The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is...

Consider the data on x taking the values 0, 2, 4, 8, ...., 2ⁿ with frequencies ⁿC₀, ⁿC₁, ⁿC₂,..., ⁿC_n respectively. If the mean of this data is $\frac{728}{2^{\mathrm{n}}}$, then n is equal to....

23.

Suppose that a function f : R $\to$ R satisfies f(x + y) = f(x)f(y) for all x, y $\in$ R and f(1) = 3. If $\sum _{\mathrm{i} = 1}^{\mathrm{n}}\mathrm{f}\left(\mathrm{i}\right) = 363$, then n is equal to.......

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{x}} \mathrm{and} \overrightarrow{\mathrm{y}} \mathrm{be} \mathrm{two} \mathrm{non}-\mathrm{zero} \mathrm{vectors} \mathrm{such} \mathrm{that} \left|\overrightarrow{\mathrm{x}} + \overrightarrow{\mathrm{y}}\right| = \left|\overrightarrow{\mathrm{x}}\right|\mathrm{and} 2\overrightarrow{\mathrm{x}} + \mathrm{\lambda }\overrightarrow{\mathrm{y}} \mathrm{is} \mathrm{perpendicular} \\ \mathrm{to} \mathrm{y}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{is}...... \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{distinct} \mathrm{values} \mathrm{of} \mathrm{l} \mathrm{for} \mathrm{which} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations}: \\ (\mathrm{\lambda } – 1)\mathrm{x} + (3\mathrm{\lambda } + 1)\mathrm{y} + 2\mathrm{\lambda z} = 0 \\ (\mathrm{\lambda }– 1)\mathrm{x} + (4\mathrm{\lambda } – 2)\mathrm{y} + (\mathrm{\lambda } + 3)\mathrm{z} = 0 \\ 2\mathrm{x} + (3\mathrm{\lambda } + 1)\mathrm{y} + 3(\mathrm{\lambda } – 1)\mathrm{z} = 0, \\ \mathrm{Has} \mathrm{non}-\mathrm{zero} \mathrm{solutions}, \mathrm{is}... \end{gathered}$$