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
Let \(\{a_n\}_{n\ge1}\) and \(\{b_n\}_{n\ge1}\) be two convergent sequences of real numbers. For \( n \geq 1 \), define \( u_n = \max\{a_n, b_n\} \) and \( v_n = \min\{a_n, b_n\} \). Then
Let \( M = \begin{bmatrix} \tfrac{1}{4} & \tfrac{3}{4} \\ \\ \tfrac{3}{5} & \tfrac{2}{5} \end{bmatrix}. \) If \( I \) is the \( 2 \times 2 \) identity matrix and \( 0 \) is the \( 2 \times 2 \) zero matrix, then
Let \( f : [-1, 1] \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^2 + [\sin\pi x]}{1 + |x|}, \text{ where } [y] \text{ denotes the greatest integer less than or equal to } y. \) Then
Consider the domain \( D = \{ (x, y) \in \mathbb{R}^2 : x \leq y \} \) and the function \( h : D \to \mathbb{R} \) defined by \[ h((x, y)) = (x - 2)^4 + (y - 1)^4, (x, y) \in D. \] Then the minimum value of \( h \) on \( D \) equals
Let \( M = [X \ Y \ Z] \) be an orthogonal matrix with \( X, Y, Z \in \mathbb{R}^3 \) as its column vectors. Then \[ Q = X X^T + Y Y^T \]
Let \( f : [0,3] \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} 0, & 0 \leq x < 1, \\ e^{x^2} - e, & 1 \leq x < 2, \\ e^{x^2} + 1, & 2 \leq x \leq 3. \end{cases} \] Now, define \( F : [0, 3] \to \mathbb{R} \) by \[ F(0) = 0 \,\,\text{and}\,\, F(x) = \int_0^x f(t) \, dt, \text{ for } 0 < x \leq 3. \] Then
If \( x, y, z \) are real numbers such that \( 4x + 2y + z = 31 \,\,\text{and}\,\, 2x + 4y - z = 19, \) then the value of \( 9x + 7y + z \) is
Let \[ M = \begin{pmatrix} 1 & -1 & 1 \\ 1 & -1 & -1 \end{pmatrix}. \] If \[ V = \{ (x, y, 0) \in \mathbb{R}^3 : M \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \}, \] \[ W = \{ (x, y, z) \in \mathbb{R}^3 : M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \}, \] then
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} x^4(2 + \sin \frac{1}{x}), & x \neq 0, \\ 0, & x = 0. \end{cases} \] Then which of the following statement(s) is (are) true?