List of top Engineering Mathematics Questions on Calculus asked in GATE PE

Let \(\vec{A} = 2\hat{i} - \hat{j} + \hat{k}\) and \(\vec{B} = \hat{i} + \hat{j}\), where \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors. The projection of \(\vec{B}\) on \(\vec{A}\) is:
Which of the following statement(s) is/are CORRECT?
Let \( \vec{A} = 2\ell - \mathbf{j} + \mathbf{k} \) and \( \vec{B} = \ell + \mathbf{f} \), where \( \ell, \mathbf{f}, \) and \( \mathbf{k} \) are unit vectors. The projection of \( \vec{B} \) on \( \vec{A} \) is:
The value of \( \lim_{x \to \frac{\pi}{2}} \left( \frac{\cos x}{x - \frac{\pi}{2}} \right) \) is:
The directional derivative of $f = x^{3} + 4y^{2} + z^{2}$ at the point $P(2,1,3)$ in the direction of the vector $\vec{V} = 3\hat{\imath} - 4\hat{k}$ is ............. (rounded to one decimal place).
The value of $\displaystyle \int_{0}^{\pi} \int_{-1}^{1} r^{2}\sin^{2}\theta \, dr \, d\theta$ is
Consider a vector field $\vec{V} = x^{3}\,\hat{\imath} + 2y^{2}x\,\hat{\jmath} + 0.5\,z\,\hat{k}$, where $\hat{\imath}$, $\hat{\jmath}$, and $\hat{k}$ are the unit vectors in $x$, $y$, and $z$ directions, respectively. The divergence of $\vec{V}$ at the point $(1,2,1)$ is ............. (rounded to one decimal place).