Class 12 Maths – Nishad Formula

Relations and Functions (सम्बन्ध एवं फलन) -

1. Cartesian Product (कार्तीय गुणन)

If A = {a, b} and B = {1, 2}

Then A × B = {(a,1), (a,2), (b,1), (b,2)}

2. Relation (सम्बन्ध)

A relation R from A to B is a subset of A × B

3. Types of Relations (सम्बन्ध के प्रकार)

Reflexive Relation (प्रतिवर्ती):
(a,a) ∈ R for all a ∈ A

Symmetric Relation (सममित):
If (a,b) ∈ R ⇒ (b,a) ∈ R

Transitive Relation (सांक्रामक):
If (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R

Equivalence Relation:
Reflexive + Symmetric + Transitive

4. Function (फलन)

f : A → B

y = f(x)

5. Types of Functions (फलनों के प्रकार)

One-One Function (Injective):
f(x₁) = f(x₂) ⇒ x₁ = x₂

Onto Function (Surjective):
Range = Co-domain

Bijective Function:
One-One + Onto

6. Composite Function (संयुक्त फलन)

(f ∘ g)(x) = f(g(x))

7. Identity Function (परिचय फलन)

I(x) = x

8. Inverse Function (प्रतिलोम फलन)

f⁻¹(x)

f(f⁻¹(x)) = x

f⁻¹(f(x)) = x

Inverse Trigonometric Functions Class 12

Inverse Trigonometric Functions (प्रतिलोम त्रिकोणमितीय फलन)

1. Basic Inverse Functions (मूल प्रतिलोम फलन)

Function (फलन)	Meaning (अर्थ)
sin^-1x	Inverse Sine
cos^-1x	Inverse Cosine
tan^-1x	Inverse Tangent
cot^-1x	Inverse Cotangent
sec^-1x	Inverse Secant
cosec^-1x	Inverse Cosecant

2. Domain and Range (परिभाषा क्षेत्र और मान क्षेत्र)

Function	Domain	Range
sin^-1x	-1 ≤ x ≤ 1	-π/2 ≤ y ≤ π/2
cos^-1x	-1 ≤ x ≤ 1	0 ≤ y ≤ π
tan^-1x	All Real Numbers	-π/2 < y < π/2
cot^-1x	All Real Numbers	0 < y < π
sec^-1x	\|x\| ≥ 1	0 ≤ y ≤ π, y ≠ π/2
cosec^-1x	\|x\| ≥ 1	-π/2 ≤ y ≤ π/2, y ≠ 0

3. Important Identities (महत्वपूर्ण सूत्र)

sin^-1x + cos^-1x = π/2
tan^-1x + cot^-1x = π/2
sec^-1x + cosec^-1x = π/2

4. Negative Property

sin^-1(-x) = -sin^-1x
tan^-1(-x) = -tan^-1x
cos^-1(-x) = π − cos^-1x
cot^-1(-x) = π − cot^-1x

5. Standard Results

sin^-1(1) = π/2
sin^-1(0) = 0
cos^-1(1) = 0
cos^-1(0) = π/2
tan^-1(1) = π/4
tan^-1(0) = 0

6. Important Formula

tan^-1x + tan^-1y = tan^-1[(x+y)/(1−xy)]
tan^-1x − tan^-1y = tan^-1[(x−y)/(1+xy)]

20 Matrix Formulas Class 12

Matrices (आव्यूह) – 20 Important Formulas

1. Order of Matrix = m × n
2. A = [a_ij]
3. A + B = [a_ij + b_ij]
4. A − B = [a_ij − b_ij]
5. kA = [ka_ij]
6. AB ≠ BA (Matrix multiplication not commutative)
7. A(BC) = (AB)C
8. A(B + C) = AB + AC
9. (A + B)C = AC + BC
10. A + B = B + A
11. A + (B + C) = (A + B) + C
12. A + O = A
13. AI = IA = A
14. (A^T)^T = A
15. (A + B)^T = A^T + B^T
16. (AB)^T = B^TA^T
17. (kA)^T = kA^T
18. Symmetric Matrix: A^T = A
19. Skew Symmetric Matrix: A^T = −A
20. If A is symmetric and B is skew symmetric then A + B is square matrix

Determinants Class 12

Determinants (सारणिक) – 20 Important Formulas

1. Determinant of Matrix A = |A|
2. |A| = ∑a_ijC_ij
3. Minor M_ij = determinant after deleting ith row & jth column
4. Cofactor A_ij = (−1)^i+jM_ij
5. |A^T| = |A|
6. |AB| = |A||B|
7. |kA| = kⁿ|A|
8. |I| = 1
9. |O| = 0
10. If any row is zero then |A| = 0
11. If any two rows are equal then |A| = 0
12. If any two rows are proportional then |A| = 0
13. Interchanging two rows changes sign
14. If two rows are identical then determinant is zero
15. Adding multiple of row to another row does not change value
16. If one row is multiplied by k then determinant is multiplied by k
17. |adj A| = |A|^n-1
18. A(adj A) = |A|I
19. If |A| ≠ 0 then A^-1 exists
20. A^-1 = adj A / |A|

Continuity and Differentiability Class 12

Continuity and Differentiability (सततता एवं अवकलनीयता) – Important Formulas

1. lim x→a f(x) = f(a) → Continuous Function
2. d/dx (c) = 0
3. d/dx (xⁿ) = nx^n-1
4. d/dx (sinx) = cosx
5. d/dx (cosx) = -sinx
6. d/dx (tanx) = sec²x
7. d/dx (cotx) = -cosec²x
8. d/dx (secx) = secx tanx
9. d/dx (cosecx) = -cosecx cotx
10. d/dx (e^x) = e^x
11. d/dx (a^x) = a^x ln a
12. d/dx (lnx) = 1/x
13. d/dx (logx) = 1/(x ln10)
14. (uv)' = u'v + uv'
15. (u/v)' = (vu' - uv') / v²
16. Chain Rule: dy/dx = (dy/du)(du/dx)
17. d/dx (sin^-1x) = 1/√(1-x²)
18. d/dx (cos^-1x) = -1/√(1-x²)
19. d/dx (tan^-1x) = 1/(1+x²)
20. d/dx (cot^-1x) = -1/(1+x²)

Applications of Derivatives Class 12

Applications of Derivatives (अवकलज के अनुप्रयोग) – Important Formulas

1. dy/dx = Rate of change of y w.r.t x
2. Increasing Function: dy/dx > 0
3. Decreasing Function: dy/dx < 0
4. Second Derivative: d²y/dx²
5. Maxima Condition: dy/dx = 0 and d²y/dx² < 0
6. Minima Condition: dy/dx = 0 and d²y/dx² > 0
7. Slope of Tangent = dy/dx
8. Slope of Normal = -1/(dy/dx)
9. Equation of Tangent: y - y₁ = m(x - x₁)
10. Equation of Normal: y - y₁ = -1/m(x - x₁)
11. Velocity = ds/dt
12. Acceleration = d²s/dt²
13. dy/dx = 0 → Stationary Point
14. d²y/dx² = 0 → Point of Inflection
15. Maximum Value → f''(x) < 0
16. Minimum Value → f''(x) > 0
17. Chain Rule: dy/dx = (dy/du)(du/dx)
18. Product Rule: (uv)' = u'v + uv'
19. Quotient Rule: (u/v)' = (vu' - uv') / v²
20. Horizontal Tangent: dy/dx = 0

Integrals Class 12

Integrals (समाकलन) – Important Formulas

1. ∫dx = x + C
2. ∫xⁿdx = xⁿ⁺¹/(n+1) + C
3. ∫1/x dx = ln|x| + C
4. ∫e^x dx = e^x + C
5. ∫a^x dx = a^x/ln(a) + C
6. ∫sinx dx = -cosx + C
7. ∫cosx dx = sinx + C
8. ∫sec²x dx = tanx + C
9. ∫cosec²x dx = -cotx + C
10. ∫secx tanx dx = secx + C
11. ∫cosecx cotx dx = -cosecx + C
12. ∫1/√(1-x²) dx = sin^-1x + C
13. ∫1/(1+x²) dx = tan^-1x + C
14. ∫1/|x|√(x²-1) dx = sec^-1x + C
15. ∫(f'(x)/f(x)) dx = ln|f(x)| + C
16. ∫u dv = uv - ∫v du
17. ∫(dx/(a²+x²)) = (1/a)tan^-1(x/a) + C
18. ∫(dx/√(a²-x²)) = sin^-1(x/a) + C
19. ∫(dx/(x√(x²-a²))) = sec^-1(x/a) + C
20. ∫(dx/√(x²+a²)) = ln|x+√(x²+a²)| + C

Differential Equations Class 12

Differential Equations (अवकल समीकरण) –Important Formulas

1. dy/dx = f(x)
2. dy = f(x) dx
3. ∫dy = ∫f(x)dx
4. General Solution: y = ∫f(x)dx + C
5. Variable Separable Form: dy/dx = g(x)h(y)
6. dy/h(y) = g(x)dx
7. ∫dy/h(y) = ∫g(x)dx
8. Homogeneous Equation: dy/dx = F(y/x)
9. Put y = vx
10. dy/dx = v + x dv/dx
11. Linear Equation: dy/dx + Py = Q
12. Integrating Factor (I.F) = e^(∫P dx)
13. Solution: y(IF) = ∫Q(IF)dx + C
14. Exact Equation: Mdx + Ndy = 0
15. ∂M/∂y = ∂N/∂x
16. ∫M dx + ∫N dy = C
17. Order of D.E = highest derivative
18. Degree of D.E = power of highest derivative
19. Particular Solution → put given condition
20. Complementary Function + Particular Integral = General Solution

Vector Algebra Class 12

Vector Algebra (सदिश बीजगणित) – Important Formulas

1. Vector: a = ai + bj + ck
2. |a| = √(a² + b² + c²)
3. Unit Vector: â = a/|a|
4. a + b = (a₁+a₂)i + (b₁+b₂)j + (c₁+c₂)k
5. a − b = (a₁−a₂)i + (b₁−b₂)j + (c₁−c₂)k
6. Dot Product: a·b = |a||b|cosθ
7. a·b = a₁a₂ + b₁b₂ + c₁c₂
8. cosθ = (a·b)/(|a||b|)
9. If a·b = 0 → vectors are perpendicular
10. Cross Product: a×b = |a||b|sinθ n̂
11. |a×b| = |a||b|sinθ
12. Area of parallelogram = |a×b|
13. Area of triangle = ½|a×b|
14. a×b = determinant
15. a×b = -b×a
16. a·(b×c) = scalar triple product
17. Volume = |a·(b×c)|
18. Coplanar vectors → a·(b×c) = 0
19. Projection of a on b = (a·b)/|b|
20. Vector equation of line = a + λb

3D Geometry Class 12

Three Dimensional Geometry (त्रिविमीय ज्यामिति) – Important Formulas

1. Distance between two points = √[(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²]
2. Section Formula (Internal) = [(mx₂+nx₁)/(m+n)]
3. Direction Cosines: l²+m²+n² = 1
4. Direction Ratios: a, b, c
5. Equation of Line: r = a + λb
6. Cartesian Form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
7. Angle between lines: cosθ = (a₁a₂+b₁b₂+c₁c₂)/(|b₁||b₂|)
8. Shortest distance between lines
9. Equation of Plane: ax+by+cz+d = 0
10. Normal Form: lx+my+nz = p
11. Distance of point from plane
12. Angle between planes
13. Angle between line & plane
14. Parallel planes: a₁/a₂=b₁/b₂=c₁/c₂
15. Perpendicular planes: a₁a₂+b₁b₂+c₁c₂=0
16. Coplanar lines condition
17. Image of point in plane
18. Distance between parallel planes
19. Intercept form: x/a + y/b + z/c = 1
20. Vector form of plane: r·n = d

Linear Programming Class 12

Linear Programming (रैखिक प्रोग्रामन)– Important Formulas

1. Objective Function: Z = ax + by
2. Constraints: ax + by ≤ c
3. Non-negative condition: x ≥ 0, y ≥ 0
4. Feasible Region
5. Corner Point Method
6. Optimal Solution
7. Maximization Problem
8. Minimization Problem
9. Feasible Solution
10. Infeasible Solution
11. Bounded Region
12. Unbounded Region
13. Extreme Points
14. Solution at corner points
15. Graphical Method
16. Z max or min at corner points
17. Parallel Lines Condition
18. Multiple Optimal Solutions
19. Unique Solution
20. No Solution

Probability Class 12

Probability (प्रायिकता) – Important Formulas

1. Probability: P(A) = n(A)/n(S)
2. 0 ≤ P(A) ≤ 1
3. P(S) = 1
4. P(Φ) = 0
5. Complementary Event: P(A') = 1 − P(A)
6. P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
7. If A and B are mutually exclusive: P(A ∩ B) = 0
8. Conditional Probability: P(A|B) = P(A ∩ B)/P(B)
9. P(A ∩ B) = P(B)P(A|B)
10. P(A ∩ B) = P(A)P(B|A)
11. Independent Events: P(A ∩ B) = P(A)P(B)
12. Total Probability Theorem
13. Bayes Theorem
14. Random Variable
15. Probability Distribution
16. Mean = ∑xP(x)
17. Variance = ∑x²P(x) − [∑xP(x)]²
18. Standard Deviation = √Variance
19. Binomial Distribution
20. Normal Distribution