The angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=6$ is $\cos ^{-1} \sqrt{\frac{5}{14}}$, then the value of $\lambda$ is

The length and breadth of a rectangle are $x_{x \mathrm{~cm}}$ and $y \mathrm{~cm}$ respectively. If the length decreases at the rate of $5 \mathrm{~cm} /$ minute and the breadth increases at the rate of $3 \mathrm{~cm} /$ minute, then the rates of change of the perimeter and area respectively when the length is 5 cm and breadth is 2 cm , are

If $\mathrm{f}(x)=3 x^3+2 x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+\mathrm{f}^{\prime \prime \prime}(3)$ then $\mathrm{f}(x)=$ __________ .

If $x=\sin \theta, y=\sin ^3 \theta$, then $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{6}$ is