21.

The differential equation for the family of curves,x² +y² -2ay = 0  where a is an arbitrary constant is

• 2(x²-y²)y' = xy

• (x²+y²)y' = xy

• 2(x²+y²)y' = xy

• 2(x²+y²)y' = xy

The solution of the differential equation ydx + (x + x²y) dy = 0 is

• -1/ XY  =C

• -1/XY + log y = C

• 1/XY + log y = C

• 1/XY + log y = C

If $\sum _{i=1}^9 (x_i -5) = 9 and \sum _{i = 1}^9(x_i - 5{)}^2 = 45$ then the
standard deviation of the 9 items x₁, x₂, ...., x₉ is

• 3

• 9

• 4

• 2

Let f(x) = $$\begin{gathered} \left\{\frac{{\mathrm{x}}^{\mathrm{p}}}{{\left(\sin \left(\mathrm{x}\right)\right)}^4}, \mathrm{if} 0 < \mathrm{x} \le \frac{\mathrm{\pi }}{2} \\ 0 , \mathrm{if} \mathrm{x} = 0\right\} \end{gathered}$$

(p, q $\in$ R). Then,  Lagrange's mean value theorem is applicable tof(x) in closed interval [ 0, x]

• for all p, q

• only when p > q

• only when p < q

• for no value of p, q

$\underset{\mathrm{x}\to 0}{\lim }{\left(\sin \left(\mathrm{x}\right)\right)}^{2\tan \left(\mathrm{x}\right)}$ is equal to

• 2

• 1

• 0

• does not exist

Let for all x> 0, f(x)=$\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left({\mathrm{x}}^{1/\mathrm{n}} - 1\right)$, then

• $\mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(\frac{1}{\mathrm{x}}\right) = 1$

• $\mathrm{f}\left(\mathrm{xy}\right) = \mathrm{f}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{f}\left(\mathrm{y}\right)$

• $\mathrm{f}\left(\mathrm{xy}\right) = \mathrm{xf}\left(\mathrm{y}\right) +\hspace{0.17em}\mathrm{yf}\left(\mathrm{x}\right)$

• f(xy) = xf(x) + yf(y)

The value of K in order that f (x) = sin(x) - cos(x) - Kx + 5 decreases for all positive real values of x is given by

• K<1

• $\mathrm{K} \ge 1$

• $\mathrm{K} > \sqrt{2}$

• K < $\sqrt{2}$

Let f : R ➔ R be twice continuously differentiable. Let f(0) = f(D) = f'(0) = 0. Then,

• f''(x) $\ne$ 0 for all x

• f''(c) = 0 for some $\mathrm{c} \in \mathrm{R}$

• f''(x) $\ne 0 \mathrm{if} \mathrm{x} \ne 0$

• f'(x) > 0 for all x

If f(x) =xⁿ, n being a non-negative integer, then the values of n for which $\mathrm{f}\text{'}(\mathrm{\alpha } + \mathrm{\beta }) = \mathrm{f}\text{'}\left(\mathrm{\alpha }\right) + \mathrm{f}\text{'}\left(\mathrm{\beta }\right)$ for all $\mathrm{\alpha }, \mathrm{\beta } > 0$ is

• 1

• 2

• 0

• 5

If y = (1 + x)(1 + x²)(1 + x⁴)...(1 + x²ⁿ), then the value of $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$ at x = 0 is

• 0

• - 1

• 1

• 2