71.

$\mathrm{If} \frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xtan}\left(\mathrm{x} - \mathrm{y}\right) = 1, \mathrm{then} \sin \left(\mathrm{x} - \mathrm{y}\right) = ?$

• ${\mathrm{Ae}}^{- {\mathrm{x}}^2}$

• Ae^2x

• ${\mathrm{Ae}}^{{\mathrm{x}}^2}$

• Ae ^{- 2x}

$$\begin{gathered} \mathrm{An} \mathrm{integrating} \mathrm{factor} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \left(1 - {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = \frac{{\mathrm{x}}^4}{\left(1 + {\mathrm{x}}^5\right)}{\left(\sqrt{1 - {\mathrm{x}}^2}\right)}^3 \mathrm{is} \end{gathered}$$

• $\sqrt{1 - {\mathrm{x}}^2}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{{\mathrm{x}}^2}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left|\begin{array}{cc}2& {\mathrm{e}}^{\mathrm{i\pi }}\\ - 1& {\mathrm{i}}^{2012}\end{array}\right|, \mathrm{C}= \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)\mathrm{x} = 1, \\ \mathrm{D} = {\int }_{{\mathrm{e}}^2}^1\frac{\mathrm{dx}}{\mathrm{x}}. \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{Ax}}^3 + {\mathrm{Bx}}^2 + \mathrm{Cx} - \mathrm{D} = 0 \\ \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{zero}, \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• - 1

• 0

• 1

• 2

$\mathrm{a} = \mathrm{i} + \mathrm{j} - 2\mathrm{k} \Rightarrow \sum _{}{\left\{\left(\mathrm{a} \times \mathrm{i}\right) \times \mathrm{j}\right\}}^2 = ?$

• $\sqrt{6}$

• 6

• 36

• $6\sqrt{6}$

Let a, b and c be three non-coplanar vectors and let p, q and r be the vectors defined by

$$\begin{gathered} \mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right]}, \mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right]}, \mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{abc}\right]} \mathrm{Then}, \\ \left(\mathrm{a} + \mathrm{b}\right) . \mathrm{p} + \left(\mathrm{b} + \mathrm{c}\right) . \mathrm{q} + \left(\mathrm{c} + \mathrm{a}\right) . \mathrm{r} = ? \end{gathered}$$

• 0

• 1

• 2

• 3

Let a = i + 2j + k, b = i - j + k, c = i + j - k.

A vector in the plane of a and b has projection $\frac{1}{\sqrt{3}}$ on c. Then, one such vector is

• 4i + j - 4k

• $3\mathrm{i} + \mathrm{j} - 3\mathrm{k}$

• 4i - j + 4k

• 2i + j + 2k

The point if intersection of the lines

l₁ : r(t) = (i - 6j + 2k) + t(i + 2j + k)

l₂ : R(u) = (4j + k) + u(2i + j + 2k) is

• (10, 12, 11)

• (4, 4, 5)

• (6, 4, 7)

• (8, 8, 9)

The vectors AB = 3i - 2j + 2k and BC = i - 2k are the adjacent sides of a parallelogram. The angle between its diagonals is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3} \mathrm{or} \frac{2\mathrm{\pi }}{3}$

• $\frac{3\mathrm{\pi }}{4} \mathrm{or} \frac{\mathrm{\pi }}{4}$

• None of these

Consider the circle x² + y - 4x - 2y + c = 0 whose centre is A(2, 1). If the point P(10, 7) is such that the line segment PA meets the circle in Q with PQ = 5, then c is equal to

• - 15

• 20

• 30

• - 20

A vertical pole subtends an angle ${\tan }^{-1}\left(\frac{1}{2}\right)$ at a point P on the ground. If the angles substended by the upper half and the lower half of the pole at P are respectively $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ , then $\left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\beta }\right)\right)$ is equal to

• $\left(\frac{1}{4}, \frac{1}{5}\right)$

• $\left(\frac{1}{5}, \frac{2}{9}\right)$

• $\left(\frac{2}{9}, \frac{1}{4}\right)$

• $\left(\frac{1}{4}, \frac{2}{9}\right)$