The point to which the origin is to be shifted to remove the first degree terms from the equation 2x² + 4xy - 6y² + 2x + 8y + 1 = 0 is

• $\left(\frac{7}{8}, \frac{3}{8}\right)$

• $\left(\frac{- 7}{8}, \frac{ - 3}{8}\right)$

• $\left(\frac{- 7}{8}, \frac{3}{8}\right)$

• $\left(\frac{7}{8}, \frac{ - 3}{8}\right)$

If lx + my = 1 is a normal to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$, then a²m² - b²l² = ?

• $\frac{{\mathrm{m}}^2}{{\mathrm{l}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l² + m²(a² + b²)²

• $\frac{{\mathrm{l}}^2}{{\mathrm{m}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l²m²(a² + b²)²

$\frac{\mathrm{x} - 1}{3\mathrm{x} + 4} < \frac{\mathrm{x} - 3}{3\mathrm{x} - 2} \mathrm{holds}, \mathrm{for} \mathrm{all} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{internal}$

• $\left(\frac{ - 4}{3}, \frac{2}{3}\right)$

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

• $\left(\frac{3}{3}, \infty \right)$

• $\left( - \infty , \frac{ - 5}{4}\right) \cup \left(\frac{3}{4}, - \infty \right)$

There are 10 intermediate stations on a railway line between two particular stations. The number ofways that a train can be made to stop at 3 of these intermediate stations so that no two of these halting stations are consecutive is

• 56

• 20

• 126

• 120

The figure formed by the pairs of lines
$6{\mathrm{x}}^2 + 13\mathrm{xy} + 6{\mathrm{y}}^2 = 0$ and
6x² + 13xy + 6y + 10x + 10y + 4 = 0, is a

• Square

• Parallelogram

• Rhombus

• Rectangle

If the point of intersection of the tangents drawn at the points where the line 5x + y + 1 = 0 cuts the circle x² + y² - zx - 6y - 8 = 0 is (a, b), then 5a + b =

• 3

•  - 44

•  - 1

• 4

$\underset{\mathrm{x}\to 0}{\lim } \frac{\left(1 - \cos \left(2\mathrm{x}\right)\right)\left(3 + \cos \left(\mathrm{x}\right)\right)}{\mathrm{xtan}\left(4\mathrm{x}\right)} = ?$

• $\frac{ - 1}{4}$

• $\frac{1}{2}$

• 1

• 2

28.

$\mathrm{An} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{curves} {\mathrm{x}}^2=3\mathrm{y} \mathrm{and} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 4 \mathrm{is}$

• ${\tan }^{-1}\left(\frac{5}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\sqrt{\frac{5}{3}}\right)$

• ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• $\frac{\mathrm{\pi }}{3}$

The number of different ways of preparing a garland using 6 distinct white roses and 5 distinct red roses such that no two red roses come together is

• 21600

• 43200

• 86400

• 151200

$$\begin{gathered} \mathrm{If} \tan \left(\mathrm{\alpha }\right) \mathrm{and} \tan \left(\mathrm{\beta }\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + \mathrm{px} + \mathrm{q} = 0, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \\ {\sin }^2\left(\mathrm{\alpha } + \mathrm{\beta }\right) + \mathrm{pcos}\left(\mathrm{\alpha } + \mathrm{\beta }\right)\sin \left(\mathrm{\alpha } + \mathrm{\beta }\right) + {\mathrm{qcos}}^2\left(\mathrm{\alpha } + \mathrm{\beta }\right) \end{gathered}$$

• p + q

• p

• q

• $\frac{\mathrm{p}}{\mathrm{p} + \mathrm{q}}$