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The number of integral values of 'a' for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions is
- a)2
- b)3
- c)4
- d)5
Correct answer is option 'D'. Can you explain this answer?
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The number of integral values of a for which the equation cos 2x + a s...
Explanation:
Given equation: cos 2x + a sin x = 2a – 7
To find integral values of 'a' for which the equation possesses possible solutions, we need to analyze the conditions under which the equation has real roots.
Key Points:
- For real roots to exist, the discriminant of the quadratic equation cos 2x + a sin x - (2a - 7) = 0 should be greater than or equal to 0.
- The discriminant of the quadratic equation Ax^2 + Bx + C = 0 is given by Δ = B^2 - 4AC.
Solving the Equation:
- The given equation can be rewritten as cos 2x + a sin x - 2a + 7 = 0.
- Comparing with the standard form, we have A = 1, B = a, and C = -2a + 7.
- The discriminant of this quadratic equation is Δ = a^2 - 4(1)(-2a + 7) = a^2 + 8a - 28.
Conditions for Real Roots:
- For real roots, Δ ≥ 0.
- Therefore, a^2 + 8a - 28 ≥ 0.
- Solving this inequality, we get a ≤ -14 or a ≥ 2.
Integral Values of 'a':
- We need to find integral values of 'a' that satisfy the inequality a ≤ -14 or a ≥ 2.
- The integral values of 'a' satisfying these conditions are -14, -13, -12, ..., 2.
- There are a total of 5 integral values of 'a' that satisfy the given conditions.
Therefore, the correct answer is option 'D' (5 integral values of 'a').
Given equation: cos 2x + a sin x = 2a – 7
To find integral values of 'a' for which the equation possesses possible solutions, we need to analyze the conditions under which the equation has real roots.
Key Points:
- For real roots to exist, the discriminant of the quadratic equation cos 2x + a sin x - (2a - 7) = 0 should be greater than or equal to 0.
- The discriminant of the quadratic equation Ax^2 + Bx + C = 0 is given by Δ = B^2 - 4AC.
Solving the Equation:
- The given equation can be rewritten as cos 2x + a sin x - 2a + 7 = 0.
- Comparing with the standard form, we have A = 1, B = a, and C = -2a + 7.
- The discriminant of this quadratic equation is Δ = a^2 - 4(1)(-2a + 7) = a^2 + 8a - 28.
Conditions for Real Roots:
- For real roots, Δ ≥ 0.
- Therefore, a^2 + 8a - 28 ≥ 0.
- Solving this inequality, we get a ≤ -14 or a ≥ 2.
Integral Values of 'a':
- We need to find integral values of 'a' that satisfy the inequality a ≤ -14 or a ≥ 2.
- The integral values of 'a' satisfying these conditions are -14, -13, -12, ..., 2.
- There are a total of 5 integral values of 'a' that satisfy the given conditions.
Therefore, the correct answer is option 'D' (5 integral values of 'a').
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The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer?
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The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer?.
The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of integral values of a for which the equation cos 2x + a sin x = 2a – 7 possesses possible solutions isa)2b)3c)4d)5Correct answer is option 'D'. Can you explain this answer?.
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