The domain of defined of the function \(f(x)=\sqrt{\frac{1-|x|}{2-|x|}}\) is

The system of equations \(x+2 y=3\) and \(3 x+6 y=a-2\) has no solution

\[
\lim _{n \rightarrow \infty} \sum_{n=5}^{5 n+5} \frac{n^2}{2^{\frac{3}{2}}}=
\]

Suppose $f^{\text{'}\text{'}}\left(x\right)$ exists for all real $x$. If $f\left(2\right)=2, f\left(3\right)=5$ and $f\left(4\right)=10,$ then which one among the following statements is definitely true?

If the slope of the tangent at a point \((\mathrm{x}, \mathrm{y})\) on a curve is \(\frac{y-4}{x-3}\) and the curve passes through \((4,3)\), then the point where it cuts the line \(\mathrm{y}=\mathrm{x}\) is

If a point \(\mathrm{C}\) divides the line segment joining the points with the position vectors \(2 \hat{i}-3 \hat{j}+2 \hat{k}\) and \(3 \hat{i}-\hat{j}-2 \hat{k}\) in the ratio \(2: 3\), then the distance of \(\mathrm{C}\) from the point with position vector \(2 \hat{i}-\hat{j}+\hat{k}\) is

If \(f: R \rightarrow R\) is defined as \(f(x+y)=f(x)+f(y)\), \(\forall x, y \in R\) and \(f(\mathrm{l})=5\), then find the value of the following \(\sum_{r=1}^n f(r)\) is equal to

The value of ${\mathrm{K}}_{\mathrm{C}}$ for the equilibrium reaction:
${\mathrm{CO}}_{2\left(\mathrm{g}\right)}+{\mathrm{C}}_{\left(\mathrm{s}\right)}\rightleftharpoons 2{\mathrm{CO}}_{\left(\mathrm{g}\right)}$
at $\mathrm{T}\left(\mathrm{K}\right)$ is $0.036.$ If the equilibrium concentration of ${\mathrm{CO}}_{2( \mathrm{g})}$ is $0.004\mathrm{M},$ the concentration of ${\mathrm{CO}}_{\left(\mathrm{g}\right)}$ in ${\mathrm{molL}}^{-1}$ is

Three charged particles of each mass \(0.1 \mathrm{~g}\) and charge ' \(\mathrm{q}\) ' are suspended from a common rigid point by insulated massless threads of each \(1 \mathrm{~m}\) long. If the three particles are in equilibrium and are located at the corners of an equilateral triangle of side \(3 \mathrm{~cm}\), the charge ' \(q\) ' on each particle is ____ \(\mathrm{nC}\). (The angle made by the line joining the centroid of the triangle and the point of suspension with the vertical is very small).
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) and \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\) )

If \(a, b, c\) and \(d\) are real numbers such that \(a^2+b^2+c^2+d^2=1 \quad\) and \(A=\left[\begin{array}{c}a+i b c+i d \\ -c+i d a-i b\end{array}\right]\), then \(A^{-1}\) equals to

The value of \(c\) such that the straight line joining the points \((0,3)\) and \((5,-2)\) is tangent to the curve \(y=\frac{c}{x+1}\) is

The greatest integer \(\mathrm{r}\) such that \(30^{\mathrm{r}}\) divides 30 ! is

A circular coil of radius \(10 \mathrm{~cm}\) and resistance of \(2 \Omega\) is placed with its plane perpendicular to the horizontal component of the earth's magnetic field. It is rotated about its vertical diameter through \(180^{\circ}\) in \(0.25 \mathrm{~s}\). If the magnitude of the induced emf is \(3.8 \times 10^{-3} \mathrm{~V}\), then the number of turns of the coil is
(Horizontal component of earth's magnetic field at the place is \(3 \times 10^{-5} \mathrm{~T}\) )