79241 A cubic equation \(x^{3}+r x-p=0\) has roots \(a, b\) and \(c\), A square matrix \(M=\left[m_{i j}\right], i, j=0,1\) and 2 , of size \(3 \times 3\) is made such that \(\mathrm{m}_{00}=a, \mathrm{~m}_{11}=\mathrm{b}\) and \(m_{22}=c\), All other elements of \(M\) are 1 . What should be the least value of \(p\) so that \(|M|\) is an odd prime?

79242 If \(y=\sin m x\) and \(y_{n}=\frac{d^{n} y}{d x^{n}}\), then \(\left|\begin{array}{lll}y_{1} & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8}\end{array}\right|=\)

79243 If \(A(x)=\left|\begin{array}{ccc}x+1 & 2 x+1 & 3 x+1 \\ 2 x+1 & 3 x+1 & x+1 \\ 3 x+1 & x+1 & 2 x+1\end{array}\right|\), then \(\int_{0}^{1} A(x) d x\)
is equal to

79244 If \(A(x)\)
\(=\left[\begin{array}{ccc} 1 & 2 & 3 \\ x+1 & 2 x+1 & 3 x+1 \\ x^{2}+1 & 2 x^{2}+1 & 3 x^{2}+1 \end{array}\right] \text { then } \int_{0}^{1} A(x) d x \text { equals }\)

79241 A cubic equation \(x^{3}+r x-p=0\) has roots \(a, b\) and \(c\), A square matrix \(M=\left[m_{i j}\right], i, j=0,1\) and 2 , of size \(3 \times 3\) is made such that \(\mathrm{m}_{00}=a, \mathrm{~m}_{11}=\mathrm{b}\) and \(m_{22}=c\), All other elements of \(M\) are 1 . What should be the least value of \(p\) so that \(|M|\) is an odd prime?

79242 If \(y=\sin m x\) and \(y_{n}=\frac{d^{n} y}{d x^{n}}\), then \(\left|\begin{array}{lll}y_{1} & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8}\end{array}\right|=\)

79243 If \(A(x)=\left|\begin{array}{ccc}x+1 & 2 x+1 & 3 x+1 \\ 2 x+1 & 3 x+1 & x+1 \\ 3 x+1 & x+1 & 2 x+1\end{array}\right|\), then \(\int_{0}^{1} A(x) d x\)
is equal to

79244 If \(A(x)\)
\(=\left[\begin{array}{ccc} 1 & 2 & 3 \\ x+1 & 2 x+1 & 3 x+1 \\ x^{2}+1 & 2 x^{2}+1 & 3 x^{2}+1 \end{array}\right] \text { then } \int_{0}^{1} A(x) d x \text { equals }\)

79241 A cubic equation \(x^{3}+r x-p=0\) has roots \(a, b\) and \(c\), A square matrix \(M=\left[m_{i j}\right], i, j=0,1\) and 2 , of size \(3 \times 3\) is made such that \(\mathrm{m}_{00}=a, \mathrm{~m}_{11}=\mathrm{b}\) and \(m_{22}=c\), All other elements of \(M\) are 1 . What should be the least value of \(p\) so that \(|M|\) is an odd prime?

79242 If \(y=\sin m x\) and \(y_{n}=\frac{d^{n} y}{d x^{n}}\), then \(\left|\begin{array}{lll}y_{1} & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8}\end{array}\right|=\)

79243 If \(A(x)=\left|\begin{array}{ccc}x+1 & 2 x+1 & 3 x+1 \\ 2 x+1 & 3 x+1 & x+1 \\ 3 x+1 & x+1 & 2 x+1\end{array}\right|\), then \(\int_{0}^{1} A(x) d x\)
is equal to

79244 If \(A(x)\)
\(=\left[\begin{array}{ccc} 1 & 2 & 3 \\ x+1 & 2 x+1 & 3 x+1 \\ x^{2}+1 & 2 x^{2}+1 & 3 x^{2}+1 \end{array}\right] \text { then } \int_{0}^{1} A(x) d x \text { equals }\)

79241 A cubic equation \(x^{3}+r x-p=0\) has roots \(a, b\) and \(c\), A square matrix \(M=\left[m_{i j}\right], i, j=0,1\) and 2 , of size \(3 \times 3\) is made such that \(\mathrm{m}_{00}=a, \mathrm{~m}_{11}=\mathrm{b}\) and \(m_{22}=c\), All other elements of \(M\) are 1 . What should be the least value of \(p\) so that \(|M|\) is an odd prime?

79242 If \(y=\sin m x\) and \(y_{n}=\frac{d^{n} y}{d x^{n}}\), then \(\left|\begin{array}{lll}y_{1} & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8}\end{array}\right|=\)

79243 If \(A(x)=\left|\begin{array}{ccc}x+1 & 2 x+1 & 3 x+1 \\ 2 x+1 & 3 x+1 & x+1 \\ 3 x+1 & x+1 & 2 x+1\end{array}\right|\), then \(\int_{0}^{1} A(x) d x\)
is equal to

79244 If \(A(x)\)
\(=\left[\begin{array}{ccc} 1 & 2 & 3 \\ x+1 & 2 x+1 & 3 x+1 \\ x^{2}+1 & 2 x^{2}+1 & 3 x^{2}+1 \end{array}\right] \text { then } \int_{0}^{1} A(x) d x \text { equals }\)