If the zeroes of p(x) = x² − k are irrational real, which condition on k is correct?
Answer and explanation
Correct answer: k is positive but not a perfect square
To determine the zeroes, set x² − k = 0. This gives x² = k and hence x = ±√k. For the zeroes to be real, k must be non-negative; for them to be irrational, k must be positive and not a perfect square. If k is a positive perfect square, √k is an integer and therefore rational. If k = 0, both zeroes are 0, which are rational and equal. If k is negative, the square root is not real. Consequently, the necessary and sufficient condition is that k be positive but not a perfect square. Thus option B is correct; the other options each violate either reality or irrationality.
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What is the correct answer to this question?
k is positive but not a perfect square
Why is this the correct answer?
To determine the zeroes, set x² − k = 0. This gives x² = k and hence x = ±√k. For the zeroes to be real, k must be non-negative; for them to be irrational, k must be positive and not a perfect square. If k is a positive perfect square, √k is an integer and therefore rational. If k = 0, both zeroes are 0, which are rational and equal. If k is negative, the square root is not real. Consequently, the necessary and sufficient condition is that k be positive but not a perfect square. Thus option B is correct; the other options each violate either reality or irrationality.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Irrational numbers.
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