31.

$\mathrm{y} = {\mathrm{e}}^{{\mathrm{asin}}^{-1}\left(\mathrm{x}\right)} \Rightarrow \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_{\mathrm{n} + 2} - \left(2\mathrm{n} + 1\right){\mathrm{xy}}_{\mathrm{n} + 1}$ is equal to

• $- \left({\mathrm{n}}^2 + {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 - {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 + {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $- \left({\mathrm{n}}^2 - {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

The function f(x) = x³ + ax + bx + c, a² $\le$ 3b has

• one maximum value

• one minimum value

• no extreme value

• one maximum and one minimum value

In a $∆\mathrm{ABC}$, $\frac{\left(\mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} + \mathrm{a} - \mathrm{b}\right)\left(\mathrm{a} + \mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2}$ equals

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

• ${\sin }^2\left(\mathrm{B}\right)$

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l² + m² - n² = 0 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

If m₁, m₂, m₃ and m₄ are respectively the magnitudes of the vectors

$$\begin{gathered} \overrightarrow{{\mathrm{a}}_1} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{{\mathrm{a}}_2} = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ \overrightarrow{{\mathrm{a}}_3} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{{\mathrm{a}}_4} = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \end{gathered}$$

then the correct order of m₁, m₂, m₃ and m₄ is

• m₃ < m₁ < m₄ < m₂

• m₃ < m₁ < m₂ < m₄

• m₃ < m₄ < m₁ < m₂

• m₃ < m₄ < m₂ < m₁

$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{atan}}^{-1}\left(\mathrm{x}\right) + \mathrm{blog}\left(\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right)\right] = \frac{1}{{\mathrm{x}}^4 - 1}$ $\Rightarrow \mathrm{a} - 2\mathrm{b}$ is equal to

• 1

• - 1

• 0

• 2

$\int \left(\frac{2 - \sin \left(2\mathrm{x}\right)}{1 - \cos \left(2\mathrm{x}\right)}\right){\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to

• $- {\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

• $2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

• $- 2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int {\sin }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}, \mathrm{then} {\mathrm{nI}}_{\mathrm{n}} - \left(\mathrm{n} - 1\right){\mathrm{I}}_{\mathrm{n} - 2}$ equals

• ${\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\mathrm{xcos}\left(\mathrm{x}\right)$

• ${\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

• $- {\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $- {\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

The line $\mathrm{x} = \frac{\mathrm{\pi }}{4}$ divides the area of the region bounded by y = sin(x), y = cos(x) and x - axis $\left(0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2}\right)$ into two regions of areas A₁ and A₂. Then, A₁ : A₂ equals

• 4 : 1

• 3 : 1

• 2 : 1

• 1 : 1

The solution of the differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} = \sin \left(\mathrm{x} + \mathrm{y}\right)\tan \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) - 1$ is

• $\csc \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \tan \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{x} +\hspace{0.17em}\mathrm{c}$

• $\mathrm{x} + \csc \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{c}$

• $\mathrm{x} + \tan \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{c}$

• $\mathrm{x} + \sec \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{c}$