Dear student The given system of equations will have non trivial solution ifD=0 ie1k3k22234=0⇒18-6-k4k-4+33k-4=0⇒2-4k2+4k+9k-12=0⇒-4k2+13k-10=0⇒4k2-13k+10=0k=-b±b2-4ac2a, here a=4, b=-13 and c=10k=13±169-1608k=13±38k=168, 108k=2,54 Regards

Q64

Q64 x +2)' —3z +2)z +1)y z —2 isinconsistent, the valueofpis (B)-1/2 Q.63. Let x2 x 6 +Bx3 +Cx2+Dx+E . Then the valueof 5A+4B+3C+2D+E is equalto (B) -16 (C) 16 (D)-ll Q . The value of k e R for which the following system of equations admits of non-trivial solution is x + ky +3z = O,kx + 2y + 2z = O,2x +3y + 4z O (A) 2, 5/2 (B) 3, 5/4 0 2b c (D) None of these !.65. The values of a,b,cif a b -c is orthogonal are 1

Dear student

The given system of equations will have non trivial solution if D = 0 ie |\begin{matrix} 1 & k & 3 \\ k & 2 & 2 \\ 2 & 3 & 4 \end{matrix}| = 0 \Rightarrow 1 (8 - 6) - k (4 k - 4) + 3 (3 k - 4) = 0 \Rightarrow 2 - 4 k^{2} + 4 k + 9 k - 12 = 0 \Rightarrow - 4 k^{2} + 13 k - 10 = 0 \Rightarrow 4 k^{2} - 13 k + 10 = 0 k = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}, here a = 4, b = - 13 and c = 10 k = \frac{13 \pm \sqrt{169 - 160}}{8} k = \frac{13 \pm 3}{8} k = \frac{16}{8}, \frac{10}{8} k = 2, \frac{5}{4}

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