If \(\operatorname{gcd}( m , n )=1\) and \(1^2-2^2+3^2-4^2+\ldots \ldots\) \(+(2021)^2-(2022)^2+(2023)^2=1012 m ^2 n\), then \(m ^2- n ^2\) is equal to

If \({ }^{2n } C _3:{ }^{n } C _3=10: 1\), then the ratio \(\left(n^2+3 n\right):\left(n^2-3 n+4\right)\) is

Let \(A _{1}\) be the area of the region bounded by the curves \(y =\sin x , y =\cos x\) and \(y\) -axis in the first quadrant. Also, let \(A _{2}\) be the area of the region bounded by the curves \(y=\sin x\) \(y =\cos x , x\) -axis and \(x =\frac{\pi}{2}\) in the first quadrant. Then ..... .

If two tangents drawn from a point \(\mathrm{P}\) to the parabola \(\mathrm{y}^{2}=16(\mathrm{x}-3)\) are at right angles, then the locus of point \(\mathrm{P}\) is :

If the lines \(\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}\) intersect, then the magnitude of the minimum value of \(8 \alpha \beta\) is \(...............\).

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

Let the solution curve \(y = y ( x )\) of the differential equation \(\quad \frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{\frac{3}{2}}} y=2 x\) \(\exp \frac{x^3-\tan ^{-1} x^3}{\sqrt{(1+x)^6}}\) pass through the origin. Then \(y (1)\) is equal to:

Let \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right)\). Then the sum of the diagonal elements of the matrix \(( A + I )^{11}\) is equal to:

Let \(\vec a = \hat i - \hat j,\) \(\vec b = \hat i + \hat j + \hat k\) and \(\vec c\) be a vector such that \(\vec a \times \vec c + \vec b = 0\) and \(\vec a.\vec c = 4\), then \({\left| {\vec c} \right|^2}\) is equal to

If \(\alpha+\beta+\gamma=2 \pi\), then the system of equations \(x+(\cos \gamma) y+(\cos \beta) z=0\) \((\cos \gamma) x+y+(\cos \alpha) z=0\) \((\cos \beta) x+(\cos \alpha) y+z=0\) has :

\(\lim _{x \rightarrow 0} \frac{48}{x^4} \int \limits_0^x \frac{t^3}{t^6+1} d t\) is equal to \(.......\).

If the line \(y=m x+c\) is a common tangent to the hyperbola \(\frac{x^{2}}{100}-\frac{y^{2}}{64}=1\) and the circle \(x^{2}+y^{2}=36,\) then which one of the following is true?

If \( \alpha \) and \( \beta \) \( (\alpha < \beta) \) are the roots of the equation \( (-2+\sqrt{3})(|\sqrt{x}-3|) + (x-6\sqrt{x}) + (9-2\sqrt{3}) = 0 \), \( x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta} \) is equal to: