The mid point of the chord 4x - 3y = 5 of the hyperbola 2x² - 3y² = 12 is

• $\left(0, - \frac{5}{3}\right)$

• (2, 1)

• $\left(\frac{5}{4}, 0\right)$

• $\left(\frac{11}{4}, 2\right)$

42.

$$\begin{gathered} \mathrm{The} \mathrm{eccentricity} \mathrm{of} \mathrm{conic} \\ \frac{5}{\mathrm{r}} = 2 +\hspace{0.17em}3\cos \left(\mathrm{\theta }\right) + 4\sin \left(\mathrm{\theta }\right) \mathrm{is} \end{gathered}$$

• $\frac{1}{2}$

• 1

• $\frac{3}{2}$

• $\frac{5}{2}$

$\mathrm{The} \mathrm{perimeter} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{with} \mathrm{vertices} \mathrm{at} \left(1, 0, 0\right), \left(0, 1, 0\right) \mathrm{and} \left(0, 0, 1\right) \mathrm{is}$

• 3

• 2

• $2\sqrt{2}$

• $3\sqrt{2}$

$$\begin{gathered} \mathrm{If} \mathrm{a} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{space} \mathrm{makes} \mathrm{angle} \mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma } \mathrm{with} \mathrm{the} \mathrm{coordinate} \mathrm{axes}, \mathrm{then} \\ \cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) +\hspace{0.17em}{\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right) =? \end{gathered}$$

• - 1

• 0

• 1

• 2

$\mathrm{The} \mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{sphere} {\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 = 12\mathrm{x} + 4\mathrm{y} +\hspace{0.17em}3\mathrm{z} \mathrm{is}$

• $\frac{13}{2}$

• 13

• 26

• 52

$\mathrm{x} = \frac{1 - \sqrt{\mathrm{y}}}{1 + \sqrt{\mathrm{y}}} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{4}{{\left(\mathrm{x} + 1\right)}^2}$

• $\frac{4\left(\mathrm{x} - 1\right)}{{\left(\mathrm{x} + 1\right)}^3}$

• $\frac{\left(\mathrm{x} - 1\right)}{{\left(1 + \mathrm{x}\right)}^3}$

• $\frac{4}{{\left(1 + \mathrm{x}\right)}^3}$

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + 5}{\mathrm{x} +\hspace{0.17em}2}\right)}^{\mathrm{x} + 3}$ equals

• e

• e²

• e³

• e⁵

If f(x) = 2x⁴ - 13x² + ax + b is divisible by x² - 3x + 2, then (a, b) is equal to

• (- 9, - 2)

• (6, 4)

• (9, 2)

• (2, 9)

Let A and B be two symmetric matrices of same order. Then, the matrix AB - BA is

• a symmetnc matrix

• a skew-symmetric matrix

• a null matrix

• the identity matrix

If one of the roots of $\left|\begin{array}{ccc}3& 5& \mathrm{x}\\ 7& \mathrm{x}& 7\\ \mathrm{x}& 5& 3\end{array}\right|$ = 0 is - 10, then the other roots are

• 3, 7

• 4, 7

• 3, 9

• 3, 4