\(\begin{aligned} & \int(1+x) \log \left(1+x^2\right) d x=\left(x+\frac{x^2}{2}+\frac{1}{2}\right) \\ & \log \left(1+x^2\right)+g(x)+C, \text { then } g(x)=\end{aligned}\)

The equation of a line though the point (1, 2) whose distance from the point (3, 1) has the greatest value is

In a shoe rack there are 4 pairs of shoes and 4 shoes are drawn one after the other at random without replacement. Then the probability of getting atleast one correct pair of shoes among the four shoes drawn is

The distance (s) travelled by a particle in time \(t\) is given by \(s=4 t^2+2 t+3\). The velocity of the particle when \(t=3\) seconds is

\(\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\) \(\times\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \ldots \ldots \infty=\)

If \(\alpha\) is a common root of \(x^2-5 x+\lambda=0\) and \(x^2-8 x-2 \lambda\) \(=0(\lambda \neq 0)\) and \(\beta, \gamma\) are the other roots of them, then \(\alpha+\beta+\gamma+\lambda=\)

The equations \(x-y=4\) and \(x^2+4 x y+y^2=0\) represent the sides of a/an

The magnifying power of a telescope with tube length 60 \(\mathrm{cm}\) is 5 . Then the focal length of its eye piece is

If \(\mathrm{A}(2, c)\) and \(\mathrm{B}(d, 2)\) are two points such that the polar of one point with respect to the circle \(x^2+y^2=16\) passes through the other, then \(c+d=\)

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
\(\begin{array}{llll}A & B & C & D\end{array}\)

If \(C, R, L\) and \(I\) denote capacity, resistance, inductance and electric current respectively, the quantities having the same dimensions of time are :
(1) \(C R\)
(2) \(\frac{L}{R}\)
(3) \(\sqrt{L C}\)
(4) \(L I^2\)

If \(\vec{a}=(2 x+y) \hat{i}+3 \hat{j}+9 \hat{k}\) and \(\vec{b}=2 \hat{i}+\hat{j}-(x-y) \hat{k}\) are two collinear vectors, then \(x^3+27 y^3=\)

\(Z\) is an aromatic compound with substituents \(P\) and \(Q\). What are \(P\) and \(Q\) ?