Let \(A=\{1,2,3\}\). The number of relations on \(A\), containing \((1,2)\) and \((2,3)\), which are reflexive and transitive but not symmetric, is ______ -

Let the Mean and Variance of five observations \(x_1=1, x_2=3, x_3=a, x_4=7\) and \(x_5=b, a \gt b\), be 5 and 10 respectively. Then the Variance of the observations \(n+x_n, n=1,2, \ldots \ldots . .5\) is

Let \(f:[0,3] \rightarrow\) A be defined by \(f(x)=2 x^3-15 x^2+36 x+7\) and \(g:[0, \infty) \rightarrow B\) be defined by \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in \mathrm{~B}\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to :

Let \(A=\{z \in C:|z-2-i|=3\}\), \(B=\{z \in C: \operatorname{Re}(z-i z)=2\}\) and \(S=A \cap B\). Then \(\sum_{z \in S}|z|^2\) is equal to ________ .

Let the mirror image of the point \((a, b\), c) with respect to the plane \(3 x-4 y+12 z+19=0\) be (a- \(6, \beta, \gamma)\). If \(a+b+c=5\), then \(7 \beta-9 \gamma\) is equal to

Let the circle \(S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0\) be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \(x-2 y=4\) and \(2 x-y=5\) lies inside the circle \(S\), then :

If \(y =\left(\frac{2}{\pi} x -1\right) \operatorname{cosec} x\) is the solution of the differential equation, \(\frac{d y}{d x}+p(x) y=\frac{2}{\pi} \operatorname{cosec} x, 0 < x < \frac{\pi}{2},\) then the function \(p ( x )\) is equal to

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

If \(\left| {z - 3 + 2i} \right| \leq 4\) then the difference between the greatest value and the least value of \(\left| z \right|\) is

For \(x \in \left( {0,\frac{3}{2}} \right)\), let \(f\left( x \right) = \sqrt x \), \(g\left( x \right) = \tan \,x\) and \(h\left( x \right) = \frac{{1 - {x^2}}}{{1 + {x^2}}}\). If \(\phi \left( x \right) = \left( {\left( {hof} \right)og} \right)\left( x \right)\), then \(\phi \left( {\frac{\pi }{3}} \right)\) is equal to

If vertex of a parabola is \((2,-1)\) and the equation of its directrix is \(4 x-3 y=21\), then the length of its latus rectum is

If the remainder when \(x\) is divided by \(4\) is \(3 ,\) then the remainder when \((2020+ x )^{2022}\) is divided by \(8\) is ....... .

Suppose \(a_1, a_2, 2, a_3, a_4\) be in an arithmeticogeometric progression. If the common ratio of the corresponding geometric progression is \(2\) and the sum of all \(5\) terms of the arithmetico-geometric progression is \(\frac{49}{2}\), then \(a_4\) is equal to \(...........\).