Three rods of the same dimensions have thermal conductivities $3 K, 2 K$,and $K$. They are arranged as shown in the figure below. The temperature of the junction in the steady state is:

If $y=\sin (\log _e x)$,then $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is equal to

$A$ point moves with uniform acceleration and $v_1, v_2$ and $v_3$ denote the average velocities in the three successive intervals of time $t_1, t_2$ and $t_3$. Which of the following relations is correct?

An auditorium has a volume of $10^5 \ m^3$ and a total absorption area of $2 \times 10^4 \ m^2$. Its average absorption coefficient is $0.2$. The reverberation time of the auditorium in seconds is:

In an experiment,four quantities $a, b, c$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $P$ is calculated as $P = \frac{a^3 b^2}{cd}$. The percentage error in the measurement of $P$ is .......... $\%$

In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.