List of top Statistics Questions on Calculus asked in GATE ST

Consider the polynomial \[ p(t)=t(t-1)(t-2) \] and define \[ F(x)=\int_{1/2}^{x}\frac{1}{p(t)}\,dt,\qquad x\in\left[\frac{1}{2},1\right). \] Which of the following statements is correct?

Let \(f:(-1,1)\to\mathbb{R}\) be a differentiable function. Consider the following statements:
(I) Suppose \(f(0)\geq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\geq0\). Then \(f(x)>0\), for any \(x>0\).
(II) Suppose \(f(0)\leq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\leq0\). Then \(f(x)<0\), for any \(x<0\).
Which of the following statements is correct?

The value of
\[ \lim_{n\to\infty} n\int_{1-\frac{1}{2n}}^{1+\frac{1}{2n}} e^{t^2-1}\,dt \]equals _______ (answer in integer).

Consider the function \(f:\mathbb{R}^2 \to \mathbb{R}\) defined by
\[ f(x_1,x_2)=2x_1^4+x_2^2+x_2x_1^2. \]
Which of the following statements is correct?

Let \(x_1\in(0,4)\) and consider the sequence \(\{x_n\}_{n\geq1}\) defined iteratively by
\[ x_{n+1}=2-(4-x_n)^{\frac{1}{2}},\quad n\geq1. \]
Consider the following statements:
(I) \(\{x_n\}\) converges to \(0\).
(II) \(\left\{\dfrac{x_{n+1}}{x_n}\right\}\) converges to \(\dfrac{1}{4}\).

Which of the following statements is correct?

Consider the function \(f:\mathbb{R}^2\to\mathbb{R}\) defined by
\[ f(x,y)= \begin{cases} \dfrac{x^3+y^3}{\sqrt{x^2+2y^2}} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0). \end{cases} \]
Which of the following statements is/are correct?

Let \(D=\{(x,y)\in\mathbb{R}^2:0<x^2+y^2\leq1\}\). For \(\alpha\geq0\), consider the integral
\[ I_\alpha=\iint_D \frac{1}{(x^2+y^2)^\alpha}\,dx\,dy. \]
Let
\[ N_0=\sup\{\alpha\geq0:I_\alpha<\infty\}. \]
Then the value of \(N_0\) equals ______ (answer in integer).