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← IIT-JEE 2008 · Paper 2
MathematicsMatrix match+6Question 20 of 66

IIT-JEE 2008 · Paper 2, Mathematics question 20

Consider the lines given by
L1:x+3y−5=0,L2:3x−ky−1=0,L3:5x+2y−12=0.L_1:x+3y-5=0,\qquad L_2:3x-ky-1=0,\qquad L_3:5x+2y-12=0.
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I

  • A
    L1,L2,L3L_1,L_2,L_3 are concurrent, if
  • B
    One of L1,L2,L3L_1,L_2,L_3 is parallel to at least one of the other two, if
  • C
    L1,L2,L3L_1,L_2,L_3 form a triangle, if
  • D
    L1,L2,L3L_1,L_2,L_3 do not form a triangle, if

Column II

  • P
    k=−9k=-9
  • Q
    k=−65k=-\frac65
  • R
    k=56k=\frac56
  • S
    k=5k=5
Mark every match for each entry
PQRS
A
B
C
D
Answer and worked solution

Official answer: A → S; B → P, Q; C → R; D → P, Q, S

Concurrency. L1L_1 and L3L_3 meet at (2,1)(2,1). L2L_2 passes through it when 6−k−1=06-k-1=0, i.e. k=5k=5.

Parallel lines. Slopes: L1L_1: −13-\frac13; L3L_3: −52-\frac52; L2L_2: 3k\frac3k.

  • L2∥L1L_2\parallel L_1: 3k=−13\frac3k=-\frac13, k=−9k=-9.
  • L2∥L3L_2\parallel L_3: 3k=−52\frac3k=-\frac52, k=−65k=-\frac65.

Triangle. The lines form a triangle unless they are concurrent or two are parallel: so for k=56k=\frac56 but not for k=−9,−65,5k=-9,-\frac65,5.

Answer: A → S; B → P, Q; C → R; D → P, Q, S

Topics:Family of lines and pair of linesAngle between lines
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