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Three Dimensional Geometry · Question 4

JEE Advanced 2024 Paper 1Multiple CorrectMedium📈 Topic in 17% papers+4−1
1

Let R3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let dist(X,Y) denote the distance between two points X and Y in R3. Let S= \X R3 : (dist(X,P) )2- (dist(X,Q) )2=50 \ and T= \Y R3 : (dist(Y,Q) )2- (dist(Y,P) )2=50 \. Then which of the following statements is (are) TRUE?

  1. There is a triangle whose area is 1 and all of whose vertices are from S.
  2. There are two distinct points L and M in T such that each point on the line segment LM is also in T.
  3. There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.
  4. There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

Answer

(1) There is a triangle whose area is 1 and all of whose vertices are from S., (2) There are two distinct points L and M in T such that each point on the line segment LM is also in T., (3) There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T., (4) There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

Solution

|X-P|2-|X-Q|2=2X (Q-P)+|P|2-|Q|2, with Q-P=(3,0,4), |P|2=14, |Q|2=69.

S: 2(3x+4z)-55=50 3x+4z= 1052 (a plane)

T: -2(3x+4z)+55=50 3x+4z= 52 (a parallel plane)

Distance between planes = 105/2-5/25=10.

(A), (B) True: each set is a whole plane.

(C) Sides 10 (normal to planes) and 14 (in a plane) give perimeter 48; such rectangles can be placed anywhere and rotated, so infinitely many. True.

(D) Side 12: take one side from a point of S to a point of T of length 12 (>10), and the adjacent side of length 12 inside the plane, perpendicular to the first side's in-plane component. True.

Answer: (A), (B), (C), (D)

JEE Advanced 2024 Paper 1

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2

Let R be such that the lines L1: x+111= y+212= z+293 and L2: x+163= y+112= z+4 intersect. Let R1 be the point of intersection of L1 and L2. Let O=(0,0,0), and n denote a unit normal vector to the plane containing both the lines L1 and L2. Match each entry in List-I to the correct entry in List-II.

  1. (P) → (3) (Q) → (4) (R) → (1) (S) → (2)
  2. (P) → (5) (Q) → (4) (R) → (1) (S) → (2)
  3. (P) → (3) (Q) → (4) (R) → (1) (S) → (5)
  4. (P) → (3) (Q) → (1) (R) → (4) (S) → (5)
JEE Advanced 2024 Paper 1
3

A straight line drawn from the point P(1,3,2), parallel to the line x-21= y-42= z-61, intersects the plane L1: x-y+3z=6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L1 intersects the plane L2: 2x-y+z=-4 at the point R. Then which of the following statements is (are) TRUE?

  1. The length of the line segment PQ is 6
  2. The coordinates of R are (1,6,3)
  3. The centroid of the triangle PQR is ( 43, 143, 53 )
  4. The perimeter of the triangle PQR is 2+ 6+ 11
JEE Advanced 2024 Paper 2

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