List of practice Questions

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y) = \begin{cases} \dfrac{2x^3 + 3y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \] 

Let $f_x(0,0)$ and $f_y(0,0)$ denote first order partial derivatives of $f(x,y)$ at $(0,0)$. Which one of the following statements is TRUE? 
 

Let $\{a_n\}_{n \ge 1}$ be a sequence of real numbers such that $a_1 = 1, a_2 = 7$, and $a_{n+1} = \dfrac{a_n + a_{n-1}}{2}$, $n \ge 2$. Assuming that $\lim_{n \to \infty} a_n$ exists, the value of $\lim_{n \to \infty} a_n$ is
 

Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Define \[ W = \frac{1}{2n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} (X_i - X_j)^2. \] Then $W$, as an estimator of $\sigma^2$, is 
 

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that $E(X_i) = 1$ and $\text{Var}(X_i) = 1$. Then the approximate distribution of $\dfrac{1}{\sqrt{n}} \sum_{i=1}^n (X_{2i} - X_{2i-1})$, for large $n$, is
 

Let $X_1, X_2, \dots, X_{10}$ be a random sample from $N(1,2)$ distribution. If $\bar{X} = \dfrac{1}{10} \sum_{i=1}^{10} X_i$ and $S^2 = \dfrac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2$, then $\text{Var}(S^2)$ equals
 

Consider the following system of linear equations: 
\[ \begin{cases} ax + 2y + z = 0 \\ y + 5z = 1  \\ by - 5z = -1 \end{cases} \] 

Which one of the following statements is TRUE? 
 

The area of the region bounded by the curves $y_1(x) = x^4 - 2x^2$ and $y_2(x) = 2x^2$, $x \in \mathbb{R}$, is
 

For real constants $a$ and $b$, let \[ f(x) = \begin{cases} \frac{a \sin x - 2x}{x}, & x < 0 \\ bx, & x \ge 0 \end{cases} \] 

If $f$ is a differentiable function, then the value of $a + b$ is 
 

If $\{x_n\}_{n \ge 1}$ is a sequence of real numbers such that $\lim_{n \to \infty} \frac{x_n}{n} = 0.001$, then