Find the values of k for which the following equations have real and equal roots.
$(i) x^{2} - 2 (k + 1) x + k^{2} = 0 (i i) k^{2} x^{2} - 2 (2 k - 1) x + 4 = 0 (i i i) (k + 1) x^{2} - 2 (k - 1) x + 1 = 0 (i v) x^{2} + k (2 x + k - 1) + 2 = 0$

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Solution

$(i) k = \frac{- 1}{2} (i i) k = \frac{1}{4} (i i i) k = 0, 3 (v) k = 2$

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Find the values of k for which the roots are real and equal in each of the following equations.
$(i) k x^{2} + 4 x + 1 = 0 (i i) k x^{2} - 2 \sqrt{5} x + 4 = 0 (i i i) 3 x^{2} - 5 x + 2 k = 0 (i v) 4 x^{2} + k x + 9 = 0 (v) 2 k x^{2} - 40 x + 25 = 0 (v i) 9 x^{2} - 24 x + k = 0 (v i i) 4 x^{2} - 2 k x + 1 = 0 (v i i i) x^{2} - 2 (5 + 2 k) x + 3 (7 + 10 k) = 0 (i x) (3 k + 1) x^{2} + 2 (k + 1) x + k = 0 (x) k x^{2} + k x + 1 = - 4 x^{2} - x (x i) (k + 1) x^{2} + 2 (k + 3) x (k + 8) = 0 (x i i) x^{2} - 2 k x + 7 k - 12 = 0 (x i i i) (k + 1) x^{2} - 2 (3 k + 1) x + 8 k + 1 = 0 (x i v) 5 x^{2} - 4 x + 2 + k (4 x^{2} 1 - 2 x - 1) = 0 (x v) (4 - k) x^{2} + (2 k + 4) x + (8 k + 1) = 0 (x v i) (2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0 (x v i i) 4 x^{2} - 2 (k + 1) x + (k + 4) = 0 (x v i i i) 4 x^{2} - 2 (k + 1) x + (k + 1) = 0$

Q. (i) Find the values of k for which the quadratic equation

(3 k + 1) x^{2} + 2 (k + 1) x + 1 = 0

has real and equal roots. [CBSE 2014]
(ii) Find the value of k for which the equation

x^{2} + k (2 x + k - 1) + 2 = 0

has real and equal roots. [CBSE 2017]

(i) Find the values of k for which the quadratic equation $(3 k + 1) x^{2} + 2 (k + 1) x + 1 = 0$ has real and equal roots.
(ii) Find the value of k for which the equations $x^{2} + k (2 x + k - 1) + 2 = 0$ has real and equal roots.