81.

The area of the region bounded by y² = 4ax and x² = 4ay, a > 0 in sq unit, is :

• 16$\frac{{\mathrm{a}}^2}{3}$

• $14\frac{{\mathrm{a}}^2}{3}$

• $13\frac{{\mathrm{a}}^2}{3}$

• 16a²

The area (in sq. units) of the region A = $\left\{\left(\mathrm{x}, \mathrm{y}\right) \ge \mathrm{R} \times \mathrm{R} | 0 \le \mathrm{x} \le 3, 0 \le \mathrm{y} \le 4, \mathrm{y} \le {\mathrm{x}}^2 + 3\mathrm{x}\right\}$ is :

• $\frac{59}{6}$

• $\frac{53}{6}$

• 8

• $\frac{26}{3}$

Let S$\left(\mathrm{\alpha }\right) = \left\{\left(\mathrm{x}, \mathrm{y}\right) : {\mathrm{y}}^2 \le \mathrm{x}, 0 \le \mathrm{x} \le \mathrm{\alpha }\right\}$ and $\mathrm{A}\left(\mathrm{\alpha }\right)$is area of the region $\mathrm{S}\left(\mathrm{\alpha }\right)$. If for a $\mathrm{\lambda }, 0 \mathrm{n}< \mathrm{\lambda } < 4, \mathrm{A}\left(\mathrm{\lambda }\right) : \mathrm{A}\left(4\right) = 2 : 5$, then $\mathrm{\lambda }$ equals :

• $2{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$

• $4{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$

• $2{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$

• $4{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$

The area (in sq.units) of the region A = $\left\{\left(\mathrm{x}, \mathrm{y}\right) : \frac{{\mathrm{y}}^2}{2} \le \mathrm{x} \le \mathrm{y} +\hspace{0.17em}4\right\}$ is

• 18

• 16

• $\frac{53}{3}$

• 30

The area (in sq. units) of the region {9x, y) : x² $\le \mathrm{y} \le \mathrm{x}$ + 2} is :

• $\frac{9}{2}$

• $\frac{10}{3}$

• $\frac{13}{6}$

• $\frac{31}{6}$

The area (in sq.units ) of the region bounded by the curves y = 2^x and $\mathrm{y} = \left|\mathrm{x} + 1\right|$,in the first quadrant is:

• $\frac{3}{2} - \frac{1}{{\log }_{\mathrm{e}}\left(2\right)}$

• ${\log }_{\mathrm{e}}\left(2\right) +\hspace{0.17em}\frac{3}{2}$

• $\frac{1}{2}$

• $\frac{3}{2}$

Area bounded by the lines y = x, x = - 1, x = 2 and x - axis is

• 5/2 sq unit

• 3/2 sq unit

• 1/2 sq unit

• None of the above

The volume of solid generated by revolving about the y-axis the figure bounded by the parabola y = x and x = y² is

• $\frac{21}{5}\mathrm{\pi }$

• $\frac{24}{5}\mathrm{\pi }$

• $\frac{2}{15}\mathrm{\pi }$

• $\frac{5}{24}\mathrm{\pi }$

Area bounded between the curve x² = y and the line y = 4x is

• $\frac{32}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{8}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{16}{3} \mathrm{sq} \mathrm{unit}$

   

The volume of the solid formed by rotating the area enclosed between the curve y² = 4x, x = 4 and x = 5 about x-axis is (in cubic units)

• $18\mathrm{\pi }$

• $36\mathrm{\pi }$

• $9\mathrm{\pi }$

• $24\mathrm{\pi }$