The lines represented by $ax^2 + 2hxy + by^2 = 0$ are perpendicular to each other,if ........

Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$,the ratio of the lengths of the strings is :

The product $(A)$ is:

When a spring is stretched by $2 \ cm$,it stores $100 \ J$ of energy. If it is stretched further by $2 \ cm$,the stored energy will be increased by ............. $J$.

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

Dividing the interval $[0,6]$ into $6$ equal parts and by using the trapezoidal rule, the value of $\int_0^6 x^3 dx$ is approximately: