3D Geometry · JEE Advanced previous year questions

2025 2 questions

1. Q52025 · 18 May 2025 · Paper 1· One or more correct

   Let $L_1$ be the line of intersection of the planes given by the equations $$2x+3y+z=4\text{and}x+2y+z=5.$$ Let $L_2$ be the line passing through the point $P(2,-1,3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equation $$2x+y-2z=6.$$ Suppose that the line $L_2$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$. Then which of the following statements is (are) TRUE?

   1. A.The length of the line segment $PQ$ is $9\sqrt{3}$
   2. B.The length of the line segment $QR$ is 15
   3. C.The area of $\Delta PQR$ is $\frac{3}{2}\sqrt{234}$
   4. D.The acute angle between the line segments $PQ$ and $PR$ is ${\cos }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$
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   Answer: A, C

2. Q92025 · 18 May 2025 · Paper 1· Numerical answer

   For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P,Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that $$\overrightarrow{SP}+5\overrightarrow{SQ}+6\overrightarrow{SR}=\vec{0}.$$ Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of $$\frac{\text{length of the line segment }EF}{\text{length of the line segment }ES}$$ is ____.

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   Answer: 1.15 to 1.25