Assertion -In a -ABC- if 2a2 - 4b2 - c2 - 4ab - 2ac then cos A - 1-4 Reason- In a - if cos A - 1-4 then a - b - cb - c - a - 5-2 bc

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Assertion :In...

Question

Assertion :In a $Δ$ ABC, if $2 a^{2} + 4 b^{2} + c^{2} = 4 a b + 2 a c$ then $c o s A = 1 / 4$ Reason: In a $Δ$ , if $c o s A = 1 / 4$ then $(a + b + c) (b + c - a) = (5 / 2) b c$

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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
$2 a^{2} + 4 b^{2} + c^{2} = 4 a b + 2 a c \Rightarrow {(a - 2 b)}^{2} + {(a - c)}^{2} = 0 \Rightarrow a = 2 b = c$
$\Rightarrow \frac{a}{1} = \frac{b}{1 / 2} = \frac{c}{1}$
Now
$cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} = \frac{1 / 4 + 1 - 1}{1} = \frac{1}{4}$
Reason
$(a + b + c) (b + c - a) = \frac{5 b c}{2} \Rightarrow b^{2} + c^{2} - a^{2} = \frac{5 b c}{2} - 2 b c$
$\Rightarrow \frac{b^{2} + c^{2} - a^{2}}{2 b c} = \frac{1}{4} \Rightarrow cos A = \frac{1}{4}$

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Assertion :In a ΔABC, if 2a2+4b2+c2=4ab+2ac then cosA=1/4 Reason: In a Δ, if cosA=1/4 then (a+b+c)(b+c−a)=(5/2)bc

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion2a2+4b2+c2=4ab+2ac⇒(a−2b)2+(a−c)2=0⇒a=2b=c⇒a1=b1/2=c1NowcosA=b2+c2−a22bc=1/4+1−11=14Reason (a+b+c)(b+c−a)=5bc2⇒b2+c2−a2=5bc2−2bc⇒b2+c2−a22bc=14⇒cosA=14

Assertion :In a $Δ$ ABC, if $2 a^{2} + 4 b^{2} + c^{2} = 4 a b + 2 a c$ then $c o s A = 1 / 4$ Reason: In a $Δ$ , if $c o s A = 1 / 4$ then $(a + b + c) (b + c - a) = (5 / 2) b c$