List of practice Questions

Evaluate the integral: \(\int \frac{x}{x + 2} \, dx\).
What is the value of the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \)?
The equation \(x^2 - Ky^2 - 4x + 6y - 5 = 0\) represents a pair of straight lines. Find the point of intersection.
Find the approximate value of \(\sqrt[3]{63}\).
Find the eigenvalues of the matrix \(A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}\).
Evaluate the integral: \(\int \frac{2x}{x^2 - 5x + 4} \, dx\)
Evaluate: \(\int_0^3 \sqrt{9 - x^2} \, dx\).
If \(\tan^{-1}(-1) + \tan^{-1}(5) + \tan^{-1}(3) + \tan^{-1}\left(\frac{1}{4}\right) = \pi + \tan^{-1}\left(\frac{\alpha}{2}\right)\), find \(\alpha\).
Given \(\int_1^a (2x + 1) \, dx = 5\), find the sum of all values of \(a\).
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?
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