A die is rolled. If $X$ denotes the number of positive divisors of the outcome then the range of the random variable $X$ is

If one of the lines given by the equation \(x^{2}+k x y+2 y^{2}=0\) is \(x+2 y=0\), then \(\mathrm{k}=\)

Find a polynomial \(f(x)\) of degree 2 where \(f(0)=8, f(1)=12, f(2)=18\)

If \(y=\tan ^{-1}\left[\sqrt{\frac{1+\cos \frac{x}{2}}{1-\cos \frac{x}{2}}}\right]\), then \(\frac{d y}{d x}=\)

If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are three non coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}, \overrightarrow{\mathbf{r}}\) are defined by the relations
\(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \quad \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\)
then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{p}}+\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{q}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{r}}\) is equal to

If the population grows at the rate \(5 \%\) peryear, then the time taken for the population
to become double is \(\quad\) (Given \(\log 2=0.6912\) )

The vector projection of \(\overline{\mathrm{AB}}\) on \(\overline{\mathrm{CD}}\), where \(A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)\) and \(\mathrm{D} \equiv(7,0,10)\), is

\(\int \frac{\operatorname{cosec} x \mathrm{~d} x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)}=\)

Calculate the radius of an atom of an element in pm if it forms bcc unit cell structure having edge length \(4.3 \times 10^{-8} \mathrm{~cm}\).

_________ is an intragenic interaction .

Identify the structure of \(\mathrm{XeF}_4\) molecule from following.

If a colour blind man marries a normal visioned woman, what is the percentage of offsprings showing colour blindness phenotypically?

In biprism experiment, if \(5^{\text {th }}\) bright band with wavelength \({ }^{\prime} \lambda_{1}{ }^{\prime}\) coincides with \(6^{\text {th }}\) dark band with wavelength \(^{\prime} \lambda_{2}\) ' then the ratio \(\left(\frac{\lambda_{2}}{\lambda_{1}}\right)\) is