Verified Solution • 2025–26 Session moderate

QJEEM24Ma346c: JEE Main 2024 Mathematics - Class 12 JEEM24Ma346c Solution: Let and be opposite vertices of a parallelogram . If the…

Let A(2,3,5)A(2,3,5) and C(−3,4,−2)C(-3,4,-2) be opposite vertices of a parallelogram ABCDA B C D . If the diagonal BD→=i^+2j^+3k^\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k} , then the area of the parallelogram is equal to :

Multiple Choice Options

  • A 12410\frac{1}{2} \sqrt{410}
  • B 12306\frac{1}{2} \sqrt{306}
  • C 12586\frac{1}{2} \sqrt{586}
  • D 12474\frac{1}{2} \sqrt{474}

Step-by-Step Verified Solution

amp; Area =∣AC→×BD→∣amp;=∣i^amp;j^amp;k^5amp;−1amp;71amp;2amp;3∣amp;=12∣−17i^−8j^+11k^∣=12474\begin{aligned}& \text { Area }=|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{BD}}| \\& =\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\5 & -1 & 7 \\1 & 2 & 3\end{array}\right| \\& =\frac{1}{2}|-17 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}|=\frac{1}{2} \sqrt{474}\end{aligned}

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