\(\int \frac{e^{2x}-1}{e^{2x}+1}\,dx=\)

Match List-I with List-II List-I | List-II (A) Line : \(x=2y+1=z-1\) | (I) Crosses xz plane at (1, 0, 1) (B) Line : \(x+1=2y+1=z\) | (II) Crosses xz plane at (0, 0, 1) (C) Line : \(x-1=2y=z+1\) | (III) Crosses xz plane at (1, 0, -1) (D) Line : \(x-1=2y=z-1\) | (IV) Crosses xz pla…

If \(\begin{vmatrix}p-a&0&c-r\\0&q-b&c-r\\a&b&r\end{vmatrix}=0\) then the value of \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\) is

The following system of equations: \(x+y-z=7\) \(4x+\lambda y-\lambda z=3\) \(3x+2y-4z=5\) does not possess a solution if the value of \(\lambda\) is:

\(\int e^{-x}(\cot x+\operatorname{cosec}^2x)\,dx=\)

The solution of the differential equation \((x^2+xy)dy=(x^2+y^2)dx\) is

The area of the region bounded by the curves \(y=x\) and \(y=x^3\) is:

If \(c_{ij}\) denotes the cofactor of element \(a_{ij}\) of the matrix \(A=\begin{bmatrix}1&2&-1\\0&-3&2\\4&2&3\end{bmatrix}\), then the value of \(c_{21}\cdot c_{33}\) is

The corner points of the bounded feasible region determined by a set of constraints in an LPP are P(0, 5) Q(3, 5), R(5, 0) and S(4,1). If the objective function is \(z=ax+2by\), where, \(a,b>0\), then the condition on \(a\) and \(b\) such that the maximum value of \(z\) occurs at…