Let \(f:(0, \infty) \rightarrow R\) and \(F(x)=\int_0^x t f(t) d t\). If \(F\left(x^2\right)=\) \(x^4+x^5\), then \(\sum_{r=1}^{12} f\left(r^2\right)\) is equal to :

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

Let \(f :R \to R\) be defined by \(f(x)\,\, = \,\,\frac{x}{{1 + {x^2}}},\,x\, \in \,R.\) Then the range of \(f\) is

The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) is

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

The tangent to the curve \(y = {x^2} - 5x + 5,\) parallel to the line \(2y=4x+1,\) also passes through the point

If the shortest distance between the line joining the points \((1, 2, 3)\) and \((2,3,4)\), and the line \(\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}\) is \(\alpha\), then \(28 \alpha^2\) is equal to \(........\).

The equation \(\arg \left(\frac{\mathrm{z}-1}{\mathrm{z}+1}\right)=\frac{\pi}{4}\) represents a circle with:

Given below are two statements :
Statement I : \(\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}\)
Statement II : \(\lim _{\mathrm{x} \rightarrow 1}\left(\mathrm{x}^{\frac{2}{1-\mathrm{x}}}\right)=\frac{1}{\mathrm{e}^2}\)
In the light of the above statements, choose the correct answer from the options given below :

If a variable line drawn through the intersection of the lines \(\frac{x}{3} + \frac{y}{4} = 1\)  and \(\frac{x}{4} + \frac{y}{3} = 1\) , meets the coordinate axes at \(A\) and \(B,\) \((A \ne B),\)  then the locus of the midpoint of \(AB\)  is

A vertical pole fixed to the horizontal ground is divided in the ratio \(3: 7\) by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground \(18\, \mathrm{~m}\) away from the base of the pole, then the height of the pole (in \(meters\)) is :

Let \(Z\) be the set of integers. If \(A\, = \,\{ x\, \in \,Z\,:\,{2^{(x + 2)({x^2} - 5x + 6)}} = 1\} \) and \(B\, = \,\{ x\, \in \,Z\,:\, - 3\, < \,2x\, - 1\, < \,9\} ,\) then the number of subsets of the set \(A \times  B\) is 

The shortest distance between lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\), where \(L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}\) and \(L_2\) is the line passing through the points \(\mathrm{A}(-4,4,3) \cdot \mathrm{B}(-1,6,3)\) and perpendicular to the line \(\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}\), is