The radius of the circle $r = 12 \cos \theta + 5 \sin \theta$ is

If $x$ is numerically so small that $x^2$ and higher powers of $x$ can be neglected,then $\left(1+\frac{2 x}{3}\right)^{3 / 2} \cdot(32+5 x)^{-1 / 5}$ is approximately equal to

Observe the following reaction: $2 A + B \longrightarrow C$. The rate of formation of $C$ is $2.2 \times 10^{-3} \ mol \ L^{-1} \ min^{-1}$. What is the value of $-\frac{d[A]}{dt}$ (in $mol \ L^{-1} \ min^{-1}$)?

The equation of the tangent at $(-4, -4)$ on the curve $x^2 = -4y$ is

In a compound microscope,the focal lengths of the objective and the eye lens are $2.5 \, cm$ and $5 \, cm$ respectively. An object is placed at $3.75 \, cm$ before the objective and the image is formed at the least distance of distinct vision. Then,the distance between the two lenses (i.e.,the length of the microscope tube) will be......$cm$.

The position vector of a point lying on the line joining the points whose position vectors are $\hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is: