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# Hemant

∫ ∮ ⋀ ⋁ ¬ ∠ ∡ ⊥ ± ≠ ° ¹ ² ³ ⁴ ª ⁿ ₁ ₂ ⇒ ∀ ∃ ∇ ∂ ∑ ∞ ≅ ≈ ≠ ½ ⅓ ⅔ ¼ ¾ ⅛ ⅜ ⅝ ⅞ ° ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁼ ⁽ ⁾ ⁿ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋ ₌ ₍ ₎ α β γ δ ε ζ η θ λ μ ξ ρ Σ σ φ ψ ω Π ∂ Δ Ө Φ 𝑥 𝑦 𝑎 𝑏 𝑐 ∵∴ ∫ֿ¹⁺∑ℯֿˣ≠⇒ θ ¹ʹ²ʹ³ʹ² ☚ₐ田 x̅ ∀∈ ℤ ℕ ℝ⁺ֿ ∵∴ ∫⇒ ∋ ⇒ ∀ ∃∈ ∉ ∋ ∛ ∜ ⇒ ∀ ∃ ∋ ∄ ∩ ∊ ∪ ⊂ ⊆ ⊃ ⊇ ∈ ∉ ≢ ≠ ⊢ ❶ ❷ ➌ ★ ✰ ☚ � √[ (x₂ - x₁)² + ( y₂ - y₁ )² ] y - y₁ = m ( x - x₁ ) ℯˣ ℯֿˣ ( Σ a )³ = Σ (a³) + 3[ Σ ab(a+b) ] + 6abc ∩ ∊ ∪ ⊂ ⊆ ⊃ ⊇ − ∉ ∈ ∋ ∋ → ⇒ ∀ ∃ ∄ ... ................................................................... I believe in " Cogito Ergo Sum ". Obsessed with making Maths simple and accessible to students who are putting in an honest effort. I specialize in Multiple Choice Questions in Mathematics for various Engineering Entrance Examinations in India.

• ### Integrate the following function w.r.t. x :?

f (x) = ( x^3 + a^3 + b^3 - 3abx ) / ( x + a + b ).

• ### Betweenness of Three Roots of Three Quadratic Equations :?

If X1 is a root of an equation

ax^2 + bx + c = 0

and X2 is a root of the equation

- ax^2 + bx + c = 0,

then, show that there is a root X3 of the equation

( a/2 )x^2 + bx + c = 0

such that X3 lies between X1 and X2.

• ### Prove: Amongst any 6 people, there are 3 people pairwise acquainted or 3 people pairwise unaquainted.?

This is a question on combinatorial reasoning.

I have a vague idea of its solution.

But I want a systematic unassailable reasoning

which can be explained to a beginner in

combinatorics.