The distance between the lines \(\vec{r} = \hat{i} - 2\hat{j} + 3\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})\) and \(\vec{r} = 3\hat{i} - 2\hat{j} + 1\hat{k} + \mu(4\hat{i} + 6\hat{j} + 12\hat{k})\) is :

The unit vector perpendicular to each of the vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\), where \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}\), is:

If \(f(x) = 2\left(\tan^{-1}(e^x) - \frac{\pi}{4}\right)\), then \(f(x)\) is :

\(\sin y = x\sin(a + y)\), then \(\frac{dy}{dx}\) is :

For the differential equation \((x\log_e x)dy = (\log_e x - y)dx\) (A) Degree of the given differential equation is 1. (B) It is a homogeneous differential equation. (C) Solution is \(2y\log_e x + A = (\log_e x)^2\), where A is an arbitrary constant (D) Solution is…

Let \(X\) denote the number of hours you play during a randomly selected day. The probability that \(X\) can take values \(x\) has the following form, where \(c\) is some constant. \(P(X = x) = 0.1\) if \(x = 0\); \(cx\) if \(x = 1\) or \(x = 2\); \(c(5 - x)\) if \(x = 3\) or…

There are two bags. Bag-1 contains 4 white and 6 black balls and Bag-2 contains 5 white and 5 black balls. A die is rolled, if it shows a number divisible by 3, a ball is drawn from Bag-1, else a ball is drawn from Bag-2. If the ball drawn is not black in colour, the probability…

The direction cosines of the line which is perpendicular to the lines with direction ratios \(1, -2, -2\) and \(0, 2, 1\) are:

Which of the following cannot be the direction ratios of the straight line \(\frac{x-3}{2} = \frac{2-y}{3} = \frac{z+4}{-1}\) ?