Definite Integral Properties

    1. ∫abf(x) dx=F(b)−F(a), where F(x) is the antiderivative of f(x).

    2. ∫aaf(x) dx=0

    3. ∫abf(x) dx=−∫baf(x) dx

  6. 6.Exponent Laws

    Exponent Laws

    1. a0=1

      • Explanation: Any non-zero number raised to the power of 0 is always 1.

    2. ap⋅aq=ap+q

      • Explanation: When multiplying two numbers with the same base, add the exponents.

    3. (ap)q=ap⋅q

      • Explanation: When raising a power to another power, multiply the exponents.

    4. apaq=ap−q

      • Explanation: When dividing two numbers with the same base, subtract the exponents.

    5. a−n=1an

      • Explanation: A negative exponent means the reciprocal of the number raised to the positive exponent.

    6. a=a1/2

      • Explanation: The square root of a number can be expressed as the number raised to the power 1/2.

    7. a3=a1/3

      • Explanation: The cube root of a number can be expressed as the number raised to the power 1/3.

    8. an=a1/n

      • Explanation: The n-th root of a number can be expressed as the number raised to the power 1/n.

    9. a1=a

      • Explanation: Any number raised to the power 1 is the number itself.

More Class 11 Maths chapters