$\int \frac{3^x}{\sqrt{1-9^x}}dx=$

The required percentage increase in the energy of the earth's satellite to shift it from an orbit of radius \(r\) to \(\frac{3 r}{2}\).

A small electric heater is used to heat \(200 \mathrm{~g}\) of water. The time required to bring all this water from \(40^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) is \(200 \mathrm{~s}\). If specific heat of the water is \(4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) then the power supplied by the heater is

When a mass ' \(m\) ' is connected individually to the springs \(s_1\) and \(s_2\), the oscillation frequencies are \(v_1\) and \(v_2\). If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be

A long straight wire carrying electric current $\text{'}i \text{'}$is bent at its mid point to form an angle of $45°$ as shown in the figure. Magnetic field at a point $P$ at a distance $d$ from the point $Q$ of bending is

Match of the following.
\(\begin{array}{ll}
\hline \text{ Column I } & \text{ Column II } \\
\hline \begin{array}{ll}
\text{A. Ratio of } \frac{\Delta Q}{\Delta U} \text{ in an isobaric} \\
\text{process}
\end{array} & 1. \frac{T_1}{\left(T_1-T_2\right)} \\
\hline \begin{array}{ll}
\text{B. Ratio of } \frac{\Delta Q}{\Delta W} \text{ in an isobaric} \\
\text{process}
\end{array} & 2. \frac{T_2}{\left(T_1-T_2\right)} \\
\hline \begin{array}{l}
\text{C. Coefficient of performance} \\
\text{of a refrigerator}
\end{array} & 3. \frac{\gamma}{\gamma-1} \\
\hline \begin{array}{l}
\text{D. Coefficient of performance} \\
\text{of a heat pump}
\end{array} & 4. \gamma \\
\hline
\end{array}\)
Codes
\(\begin{array}{llll}A & B & C & D\end{array}\)

A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is

Suppose the vertices of a triangle are given by \(\mathrm{A}(0,3)\), \(\mathrm{B}(-2,0)\) and \(\mathrm{C}(6,1)\). For \((\alpha, \alpha+1)\) to lie inside the triangle, \(\alpha\) should lie in the interval

Which among the following isoelectronic species has the smallest size?
\[
\mathrm{O}^{2-}, \mathrm{F}^{-}, \mathrm{Ne}, \mathrm{Na}^{+}, \mathrm{Mg}^{2+}, \mathrm{Al}^{3+}, \mathrm{Si}^{4+}
\]

Assertion (A) At critical temperature, surface tension of liquids become zero.
Reason (R) At critical temperature, intermolecular forces for liquids and gases become equal.

The value of \(\cos ^4 x\) is

The displacement of a particle executing simple harmonic motion is \(y=\mathrm{A} \sin (2 \mathrm{t}+\phi) m\), where \(t\) is time in second and \(\phi\) is phase angle. At time \(t=0\), the displacement and velocity of the particle are 2 m and \(4 \mathrm{~ms}^{-1}\). The phase angle, \(\phi=\)

An ellipse having the coordinate axes as its axes and its major axis along \(Y\)-axis, passes through the point \((-3,1)\) and has eccentricity \(\sqrt{\frac{2}{5}}\). Then its equation is