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MathematicsList match+4 / -1Question 16 of 48

JEE Advanced 2025 · Paper 1, Mathematics question 16

Let w⃗=i^+j^−2k^\vec w = \hat i + \hat j - 2\hat k, and u⃗\vec u and v⃗\vec v be two vectors, such that u⃗×v⃗=w⃗\vec u \times \vec v = \vec w and v⃗×w⃗=u⃗\vec v \times \vec w = \vec u. Let α,β,γ\alpha, \beta, \gamma, and tt be real numbers such that

u⃗=αi^+βj^+γk^,−tα+β+γ=0,α−tβ+γ=0,andα+β−tγ=0.\vec u = \alpha\hat i + \beta\hat j + \gamma\hat k,\quad -t\alpha + \beta + \gamma = 0,\quad \alpha - t\beta + \gamma = 0,\quad\text{and}\quad \alpha + \beta - t\gamma = 0.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

  • P
    ∣v⃗∣2|\vec v|^2 is equal to
  • Q
    If α=3\alpha = \sqrt3, then γ2\gamma^2 is equal to
  • R
    If α=3\alpha = \sqrt3, then (β+γ)2(\beta + \gamma)^2 is equal to
  • S
    If α=2\alpha = \sqrt2, then t+3t + 3 is equal to

List-II

  • 1
    00
  • 2
    11
  • 3
    22
  • 4
    33
  • 5
    55
Choose one option
Answer and worked solution

Official answer: A

Perpendicularity. u⃗×v⃗=w⃗\vec u\times\vec v=\vec w and v⃗×w⃗=u⃗\vec v\times\vec w=\vec u make u⃗,v⃗,w⃗\vec u,\vec v,\vec w mutually perpendicular.

Lengths.

  • ∣u⃗∣∣v⃗∣=∣w⃗∣=6|\vec u||\vec v|=|\vec w|=\sqrt6.
  • ∣v⃗∣∣w⃗∣=∣u⃗∣|\vec v||\vec w|=|\vec u|.

Together these give ∣v⃗∣2=1|\vec v|^2=1 and ∣u⃗∣2=6|\vec u|^2=6. So (P) → (2).

The linear system. A nonzero (α,β,γ)(\alpha,\beta,\gamma) requires
det⁡(−t111−t111−t)=−(t−2)(t+1)2=0.\det\begin{pmatrix}-t&1&1\\1&-t&1\\1&1&-t\end{pmatrix}=-(t-2)(t+1)^2=0.
Also u⃗⊥w⃗\vec u\perp\vec w gives α+β−2γ=0\alpha+\beta-2\gamma=0.

  • t=2t=2: α=β=γ\alpha=\beta=\gamma and 3α2=63\alpha^2=6, so α=2\alpha=\sqrt2.
  • t=−1t=-1: α+β+γ=0\alpha+\beta+\gamma=0. Combined with α+β=2γ\alpha+\beta=2\gamma this gives γ=0\gamma=0, β=−α\beta=-\alpha, and 2α2=62\alpha^2=6, so α=3\alpha=\sqrt3.

Matches.

  • (Q) α=3\alpha=\sqrt3 gives γ2=0\gamma^2=0 → (1).
  • (R) (β+γ)2=α2=3(\beta+\gamma)^2=\alpha^2=3 → (4).
  • (S) α=2\alpha=\sqrt2 gives t=2t=2, so t+3=5t+3=5 → (5).

The match is P→2, Q→1, R→4, S→5, which is option (A).

Topics:Cross productSystems of linear equations
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