List of practice Questions

Which partial differential equation represents the Laplace equation in two dimensions?
In a metric space, is every Cauchy sequence necessarily a convergent sequence?
What is the value of the integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \)?
What is the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)?
Which theorem states that every bounded sequence in \( \mathbb{R}^n \) has a convergent subsequence?
What is the value of the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \)?
If a function \( f(x) \) is continuous on a closed interval \( [a,b] \), is it necessarily uniformly continuous?
For the function \( f(x, y) = x^3 + y^3 - 3x - 12y + 12 \), which of the following are correct:
A. minima at (1,2)
B. maxima at (-1,-2)
C. neither a maxima nor a minima at (1,-2) and (-1,2)
D. the saddle points are (-1,2) and (1,-2)
The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:
A quadratic function of two variables is given as \( f(x_1, x_2) = x_1^2 + 2x_2^2 + 3x_1 + 3x_2 + 1 \). The magnitude of maximum rate of change of the function at the point (1,1) is
The value of \(\lim_{x \to 1} \frac{x^3-1}{x-1}\) is
The surface area of the solid generated by revolving the curve \(x = e^t \cos t, y = e^t \sin t\) about y-axis for \(0 \le t \le \pi/2\) is
Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \lim_{x \to 0} \frac{\ln(1+x)}{\sin x} & (I) \; 1 \\ (B) \; \lim_{x \to \infty} 2x \tan\left(\tfrac{1}{x}\right) & (II) \; 0 \\ (C) \; \lim_{x \to \infty} \frac{x^2}{e^x} & (III) \; 2 \\ (D) \; \lim_{x \to 1} x^{\tfrac{1}{x-1}} & (IV) \; e \\ \hline \end{array} \]
Choose the correct answer from the options given below:
For the function \(f(x) = 2x^3 - 15x^2 + 36x + 10\), the local maxima and local minima occur respectively at:
If \(f'(x) = 3x^2 - \frac{2}{x^2}\), \(f(1) = 0\) then, \(f(x)\) is
For Lagrange's mean value theorem, the value of 'c' for the function \(f(x) = px^2+qx+r, p\neq 0\) in the interval \([1, b]\) and \(c \in ]1, b[\), is:
The value of \( \lim_{h \to 0} \left(\frac{1}{h} \int_{4}^{4+h} e^{t^2} dt \right) \) is
The volume of the solid for the region enclosed by the curves \(X = \sqrt{Y}\), \(X = \frac{Y}{4}\) revolve about x-axis, is
The area of the surface generated by revolving the curve \(X = \sqrt{9-Y^2}, -2 \leq Y \leq 2\) about the y-axis, is
What are the absolute maximum value and the absolute minimum value of a function \( f(x) = \sin x + \cos x \) in the interval \( [0, \pi] \)?
If \( y = e^{(x+e)^{(x+e)^{(x+\cdots)}}} \), what is the value of \( \frac{d}{dx}(y) \)?
The value of the integral \( \int_C \frac{3\sigma^2 + x}{z^2 - 1} \, dz \), where \( C \) is the circle \( |z - 1| = 1 \), is
Evaluate the following limit: \[ \lim_{t \to \infty} \sqrt{t^2 + t - t} \]
Consider two functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to (1, \infty) \). Both functions are differentiable at a point \( c \). Which of the following functions is/are ALWAYS differentiable at \( c \)? The symbol \( \cdot \) denotes product and the symbol \( \circ \) denotes composition of functions.
The value of \( \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x} dx \) is: