List of top Mathematics Questions

$\displaystyle\lim_{x\to0} \sqrt{\frac{x-\sin x}{x+\sin^{2}x}} $ is equal to

Let \( a \) be a positive real number. If \( f \) is a continuous and even function defined on the interval \( [-a, a] \), then \( \int_{-a}^{a} \dfrac{f(x)}{1+e^x} dx \) is equal to 
 

Which one of the following is TRUE?
Suppose \( \alpha, \beta, \gamma \in \mathbb{R} \). Consider the following system of linear equations.
\[ x + y + z = \alpha, x + \beta y + z = \gamma, x + y + \alpha z = \beta. \] If this system has at least one solution, then which of the following statements is (are) TRUE?

If \( \vec{\mathbf{F}} (x, y, z) = (2x + 3yz) \hat{i} + (3xz + 2y) \hat{j} + (3xy + 2z) \hat{k} \) for \( (x, y, z) \in \mathbb{R}^3 \), then which among the following is (are) TRUE? 
 

Which of the following subsets of \( \mathbb{R} \) is (are) connected?
Let \( S \) be a subset of \( \mathbb{R} \) such that 2018 is an interior point of \( S \). Which of the following is (are) TRUE?
The order of the element \( (1 \ 2 \ 3)(2 \ 4 \ 5)(4 \ 5 \ 6) \) in the group \( S_6 \) is ..............
Let \( A_6 \) be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in \( A_6 \) is ..................
Suppose \( x, y, z \) are positive real numbers such that \( x + 2y + 3z = 1 \). If \( M \) is the maximum value of \( xyz^2 \), then the value of \( \frac{1}{M} \) is ..................

If \( a = \int_{\pi/3}^{\pi/6} \dfrac{\sin t + \cos t}{\sqrt{\sin 2t}} \, dt, \) then the value of \( \left( 2 \sin \frac{\alpha}{2} + 1 \right)^2 \text{ is ............}. \)

Which one of the following curves correctly represents (schematically) the solution for the equation $\dfrac{df}{dx}+2f=3;\; f(0)=0$ ?

Let matrix $M=\begin{pmatrix} 4 & x \\ 6 & 9\end{pmatrix}$. If $\det(M)=0$, then

The coefficient of $x^3$ in the Taylor expansion of $\sin(\sin x)$ around $x = 0$ is .............. (Specify your answer up to two digits after the decimal point.)
Let $f(x)=3x^6 - 2x^2 - 8$. Which of the following statements is (are) true?
The value of \[ \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^{n^2} e^{-2n} \text{ is } \]

The value of \[ \lim_{n \to \infty} \frac{3n^2 + 5n + 4}{4 + 2n^2} \] is 
 

If \( \varphi(x) = x^2 \) and \( \psi(x) = 2^x \), then \( \psi(\varphi(x)) \) is
Consider the equation \( x^3 - 1 = 0 \). If one of the solutions to this equation is 1, the other solution(s) is/are
The value of \( \log_n 4^{16} = -32 \). The value of \( n \) is ................

The determinant of the matrix \[ \left[ \begin{array}{ccc} 1 & 3 & 0 \\ 2 & 6 & 4 \\ -1 & -1 & 2 \end{array} \right] \] is ........... 
 

For \( a = \) ..............., the following simultaneous equations have an infinite number of solutions: \[ 10x + 13y = 6 \] \[ ax + 32.5y = 15 \]
If A and B are events such that \( P(A) = 0.3, P(B) = 0.2 \) and \( P(A \cup B) = 0.45 \), the value of \( P(A \cap B) \) is .............

What is the solution of \( \int x^2 \ln x \, dx \)? Given \( C \) is an arbitrary constant. 
 

Let \( H \) be the quotient group \( \mathbb{Q} / \mathbb{Z} \). Consider the following statements.
I. Every cyclic subgroup of \( H \) is finite.
II. Every finite cyclic group is isomorphic to a subgroup of \( H \).
Which one of the following holds?