79419 A non-trivial solution of the system of equations \(x+\lambda y+2 z=0,2 x+\lambda z=0\)
\(2 \lambda x-2 y+3 z=0\) is given by \(x: y: z=\)

79420 If \(\mathbf{a} \neq \mathbf{p}, \mathbf{b} \neq \mathbf{q}, \mathbf{c} \neq \mathbf{r}\) and \(\left|\begin{array}{lll}p & b & c \\ \mathbf{a} & \mathbf{q} & \mathbf{c} \\ \mathbf{a} & \boldsymbol{b} & \mathbf{r}\end{array}\right|=0\), then the value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79421 If the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\) and \(\left(a_{1}+a_{2}, b_{1}+b_{2}\right)\) are collinear, then

79422 For how many values of \(x\) in the closed interval \([-4,-1]\), the matrix
\(\left[\begin{array}{ccc}3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2\end{array}\right]\) is singular?

79423 If \(a, b, c\) are in A.P., then the value of
\(\left|\begin{array}{lll}
\mathbf{x}+1 & \mathbf{x}+2 & \mathbf{x}+\mathbf{a} \\ \mathbf{x}+2 & \mathbf{x}+3 & \mathbf{x}+\mathbf{b} \\ \mathbf{x}+3 & \mathbf{x}+4 & \mathbf{x}+\mathbf{c} \end{array}\right| \text { is }\)

79419 A non-trivial solution of the system of equations \(x+\lambda y+2 z=0,2 x+\lambda z=0\)
\(2 \lambda x-2 y+3 z=0\) is given by \(x: y: z=\)

79420 If \(\mathbf{a} \neq \mathbf{p}, \mathbf{b} \neq \mathbf{q}, \mathbf{c} \neq \mathbf{r}\) and \(\left|\begin{array}{lll}p & b & c \\ \mathbf{a} & \mathbf{q} & \mathbf{c} \\ \mathbf{a} & \boldsymbol{b} & \mathbf{r}\end{array}\right|=0\), then the value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79421 If the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\) and \(\left(a_{1}+a_{2}, b_{1}+b_{2}\right)\) are collinear, then

79422 For how many values of \(x\) in the closed interval \([-4,-1]\), the matrix
\(\left[\begin{array}{ccc}3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2\end{array}\right]\) is singular?

79423 If \(a, b, c\) are in A.P., then the value of
\(\left|\begin{array}{lll}
\mathbf{x}+1 & \mathbf{x}+2 & \mathbf{x}+\mathbf{a} \\ \mathbf{x}+2 & \mathbf{x}+3 & \mathbf{x}+\mathbf{b} \\ \mathbf{x}+3 & \mathbf{x}+4 & \mathbf{x}+\mathbf{c} \end{array}\right| \text { is }\)

79419 A non-trivial solution of the system of equations \(x+\lambda y+2 z=0,2 x+\lambda z=0\)
\(2 \lambda x-2 y+3 z=0\) is given by \(x: y: z=\)

79420 If \(\mathbf{a} \neq \mathbf{p}, \mathbf{b} \neq \mathbf{q}, \mathbf{c} \neq \mathbf{r}\) and \(\left|\begin{array}{lll}p & b & c \\ \mathbf{a} & \mathbf{q} & \mathbf{c} \\ \mathbf{a} & \boldsymbol{b} & \mathbf{r}\end{array}\right|=0\), then the value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79421 If the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\) and \(\left(a_{1}+a_{2}, b_{1}+b_{2}\right)\) are collinear, then

79422 For how many values of \(x\) in the closed interval \([-4,-1]\), the matrix
\(\left[\begin{array}{ccc}3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2\end{array}\right]\) is singular?

79423 If \(a, b, c\) are in A.P., then the value of
\(\left|\begin{array}{lll}
\mathbf{x}+1 & \mathbf{x}+2 & \mathbf{x}+\mathbf{a} \\ \mathbf{x}+2 & \mathbf{x}+3 & \mathbf{x}+\mathbf{b} \\ \mathbf{x}+3 & \mathbf{x}+4 & \mathbf{x}+\mathbf{c} \end{array}\right| \text { is }\)

79419 A non-trivial solution of the system of equations \(x+\lambda y+2 z=0,2 x+\lambda z=0\)
\(2 \lambda x-2 y+3 z=0\) is given by \(x: y: z=\)

79420 If \(\mathbf{a} \neq \mathbf{p}, \mathbf{b} \neq \mathbf{q}, \mathbf{c} \neq \mathbf{r}\) and \(\left|\begin{array}{lll}p & b & c \\ \mathbf{a} & \mathbf{q} & \mathbf{c} \\ \mathbf{a} & \boldsymbol{b} & \mathbf{r}\end{array}\right|=0\), then the value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79421 If the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\) and \(\left(a_{1}+a_{2}, b_{1}+b_{2}\right)\) are collinear, then

79422 For how many values of \(x\) in the closed interval \([-4,-1]\), the matrix
\(\left[\begin{array}{ccc}3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2\end{array}\right]\) is singular?

79423 If \(a, b, c\) are in A.P., then the value of
\(\left|\begin{array}{lll}
\mathbf{x}+1 & \mathbf{x}+2 & \mathbf{x}+\mathbf{a} \\ \mathbf{x}+2 & \mathbf{x}+3 & \mathbf{x}+\mathbf{b} \\ \mathbf{x}+3 & \mathbf{x}+4 & \mathbf{x}+\mathbf{c} \end{array}\right| \text { is }\)

79419 A non-trivial solution of the system of equations \(x+\lambda y+2 z=0,2 x+\lambda z=0\)
\(2 \lambda x-2 y+3 z=0\) is given by \(x: y: z=\)

79420 If \(\mathbf{a} \neq \mathbf{p}, \mathbf{b} \neq \mathbf{q}, \mathbf{c} \neq \mathbf{r}\) and \(\left|\begin{array}{lll}p & b & c \\ \mathbf{a} & \mathbf{q} & \mathbf{c} \\ \mathbf{a} & \boldsymbol{b} & \mathbf{r}\end{array}\right|=0\), then the value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79421 If the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\) and \(\left(a_{1}+a_{2}, b_{1}+b_{2}\right)\) are collinear, then

79422 For how many values of \(x\) in the closed interval \([-4,-1]\), the matrix
\(\left[\begin{array}{ccc}3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2\end{array}\right]\) is singular?

79423 If \(a, b, c\) are in A.P., then the value of
\(\left|\begin{array}{lll}
\mathbf{x}+1 & \mathbf{x}+2 & \mathbf{x}+\mathbf{a} \\ \mathbf{x}+2 & \mathbf{x}+3 & \mathbf{x}+\mathbf{b} \\ \mathbf{x}+3 & \mathbf{x}+4 & \mathbf{x}+\mathbf{c} \end{array}\right| \text { is }\)