Score-Boosting Class 12 Maths Notes – MP Board 2025

 

Class 12th Math's Important Notes – MP Board      (Updated 2025)

Chapter 1: Relations and Functions (Class 12 MP Board Notes)

Key Concepts

  • Relation = Subset of A×BA \times B

  • Function = Special type of relation

  • Types: One-one, Onto, Bijective

  • Composite Function: f(g(x))f(g(x))

  • Inverse of a function exists only for bijective functions.

Important Formula

  • Domain of composite function:
    Domain(f∘g)={x:g(x)∈Domain(f)}\text{Domain}(f \circ g) = \{ x : g(x) \in \text{Domain}(f) \}

Chapter 2: Inverse Trigonometric Functions – Important Notes

Principal Value Branches

  • sin⁡−1x:[−π2,π2]\sin^{-1}x: \left[-\frac{\pi}{2},\frac{\pi}{2}\right]

  • cos⁡−1x:[0,π]\cos^{-1}x: [0,\pi]

  • tan⁡−1x:(−π2,π2)\tan^{-1}x: \left(-\frac{\pi}{2},\frac{\pi}{2}\right)

Important Formulas

  • sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}

  • tan⁡−1x+cot⁡−1x=π2\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}

Chapter 3: Matrices – MP Board Class 12 Notes

Key Concepts

  • Types: Row, Column, Identity, Zero

  • 2×2 Determinant: ad−bcad - bc

  • Properties:

    • Interchange rows → sign changes

    • Equal rows → determinant = 0

    • ∣AB∣=∣A∣∣B∣|AB| = |A||B|

Inverse of Matrix

  • A−1=1∣A∣adj(A)A^{-1} = \frac{1}{|A|} \text{adj}(A)

Chapter 4: Determinants – Important Notes

Key Formula

  • Area of Triangle
    12∣x1y11x2y21x3y31∣\frac{1}{2} \left| \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right|

Chapter 5: Continuity and Differentiability

Continuity Condition

  • lim⁡x→af(x)=f(a)\lim_{x\to a} f(x) = f(a)

Differentiability

  • Differentiable ⇒ Continuous

Chain Rule

  • dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Chapter 6: Applications of Derivatives – Class 12 MP Board

Important Formulas

  • Increasing function: f′(x)>0f'(x) > 0

  • Decreasing function: f′(x)<0f'(x) < 0

  • Maxima/Minima:

    • f′(x)=0f'(x)=0

    • f′′(x)>0f''(x)>0 ⇒ Min

    • f′′(x)<0f''(x)<0 ⇒ Max

Chapter 7: Integrals – Class 12 Maths Notes

Basic Integrals

  • ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C

  • ∫exdx=ex+C\int e^x dx = e^x + C

  • ∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C

Integration by Parts

  • ∫uvdx=u∫vdx−∫(dudx∫vdx)dx\int uv dx = u\int vdx - \int \left(\frac{du}{dx} \int v dx\right) dx

Chapter 8: Application of Integrals – MP Board

Area Under a Curve

  • ∫ab(f(x)−g(x))dx\int_a^b (f(x) - g(x)) dx

Chapter 9: Differential Equations

Key Terms

  • Order = Highest derivative

  • Degree = Power of highest order derivative

Solution Methods

  • Variable separable

  • Linear equation:

    • dydx+Py=Q\frac{dy}{dx} + Py = Q

    • IF = e∫Pdxe^{\int Pdx}

Chapter 10: Vector Algebra

Formulas

  • Magnitude: ∣a⃗∣=a12+a22+a32|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}

  • Dot Product: a⋅b=abcos⁡θa \cdot b = ab\cos\theta

  • Cross Product:
    ∣ijka1a2a3b1b2b3∣\begin{vmatrix} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Chapter 11: 3D Geometry – Important Notes

Distance Formula

  • (x2−x1)2+(y2−y1)2+(z2−z1)2\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Direction Cosines

  • l=aa2+b2+c2l = \frac{a}{\sqrt{a^2+b^2+c^2}}

Chapter 12: Linear Programming

Key Concepts

  • Constraints

  • Feasible region

  • Corner point method

Chapter 13: Probability – 

Important Formulas

  • P(A)+P(Aˉ)=1P(A) + P(\bar{A}) = 1

  • Bayes Theorem:
    P(A∣B)=P(A)P(B∣A)P(A)P(B∣A)+P(Aˉ)P(B∣Aˉ)P(A|B) = \frac{P(A)P(B|A)}{P(A)P(B|A) + P(\bar{A})P(B|\bar{A})}