The displacement of a particle executing $SHM$ is given by $y=5 \sin \left(4 t+\frac{\pi}{3}\right)$. If $T$ is the time period and the mass of the particle is $2 \text{ g}$,the kinetic energy of the particle when $t=\frac{T}{4}$ is given by (in $J$)

Evaluate $\int_{-2}^1 f(x) dx$,where $f(x)= \begin{cases} 1-2x, & x \leq 0 \\ 1+2x, & x > 0 \end{cases}$

$A$ cylinder of height $h$ is filled with water and is kept on a block of height $h / 2$. The level of water in the cylinder is kept constant. Four holes numbered $1, 2, 3$ and $4$ are at the side of the cylinder and at heights $0, h / 4, h / 2$ and $3h / 4$ respectively from the base of the cylinder. When all four holes are opened together,the hole from which water will reach the farthest distance on the plane $PQ$ is the hole number

If $y=(1+x)(1+x^2)(1+x^4) \dots (1+x^{2^n})$,then $(\frac{dy}{dx})_{x=0}$ is equal to

$A$ point is moving on $y=4-2x^2$. The $x$-coordinate of the point is decreasing at the rate of $5 \text{ units/s}$. Then,the rate at which the $y$-coordinate of the point is changing when the point is at $(1,2)$ is

Which of the following is an example of a heterogeneous catalysis reaction?