31.

The area in the first quadrant between x² + y² = ${\mathrm{\pi }}^2$ and y = sin(x) is

• 

• 

• 

• 

The area bounded by y = xe^lxl and lines lxl = 1, y = 0 is,

• 4 sq units

• 6 sq units

• 1 sq unit

• 2 sq unit

The solution of $\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{{\mathrm{x}}^2 + {\mathrm{y}}^2 + 1}{2\mathrm{xy}}$, satisfying y(1) = 0 is given by

• hyperbola

• circle

• ellipse

• parabola

If $\mathrm{x} . \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = \mathrm{x} . \frac{\mathrm{f}\left(\mathrm{xy}\right)}{\mathrm{f}\text{'}\left(\mathrm{xy}\right)}$, then f(xy) is equal to

• $\mathrm{k} . {\mathrm{e}}^{\frac{{\mathrm{x}}^2}{2}}$

• $\mathrm{k} . {\mathrm{e}}^{\raisebox{1ex}{${\mathrm{y}}^2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• $\mathrm{k} . {\mathrm{e}}^{x^2}$

• $\mathrm{k} . {\mathrm{e}}^{\frac{\mathrm{xy}}{2}}$

The differential equation of the rectangular hyperbola, where axes are the asymptotes of the hyperbola, is

• $\mathrm{y}\frac{d\mathrm{u}}{d\mathrm{x}} = \mathrm{x}$

• $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = - \mathrm{y}$

• $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{y}$

• xdy + ydx = c

The length of longer diagonal of the parallelogram constructed on 5a + 2b and a - 3b, if it is given that $\left|\mathrm{a}\right| = 2\sqrt{2}, \left|\mathrm{b}\right| = 3$and the angle between a and b is $\frac{\mathrm{\pi }}{4}$, is

• 15

• $\sqrt{113}$

• $\sqrt{593}$

• $\sqrt{369}$

If $\mathrm{r} = \mathrm{\alpha b} \times \mathrm{c} + \mathrm{\beta c} \times \mathrm{a} + \mathrm{\gamma a} \times \mathrm{b}$ and [a b c] = 2, then $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma }$ is equal to

• $\mathrm{r} . \left[\mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} + \mathrm{a} \times \mathrm{b}\right]$

• $\frac{1}{2}\mathrm{r} . \left(\mathrm{a} +\hspace{0.17em}\mathrm{b} + \mathrm{c}\right)$

• 2r . (a + b + c)

• 4

If a, b, c are three non-coplanar vectors and p, q, rare reciprocal vectors, then (la + mb + nc) · (lp + mq + nr) is equal to

• l + m + n

• l³ + m³ + n³

• l² + m² + n²

• None of these

If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7ⁿ + 7^m is divisible by 5, equals to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\frac{1}{3}$

Let X denote the sum of the numbers obtained when two fair dice are rolled. The variance and standard deviation of X are

• $\frac{31}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 31}}{{\displaystyle 6}}}$

• $\frac{35}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 35}}{{\displaystyle 6}}}$

• $\frac{17}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 17}}{{\displaystyle 6}}}$

• $\frac{31}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 35}}{{\displaystyle 6}}}$