Answer :

(i)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = 9 – 4 × 2 × k


9 – 8k ≥ 0


k ≤ 9/8


(ii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = k2 – 4 × 2 × 3


D ≥ 0


k2 – 24 ≥ 0


(k + 2√6)(k – 2√6) ≥ 0


Thus, k ≤ - 2√6 or k ≥ 2√6


(iii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = 25 – 8k


D ≥ 0


25 – 8k ≥ 0


k ≤ 25/8


(iv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = 36 – 4k


36 – 4k ≥ 0


k ≤ 9


(v)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = k2 – 36


k2 – 36 ≥ 0


(k – 6)(k + 6) ≥ 0


k ≥ 6 or k ≤ -6


(vi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = k2 – 4 × 4


k2 – 16 ≥ 0


(k + 4)(k – 4) ≥ 0


k ≥ 4 or k ≤ -4


(vii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = 4 – 12k


4 – 12k ≥ 0


k ≤ 1/3


(viii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = 9k2 – 16


9k2 – 16 ≥ 0


(3k – 4)(3k + 4) ≥ 0


k ≤ -4/3 or k ≥ (4/3)


(ix)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D ≥ 0, roots are real



D = k2 + 4 × 2 × 4 = k2 + 32


Thus, D is always greater than 0 for all values of k.


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