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If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)
(1) b2 - b = 30
(2) b2 + b = 72
(3) b2 - b = 42
(4) b2 + b = 12?
what to do with Z?
(1) b2 - b = 30
(2) b2 + b = 72
(3) b2 - b = 42
(4) b2 + b = 12?
what to do with Z?
Most Upvoted Answer
If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots a...
Solution:
Given equation is x2 - bx + 45 = 0. Let the roots of the equation be z1 and z2. As the roots are conjugate complex, they can be written as z1 = a + ib and z2 = a - ib.
Using sum and product of roots, we can write:
z1 + z2 = 2a = b (i)
z1z2 = a2 + b2 = 45 (ii)
Also, |z1| = |z2| = 2√10.
Using modulus of complex numbers, we get:
|z1| = √(a2 + b2) = 2√10
Squaring both sides, we get:
a2 + b2 = 40 (iii)
Solving equations (ii) and (iii), we get:
a2 = 5 and b2 = 35
Substituting these values in equation (i), we get:
b = ±√(5×35) = ±√175
As the roots are conjugate complex, b must be negative. Therefore, b = -√175.
Now, we can check which option satisfies this value of b:
Option (2) b2 - b = 72
Substituting b = -√175, we get:
175 + √175 ≠ 72
Therefore, option (2) is not correct.
Option (1) b2 + b = 30
Substituting b = -√175, we get:
175 - √175 = 30
Option (3) b2 - b = 42
Substituting b = -√175, we get:
175 + √175 ≠ 42
Therefore, option (3) is not correct.
Option (4) b2 + b = 12
Substituting b = -√175, we get:
175 - √175 ≠ 12
Therefore, option (4) is not correct.
Hence, the correct option is (1) b2 + b = 30.
Given equation is x2 - bx + 45 = 0. Let the roots of the equation be z1 and z2. As the roots are conjugate complex, they can be written as z1 = a + ib and z2 = a - ib.
Using sum and product of roots, we can write:
z1 + z2 = 2a = b (i)
z1z2 = a2 + b2 = 45 (ii)
Also, |z1| = |z2| = 2√10.
Using modulus of complex numbers, we get:
|z1| = √(a2 + b2) = 2√10
Squaring both sides, we get:
a2 + b2 = 40 (iii)
Solving equations (ii) and (iii), we get:
a2 = 5 and b2 = 35
Substituting these values in equation (i), we get:
b = ±√(5×35) = ±√175
As the roots are conjugate complex, b must be negative. Therefore, b = -√175.
Now, we can check which option satisfies this value of b:
Option (2) b2 - b = 72
Substituting b = -√175, we get:
175 + √175 ≠ 72
Therefore, option (2) is not correct.
Option (1) b2 + b = 30
Substituting b = -√175, we get:
175 - √175 = 30
Option (3) b2 - b = 42
Substituting b = -√175, we get:
175 + √175 ≠ 42
Therefore, option (3) is not correct.
Option (4) b2 + b = 12
Substituting b = -√175, we get:
175 - √175 ≠ 12
Therefore, option (4) is not correct.
Hence, the correct option is (1) b2 + b = 30.
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If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z?.
If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the equation x2 + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy |z+1| = 2√10, then (2020)(1) b2 - b = 30(2) b2 + b = 72(3) b2 - b = 42(4) b2 + b = 12?what to do with Z?.
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