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Question

If 1,5,35,7,5,5,1,λ,7 and 2λ,1,2 are co-planar, then the sum of all possible values of λ is:


A

-445

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B

395

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C

-395

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D

445

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Solution

The correct option is D

445


Explanation for correct option:

Step-1: Apply condition for four points are co-planer.

Let the given points, 1,5,35,7,5,5,1,λ,7

Given four points are 1,5,35,7,5,5,1,λ,7 and 2λ,1,2.

We know that, four points are co-planer iffx2-x1y2-y1z2-z1x3-x1y3-y1z3-z1x4-x1y4-y1z4-z1=0

7-15-55-351-1λ-57-352λ-11-52-35=0

60-300λ-5-282λ-1-4-33=0

Step-2: Solve determinant by sarrus rule.

6×-33×λ-5+0+0-2λ-1λ-5-30+0+6×-4×-28=0

-198λ+990-2λ2-10λ-λ+5-30+0+672=0

-198λ+990--60λ2+330λ+150+672=0

60λ2-528λ+468=0

10λ2-88λ+78=0

Step-3: Use formula for sum of roots of quadratic equation -ba.

10λ2-88λ+78=0

Sum of all value of λ is --8810=445

Hence, correct answer is option (D).


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