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Consider the following statements in respect of the given equation:
(x2 + 2)2 + 8x2 = 6x(x2 + 2)
1. All the roots of the equation are complex.
2. The sum of all the roots of the equation is 6.
(x2 + 2)2 + 8x2 = 6x(x2 + 2)
1. All the roots of the equation are complex.
2. The sum of all the roots of the equation is 6.
Q. Which of the above statements is/are correct?
- a)1 only
- b)2 only
- c)Both 1 and 2
- d)Neither 1 nor 2
Correct answer is option 'B'. Can you explain this answer?
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Consider the following statements in respect of the given equation:(x2...
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Consider the following statements in respect of the given equation:(x2...
Statement 1: All the roots of the equation are complex.
Statement 2: The sum of all the roots of the equation is 6.
To determine the correctness of these statements, let's analyze the given equation and its roots.
The given equation is:
(x^2 - 2)^2 - 8x^2 = 6x(x^2 - 2)
Simplifying the equation:
(x^4 - 4x^2 + 4) - 8x^2 = 6x^3 - 12x
Rearranging the equation:
x^4 - 12x^3 + 4x^2 + 12x + 4 = 0
Now, let's analyze each statement separately:
Statement 1: All the roots of the equation are complex.
To determine the nature of the roots, we can consider the discriminant of the equation. For a quartic equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0, the discriminant is given by:
Δ = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd
If Δ > 0, the equation has two complex roots and two real roots.
If Δ = 0, the equation has two repeated roots and two real roots.
If Δ < 0,="" the="" equation="" has="" four="" complex="" />
Let's calculate the discriminant for the given equation:
a = 1, b = -12, c = 4, d = 12
Δ = (-12)^2(4)^2 - 4(1)(4)^3 - 4(-12)^3(12) - 27(1)^2(12)^2 + 18(1)(-12)(4)(12)
After simplification, Δ = -3456
Since the discriminant Δ < 0,="" it="" implies="" that="" all="" the="" roots="" of="" the="" equation="" are="" complex.="" hence,="" />Statement 1 is correct.
Statement 2: The sum of all the roots of the equation is 6.
To find the sum of the roots, we can use Vieta's formulas. For a quartic equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0 with roots α, β, γ, and δ, the sum of the roots is given by:
α + β + γ + δ = -b/a
In our equation, a = 1 and b = -12. Therefore, the sum of the roots is:
α + β + γ + δ = -(-12)/1 = 12
Since the sum of the roots is 12, which is not equal to 6, Statement 2 is incorrect.
Hence, the correct answer is Option B: 2 only.
Statement 2: The sum of all the roots of the equation is 6.
To determine the correctness of these statements, let's analyze the given equation and its roots.
The given equation is:
(x^2 - 2)^2 - 8x^2 = 6x(x^2 - 2)
Simplifying the equation:
(x^4 - 4x^2 + 4) - 8x^2 = 6x^3 - 12x
Rearranging the equation:
x^4 - 12x^3 + 4x^2 + 12x + 4 = 0
Now, let's analyze each statement separately:
Statement 1: All the roots of the equation are complex.
To determine the nature of the roots, we can consider the discriminant of the equation. For a quartic equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0, the discriminant is given by:
Δ = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd
If Δ > 0, the equation has two complex roots and two real roots.
If Δ = 0, the equation has two repeated roots and two real roots.
If Δ < 0,="" the="" equation="" has="" four="" complex="" />
Let's calculate the discriminant for the given equation:
a = 1, b = -12, c = 4, d = 12
Δ = (-12)^2(4)^2 - 4(1)(4)^3 - 4(-12)^3(12) - 27(1)^2(12)^2 + 18(1)(-12)(4)(12)
After simplification, Δ = -3456
Since the discriminant Δ < 0,="" it="" implies="" that="" all="" the="" roots="" of="" the="" equation="" are="" complex.="" hence,="" />Statement 1 is correct.
Statement 2: The sum of all the roots of the equation is 6.
To find the sum of the roots, we can use Vieta's formulas. For a quartic equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0 with roots α, β, γ, and δ, the sum of the roots is given by:
α + β + γ + δ = -b/a
In our equation, a = 1 and b = -12. Therefore, the sum of the roots is:
α + β + γ + δ = -(-12)/1 = 12
Since the sum of the roots is 12, which is not equal to 6, Statement 2 is incorrect.
Hence, the correct answer is Option B: 2 only.
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Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer?
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Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer?.
Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements in respect of the given equation:(x2 + 2)2 + 8x2= 6x(x2+ 2)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Q. Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer?.
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