The equation of a stationary wave along a stretched string is given by $y = 5 \sin \frac{\pi x}{3} \cos 40 \pi t$,where $x$ and $y$ are in $cm$ and $t$ is in $s$. The separation between two adjacent nodes is .... $cm$.

Two satellites $A$ and $B$,having a ratio of masses $3 : 1$,are in circular orbits of radii $r$ and $4r$ respectively. Calculate the ratio of the total mechanical energy of $A$ and $B$.

Which one of the following is a slime mould?

Identify the blank spaces $a, b, c$ and $d$ given in the following table and select the correct answer.
| Organism | Type | Product |
| :--- | :--- | :--- |
| $a$ | $b$ | Cyclosporin-$A$ |
| $c$ | Bacterium | Butyric acid |
| $d$ | Bacterium | Lactic acid |

The true solution set of the inequality,$\sqrt{5x - 6 - x^2} + \frac{\pi}{2} \int_{0}^{x} dz > x \int_{0}^{\pi} \sin^2 x \, dx$ is :

An ideal heat engine operates on a Carnot cycle between $227\,^{\circ}C$ and $127\,^{\circ}C$. It absorbs $6 \times 10^4\, \text{cal}$ at the higher temperature. The amount of heat converted into work is equal to: