Consider a system of linear equation in three variables x,y,z
`a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3`
The systems can be expressed by matrix equation `[(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)]`
if A is non-singular matrix then the solution of above system can be found by X =`A^(-1)B`, the solution in this case is unique.
if A is a singular matrix i.e. then the system will have
no solution (i.e. it is inconsistent) if
Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors.
Now consider the following matrix.
`A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)]`
The system AX=U has infinitely many solutions if :

Options

A. `((4)/sqrt(3),oo)`
B. `[(4)/sqrt(3),oo]`
C. `[0,(4)/sqrt(3)]`
D. `(0,(4)/sqrt(3))`

Answer

B

Explanation

if a,b `gt0`
Using AM `ge` GM we get
`(1)/(a)+(1)/(b)ge(2)/sqrt(ab)`
`rArr f(x)ge-(2)/(sqrt(cos((pi)/(6)-x)cos((pi)/(6)+x)))`
`=(2)/(sqrt(cos^(2)((pi)/(6))-sin^(2)x))`
`=(2)/(sqrt((3)/(4)-(1-cos2x)/(2)))`
`=(2)/(sqrt((1)/(4)+(cos2x)/(2)))`
Now for `0lexle(pi)/(3),(-1)/(2)lecos 2xle1`
`rArr 0lesqrt((1)/(4)+(cos2x)/(2))lesqrt(3)/(2)`
`rArr f(x)ge(4)/sqrt(3)`
Since f is continous rang of f is `[(4)/sqrt(3),oo]`