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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
- a)8
- b)7
- c)9
- d)13
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Suppose k is any integer such that the equation 2x2+ kx + 5 = 0 has no...
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)
⇒ 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)
⇒ (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.
-6, -5, -4, -3, -2, -1, 0, 1, 2 i.e., 9 values.
Hence, option (c).
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Community Answer
Suppose k is any integer such that the equation 2x2+ kx + 5 = 0 has no...
Understanding the Problem
To solve for the integer values of k, we need to analyze two quadratic equations:
1. Equation 1: 2x² + kx + 5 = 0
2. Equation 2: x² + (k - 5)x + 1 = 0
Condition for No Real Roots
For the first equation to have no real roots, the discriminant must be less than zero.
- Discriminant (D1) = k² - 4(2)(5) = k² - 40
- Condition: k² - 40 < 0="" />
- This leads to: -√40 < k="" />< />
- Thus, k must fall within the interval: -6.32 < k="" />< />
Since k is an integer, possible values are: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. (Total 13 values)
Condition for Distinct Real Roots
For the second equation to have two distinct real roots, the discriminant must be greater than zero.
- Discriminant (D2) = (k - 5)² - 4(1)(1) = (k - 5)² - 4
- Condition: (k - 5)² - 4 > 0
- This leads to: (k - 5) > 2 or (k - 5) < />
- Thus, k > 7 or k < />
Combining Conditions
Now, we combine both conditions:
1. From the first condition: -6 < k="" />< />
2. From the second condition: k < 3="" or="" k="" /> 7
The overlapping range for k is: -6 < k="" />< />
Valid Integer Solutions
The integer values satisfying both conditions are: -5, -4, -3, -2, -1, 0, 1, 2. (Total 8 values)
Since the first condition allows 13 values, but the second limits them to 8, the final number of possible values for k is 8.
Thus, the correct answer is option 'C' which states that there are 9 possible values for k, but through our analysis, we find that 8 values are valid.
To solve for the integer values of k, we need to analyze two quadratic equations:
1. Equation 1: 2x² + kx + 5 = 0
2. Equation 2: x² + (k - 5)x + 1 = 0
Condition for No Real Roots
For the first equation to have no real roots, the discriminant must be less than zero.
- Discriminant (D1) = k² - 4(2)(5) = k² - 40
- Condition: k² - 40 < 0="" />
- This leads to: -√40 < k="" />< />
- Thus, k must fall within the interval: -6.32 < k="" />< />
Since k is an integer, possible values are: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. (Total 13 values)
Condition for Distinct Real Roots
For the second equation to have two distinct real roots, the discriminant must be greater than zero.
- Discriminant (D2) = (k - 5)² - 4(1)(1) = (k - 5)² - 4
- Condition: (k - 5)² - 4 > 0
- This leads to: (k - 5) > 2 or (k - 5) < />
- Thus, k > 7 or k < />
Combining Conditions
Now, we combine both conditions:
1. From the first condition: -6 < k="" />< />
2. From the second condition: k < 3="" or="" k="" /> 7
The overlapping range for k is: -6 < k="" />< />
Valid Integer Solutions
The integer values satisfying both conditions are: -5, -4, -3, -2, -1, 0, 1, 2. (Total 8 values)
Since the first condition allows 13 values, but the second limits them to 8, the final number of possible values for k is 8.
Thus, the correct answer is option 'C' which states that there are 9 possible values for k, but through our analysis, we find that 8 values are valid.
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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 isa)8b)7c)9d)13Correct answer is option 'C'. Can you explain this answer?
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