If [.] denotes the greatest integer function, then \(f(x)=[x]^2-\left[x^2\right]\) is discontinuous at

If a polygon of $n$ sides has $560$ diagonals, then $n=$

\(\frac{d}{d x}\left(e^{\log _e \sqrt{1+\tan ^2 x}}\right)=\)

The sum of the rational term in the binomial expansion of \(\left(\sqrt{2}+3^{1 / 5}\right)^{10}\) is

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

The semivertical angle of a cone is \(45^{\circ}\). If the height of the cone is \(20.025 \mathrm{~cm}\), then the approximate value of its lateral surface area (in sq. \(\mathrm{cm}\) ) is

For how many values \(a \in \mathbf{C}\), the equations \(x^2-8 x+7=0\) and \(x^2-2 a x+49=0\) have a common root?

The centripetal acceleration of a particle in uniform circular motion is \(18 \mathrm{~ms}^{-2}\). If the radius of the circular path is 50 cm , the change in velocity of the particle in a time of \(\frac{\pi}{18} \mathrm{~s}\) is

The coil of a moving coil galvanometer has an effective area of \(4 \times 10^{-2} \mathrm{~m}^2\). It is suspended in a magnetic field of \(5 \times 10^{-2} \mathrm{~Wb} \mathrm{~m}^{-2}\). If the deflection in the galvanometer coil is \(0.2 \mathrm{rad}\), when a current of \(5 \mathrm{~mA}\) is passed through it, then

The number of unit cells present in \(39 \mathrm{~g}\) of potassium if it crystallizes as body centred cube is ( \(N=\) Avogadro number, At. wt. of potassium \(=39\) )

Tangents are drawn at three points \(\mathrm{P}\left(\mathrm{t}_1\right), \mathrm{Q}\left(\mathrm{t}_2\right), \mathrm{R}\left(\mathrm{t}_3\right)\) on the parabola \(\mathrm{y}^2=\mathrm{x}\). Let these tangents intersect each other at the points \(\mathrm{L}, \mathrm{M}, \mathrm{N}\). If \(\mathrm{t}_1=2, \mathrm{t}_2=-4, \mathrm{t}_3=6\), then the area of the triangle LMN is

The formation of oxide ion \(\mathrm{O}^{2-}\) from oxygen atom requires an exothermic reaction, followed by an endothermic step as shown below. The process of formation of \(\mathrm{O}^{2-}\) in gas phase is unfavorable \(\left(\Delta H^{\ominus}=+\right.\) ve \()\), even though it has stable configuration of the nearest noble gas neon. This is because
\(\begin{aligned} & \mathrm{O}(g)+e^{-} \longrightarrow \mathrm{O}^{-}(g) ; \Delta H^{\ominus}=-141 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & \mathrm{O}^{-}(g)+e^{-} \longrightarrow \mathrm{O}^{2-}(g) ; \Delta H^{\ominus}=+760 \mathrm{~kJ} \mathrm{~mol}^{-1}\end{aligned}\)

Which of the following exhibit ionization isomerism ?
I) \(\left[\mathrm{Cr}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] \mathrm{Cl}\)
II) \(\left[\mathrm{Ti}\left(\mathrm{H}_2 \mathrm{O}\right)_5 \mathrm{Cl}\right]\left(\mathrm{NO}_3\right)_2\)
III) \(\left[\mathrm{Pt}(\mathrm{en})\left(\mathrm{NH}_3\right) \mathrm{Cl}\right] \mathrm{NO}_3\)
IV) \(\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4\left(\mathrm{NO}_3\right)_2\right] \mathrm{NO}_3\)