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 \(\frac{\vec a\cdot\vec b}{|\vec a|}\) (C) If \(\vec a=\hat i-2\hat j+\hat k\) and \(\vec b=-2\hat i+4\hat j+5\hat k\) then \(\vec a+\vec b=-\hat i+2\hat j+6\hat k\) (D) If \(\theta\) is the angle between \(\vec a\) and \(\vec b\) then \(\cos\theta=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}\) Choose the correct answer from the options below:

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) \(L_1\)…

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

For the LPP: minimize \(z=6x+3y\) subject to the constraints \(4x+y\ge80\) \(x+5y\ge115\) \(3x+2y\le150\) \(x\ge0,\ y\ge0\) then the minimum value of z is

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

If \(A\) and \(B\) square matrices of order 3 such that \(|A|=-1\), \(|B|=5\), then the value of \(|2AB|\) 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:

If \(y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\cdots+\infty}}}\) then \(\frac{dy}{dx}\) equals to

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…