JEE Advanced 2025 · Paper 1 · Mathematics Q5

MathematicsMultiple correct+4 / -2Question 5 of 48

Let $L_1$ be the line of intersection of the planes given by the equations

$2x + 3y + z = 4 \quad\text{and}\quad x + 2y + z = 5.$

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?

The length of the line segment

PQ

9\sqrt3

The length of the line segment

QR

15

The area of

\Delta PQR

\dfrac32\sqrt{234}

The acute angle between the line segments

PQ

and

PR

\cos^{-1}\left(\dfrac{1}{2\sqrt3}\right)

Answer and worked solution

Official answer: A, C

Direction of $L_1$ . $(2,3,1)\times(1,2,1)=(1,-1,1)$ .

Q. $L_2$ : $(2+t,\,-1-t,\,3+t)$ . Substituting into $2x+y-2z=6$ :
$4+2t-1-t-6-2t=6\implies t=-9,$
so $Q=(-7,8,-6)$ and $PQ=9\sqrt3$ .

R. $2(2)+(-1)-2(3)-6=-9$ , so $R=P+\frac{9}{9}(2,1,-2)=(4,0,1)$ and $PR=3$ .

(A) True: $PQ=9\sqrt3$ . ✓
(B) False: $QR=|(-11,8,-7)|=\sqrt{234}$ , not 15. ✗
(C) True: $PR\perp$ plane, so the triangle is right-angled at $R$ . Area $=\frac12\cdot3\cdot\sqrt{234}=\frac32\sqrt{234}$ . ✓
(D) False: $\cos\angle QPR=\frac{PR}{PQ}=\frac{1}{3\sqrt3}$ , not $\frac1{2\sqrt3}$ . ✗

JEE Advanced 2025 · Paper 1, Mathematics question 5