Let $\alpha$ be the real number such that the coefficient of $x^{125}$ in Maclaurin's series of $(x + \alpha^3)^3 e^x$ is $\dfrac{28}{124!}$. Then $\alpha$ equals ..............
Consider the matrix $M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & 1 & 4 \end{bmatrix}$. Let $P$ be a nonsingular matrix such that $P^{-1}MP$ is a diagonal matrix. Then the trace of the matrix $P^{-1}M^3P$ equals ...........
Let $P$ be a $3 \times 3$ matrix having characteristic roots $\lambda_1 = -\dfrac{2}{3}$, $\lambda_2 = 0$ and $\lambda_3 = 1$. Define $Q = 3P^3 - P^2 - P + I_3$ and $R = 3P^3 - 2P$. If $\alpha = \det(Q)$ and $\beta = \text{trace}(R)$, then $\alpha + \beta$ equals .......... (round off to two decimal places).
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x, y) = x^2(2 - y) - y^3 + 3y^2 + 9y$, where $(x, y) \in \mathbb{R}^2$. Which of the following is/are saddle point(s) of $f$?
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?
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
Let $f : [-1,3] \to \mathbb{R}$ be a continuous function such that $f$ is differentiable on $(-1,3)$, $|f'(x)| \le \dfrac{3}{2}$ for all $x \in (-1,3)$, $f(-1) = 1$ and $f(3) = 7$. Then $f(1)$ equals .................
Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?
For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?
Let the sequence $\{x_n\}_{n \ge 1}$ be given by $x_n = \sin \dfrac{n\pi}{6}$, $n = 1, 2, \ldots$. Then which of the following statements is/are TRUE?
Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
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