If \(\vec{p} = \hat{i} + \hat{j} - 2\hat{k}\) and \(\vec{q} = 2\hat{i} + \hat{j} - \hat{k}\), then the area of parallelogram having diagonals \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\) is

If \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) are two non zero vectors inclined at an angle \(\theta\), then identify the correct option out of the given options (A)…

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three unit vectors such that \(\vec{a} + \vec{b} + \vec{c} = 0\) then the value of \(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}\) is

Distance between two planes \(x + 2y - z = 5\) and \(2x + 4y - 2z + 2 = 0\) is

The corner points of the feasible region for an L.P.P. are \((0, 10)\), \((5, 5)\), \((15, 15)\) and \((0, 20)\). If the objective function is \(z = px + qy\); \(p, q > 0\), then the condition on \(p\) and \(q\) so that the maximum of \(z\) occurs at \((15, 15)\) and \((0, 20)\)…

The shortest distances of the point \((1,2,3)\) from \(x, y, z\) axes respectively are

\(\int x\sqrt{x + 2}\,dx\) is equal to:

\(\int\sqrt{1 - 49x^2}\,dx\) is equal to

Three urns contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black marbles respectively. One of the urns is selected at random and a marble is drawn from it. If the marble drawn is red, then the probability that it is drawn from the first urn is