Let for some real numbers \(α\) and \(β\), \(a = α – iβ\). If the system of equations \(4ix + (1 + i) y = 0\) and \(8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0 \)has more than one solution, then \(\frac{α}{β}\) is equal to
- \(-2 + \sqrt3\)
- \(2 – \sqrt3\)
- \(2 + \sqrt3\)
- \(-2 – \sqrt3\)
The Correct Option is B
Solution and Explanation
Given \(a = α – iβ\) and
\(4ix + (1 + i)y = 0 …(i)\)
\(8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0....(ii)\)
By (i) \(\frac{x}{y}=\frac{−(1+i)}{4i}\)…(iii)
By (ii) \(\frac{x}{y}=\frac{−\bar a}{8\bigg(\frac{−1}{2}+\frac{\sqrt{3t}}{2}\bigg)}\)….(iv)
Now by \((iii)\) and \((iv)\)
\(\frac{1+i}{4i}=\frac{\bar a}{4(−1+\sqrt3i)}\)
\(⇒\bar a=(\sqrt3–1)+(\sqrt3+1)i\)
\(⇒α+iβ=(\sqrt3–1)+(\sqrt3+1)i\)
\(∴\frac{α}{β}=\frac{\sqrt3−1}{\sqrt3+1}=2–\sqrt3\)
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Concepts Used:
Operations on Real Numbers
Real numbers are the set of numbers that include rational numbers, irrational numbers, and integers. Operations on real numbers refer to the basic arithmetic operations that can be performed on real numbers, including addition, subtraction, multiplication, and division.
Addition is the operation of combining two or more numbers to obtain a sum. Subtraction is the operation of taking away one number from another to obtain the difference. Multiplication is the operation of combining two or more numbers to obtain a product. Division is the operation of splitting a number into equal parts.
When adding, subtracting, multiplying, or dividing real numbers, there are certain rules that must be followed. For example, the commutative property of addition states that changing the order of the addends does not change the sum. The associative property of addition states that changing the grouping of the addends does not change the sum. The distributive property of multiplication over addition states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.
Real numbers also follow certain rules when it comes to negative numbers. For example, adding a negative number is the same as subtracting the absolute value of that number. Subtracting a negative number is the same as adding the absolute value of that number. Multiplying two negative numbers results in a positive number, while multiplying a negative and positive number results in a negative number.
Understanding the rules and properties of operations on real numbers is important in mathematics and various real-world applications, such as in finance, engineering, and physics.