Untitled

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

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, how many possible real values can \( r \ ) have?

If one root of the equation \( x^{2} - kx + k = 0 \) is \( 2 \sqrt{3 } \) more than the other , then what is the value of \( k \ ) among the following ?

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

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

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

If \( x=(111 1)_{2}, y=( 1001)_{2} \ ) and \( z =(110)_{2} \), then what is the value of \( x^{ 3}-y^{3}-z ^{3}-3xyz \ )?

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 cof actors of the elements \( a, b, c, d , g\) respectively , then what is the value of \(bB + cC - dD - gG\)?

Summary \ [\Delta=\left |\begin{array }{ccc}k(k+ 2) & 2 k+1 & 1 \\ 2 k +1 & k+2 & 1 \\ 3 & 3 & 1\end{ array}\right |\]Consider the following statements in the context of: 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 above statements are correct ?

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 the value of \( A+B\)?

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

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

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

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