Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has no real roots and the equation x2 + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is
Explanation:
2x2 + kx + 5 = 0 has no real roots ⇒ D < 0 ⇒ k2 – 4 × 2 × 5 < 0 ⇒ k2 < 40 ⇒ -√40 < k < √40 ∴ Possible integral values of k are -6, -5, -4, …, 0, …4, 5, 6 …(1)
Also, x2 + (k - 5)x + 1 = 0 has two distinct roots ⇒ D > 0 ⇒ (k - 5)2 – 4 × 1 × 1 > 0 ⇒ k2 + 25 – 10k – 4 > 0 ⇒ k2 – 10k + 21 > 0 ⇒ (k - 7)(k - 3) > 0 ⇒ k ∈ (-∞, 3) ∪ (7, ∞) …(2)
The integral value of k satisfying both (1) and (2) are -6, -5, -4, -3, -2, -1, 0, 1, 2 i.e., 9 values.
Hence, option (c).
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