Question

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank
spaces:
(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A
(iv) 4. . . A (v) 2. . .A (vi) 10. . .A

Answer 1

$\large \bf \clubs \: Given :-$

$α \: and \: β \: are \: the \: roots \: of \: the \: equation$

x² + 5x + 5 =0

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$\large \bf \clubs \: To \: Find :-$

• The Equation whose roots are (α + 1) and (β + 1).

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$\large \bf \clubs \: Main \: Concepts :-$

1》 For a qudratic Equation of the Form ax² + bx + c = 0

$Sum of Roots = \sf-\dfrac{b}{a} −$

$Product of Roots = \sf\dfrac{c}{a}$

2》 A Quadratic Equation whose sum and product of Roots are S and P respectively is given by x² - Sx + P = 0

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$\large \bf \clubs \: Solution$

As α and β are the roots of the equation

x² + 5x + 5 =0

Hence ,

$\begin{gathered} \pmb{ \alpha + \beta = - 5 } \: \: - - - - (1)\\ \bf and \\ \pmb{ \alpha \beta = 5} \: \: - - - - (2)\end{gathered}$

For The Equation whose Roots are

(α + 1) and (β + 1).

$\begin{gathered} \sf S um \: of \: Roots = S \\ \\ = \sf \alpha + 1 + \beta + 1 \\ \\ = \alpha + \beta + 2 \\ \\ \bf \: \: \: \{using \: \: (1) \} \\ \\ \sf S = - 5 + 2\\ \\ \large:\longmapsto \boxed{\pmb{\boxed{S = - 3}}}\end{gathered}$

$\begin{gathered} \sf P roduct \: of \: Roots = P \\ \\ = ( \alpha + 1)( \beta + 1) \\ \\ = \alpha \beta + \alpha + \beta + 1 \\ \\ \bf \: \: \: \{using \: (1) \: and \: (2) \} \\ \\ \sf P = 5 - 5 + 1 \\ \\ \large :\longmapsto\boxed{ \pmb{ \boxed{P = 1}}}\end{gathered}$

Hence The Equation whose Roots are

(α + 1) and (β + 1) will be x² - Sx + P = 0

Where S and P are Sum and Product of roots.

That is ,

The Required Equation is

x² - ( - 3) x + 1 =0

$\large \pink{ : \longmapsto \bf {x}^{2} + 3x + 1 = 0 }:$

$\begin{gathered} \LARGE\red{\mathfrak{ \text{W}hich \:\:is\:\: the\:\: required} }\\ \Huge \red{\mathfrak{ \text{ A}nswer.}}\end{gathered}$

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