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Definition of a Determinant

Match List I ...

Question

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

(C) \to (Q), (D) \to (P)

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B

(C) \to (T), (D) \to (R)

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C

(C) \to (R), (D) \to (P)

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D

(C) \to (Q), (D) \to (R)

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Solution

The correct option is B $(C) \to (T), (D) \to (R)$
$(C)$
$Area = \frac{1}{2} | d_{1} \times d_{2} |$
$= \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 3 & 1 & - 2 1 & - 3 & 4 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{2} \sqrt{4 + 196 + 100}$
$= \frac{1}{2} 10 \sqrt{3} = 5 \sqrt{3}$
$\Rightarrow μ = 5$
$(C) \to (T)$

$(D)$
$\to x = λ (\to p \times \to q)$
$\Rightarrow \to x = λ (- 2^k)$
$\Rightarrow 2 | λ | = 1 (∵ | \to x | = 1)$
$\Rightarrow λ = \pm \frac{1}{2}$
$∴ \to x = \pm \to k$
Now, $∣ ∣ \to r \cdot \to x ∣ ∣ = | \pm 3 | = 3$
$(D) \to (R)$

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Similar questions

List I

List II

and select the correct answer using the code given below the lists :

\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}

Which of the following is the only CORRECT combination?

\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 . (\to b \times \to c)

\to a = 2^i +^j + 3^k, \to b = -^i + 2^j +^k a n d \to c = 3^i +^j + 2^k

\to a, \to b

\to c

are of the same length and when taken pairwise, they form equal angles. If

\to a =^i +^j

\to b =^j +^k

\to c

Let $\to a =^i +^j +^k, \to b =^i -^j +^k$ and $\to c =^i -^j -^k$ be three vectors. A vector $\to v$ in the plane of
$\to a$ and $\to b$ , whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ , is given by

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