Solving Complex Number Equation
We are given the equation: $50\left(\frac{2x}{1+3i} - \frac{y}{1-2i}\right) = 31 + 17i$ We need to find the value of $10(x-3y)$. $x$ and $y$ are real numbers.
Step 1: Simplify Complex Fractions
First, rationalize the denominators:
- $\frac{2x}{1+3i} = \frac{2x(1-3i)}{(1+3i)(1-3i)} = \frac{2x(1-3i)}{1^2 + 3^2} = \frac{2x(1-3i)}{10} = \frac{x(1-3i)}{5}$
- $\frac{y}{1-2i} = \frac{y(1+2i)}{(1-2i)(1+2i)} = \frac{y(1+2i)}{1^2 + 2^2} = \frac{y(1+2i)}{5}$
Step 2: Substitute and Simplify the Equation
Substitute the simplified fractions back into the main equation:
$50\left(\frac{x(1-3i)}{5} - \frac{y(1+2i)}{5}\right) = 31 + 17i$ $10\left( x(1-3i) - y(1+2i) \right) = 31 + 17i$ $10\left( x - 3xi - y - 2yi \right) = 31 + 17i$Group the real and imaginary terms:
$10\left( (x-y) - (3x+2y)i \right) = 31 + 17i$ $(10x - 10y) - (30x + 20y)i = 31 + 17i$Step 3: Equate Real and Imaginary Parts
Equate the real parts and the imaginary parts of the equation:
- Real part: $10x - 10y = 31$ (Equation 1)
- Imaginary part: $-(30x + 20y) = 17 \implies 30x + 20y = -17$ (Equation 2)
Step 4: Solve the System of Linear Equations
Solve the system formed by Equation 1 and Equation 2:
- Multiply Equation 1 by 3: $30x - 30y = 93$ (Equation 3)
- Subtract Equation 2 from Equation 3: $(30x - 30y) - (30x + 20y) = 93 - (-17)$ $-50y = 110$ $y = -\frac{110}{50} = -\frac{11}{5}$
- Substitute $y = -\frac{11}{5}$ into Equation 1: $10x - 10\left(-\frac{11}{5}\right) = 31$ $10x + 22 = 31$ $10x = 9$ $x = \frac{9}{10}$
Step 5: Calculate the Target Expression
Calculate the value of $10(x-3y)$ using the found values of $x$ and $y$:
$10(x-3y) = 10\left(\frac{9}{10} - 3\left(-\frac{11}{5}\right)\right)$ $10\left(\frac{9}{10} + \frac{33}{5}\right)$ $10\left(\frac{9}{10} + \frac{66}{10}\right)$ $10\left(\frac{9 + 66}{10}\right)$ $10\left(\frac{75}{10}\right)$ $10(x-3y) = 75$Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
