Mathematics Tes...

If \(\int \frac{\mathrm{xe}^{\mathrm{x}}}{\sqrt{1+\mathrm{e}^{\mathrm{x}}}} \mathrm{dx}=\mathrm{f}(\mathrm{x}) \sqrt{1+\mathrm{e}^{\mathrm{x}}}-2 \log \frac{\sqrt{1+\mathrm{e}^{\mathrm{x}}}-1}{\sqrt{1+\mathrm{e}^{\mathrm{x}}}+1}+\mathrm{C}\), then \(\mathrm{f}(\mathrm{x})\) is:

What is the \(\mathrm{n}^{\text {th }}\) term of the sequence \(1,5,9,13,17, \ldots ?\)

If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x \end{array}\right]+\left[\begin{array}{ll} y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then find the value of \(x+y\).

The direction consines of the line drawn from \(P (-5,3,1)\) to \(Q (1,5,-2)\) is:

If \(x,y,z\) are all different and not equal to zero and \(\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z\end{array}\right|=0\) then the value of \(x^{-1}+y^{-1}+z^{-1}\) is equal to:

Consider the following in respect of a complex number \(z\):

1. \(\overline{\left(z^{-1}\right)}=(\overline{z})^{-1}\)

2. \(z z^{-1}=|z|^{2}\)

Which of the above is/are correct?

In the given circle, \(\mathrm{AB}\) is a chord and \(\mathrm{OL}\) is the perpendicular drawn to it from centre \(\mathrm{O}\). If \(A L=3.2\) units, find the length of \(A B\).

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always \(10~m\) and the distance between the flag posts is \(8~m\). Find the equation of the posts traced by the man.