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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)9
- b)7
- c)8
- d)13
Correct answer is option 'A'. Can you explain this answer?
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Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has n...
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Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has n...
Conditions for No Real Roots
To determine the values of k, we analyze the two given equations:
1. Equation 1: 2x² + kx + 5 = 0
This equation has no real roots if the discriminant (D) is less than zero.
The discriminant is given by D = b² - 4ac.
For our equation, a = 2, b = k, and c = 5.
Therefore, the condition is:
k² - 4(2)(5) < 0="" />
k² - 40 < 0="" />
k² < 40="" />
This implies that -√40 < k="" />< √40,="" which="" simplifies="" to="" approximately="" -6.32="" />< k="" />< 6.32.="" />
Hence, k can take integer values: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 (total of 13 values).
Conditions for Two Distinct Real Roots
2. Equation 2: x² + (k + 5)x + 1 = 0
This equation has two distinct real roots if its discriminant is greater than zero.
For our equation, a = 1, b = k + 5, and c = 1.
Therefore, the condition is:
(k + 5)² - 4(1)(1) > 0
(k + 5)² - 4 > 0
(k + 5)² > 4
This results in two cases:
k + 5 > 2 or k + 5 < -2,="" leading="" to="" k="" /> -3 or k < -7.="" />
Thus, the valid ranges for k become: k < -7="" or="" k="" /> -3.
Finding the Intersection
Now, we need to find the integer values of k that satisfy both conditions:
- From Condition 1: -6 < k="" />< />
- From Condition 2: k < -7="" or="" k="" /> -3
The overlapping valid integer values are: -6, -5, -4, -3, 0, 1, 2, 3, 4, 5, 6.
This gives us a total of 9 valid integers.
Conclusion
The number of possible values of k is 9. Thus, the correct answer is option 'A'.
To determine the values of k, we analyze the two given equations:
1. Equation 1: 2x² + kx + 5 = 0
This equation has no real roots if the discriminant (D) is less than zero.
The discriminant is given by D = b² - 4ac.
For our equation, a = 2, b = k, and c = 5.
Therefore, the condition is:
k² - 4(2)(5) < 0="" />
k² - 40 < 0="" />
k² < 40="" />
This implies that -√40 < k="" />< √40,="" which="" simplifies="" to="" approximately="" -6.32="" />< k="" />< 6.32.="" />
Hence, k can take integer values: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 (total of 13 values).
Conditions for Two Distinct Real Roots
2. Equation 2: x² + (k + 5)x + 1 = 0
This equation has two distinct real roots if its discriminant is greater than zero.
For our equation, a = 1, b = k + 5, and c = 1.
Therefore, the condition is:
(k + 5)² - 4(1)(1) > 0
(k + 5)² - 4 > 0
(k + 5)² > 4
This results in two cases:
k + 5 > 2 or k + 5 < -2,="" leading="" to="" k="" /> -3 or k < -7.="" />
Thus, the valid ranges for k become: k < -7="" or="" k="" /> -3.
Finding the Intersection
Now, we need to find the integer values of k that satisfy both conditions:
- From Condition 1: -6 < k="" />< />
- From Condition 2: k < -7="" or="" k="" /> -3
The overlapping valid integer values are: -6, -5, -4, -3, 0, 1, 2, 3, 4, 5, 6.
This gives us a total of 9 valid integers.
Conclusion
The number of possible values of k is 9. Thus, the correct answer is option 'A'.
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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)9b)7c)8d)13Correct answer is option 'A'. Can you explain this answer?
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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)9b)7c)8d)13Correct answer is option 'A'. Can you explain this answer?, a detailed solution for 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)9b)7c)8d)13Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of 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)9b)7c)8d)13Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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