For which value of \(k\) will the zeroes of \(x^2-2x+k\) be real and irrational?
Answer and explanation
Correct answer: k = -1
Compute the discriminant: \(D=b^2-4ac=4-4k=4(1-k)\). For real roots we need \(D>0\) (so \(k<1\)). For the roots to be irrational (given rational coefficients), \(D\) must be positive but not a perfect square. For \(k=-1\), \(D=8\), which is positive and not a perfect square, so the roots are real and irrational. Why other choices fail: \(k=0\) gives \(D=4\) (a perfect square) so roots are rational; \(k=1\) gives \(D=0\) (equal rational roots); \(k=2\) gives \(D=-4\) (complex roots). Exam tip: first check sign of \(D\), then check whether \(D\) is a perfect square to decide rational vs irrational roots.
Frequently asked questions
What is the correct answer to this question?
k = -1
Why is this the correct answer?
Compute the discriminant: \(D=b^2-4ac=4-4k=4(1-k)\). For real roots we need \(D>0\) (so \(k<1\)). For the roots to be irrational (given rational coefficients), \(D\) must be positive but not a perfect square. For \(k=-1\), \(D=8\), which is positive and not a perfect square, so the roots are real and irrational. Why other choices fail: \(k=0\) gives \(D=4\) (a perfect square) so roots are rational; \(k=1\) gives \(D=0\) (equal rational roots); \(k=2\) gives \(D=-4\) (complex roots). Exam tip: first check sign of \(D\), then check whether \(D\) is a perfect square to decide rational vs irrational roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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