The product of the distinct $(2n)^{\text{th}}$ roots of $1+i\sqrt{3}$ is equal to:

If $\hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+2 \hat{j}+\hat{k}$ are sides of a parallelogram,then a unit vector parallel to one of the diagonals of the parallelogram is

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3},$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

Let $f: R \rightarrow R$ be defined by $f(x) = \begin{cases} x+2, & x \leq -1 \\ x^2, & -1 < x < 1 \\ 2-x, & x \geq 1 \end{cases}$. Then the value of $f(-1.75) + f(0.5) + f(1.5)$ is

$A$ uniform cube of side $a$ and mass $m$ rests on a rough horizontal table. $A$ horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the face,at a height $\frac{3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to tilt about the edge is (Assume that the cube does not slide):

With a standard rectangular bar magnet,the time period of a vibration magnetometer is $4 ~s$. The bar magnet is cut parallel to its length into four equal pieces. The time period of the vibration magnetometer when one piece is used (in seconds) (bar magnet breadth is small) is