For which values of \(\theta\) does the quadratic equation \(x^2-2\theta x+3\theta=0\) have equal roots?
Answer and explanation
Correct answer: \(\theta=0\) or \(\theta=3\)
For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=1\), \(b=-2\theta\), and \(c=3\theta\), so \(D=(-2\theta)^2-4(1)(3\theta)=4\theta(\theta-3)\). Thus, \(4\theta(\theta-3)=0\) gives \(\theta=0\) or \(\theta=3\). Option B is incomplete because it omits \(\theta=0\). Exam tip: For equal roots, set the discriminant directly to zero and check every resulting parameter value.
Frequently asked questions
What is the correct answer to this question?
\(\theta=0\) or \(\theta=3\)
Why is this the correct answer?
For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=1\), \(b=-2\theta\), and \(c=3\theta\), so \(D=(-2\theta)^2-4(1)(3\theta)=4\theta(\theta-3)\). Thus, \(4\theta(\theta-3)=0\) gives \(\theta=0\) or \(\theta=3\). Option B is incomplete because it omits \(\theta=0\). Exam tip: For equal roots, set the discriminant directly to zero and check every resulting parameter value.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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