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:

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

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 \(c_{ij}\) 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

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

Which of the following statements are true? (A) The vector joining the points P(2,3,0) and Q(-1,-2,-4) directed from P to Q is \(\overrightarrow{PQ}=-3\hat i-5\hat j-4\hat k\) (B) Projection of a vector \(\vec a\) on other vector \(\vec b\) is…

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

Let \(L_1\) and \(L_2\) be two lines, represented as, \(L_1: \vec r=\hat i+\hat j+\lambda(2\hat i-\hat j+\hat k)\) and \(L_2: \vec r=2\hat i+\hat j-\hat k+\mu(4\hat i-2\hat j+2\hat k)\), where \(\lambda\) and \(\mu\) are scalars. Then which of the following are true? (A) \(L_1\)…

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