Verified Solution • 2025–26 Session moderate

QJEEM24Maf755: JEE Main 2024 Mathematics - Class 11 JEEM24Maf755 Solution: Let and be two complex numbers such that and Then, equals -

Let z1z_1 and z2z_2 be two complex numbers such that z1+z2=5z_1+z_2=5 and z13+z23=20+15iz_1^3+z_2^3=20+15 i Then, ∣z14+z24∣\left|z_1^4+z_2^4\right| equals -

Multiple Choice Options

  • A 151515 \sqrt{15}
  • B 30330 \sqrt{3}
  • C 25325 \sqrt{3}
  • D 75

Step-by-Step Verified Solution

amp;z1+z2=5amp;z13+z23=20+15iamp;z13+z23=(z1+z2)3−3z1z2(z1+z2)amp;z13+z23=125−3z1⋅z2(5)amp;⇒20+15i=125−15z1z2amp;⇒3z1z2=25−4−3iamp;⇒3z1z2=21−3iamp;⇒z1⋅z2=7−iamp;⇒(z1+z2)2=25amp;⇒z12+z22=25−2(7−i)amp;⇒11+2iamp;(z12+z22)2=121−4+44iamp;⇒z14+z24+2(7−i)2=117+44iamp;⇒z14+z24=117+44i−2(49−1−14i)amp;⇒∣z14+z24∣=75\begin{aligned}& z_1+z_2=5 \\& z_1^3+z_2^3=20+15 i \\& z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right) \\& z_1^3+z_2^3=125-3 z_1 \cdot z_2(5) \\& \Rightarrow 20+15 i=125-15 z_1 z_2 \\& \Rightarrow 3 z_1 z_2=25-4-3 i \\& \Rightarrow 3 z_1 z_2=21-3 i \\& \Rightarrow z_1 \cdot z_2=7-i \\& \Rightarrow\left(z_1+z_2\right)^2=25 \\& \Rightarrow z_1^2+z_2^2=25-2(7-i) \\& \Rightarrow 11+2 i \\& \left(z_1^2+z_2^2\right)^2=121-4+44 i \\& \Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i \\& \Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i) \\& \Rightarrow\left|z_1^4+z_2^4\right|=75\end{aligned}

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring