Find the values of k for which the given quadratic equation has real and distinct roots:

(i) Given: kx^{2} + 6x + 1 = 0

Comparing with standard quadratic equation ax^{2} + bx + c = 0

a = k b = 6 c = 1

For real and distinct roots: D > 0

Discriminant D = b^{2} – 4ac > 0

6^{2} – 4k > 0

36 – 4k > 0

4k < 36

k < 9

(ii) Given: x^{2} – kx + 9 = 0

Comparing with standard quadratic equation ax^{2} + bx + c = 0

a = 1 b = – k c = 9

For real and distinct roots: D > 0

Discriminant D = b^{2} – 4ac > 0

(– k)^{2} – 4.1.9 = k^{2} – 36 > 0

k^{2} > 36

k > 6or k < – 6 taking square root both sides

(iii) 9x^{2} + 3kx + 4 = 0

Comparing with standard quadratic equation ax^{2} + bx + c = 0

a = 9 b = 3k c = 4

For real and distinct roots: D > 0

Discriminant D = b^{2} – 4ac > 0

(3k)^{2} – 4.4.9 = 9k^{2} – 144 > 0

9k^{2} > 144

k^{2} > 16

k > 4ork < – 4 taking square root both sides

(iv) 5x^{2} – kx + 1 = 0

Comparing with standard quadratic equation ax^{2} + bx + c = 0

a = 5 b = – k c = 1

For real and distinct roots: D > 0

Discriminant D = b^{2} – 4ac > 0

(– k)^{2} – 4.5.1 = k^{2} – 20 > 0

k^{2} > 20

k > 2√5 or k < –2√5 taking square root both sides

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