(i) lf $\alpha,\beta$ are the roots of $\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}=0$. lf $\mathrm{K}$ is a real number $[\mathrm{K}\neq 0,1 ]$ then the equation whose roots are $ K\alpha,\ K\beta$ is $\mathrm{a}\mathrm{x}^{2}+$ Kbx $+\mathrm{K}^{2}\mathrm{c}=0$. (ii) 8,2 are roots of $\mathrm{x}^{2}+\mathrm{a}\mathrm{x}+\beta=0$ and 3, 3 are the roots of $\mathrm{x}^{2}+\alpha \mathrm{x}+\mathrm{b}=0$ then the roots of $\mathrm{x}^{2}+\mathrm{a}\mathrm{x}+\mathrm{b}=0$ are 9, 1. Which of the above statement is correct

For statement one, substituting Kx into the equation yields a(Kx)^2 + b(Kx) + c = K^2ax^2 + Kbx + c = 0, which simplifies to ax^2 + Kbx + K^2c = 0. For statement two, if 8 and 2 are roots of x^2 + ax + beta = 0, then the sum of roots gives a = -10 and the product gives beta = 16; if 3 and 3 are roots of x^2 + alpha*x + b = 0, then alpha = -6 and b = 9. Substituting a = -10 and b = 9 into x^2 + ax + b = 0 gives x^2 - 10x + 9 = 0, which factors to (x - 9)(x - 1) = 0, confirming the roots are 9 and 1. Both statements are verified as correct.

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