The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

Let S be the sum of the first 9 term of the series :

\((x+ka)\;+\{(x^2+(k+2)\ a\}\;+\{x^3+(k+4)\ a\}+\{x^4+(k+6)\ a\}........... \;Where\;a\neq0\;and\;x\neq1\)\(if\; S={x^{10}-x+45a(x-1)\over{x-1}},\;then\;k\;is \;equal\;to\)

Let f : \((0,\infty)\rightarrow(0,\infty)\) be a differentiable function such that f(1) = \(\lim\limits_{t\to x}{t^2f^2(x)-x^2f^2(t)\over{t-x}}=0.\) if f(x) is equal to :

The imaginary part of

\(\Big(3+2\sqrt{-54}\Big)^{\frac12}-\Big(3-2\sqrt{-54}\Big)^{\frac12}\) can be

Contrapositive o the statement :

'If a function f is differentiable at a, then it is also continuous at a', is :

Let \(a,\;b,\;c,\;\in\;R\) be all non-zero satisfy \(a^3+b^3+c^3=2\). If the matrix

\(A=\begin{bmatrix}a&b&c\\b&c&a\\c&a&b\end{bmatrix}\) stratifies \(A^TA=I\) , then a value of abc can be :

If the system of equations

\(x+y+z=2\\2x+4y-z=6\\3x+2y+\lambda z=\mu\)

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