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`

Correct Answer – B
Let A,B, C and D be the points with the given position vectors. Then
`vec(AB)=-2hati+3hatj-3hatk, vec(AC)=4hati+5hatj+(lamda-10)hatk`
and `vec(AD)=6hati+2hatj-3hatk`
`:.` Volume `=11` cubic units.
`=1/6[(vec(AB), vec(AC), vec(AD))]=+-11`
`implies1/6|(-2, 3, -3),(4,5,lamda-10),(6,2,-3)|=+-11`
`implies-88+22lamda=+-66implieslamda=1` or `lamda=7`

Correct Answer – D
Let ABCD be the given tetradehron. Then
`vec(AB)=hatj,vec(AC)=hatj+hatk` and `vec(AD)=hati+3hatj+lamdahatk`
`:.` volume `=1/6`
`implies1/6[(vec(AB), vec(AC),vec(AD))]=1/6`
`implies[(vec(AB),vec(AC),vec(AD))]=1`
`=(vec(AB)xxvec(AC).vec(AD))=1`
`implieshati.(hati+K3hatj+lamda hatk)=1`, which is true for all values of `lamda`