Test o5

If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

The sum of the first \( k \) terms of a series \( S \) is \( 3 k^{2}+5 k \). Which one of the following is correct?

If \( x=(1111)_{2}, y=(1001)_{2} \) and \( z=(110)_{2} \), then what is \( x^{3}-y^{3}-z^{3}-3 x y z \) equal to?

If \( p, 1, q \) are in AP and \( p, 2, q \) are in GP, then which of the following statements is/are correct?
I. \( p, 4, q \) are in HP.
II. \( (1 / p), 1 / 4,(1 / q) \) are in AP.

If one root of the equation \( x^{2}-k x+k=0 \) exceeds the other by \( 2 \sqrt{3} \), then which one of the following is a value of \( k \) ?

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If \( r \neq 1 \) is the common ratio, then what is the number of possible real values of \( r \) ?

If \( x+\frac{5}{y}=4 \) and \( y+\frac{5}{x}=-4 \), then what is \( (x+y) \) equal to?

If \( \omega \) is a non-real cube root of unity, then what is a root of the following equation?
\[
\left|\begin{array}{ccc}
x+1 & \omega & \omega^{2} \\
\omega & x+\omega^{2} & 1 \\
\omega^{2} & 1 & x+\omega<\br>\end{array}\right|=0
\]

Consider the following statements in respect of the determinant \(\Delta=\left|\begin{array}{ccc}k(k+2) & 2k+1 & 1\\2k+1 & k+2 & 1\\3 & 3 & 1\end{array}\right|\)
I. \(\Delta\) is positive if \(k>0\).
II. \(\Delta\) is negative if \(k<0\).
III. \(\Delta\) is zero if \(k=0\).
How many of the statements given above are correct?

If \(\Delta=\left|\begin{array}{lll}a & b & c \\\ d & e & f \\\ g & h & i\end{array}\right|\) and \(A, B, C, D, G\) are the cofactors of the elements \(a, b, c, d, g\) respectively, then what is \(bB + cC - dD - gG\) equal to?

If \( x^{2}-x+1=0 \), then what is \( \left(x-\frac{1}{x}\right)^{2} + \left(x-\frac{1}{x}\right)^{4} + \left(x-\frac{1}{x}\right)^{8} \) equal to?

What is \( \left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{3} \) equal to?

If
\[
\left|\begin{array}{ccc}
2 & 3+i & -1 \\
3-i & 0 & i-1 \\
-1 & -1-i & 1
\end{array}\right|=A+iB
\]
where \(i=\sqrt{-1}\), then what is \(A+B\) equal to?

If \( A^{2}+B^{2}+C^{2}=0 \), then what is the value of the following? \[ \left|\begin{array}{ccc}1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1\end{array}\right| \]