Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals
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
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?
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
The value of
is
If f(x) = \(\left\{ \begin{aligned} & \frac{1-\cos Kx}{x\sin x} ,\ \ \text{if x}\neq0 \\ &\ \ \ \ \ \ \ \frac{1}{2}\ \ \ \ \ \ \ \ \ ,\ \ \ \text{if x=0} \end{aligned} \right.\) is continuous at x = 0, then the value of K is