If the area of the triangle formed by the positive \(x-\)axis, the normal and the tangent to the circle \((x-2)^{2}+(y-3)^{2}=25\) at the point \((5,7)\) is \(A\) then \(24 A\) is equal to ...... .

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Let \(x = x(y)\) be the solution of the differential equation \(2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0\), \(y > 1\), \(x(e) = e\). Then \(x(e^2)\) is equal to:

If \(S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\dfrac{5\theta}{2} = \cos 7\theta \cos\dfrac{7\theta}{2}\right\}\), then \(n(S)\) is equal to _______.

Let \(\overrightarrow{ a }\) be a non-zero vector parallel to the line of intersection of the two planes described by \(\hat{i}+\hat{j}, \hat{i}+\hat{k}\) and \(\hat{i}-\hat{j}, \hat{j}-\hat{k}\). If \(\theta\) is the angle between the vector \(\vec{a}\) and the vector \(\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{a} \cdot \vec{b}=6\) then the ordered pair \((\theta,|\vec{a} \times \vec{b}|)\) is equal to

Let \(A\) be a symmetric matrix of order \(2\) with integer entries. If the sum of the diagonal elements of \(A ^{2}\) is \(1,\) then the possible number of such matrices is

The sum of the solutions of the equation \(\left| {\sqrt x  - 2} \right| + \sqrt x \left( {\sqrt x  - 4} \right) + 2 = 0\left( {x > 0} \right)\) is equal to

The number of values of \(k\) for which the system of linear equations, \((k + 2) x + 10y = k,\,\,kx + (k + 3)y = k - 1\)  has no solution, is

The integral \(\int \limits_{1}^{2} e ^{ x } \cdot x ^{ x }\left(2+\log _{ e } x \right) d x\) equal

The distance of the point having position vector \( - \,\hat i\, + \,2\hat j\, + 6\hat k\) from the straight line passing through the point \((2, 3, -4)\) and parallel to the vector \(6\,\hat i\, + 3\hat j\, - 4\hat k\) is

The number of symmetric matrices of order \(3\), with all the entries from the set \(\{0,1,2,3,4,5,6,7,8,9\}\), is :

Let \(\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be three vectors such that \((\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i}) \cdot \vec{a} \cdot \vec{c}=-29\), then \(\overrightarrow{\mathrm{c}} \cdot(-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\) is equal to :

Among the statements: \((S1):\) \(2023^{2022}-1999^{2022}\) is divisible by \(8.\) \((S2)\) : \(13(13)^{ n }-11 n -13\) is divisible by \(144\) for infinitely many \(n \in N\).