JEE Advanced Model Paper 19 with solutions for online practice

Given lines can be written as

−

r

= (i + j + k) +λ {(1−α )i + 3 j − 2k}
and

−

r

= (3i + j −5k) +μ {(3−α )i + 4 j −8k}
Since these lines are coplanar, ∃ λ, μ for which
(i + j + k) + λ{(i – α)I + 3j – 2k} = (3i + j – 5k) + μ{(3 – α)i + 4j – 8k}
Companing the coefficient of i, j, k, we get
1 + λ(1 – α) = 3 + μ(3 – α) → λ(1 – α) + μ(α – 3) – 2 = 0 …… (1)
1 + 3λ = 1 + 4μ → 3λ – 4μ = 0 ………. (2) and
1 – 2λ = –5 – 8μ → –λ + 4μ + 2 = 0 ……. (3)
eliminating, λ, μ we get |

1−α	α−3	−2
3	−4	0
−1	4	2

| = 0
→ α = 3 then λ = –1.
Then point of intersection is

−

r

= 3i – 2j + 3k