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If a, b, c ...

Question

If $a, b, c$ and $d$ are linearly independent set of vectors and $K_{1} a + K_{2} b + k_{3} C + K_{4} d = 0$ , then

A

K_{1} + K_{2} + K_{3} + K_{4} = 0

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B

K_{1} + K_{2} + K_{3} + K_{4} = 1

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C

K_{1} + K_{2} = K_{3} + k_{4}

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D

None of the above

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Solution

The correct option is A $K_{1} + K_{2} + K_{3} + K_{4} = 0$
Given a, b, c, d are linearly independent
So $k_{1} a + k_{2} b + k_{3} c + k_{4} d = 0$
only when $k_{1} = 0, k_{2} = 0, k_{3} = 0, k_{4} = 0$ .

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Q. A parallel-plate capacitor of area

A

, plate separation 𝑑, and capacitance

C

is filled with four dielectric material having dielectric constants

k_{1}, k_{2}, k_{3}

k_{4}

as shown in the figure. If a single dielectric material is to be used to have the same capacitance

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N_{2} + O_{2} ⇌ 2 N O, K_{1}

;

(\frac{1}{2}) N_{2} + (\frac{1}{2}) O_{2} ⇌ N O, K_{2}

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2 N O ⇌ N_{2} + O_{2}; K_{3}

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N O ⇌ (\frac{1}{2}) N_{2} + (\frac{1}{2}) O_{2}; K_{4}

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For these chemical equilibriums:

N_{2} + O_{2} ⇋ 2 N O, K_{1}

\frac{1}{2} N_{2} + \frac{1}{2} O_{2} ⇋ N O, K_{2}

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N O ⇌ \frac{1}{2} N_{2} + \frac{1}{2} O_{2}, K_{4}

K_{1} \times K_{3} = x; \sqrt{(K_{1})} \times K_{4} = y; \sqrt{(K_{3})} \times K_{2} = z

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