List of top Calculus Questions asked in GATE ST

If $P(x,y,z)$ is the point nearest to the origin lying on the intersection of the surfaces $z=xy+5$ and $x+y+z=1$, then the distance (in integer) between the origin and $P$ is __________.

Which of the following real-valued functions is/are uniformly continuous on \([0, \infty)\)?

Consider the following infinite series: \[ S_1 := \sum_{n=0}^{\infty} (-1)^n \frac{n}{n^2 + 4} \quad \text{and} \quad S_2 := \sum_{n=0}^{\infty} (-1)^n \sqrt{n^2 + 1 - n}. \] Which of the above series is/are conditionally convergent?

If the line \( y = \alpha x, \alpha \geq \sqrt{2} \), divides the area of the region \[ R := \{(x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 2\} \] into two equal parts, then the value of \( \alpha \) is equal to

Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive real numbers satisfying \[ \frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343}, \quad n \geq 1, \] with \(a_1 = 3\) and \(a_n<7\) for all \(n \geq 2\).
Consider the following statements: \begin{itemize} \item [(I)] \(\{a_n\}\) is monotonically increasing. \item [(II)] \(\{a_n\}\) converges to a value in the interval \([3, 7]\). \end{itemize} Then which of the above statements is/are true?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function defined by \[ f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Find the value of \( \frac{\partial f}{\partial x} \) at \( (0, 0) \).