Class 11 Formula Vault — Additional Reference

THE SIMPOMATHS COLLECTION

This file contains Class 11 reference material, despite its original class12 filename. All existing formulas are preserved.

Class 11 Mathematics — Formula Vault

1. Sets

A ∪ B = {x : x ∈ A or x ∈ B}
A ∩ B = {x : x ∈ A and x ∈ B}
A − B = {x : x ∈ A and x ∉ B}
A′ = U − A

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

n(A′) = n(U) − n(A)
n(A − B) = n(A) − n(A ∩ B)

De Morgan's Laws:
(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′

Number of subsets:
If n(A) = n, then number of subsets = 2ⁿ
Number of proper subsets = 2ⁿ − 1

2. Relations & Functions

Cartesian Product:
A × B = {(a,b) : a ∈ A, b ∈ B}

If n(A) = m and n(B) = n:
n(A × B) = mn

Number of relations:
Number of relations from A to B = 2^mn

Number of functions:
Number of functions from A to B = n^m

Composition:
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))

Identity function:
f(x) = x

3. Principle of Mathematical Induction

For proving P(n):

1. Base Case: Prove P(1) is true.
2. Induction Hypothesis: Assume P(k) is true.
3. Induction Step: Prove P(k + 1) is true.

Therefore, P(n) is true for all n ∈ N.

4. Complex Numbers & Quadratic Equations

i = √−1
i² = −1
i³ = −i
i⁴ = 1

i⁴ⁿ = 1
i⁴ⁿ⁺¹ = i
i⁴ⁿ⁺² = −1
i⁴ⁿ⁺³ = −i

Complex number:
z = a + ib

Real part = Re(z) = a
Imaginary part = Im(z) = b

Conjugate:
z̄ = a − ib

Modulus:
|z| = √(a² + b²)

z × z̄ = |z|²

Quadratic equation:
ax² + bx + c = 0, a ≠ 0

Roots:
x = (−b ± √(b² − 4ac)) / 2a

Discriminant:
D = b² − 4ac

D > 0 → two distinct real roots
D = 0 → equal real roots
D < 0 → non-real roots

Sum of roots:
α + β = −b/a

Product of roots:
αβ = c/a

Quadratic equation with roots α, β:
x² − (α + β)x + αβ = 0

5. Linear Inequalities

For ax + b > 0:
If a > 0 → x > −b/a
If a < 0 → x < −b/a

Important: Multiplying or dividing an inequality by a negative number reverses the inequality sign.

a > b → −a < −b
a < b → −a > −b

6. Permutations & Combinations

Factorial:
n! = n(n − 1)(n − 2) ... 3 × 2 × 1
0! = 1

Permutation:
ⁿP_r = n!/(n − r)!

Combination:
ⁿC_r = n!/[r!(n − r)!]

ⁿP_r = r! × ⁿC_r

ⁿC_r = ⁿC_n−r

ⁿC₀ = ⁿC_n = 1

ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r

Circular permutation:
Number of arrangements of n distinct objects in a circle = (n − 1)!

7. Binomial Theorem

(a + b)ⁿ = Σ ⁿC_r a^n−rb^r

General term:
T_r+1 = ⁿC_r a^n−rb^r

First term:
T₁ = aⁿ

Last term:
T_n+1 = bⁿ

Middle term:
If n is even:
Middle term = T_{n/2 + 1}

If n is odd:
Two middle terms are
T_(n+1)/2 and T_(n+3)/2

Important:
ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ

ⁿC₀ − ⁿC₁ + ⁿC₂ − ... = 0

8. Sequences & Series

Arithmetic Progression (AP):
a, a + d, a + 2d, a + 3d, ...

n^th term:
a_n = a + (n − 1)d

Sum of first n terms:
S_n = n/2 [2a + (n − 1)d]

Also,
S_n = n/2(a + l)

where l = last term.

Arithmetic Mean:
AM = (a + b)/2

Geometric Progression (GP):
a, ar, ar², ar³, ...

n^th term:
a_n = arⁿ⁻¹

Sum of first n terms:
S_n = a(rⁿ − 1)/(r − 1), r ≠ 1

or
S_n = a(1 − rⁿ)/(1 − r)

Sum of infinite GP:
S_∞ = a/(1 − r), |r| < 1

Geometric Mean:
GM = √ab

For positive a and b:
AM ≥ GM
(a + b)/2 ≥ √ab

Useful sums:
1 + 2 + 3 + ... + n = n(n + 1)/2

1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6

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

Conditions: factorial and binomial formulas use non-negative integer n and 0 ≤ r ≤ n; the alternating binomial sum is zero for n ≥ 1. The real geometric mean √(ab) here is for non-negative a and b. Circular arrangements identify rotations, not reflections.