(a) Simplify : \(\frac{x^{2} - 8x + 16}{x^{2} - 7x + 12}\).
(b) If \(\frac{1}{2}, \frac{1}{x}, \frac{1}{3}\) are successive terms of an arithmetic progression (A.P), show that \(\frac{2 - x}{x - 3} = \frac{2}{3}\).
(b) If \(\frac{1}{2}, \frac{1}{x}, \frac{1}{3}\) are successive terms of an arithmetic progression (A.P), show that \(\frac{2 - x}{x - 3} = \frac{2}{3}\).
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Correct Answer: Option n
Explanation:
(a) \(\frac{x^{2} - 8x + 16}{x^{2} - 7x + 12}\)
= \(\frac{(x - 4)(x - 4)}{(x - 4)(x - 3)}\)
= \(\frac{(x - 4)}{(x - 3)}\)
(b) Arithmetic Progression:
\(\frac{1}{2}, \frac{1}{x}, \frac{1}{3}\)
\(\frac{1}{x} - \frac{1}{2} = \frac{1}{3} - \frac{1}{x}\)
\(\frac{2 - x}{2x} = \frac{x - 3}{3x}\)
\(\frac{2 - x}{x - 3} = \frac{2x}{3x}\)
\(\implies \frac{2 - x}{x - 3} = \frac{2}{3}\)
(a) \(\frac{x^{2} - 8x + 16}{x^{2} - 7x + 12}\)
= \(\frac{(x - 4)(x - 4)}{(x - 4)(x - 3)}\)
= \(\frac{(x - 4)}{(x - 3)}\)
(b) Arithmetic Progression:
\(\frac{1}{2}, \frac{1}{x}, \frac{1}{3}\)
\(\frac{1}{x} - \frac{1}{2} = \frac{1}{3} - \frac{1}{x}\)
\(\frac{2 - x}{2x} = \frac{x - 3}{3x}\)
\(\frac{2 - x}{x - 3} = \frac{2x}{3x}\)
\(\implies \frac{2 - x}{x - 3} = \frac{2}{3}\)