If \( x^{2}+x+1=0 \) then the value of \( (x+\frac{1}{x})^{4}+(x^{2}+\frac{1}{x^{2}})^{4}+(x^{3}+\frac{1}{x^{3}})^{4}+...+(x^{25}+\frac{1}{x^{25}})^{4} \) is:

Let \(R _{1}=\{( a , b ) \in N \times N :| a - b | \leq 13\}\) and \(R _{2}=\{( a , b ) \in N \times N :| a - b | \neq 13\} .\) Thenon \(N\)

If the sum of the first ten terms of the series \(\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\ldots\) is \(\frac{m}{n}\), where \(m\) and \(n\) are co-prime numbers, then \(m + n\) is equal to

The value of \({1^2} + {3^2} + {5^2} + ....... + {25^2}\) is

Let \(a_1,a_2,a_3,....,a_{10}\) be in \(G.P.\) with \(a_i > 0\) for \(i = 1, 2,....,10\) and \(S\) be the set of pairs \((r,k), r, k \in N\) (the set of natural numbers) for which \(\left| {\begin{array}{*{20}{c}}
  {{{\log }_e}\,a_1^ra_2^k}&{{{\log }_e}\,a_2^ra_3^k}&{{{\log }_e}\,a_3^ra_4^k} \\
  {{{\log }_e}\,a_4^ra_5^k}&{{{\log }_e}\,a_5^ra_6^k}&{{{\log }_e}\,a_6^ra_7^k} \\ 
  {{{\log }_e}\,a_7^ra_8^k}&{{{\log }_e}\,a_8^ra_9^k}&{{{\log }_e}\,a_9^ra_{10}^k} 
\end{array}} \right| = 0\) Then the number of elements in \(S\), is

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\frac{4^x}{4^x+2}\) and \(M=\int_{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x,\) \(N=\int_{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2} . \text { If }\) \(\alpha \mathrm{M}=\beta \mathrm{N}, \alpha, \beta \in \mathbb{N}\), then the least value of \(\alpha^2+\beta^2\) is equal to ...........

Let \(A\) be a \(3 \times 3\) matrix with \(\operatorname{det}( A )=4\). Let \(R _{ i }\) denote the \(i ^{\text {th }}\) row of \(A\). If a matrix \(B\) is obtained by performing the operation \(R _{2} \rightarrow 2 R _{2}+5 R _{3}\) on \(2 A ,\) then \(\operatorname{det}( B )\) is equal to ...... .

If the functions \(f ( x )=\frac{ x ^3}{3}+2 bx +\frac{a x^2}{2}\) and \(g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b\) have a common extreme point, then \(a+2 b+7\) is equal to

Let \(A=\left[a_{i j}\right]\) be \(3 \times 3\) matrix such that \(A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]\) and \(A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\), then \(a_{23}\) equals :

Let the determinant of a square matrix \(A\) of order \(m\) be \(m-n\), where \(m\) and \(n\) satisfy \(4 m+n=22\) and \(17 m +4 n =93\). If \(\operatorname{det}(n \operatorname{adj}(\operatorname{adj}( mA )))=\) \(3^{ a } 5^{ b } 6^{ c }\). then \(a + b + c\) is equal to:

If \(a\) and \(b\) are the roots of equation \(x^2-7 x-1=0\), then the value of \(\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}\) is equal to \(........\).

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

If a straight line drawn through the point of intersection of the lines \(4x+3y-1=0\) and \(3x+4y-1=0\), meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is: