List of top Mathematics Questions

The number of integral values of \( \lambda \) for which the equation \[ x^2 + y^2 - 2\lambda x + 2\lambda y + 14 = 0 \] represents a circle whose radius cannot exceed 6 is:

If $x^{2}-ax-21=0$ and $x^{2}-3ax+35=0$ with $a>0$ have a common root, then $a$ equals:

If $a,b,c$ are distinct positive real numbers and $a^2+b^2+c^2=1$, then $ab+bc+ca$ is

On dividing $x^{3} - 3x^{2} + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$ respectively. Find $g(x)$.
 

If $\, {{^n C}_{12}}$ =$\, {{^n C}_8}$ then $n$ is equal to
The solution of the differential equation log x $\frac{d y}{d x}+\frac{y}{x} =$ sin 2x is
If $z=\frac{7-i}{3-4i} \, then\, z^{14}=$
The value of x obtained from the equation $\begin{vmatrix}x+\alpha&\beta&\gamma\\ \gamma&x+\beta&\alpha\\ \alpha&\beta&x+\gamma\end{vmatrix}=0$ will be
Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :
The integral $\int\limits^{\frac{3\, \pi}{4}}_{\frac{\pi}{4}} \frac{dx}{ 1 + \cos \, x}$ is equal to :
If \[ y(x) = \int_{\sqrt{x}}^x e^t \, dt, \quad x>0 \] then \( y'(1) = \) .............
The radius of convergence of the power series \[ \sum_{n=0}^{\infty} n! x^{n^2} \] is ...........
Let \( \alpha, \beta, \gamma, \delta \) be the eigenvalues of the matrix

\[ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 2 & 0 \end{pmatrix} \] Then \( \alpha^2 + \beta^2 + \gamma^2 + \delta^2 = \) ..........
For \( x>1 \), let \[ f(x) = \int_1^x \left( \sqrt{\log t - \frac{1}{2} \log \sqrt{t}} \right) dt. \] The number of tangents to the curve \( y = f(x) \) parallel to the line \( x + y = 0 \) is ..............
For a real number \( x \), define \( \left\lceil x \right\rceil \) to be the smallest integer greater than or equal to \( x \). Then \[ \int_0^1 \int_0^1 \left( \left\lceil x \right\rceil + \left\lceil y \right\rceil + \left| z \right| \right) dx \, dy \, dz = \text{.............}. \]
Let \( a_n = \sqrt{n}, n \geq 1 \), and let \[ s_n = a_1 + a_2 + \cdots + a_n. \text{ Then} \] \[ \lim_{n \to \infty} \left( \frac{a_n}{s_n} \right) \left( -\ln \left( 1 - \frac{a_n}{s_n} \right) \right) = \text{............}. \]
The maximum order of a permutation \( \sigma \) in the symmetric group \( S_{10} \) is ................
Let \( f(x) = \frac{\sin \left( \frac{n\pi x}{\pi \sin x} \right)}{ \sin x} \), Let \( x_0 \in (0, \pi) \) and let \( f'(x_0) = 0 \). Then \[ (f(x_0))^2 \left( 1 + (\pi^2 - 1)\sin^2 x_0 \right) = \text{.........}. \]
Let \( T \) be the smallest positive real number such that the tangent to the helix \[ x = \cos t, \quad y = \sin t, \quad z = \frac{t}{\sqrt{2}} \] at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Then the line integral of \( \mathbf{F} = xj - y\mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \) is ..........
Let \( y(x), x>0 \) be the solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 4y = 0 \] satisfying the conditions \( y(1) = 1 \) and \( y'(1) = 0 \). Then the value of \( e^2y(e) \) is .............
The number of subgroups of \( \mathbb{Z}_7 \times \mathbb{Z}_7 \) of order 7 is ............
For \( x>0 \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to \infty} x \left( \lfloor x \rfloor + \lfloor x / 2 \rfloor + \lfloor x / 3 \rfloor + \cdots + \lfloor x / 10 \rfloor \right) \] is ..........
Let \( P \) be a \( 7 \times 7 \) matrix of rank 4 with real entries. Let \( a \in \mathbb{R}^7 \) be a column vector. Then the rank of \( P + aa^T \) is at least ..........
Evaluate \[ \frac{1}{2\pi} \left( \frac{\pi^3}{1 \cdot 3} - \frac{\pi^5}{3 \cdot 5} + \frac{\pi^7}{5 \cdot 7} - \cdots + (-1)^{n-1} \frac{\pi^{2n+1}}{(2n-1)!} \left( 2n+1 \right) \right). \]
Let \( v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). Let \(M\) be the matrix whose columns are \(v_1,\; v_2,\; 2v_1 - v_2,\; v_1 + 2v_2\) in that order. Then the number of linearly independent solutions of the homogeneous system \(Mx = 0\) is ...........