Q26
jee_main_2024_31_jan_morning
Properties of Modulus and Argument
If α denotes the number of solutions of |1 - i|^x = 2^x and β = ((|z|)/( (z))), where z = (π)/(4) (1 + i)⁴ ( 1 - √(π) i√(π) + i + √(π) - i1 + √(π) i), i = √(-1), then the distance of the point (α, β) from the line 4x - 3y = 7 is
Numerical Answer. Answer: 3 to 3
Solution
Core Logic
|1 - i|^x = 2^x (√(2))^x = 2^x 2x/2 = 2^xThis implies (x)/(2) = x x = 0. There is exactly 1 solution, so α = 1.
Step 1: Simplify complex number z
(1+i)⁴ = ((1+i)²)² = (1 + i² + 2i)² = (2i)² = -4Thus, z = -π ( (1-√(π)i)(√(π)-i)π + 1 + (√(π)-i)(1-√(π)i)1 + π )
Step 2: Simplify Bracket
Let's expand the terms directly: z = (π)/(4)(-4) [ √(π) - π i - i - √(π)π + 1 + √(π) - i - π i - √(π)1 + π ]
= -π [ (-i(π+1))/(π+1) + (-i(π+1))/(π+1) ] = -π [ -i - i ] = 2π iStep 3: Find beta
For z = 2π i: |z| = 2π and (z) = (π)/(2).
β = (|z|)/( (z)) = (2π)/(π/2) = 4Step 4: Distance from Line
Distance of point (α, β) = (1, 4) from the line 4x - 3y - 7 = 0:
D = |4(1) - 3(4) - 7|√(4² + (-3)²) = (|4 - 12 - 7|)/(5) = (|-15|)/(5) = 3Chapter Mix
Class 11 Maths: Complex Numbers and Quadratic Equations Class 11 Maths: Straight Lines
