InterviewSolution
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InterviewSolution
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Which one of the following is not true? The quadratic equation `x^(2) - 2x - a = 0, a ne 0`,A. cannot have a real root if `a lt -1`B. may not have a rational root even if a is a perfect squareC. cannot have an integral root if `n^(2)-1 lt a lt n^(2) + 2n`, where n = 0, 1, 2,......D. none of these |
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Answer» Correct Answer - D Let D be the discriminant of `x^(2) - 2x - a = 0`. Then, D = 4 + 4a If `a lt -1, "then" D = 4(1+a) lt 0`. So, the equation cannot have real roots. Thus, option (a) is true. If a is a perfect square square, say `a = lambda^(2), lambda in Z`, then `D = 4 + 4a = 4(1+lambda^(2))`, which is not a perfect square. Thus, roots cannot be rational. So, option (b) is not true. Let `alpha` be an integral root of the given equation. Then, `alpha^(2) - 2 alpha - a = 0 rArr a = alpha^(2) - 2 alpha rArr a = (alpha-1)^(2) -1` `therefore" "n^(2) - 1 lt a lt n^(2) + 2n` `rArr" "n^(2) - 1 lt (alpha-1)^(2) -1 lt n^(2) + 2n` `rArr" "n^(2) lt (alpha - 1)^(2) lt (n + 1)^(2)`, which is not possible. So, the equation cannot have integral roots. Thus, option (c) is true. |
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