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Lata Handa
Asked: 2022-11-09T02:10:29+05:30 In:

The volume of the tetrahedronwhose vertices are the points with position vectors `hati-5hatj+10hatk, -hati-3hatj+7hatk, 5hati-hatj+lamdahatk` and `7hati-4hatj+7hatk ` is 11 cubic units then the value of `lamda` is (A) 7 (B) 1 (C) -7 (D) -1

The volume of the tetrahedronwhose vertices are the points with position vectors `hati-5hatj+10hatk, -hati-3hatj+7hatk, 5hati-hatj+lamdahatk` and `7hati-4hatj+7hatk ` is 11 cubic units then the value of `lamda` is (A) 7 (B) 1 (C) -7 (D) -1
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  1. Correct Answer – 7
    Let the vertices be, A ,B , C , D and O be the origin.
    `vecOA=hati -6hatj+10hatk,vecOB=hati-3hatj +7hatk`,
    `vecOC= -5hati-hatj+lambdahatk,vecOD=7hati -4hatj+7hatk`
    `vecAB=vecOB-vecOA= -2hati+3hatj-3hatk`
    `vecAC=vecOC-vecOA= -4hati + 5hatj + (lambda-10)hatk`
    `vecAC=vecOC -vecOA=4hati+5hatj+(lamda-10)hatk`
    `vecAD=vecOD-vecOA = 6hati +2hatj-3hatk`
    volume of tetrahedron
    `1/6[vecAB vecAC vecAD]=1/6|{:(-2,3,-3),(4,5,lamda-10),(6,2,-3):}|`
    `1/6 {-2(-15-2lambda+20)-3(-12-6lambda+60)-3(8-30)}`
    `1/6 {4lambda- 10 -144 + 18 lambda+66}`
    `= 1/6 (22lambda – 88) =11`
    `or 2lambda -8 =6`
    `or 2lambda -8 =6`
    `or lambda=7`

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