21.

If the tangent to the curve, y = e^x at a point (c, e^c) and the normal to the parabola, y² = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c is.....

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to :

$$\begin{gathered} \mathrm{Let} \mathrm{a} \mathrm{plane} \mathrm{P} \mathrm{contain} \mathrm{two} \mathrm{lines} \\ \stackrel{\rightharpoonup }{\mathrm{r}} = \hat{\mathrm{i}} + \mathrm{\lambda }\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right), \mathrm{\lambda } \in \mathrm{R} \mathrm{and} \\ \overrightarrow{\mathrm{r}} = - \hat{\mathrm{j}} + \mathrm{\mu }\left(\hat{\mathrm{j}} - \hat{\mathrm{k}}\right), \mathrm{\mu } \in \mathrm{R} \\ \mathrm{If} \mathrm{Q}\left(\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }\right) \mathrm{is} \mathrm{the} \mathrm{foot} \mathrm{of} \mathrm{the} \mathrm{perpendicular} \mathrm{drawn} \mathrm{from} \mathrm{the} \mathrm{point} \mathrm{equals}... \end{gathered}$$

The total number of 3-digit numbers, whose sum of digits is 10, is.......

$$\begin{gathered} \mathrm{Let} \mathrm{S} \mathrm{be} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{integer} \mathrm{solutions}, (\mathrm{x}, \mathrm{y}, \mathrm{z}), \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ \mathrm{x} – 2\mathrm{y} + 5\mathrm{z} = 0 \\ – 2\mathrm{x} + 4\mathrm{y} + \mathrm{z} = 0 \\ – 7\mathrm{x} + 14\mathrm{y} + 9\mathrm{z} = 0 \\ \mathrm{Such} \mathrm{that} 15 < {\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 < 150. \mathrm{Then}, \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{the} \mathrm{set} \\ \mathrm{S} \mathrm{is} \mathrm{equal} \mathrm{to}. \end{gathered}$$