List of top Mathematics Questions on Core Mathematics

Consider the following limit: $ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. $ 
Which one of the following is correct?

Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem \[ y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \] with the boundary conditions \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]

Let \( H \) be the subset of \( S_3 \) consisting of all \( \sigma \in S_3 \) such that \[ {Trace}(A_1 A_2 A_3) = {Trace}((A_1 \sigma(A_2) A_3)), \] for all \( A_1, A_2, A_3 \in M_2(\mathbb{C}) \). The number of elements in \( H \) is equal to ……… (answer in integer).

Let \( k \in \mathbb{R} \) and \( D = \{(r, \theta) : 0<r<2, 0<\theta<\pi\ \). Let \( u(r, \theta) \) be the solution of the following boundary value problem \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad (r, \theta) \in D, \] \[ u(r, 0) = u(r, \pi) = 0, \quad 0 \leq r \leq 2, \] \[ u(2, \theta) = k \sin(2\theta), \quad 0<\theta<\pi. \] If \( u\left(\frac{1}{4}, \frac{\pi}{4}\right) = 2 \), then the value of \( k \) is equal to ………. (round off to TWO decimal places).

Consider the initial value problem (IVP): \[ \frac{dy}{dx} = e^{-y}, \quad y(0) = 0. \] 1. The IVP has a unique solution on \( {R} \).
2. Every solution of the IVP is bounded on its maximal interval of existence.
Which one of the following is correct?
If the outward flux of \( F(x, y, z) = (x^3, y^3, z^3) \) through the unit sphere \( x^2 + y^2 + z^2 = 1 \) is \( \alpha \pi \), then \( \alpha \) is equal to (round off to TWO decimal places):
Let \( S^1 = \{ z \in \mathbb{C} : |z| = 1 \} \). For which one of the following functions \( f \) does there exist a sequence of polynomials in \( z \) that uniformly converges to \( f \) on \( S^1 \)?
Let \( f(x, y) = (x^2 - y^2, 2xy) \), where \( x>0, y>0 \). Let \( g \) be the inverse of \( f \) in a neighborhood of \( f(2, 1) \). Then the determinant of the Jacobian matrix of \( g \) at \( f(2, 1) \) is equal to (round off to TWO decimal places):
Let \( C \) be the ellipse \( \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 8 \} \) traversed counter-clockwise. The value of the contour integral \[ \int_C \frac{z^2 \, dz}{z^2 - 2z + 2} \] is equal to:
The area of an equilateral triangle with sides of length \( \alpha \) is