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Exercise 3.13

1. Determine the nature of the roots for the following quadratic equations

(i) 15*x* 2 + 11*x* + 2 = 0

(ii) *x* 2 − *x* −1 = 0

(iii) √2*t*2 − 3*t* + 3√2 = 0

(iv) 9*y*2 − 6√2*y* + 2 = 0

(v) 9*a*2*b*2*x* 2 − 24*abcdx* + 16*c*2*d*2 = 0 , *a* ≠ 0 , *b* ≠ 0

2. Find the value(s) of ‘*k*’ for which the roots of the following equations are real and equal.

(i) (5*k* − 6)*x*2 + 2*kx* + 1 = 0

(ii) *kx* 2 + (6*k* + 2)*x* + 16 = 0

3. If the roots of (*a* − *b* )*x*2 + (*b* − *c* )*x* + (*c* − *a* ) = 0 are real and equal, then prove that *b,* *a, c *are in arithmetic progression.

4. If *a, b* are real then show that the roots of the equation (*a* −* b* )*x*2 − 6(*a* + *b* )*x* − 9(*a* −*b*) = 0 are real and unequal.

5. If the roots of the equation (*c*2 − *ab*)*x* 2 − 2(*a*2 − *bc*)*x* +*b*2 − *ac* = 0 are real and equal prove that either *a*=0 (or) *a* 3 + *b* 3 +*c*3 = 3*abc*

Answers:

1.(i) Real and unequal (ii) Real and unequal (iii) Not real (iv) Real and equal (v) Real and equal

2.(i) 2, 3 (ii) 1, 1/9

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10th Mathematics : UNIT 3 : Algebra : Exercise 3.13: Nature of Roots of a Quadratic Equation | Problem Questions with Answer, Solution | Mathematics

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