Match $List I$ with the $List II$ and select the correct answer using the code given below the lists :

$\begin{matrix} List I & List II (A) & If \to a = t^i - 3^j + 2 t^k, \to b =^i - 2^j + 2^k and \to c = 3^i + t^j -^k, then the & (P) & 1 value of ⎛ ⎜ ⎝ 2 \int 1 \to a \cdot (\to b \times \to c) d t + 2 ⎞ ⎟ ⎠ is equal to (B) & The value of t \in R for which the vectors \to a = (1, - 2, 3), \to b = (- 2, 3, - 4), & (Q) & 2 \to c = (1, - 1, t) form a linearly dependent system is (C) & If the area of the parallelogram whose diagonals are 3^i +^j - 2^k and & (R) & 3^i - 3^j + 4^k is μ \sqrt{3} sq. unit, then the value of μ is (D) & Let \to p =^i +^j, \to q =^i -^j and \to r =^i + 2^j + 3^k . If \to x is a unit vector & (S) & 4 such that \to p \cdot \to x = 0 and \to q \cdot \to x = 0, then | \to r \cdot \to x | is equal to (T) & 5 \end{matrix}$

Which of the following is the only CORRECT combination ?

(A) \to (R), (B) \to (T)

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(A) \to (Q), (B) \to (P)

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(A) \to (S), (B) \to (P)

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(A) \to (T), (B) \to (R)

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Solution

The correct option is B $(A) \to (Q), (B) \to (P)$
$(A)$
$[\to a \to b \to c] = ∣ ∣ ∣ ∣ \begin{matrix} t & - 3 & 2 t 1 & - 2 & 2 3 & t & - 1 \end{matrix} ∣ ∣ ∣ ∣ = t (2 - 2 t) + 3 (- 7) + 2 t (t + 6) = 7 (2 t - 3)$

$2 \int 1 \to a \cdot (\to b \times \to c) d t = 7 2 \int 1 (2 t - 3) d t$
$= 7 {[t^{2} - 3 t]}_{1}^{2} = 7 [(4 - 6) - (1 - 3)] = 0$
So, $2 \int 1 \to a \cdot (\to b \times \to c) d t + 2 = 0 + 2 = 2$
$(A) \to (Q)$

$(B)$
$∣ ∣ ∣ ∣ \begin{matrix} 1 & - 2 & 3 - 2 & 3 & - 4 1 & - 1 & t \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow 3 t - 4 + 2 (- 2 t + 4) + 3 (- 1) = 0$
$\Rightarrow t = 1$
$(B) \to (P)$

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Similar questions

List I

List II

\begin{matrix} List I & List II (A) & The possible value of a if \to r = (^i +^j) + λ (^i + 2^j -^k) & (P) & - 4 and \to r = (^i + 2^j) + μ (-^i +^j + a^k) are not consistent, where λ and μ are scalars, is (B) & The angle between vectors \to a = λ^i - 3^j -^k and & (Q) & - 2 \to b = 2 λ^i + λ^j -^k is acute, whereas vector \to b makes an obtuse angle with the axes of coordinates. Then λ can be (C) & The possible value of a such that 2^i -^j +^k, & (R) & 1^i + 2^j + (1 + a)^k and 3^i + a^j + 5^k are coplanar, is (D) & If \to A = 2^i + λ^j + 3^k, \to B = 2^i + λ^j +^k, \to C = 3^i +^j & (S) & 2 and \to A + λ \to B is perpendicular to \to C, then | 2 λ | is (T) & 3 \end{matrix}

If \to a = 2^i + 3^j +^k, \to b =^i - 2^j +^k and \to c = - 3^i +^j + 2^k,

[\to a \to b \to c] .

\to r = 3^i + 2^j - 5^k, \to a = 2^i -^j +^k, \to b =^i + 3^j - 2^k

\to c = - 2^i +^j - 3^k

\to r = x \to a + y \to b + z \to c

\to a =^i +^j +^k

\to b =^i -^j +^k

\to c =^i -^j -^k

\to v

\to a

\to b

\to c

\frac{1}{\sqrt{3}}

Find $\to a . (\to b \times \to c)$ , if $\to a = 2^i +^j + 3^k, \to b = -^i + 2^j +^k$ and $\to c = 3^i +^j + 2^k$ .

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