If u- v- w are non-coplanar vectors and p-q are real numbers- then the equality - 3u pv pw - pv qw u - 2w qv qu -0 holds for

Given: [ 3u amp;pv amp;pw ] [ pv amp;qw amp;u ] [ 2w amp;qv amp;qu ]=0 ⇒ 3p^2[ u amp;v amp;w ] pq[ v amp;w amp;u ] 2q^2[ w amp;v amp;u ...

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Standard XII

Mathematics

Intersection of Sets

If u, v, w ar...

Question

If $\to u, \to v, \to w$ are non-coplanar vectors and $p, q$ are real numbers, then the equality $[\begin{matrix} 3 \to u & p \to v & p \to w \end{matrix}] - [\begin{matrix} p \to v & q \to w & \to u \end{matrix}] - [\begin{matrix} 2 \to w & q \to v & q \to u \end{matrix}] = 0$ holds for

exactly two values of

(p, q)

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more than two but not all values of

(p, q)

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all values of

(p, q)

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exactly one value of

(p, q)

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Solution

The correct option is D exactly one value of $(p, q)$
Given:
$[\begin{matrix} 3 \to u & p \to v & p \to w \end{matrix}] - [\begin{matrix} p \to v & q \to w & \to u \end{matrix}] - [\begin{matrix} 2 \to w & q \to v & q \to u \end{matrix}] = 0$
$\Rightarrow 3 p^{2} [\begin{matrix} \to u & \to v & \to w \end{matrix}] - p q [\begin{matrix} \to v & \to w & \to u \end{matrix}] - 2 q^{2} [\begin{matrix} \to w & \to v & \to u \end{matrix}] = 0$
$\Rightarrow (3 p^{2} - p q + 2 q^{2}) [\begin{matrix} \to u & \to v & \to w \end{matrix}] = 0$
As $[\begin{matrix} \to u & \to v & \to w \end{matrix}] \neq 0$
$∴ (3 p^{2} - p q + 2 q^{2}) = 0$
$\Rightarrow 2 p^{2} + p^{2} - p q + \frac{q^{2}}{4} + \frac{7 q^{2}}{4} = 0$
$\Rightarrow 2 p^{2} + {(p - \frac{q}{2})}^{2} + \frac{7 q^{2}}{4} = 0$
$\Rightarrow p = 0, q = 0, p = \frac{q}{2}$
only possible when $p = q = 0$

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Similar questions

Q. If

\to u, \to v, \to w

are non-coplanar vectors and

p, q

are real numbers, then the equality

[3 \to u p \to v p \to w] - [p \to v \to w q \to u] - [2 \to w q \to v q \to u] = 0

holds for

Q. If

u, v, w

are non-coplanar vector and

p, q

are real numbers, then the equality

[3 u p v p w] - [p v w q w] - [2 w q v q u] = 0

holds for

Q. If

\to u, \to v, \to w

are non-coplanar vectors and

p, q

are real numbers, then the equality

[3 \to u, p \to v, p \to w] - [p \to v, \to w, q \to u] - [2 \to w, q \to v, q \to u] = 0

holds for

Q. If

\to u, \to v, \to w

are non-coplanar vectors and

p, q

are real numbers then the equality

[3 \to u p \to v p \to w] - [p \to v \to w q \to u] + [2 \to w q \to v q \to u] = 0

holds for

Q. Given vectors

\to a = \frac{1}{2}^i + \frac{\sqrt{3}}{2}^j,

\to b = \frac{\sqrt{3}}{2}^j + \frac{1}{2}^k

and

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

Then the volume of a parallelepiped with three coterminous edges as

\to u = (\to a \cdot \to a) \to a + (\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c,

\to v = (\to a \cdot \to b) \to a + (\to b \cdot \to b) \to b + (\to b \cdot \to c) \to c

and

\to w = (\to a \cdot \to c) \to a + (\to b \cdot \to c) \to b + (\to c \cdot \to c) \to c

equals

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Intersection of Sets

Standard XII Mathematics

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If →u, →v, →w are non-coplanar vectors and p,q are real numbers, then the equality [3→up→vp→w]−[p→vq→w→u]−[2→wq→vq→u]=0 holds for

If $\to u, \to v, \to w$ are non-coplanar vectors and $p, q$ are real numbers, then the equality $[\begin{matrix} 3 \to u & p \to v & p \to w \end{matrix}] - [\begin{matrix} p \to v & q \to w & \to u \end{matrix}] - [\begin{matrix} 2 \to w & q \to v & q \to u \end{matrix}] = 0$ holds for