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2. Attempt any EIGHT parts: (i) Simplify by justifying each step (ii) Factorize: a²+4b² (iii) Prove that ~(p→q)→p is a tautology. (iv) Give logical proof of (A∪B)'=A'∩B' (v) Solve the matrix equation for X 3X-2A=B if A=[

Concept: Roots of quadratic equation

Question

2. Attempt any EIGHT parts:
(i) Simplify by justifying each step

(ii) Factorize: a²+4b²
(iii) Prove that ~(p→q)→p is a tautology.
(iv) Give logical proof of (A∪B)'=A'∩B'
(v) Solve the matrix equation for X 3X-2A=B if A=[[2,3,-2],[-1,1,5]] and B=[[2,-3,1],[5,4,-1]]
(vi) Without expansion verify that |[1,2,3x],[2,3,6x],[3,5,9x]|=0
(vii) Find X if [[5,2],[-2,1]]X=[[2,1],[5,10]]
(viii) Define row matrix.
(ix) Find roots of 5x²-13x+6=0 by quadratic formula.
(x) Write statement of Remainder Theorem.
(xi) Show that x³-y³=(x-y)(x-ωy)(x-ω²y)
(xii) If α, β are the roots of 3x²-2x+4=0, find the value of α²-β²
3. Attempt any EIGHT parts:
(i) Write only partial fractions (4x)/((x+1)²(x-1))
(ii) Write (2x³+x²-x-3)/(x(2x+3)(x-1)) into proper form.
(iii) Write (2x+1)/((x-1)(x+2)(x+3)) into partial fractions.
(iv) If 1/a, 1/b, and 1/c are in G.P, show that the common ratio is ±√(a/c)
(v) Find the sum of 20 terms of the series whose rth term is 3r+1.
(vi) If 1/a, 1/b, and 1/c are in A.P, show that the common difference is (a-c)/(2ac)
(vii) Define series.
(viii) Define permutation.
(ix) If "C₈="C₁₂ find n
(x) If sample space = {1,2,3,...,9} event A = {2,4,6,8} and event B = {1,3,5}, find P(A∪B)
(xi) Write binomial theorem.
(xii) Find 6th term in the expansion of (x²-(3/(2x)))¹⁰
4. Attempt any NINE parts:
(i) Find θ, when l=3.2m, r=2m
(ii) Find the value of secθ if tanθ=8/5 and terminal arm of the angle is in III quadrant.
(iii) Show that cot⁴θ+cot²θ=cosec⁴θ-cosec²θ where θ is not an integral multiple of π/2
(iv) Express sin5x+sin7x as a product.
(v) Find the value of sin2α when sinα=12/13
α2−β2=(α−β)(α+β)\alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta)

Try it yourself first

Hint: Start by finding the sum and product of the roots using the given quadratic equation.

Step-by-step solution

  1. 1

    Identify coefficients

    ax²+bx+c=0 • a=3, b=-2, c=4
  2. 2

    Sum and Product

    α+β = -b/a • αβ = c/a
    α+β=−(−2)/3=2/3\alpha+\beta = -(-2)/3 = 2/3
  3. 3

    Calculate α-β

    (α-β)² = (α+β)² - 4αβ • α-β = ±√(4/9 - 16/3) • α-β = ±√(-44/9)
    (α−β)2=(2/3)2−4(4/3)(\alpha-\beta)^2 = (2/3)^2 - 4(4/3)
  4. 4

    Final Calculation

    α²-β² = (α-β)(α+β) • α²-β² = (±√(-44)/3)(2/3) • α²-β² = ±(2i√11)/3 * (2/3)
    α2−β2=±4i119\alpha^2 - \beta^2 = \pm \frac{4i\sqrt{11}}{9}

Final answer

10/9

Remember: Vieta's formulas and algebraic identities are key for root problems.

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If the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 are α\alpha and β\beta, what is the value of α+β\alpha + \beta?

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