Assertion :Consider $D = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ Let $B_{1}, B_{2}, B_{3}$ be the co-factors of $b_{1}, b_{2}$ , and $b_{3}$ respectively, then $a_{1} B_{1} + a_{2} B_{2} + a_{3} B_{3} = 0$ Reason: If any two rows (or columns) in a determinant are identical then value of determinant is zero.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Reason is true as let
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c a & b & c 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣$
Applying $R_{2} \to R_{2} - R_{1},$ we get
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c 0 & 0 & 0 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ = 0$

Assertion:
$B_{1} =$ cofactor of $b_{1} = - ∣ ∣ ∣ \begin{matrix} a_{2} & a_{3} c_{2} & c_{3} \end{matrix} ∣ ∣ ∣$

$B_{2} =$ cofactor of $b_{2} = ∣ ∣ ∣ \begin{matrix} a_{1} & a_{3} c_{1} & c_{3} \end{matrix} ∣ ∣ ∣$

$B_{3} =$ cofactor of $b_{3} = - ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} c_{1} & c_{2} \end{matrix} ∣ ∣ ∣$

Now $a_{1} B_{1} + a_{2} B_{2} + a_{3} B_{3} = - a_{1} ∣ ∣ ∣ \begin{matrix} a_{2} & a_{3} c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ + a_{2} ∣ ∣ ∣ \begin{matrix} a_{1} & a_{3} c_{1} & c_{3} \end{matrix} ∣ ∣ ∣ - a_{3} ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} c_{1} & c_{2} \end{matrix} ∣ ∣ ∣$
$= - ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} a_{1} & a_{2} & a_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$ ........ [As two rows are identical]
Hence, both assertion and reason are true and reason is the correct explanation for assertion.

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Similar questions

Q. Assertion: The compressibility factor for

H_{2}

and

H e

(1 + \frac{P b}{R T})

Reason: The compressibility factor for

H_{2}

and

H e

can be derived from Van der Waal's equation.

Assertion (A) Order of the reaction can be zero or fractional.

Reason (R) We cannot determine order from balanced chemical equation.

(a) Both assertion and reason are correct and the reason is correct explanation of assertion
(b) Both assertion and reason are correct, but reason does not explain assertion.
(c) Assertion is correct, but reason is incorrect
(d) Both assertion and reason are incorrect
(e) Assertion is incorrect but reason is correct.

Assertion (A) Order and molecularity are same.

Reason (R) Order is determined experimentally and molecularity is the sum of the stoichimetric coeffcient of rate determining elementary step.

Q. Assertion : Two electrons of helium atom are placed in only one shell but three electrons of lithium need two shells.
Reason: A shell can have maximum

{2n}^{2}

electrons.

Assertion (A) Rate constant determined from Arrhenius equation are fairly accurate for simple as well as complex molecules.

Reason (R) Reactant molecules undergo chemical change irrespective of their orientation during collision.

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Assertion :Consider D=∣∣ ∣∣a1a2a3b1b2b3c1c2c3∣∣ ∣∣ Let B1,B2,B3 be the co-factors of b1,b2, and b3 respectively, then a1B1+a2B2+a3B3=0 Reason: If any two rows (or columns) in a determinant are identical then value of determinant is zero.