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Question

# For ax2+bx+c=0, if D>0 and D is square of rational number and a, b, c and d which of the following statements are true.i) Roots are distinct. ii) Roots are equal. iii) Roots are rational iv) Roots are irrational.

A
Only statement (i) is true.
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B
Statements (i) and(iii) are true
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C
Only statement (iv) is true
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D
Statements (ii) and (iii) are true
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Solution

## The correct option is B Statements (i) and(iii) are trueLet us recall the general solution, α = (-b-√b2-4ac)/2a and β = (-b+√b2-4ac)/2a Case I: D^2 = b2 – 4ac > 0 When a, b, and c are real numbers, a ≠ 0 and discriminant is positive, then the roots α and β of the quadratic equation ax2 +bx+ c = 0 are real and unequal. Case II: b2– 4ac = 0 When a, b, and c are real numbers, a ≠ 0 and discriminant is zero, then the roots α and β of the quadratic equation ax2+ bx + c = 0 are real and equal. Case III: b2– 4ac < 0 When a, b, and c are real numbers, a ≠ 0 and discriminant is negative, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary. Case IV: b2 – 4ac > 0 and perfect square When a, b, and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational and unequal. Case V: b2– 4ac > 0 and not perfect square When a, b, and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal. Here the roots α and β form a pair of irrational conjugates. Case VI: b2– 4ac >0 is perfect square and a or b is irrational When a, b, and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax2 + bx + c = 0 are irrational.Given question is case no (iv) so option B is true.

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