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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, then how many possible real values of \( r \) are there?

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

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

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

If \( p, 1, q \) are in \( \mathrm{AP} \) and \( p, 2, q \) are in \( \mathrm{GP} \), then which of the following statements is/are correct?
I. \( p, 4, q \) are in \( \mathrm{HP} \).
II. \( (1/p), 1/4,(1/q) \) are in \( \mathrm{AP} \).
Choose the correct answer using the code given below.

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

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 \(bB + cC - dD - gG\) is equal to what?

Determinant \[\Delta=\left|\begin{array}{ccc}k(k+2) & 2k+1 & 1\\ 2k+1 & k+2 & 1\\ 3 & 3 & 1\end{array}\right|\]Consider the following statements with respect to the above: 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 \( \left|\begin{array}{ccc}2 & 3+i & -1\\3-i & 0 & i-1\\-1 & -1-i & 1\end{array}\right| = A + i B \), where \( i = \sqrt{-1} \), then what is \( A + B \) equal to?

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?

If \( \omega \) is a non-real cube root of unity, then in 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 one root of the equation?

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

How many 7-letter words (meaningful or meaningless) can be formed using all the letters of the word CAPITAL, such that all the consonants always come together?

If \( z \neq 0 \) is a complex number, then \( \operatorname{amp}(z)+\operatorname{amp}(\bar{z}) \) is equal to what?

A polygon with 20 diagonals has how many sides?

In how many ways can the letters of the word DELHI be arranged such that the positions of vowels and consonants remain unchanged?

How many positive integer solutions are there for \( x+y+z=5 \)?

How many rational terms are there in the expansion of \( \left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{12} \)?

If the sum of the binomial coefficients in the expansion of \( (x+y)^{n} \) is 256, then in which of the following terms will the greatest binomial coefficient occur?