If $\left[x\right]$ represents the greatest integer not greater than $x,$ then $\left[{\left(1+\frac{1}{100000}\right)}^{100000}\right]=$

The number of common tangents to the two circles \(x^2+y^2-8 x+2 y=0 \quad\) and \(x^2+y^2-2 x-16 y+25=0\) is :

If \(P\) and \(Q\) are two \(3 \times 3\) matrices such that \(|P Q|=1\) and \(|P|=9\), then the determinant of adjoint the matrix \(P\). Adj \(3 Q\) is

If the median of the data \(6,7, x-2, x, 18\) and 21 written in ascending order is 16 , then the variance of that data is

A tangent to the curve \(9 b^2 x^2-4 a^2 y^2=36 a^2 b^2\) makes intercepts of unit length on each of the coordinate axes, then the point \((a, b)\) lies on

If $S$ and $S^{\text{'}}$ are the focii of an ellipse, $B$ is one end of the minor axis and $\angle SBS\text{'}=90°$, then the eccentricity of that ellipse is

Events \(A, B\) and \(C\) are mutually exclusive events such that \(P(A)=\frac{3 x+1}{3}, P(B)=\frac{1-x}{4}\) and \(P(C)=\frac{1-2 x}{2}\). The set of possible values of \(x\) are in the interval

[.] represents greatest integer function. Let \(g(x)=1+x-[x]\) and \(f(x)=\left\{\begin{array}{cl}-3, & x < 0 \\ 0, & x=0, \text { then } \\ 5, & x>0\end{array}\right.\) \(f(g(x))\) is

In a \(\triangle A B C\) if \(r_1=2 r_2=3 r_3\) then \(a: b\) is

Let \(\pi\) be the plane passing through the point \((3,-3,1)\) and perpendicular to the line joining the points \((3,4,-1)\) and \((2,-1,5)\). If the equation of the plane containing the points \((3,4,-1),(-1,2,5)\) and perpendicular to the plane \(\pi\) is \(a x+y+c z-d=0\), then \(3(a+c)=\)

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

In a triangle, if \(b=20, c=21\) and \(\sin A=\frac{3}{5}\), then \(a\) is equal to :

Assertion (A) Molality of a solution increases with temperature.
Reason (R) Molality expression does not involve any volume term.