Assertion :If a^i+^j+^k,^i+b^j+^k,^i+^j+c^k are coplanar, then 11−a+11−b+11−c=1 provided a≠1,b≠1,c≠1. Reason: Vectors →a,→b,→c are coplanar, then →a⋅(→b×→c)=0.
A
Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
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B
Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
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C
Assertion is true but Reason is false
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D
Assertion is false but Reason is true
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Solution
The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion Three vectors ¯a,¯b,¯c are coplanar then [¯a¯b¯c]=¯a⋅(¯bׯc) ∴ Reason is true & Correct explanation for Assertion (A) Now , a^i+^j+^k,^i+b^j+^k,^i+^j+c^k are given coplanar ⇒∣∣
∣∣a111b111c∣∣
∣∣[^i^j^k]=0 ⇒∣∣
∣∣a111b111c∣∣
∣∣=0as^i⋅(^j×^k)=1 Using C2→C2−C1,C3→C3−C2 ⇒∣∣
∣∣a1−a01b−11−b10c−1∣∣
∣∣=0 ⇒a(b−1)(c−1)−(1−a)[(c−1)−(1−b)]=0 ⇒a(b−1)(c−1)−(1−a)(c−1)+(1−a)(1−b)=0 on dividing by ⇒(1−a)(1−b)(1−c)=0 ⇒1−(1−a)1−a+11−b+11−c=0 ⇒11−a+11−b+11−c=1 ⇒ Assertion (A) is true .