The differential equation of all circles having their centres on the line \(y=5\) and touching ( X -axis) is \(\qquad\)
\[ \int_0^{\frac{\pi}{4}} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x= \]
If \(y=\sin ^2\left(\cot ^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\)
\[ \int \frac{\sin x}{\sqrt{5 \sin ^2 x+6 \cos ^2 x}} d x \]
The value of \(\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ}=\)
\[ \text { The c.d.f. of a discrete random variable } \mathrm{X} \text { is } \]
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X} & -3 & -1 & 0 & 1 & 3 & 5 & 7 & 9 \\ \hline \mathrm{~F}(\mathrm{X}=x) & 0.1 & 0.3 & 0.5 & 0.65 & 0.75 & 0.85 & 0.90 & 1 \\ \hline \end{array} \]
Then \frac{P[X=-3]}{P[X<0]}=$
The distance between the lines represented by \(16 x^2+9 y^2+48 x-24 x y-36 y+35=0\) is ......... units
Methyl propanoate on hydrolysis with dil NaOH forms a salt that on further acidification with conc. HCl forms
What is the number of electrons transferred considering Mn when \(\mathrm{KMnO}_4\) is converted into \(\mathrm{Mn}_2 \mathrm{O}_3\) ?
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{3}\) and \(\cot ^{-1}\left(\frac{1}{x}\right)-\cot ^{-1}\left(\frac{1}{y}\right)=0\) then \(2 x^2+y^2-x y=\) \(\qquad\)
If \(f(x)= \begin{cases}\frac{8^x-4^x-2^x+1^x}{x^2}, & \text { if } x>0 \\ \mathrm{e}^x \sin x+\mathrm{i} x+\lambda \log 4, & \text { if } x \leqslant 0, \mathrm{i} \in \mathbb{R}\end{cases}\) continuous at \(x=0\), then the value of \(500 \mathrm{e}^\lambda\) is
The distance between the line \(\overline{\mathrm{r}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}+\lambda(\hat{\mathrm{i}}-2 \hat{\mathrm{j}})\) and the plane \(\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=4\) is
Calculate the number of \(\mathrm{Ca}^{2+}\) ion in 222 g anhydrous calcium chloride? (At. Mass \(\mathrm{Ca}=40, \mathrm{Cl}=35.5\) )
Calculate the number of atoms present per unit cell if product of density and volume of unit cell is \(1.8 \times 10^{-22} \mathrm{~g}\).
[Mass of an atom \(=4.5 \times 10^{-23} \mathrm{~g}\) ]
Heat is given to an ideal gas in an isothermal process. Then
A. internal energy of the gas will decrease.
B. internal energy of the gas will increase.
C. internal energy of the gas will not change.
D. the gas will do negative work.
Nitric oxide reacts with \(\mathrm{H}_2\) according to reaction, \(2 \mathrm{NO}_{(\mathrm{g})}+2 \mathrm{H}_{2(\mathrm{~g})} \longrightarrow \mathrm{N}_{2(\mathrm{~g})}+2 \mathrm{H}_2 \mathrm{O}_{(\mathrm{g})}\), identify the correct relationship among the following.
\[ \text { Identify ' } Z \text { ' in the following reaction. } \]
Which from following is the general formula of compound obtained when lanthanoid (Ln) reacts with carbon at elevated temperature?
An ideal inductor of \(\left(\frac{1}{\pi}\right) \mathrm{H}\) is connected in series with a \(300 \Omega\) resistor. If a \(20 \mathrm{~V}, 200 \mathrm{~Hz}\) alternating source is connected across the combination, the phase difference between the voltage and current is
If a black body at 400 K surrounded by atmosphere at 300 K has rate of cooling ' \(\mathrm{R}_0\) ', the same body at 900 K , surrounded by same atmosphere, will have rate of cooling nearly
A Carnot engine has efficiency \(\frac{1}{6}\). It becomes \(\frac{1}{3}\), when the temperature of \(\operatorname{sink}\) is lowered by
57 K . The temperature of the source is
A vehicle starts from rest and accelerates along straight path at \(2 \mathrm{~m} / \mathrm{s}^2\). At the starting point of the vehicle, there is a stationary electric siren. How far has the vehicle nearly gone when the driver hears the siren at \(94 \%\) of its value when the vehicle was at rest?
(speed of sound \(=330 \mathrm{~m} / \mathrm{s}\) )
The equation of a progressive wave is \(\mathrm{Y}=3 \sin \left[\pi\left(\frac{\mathrm{t}}{3}-\frac{\mathrm{x}}{5}\right)+\frac{\pi}{4}\right]\) where x and y are in meter and time in second. Which of the following is correct?
The radii of circular orbits of two satellites \(A\) and \(B\) of the earth are ' \(4 R\) ' and ' \(R\) ' respectively, where \(R\) is the radius of earth. If the speed of satellite \(B\) is 6 V , then the speed of satellite \(A\) will be
The two coherent sources produce interference with intensity ratio ' \(b\) '. In the interference pattern, the ratio \(\frac{I_{\text {max }}+I_{\text {min }}}{I_{\text {max }}-I_{\text {min }}}\) will be
