Let x and y be real numbers such that 50((2x)/(1+3i) - (y)/(1-2i)) = 31 + 17i, i = sqrt(-1). Then the value of 10(x-3y) is :

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q2 – JEE-MAIN 2026

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Home Maths Complex & QuadraticQuestion #67

Let

x

and

y

be real numbers such that

50 (\frac{2 x}{1 + 3 i} - \frac{y}{1 - 2 i}) = 31 + 17 i

i = \sqrt{- 1}

. Then the value of

10 (x - 3 y)

is :

Clue 1 of 3: Complex Number Simplification

To simplify complex fractions like $\frac{a + b i}{c + d i}$ , multiply the numerator and denominator by the conjugate of the denominator, which is $c - d i$ .

🧠 Full Solution Path

Step 1: Simplify the Complex Expression✦ Active

First, simplify each complex fraction by multiplying the numerator and denominator by the conjugate of the denominator:

\frac{2 x}{1 + 3 i} = \frac{2 x (1 - 3 i)}{(1 + 3 i) (1 - 3 i)} = \frac{2 x (1 - 3 i)}{1^{2} + 3^{2}} = \frac{2 x (1 - 3 i)}{10} = \frac{x}{5} - \frac{3 x}{5} i

\frac{y}{1 - 2 i} = \frac{y (1 + 2 i)}{(1 - 2 i) (1 + 2 i)} = \frac{y (1 + 2 i)}{1^{2} + (- 2)^{2}} = \frac{y (1 + 2 i)}{5} = \frac{y}{5} + \frac{2 y}{5} i

Substitute these back into the given equation:

50 ((\frac{x}{5} - \frac{3 x}{5} i) - (\frac{y}{5} + \frac{2 y}{5} i)) = 31 + 17 i

50 (\frac{x - y}{5} - \frac{3 x + 2 y}{5} i) = 31 + 17 i

10 (x - y) - 10 (3 x + 2 y) i = 31 + 17 i

Step 2: Equate Real and Imaginary Parts○ Expand

By equating the real and imaginary parts of the equation, we get a system of two linear equations:

10 (x - y) = 31 (Equation 1)

- 10 (3 x + 2 y) = 17 ⟹ 10 (3 x + 2 y) = - 17 (Equation 2)

From Equation 1: $x - y = \frac{31}{10}$

From Equation 2: $3 x + 2 y = - \frac{17}{10}$

Multiply the first equation by 2: $2 x - 2 y = \frac{62}{10} = \frac{31}{5}$

Add this to the second equation: $(2 x - 2 y) + (3 x + 2 y) = \frac{31}{5} - \frac{17}{10}$

5 x = \frac{62 - 17}{10} = \frac{45}{10} = \frac{9}{2}

So, $x = \frac{9}{10}$ .

Substitute $x$ back into $x - y = \frac{31}{10}$ :

\frac{9}{10} - y = \frac{31}{10} ⟹ y = \frac{9}{10} - \frac{31}{10} = - \frac{22}{10} = - \frac{11}{5}

💡 Teacher's Secret Hint

Ensure careful algebraic manipulation when solving the system of equations to avoid sign errors.

Step 3: Calculate the Required Value○ Expand

Now, calculate the value of $10 (x - 3 y)$ using the obtained values of $x$ and $y$ :

10 (x - 3 y) = 10 (\frac{9}{10} - 3 (- \frac{11}{5}))

= 10 (\frac{9}{10} + \frac{33}{5})

= 10 (\frac{9}{10} + \frac{66}{10})

= 10 (\frac{9 + 66}{10})

= 10 (\frac{75}{10})

= 75

💡 Teacher's Secret Hint

Double-check your arithmetic, especially when dealing with fractions and negative signs.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 2 – JEE-MAIN 2026

Clue 1 of 3: Complex Number Simplification

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