Which of the following options is not the asymptote of the curve 3x³ + 2x²y- 7xy² + 2y³ - 14xy + 7y² + 4x + 5y = 0?

• y = $\frac{- 1}{2}\mathrm{x} - \frac{5}{6}$

• y = $\mathrm{x} - \frac{7}{6}$

• y = $2\mathrm{x} + \frac{3}{7}$

• y = $3\mathrm{x} - \frac{3}{2}$

The two lines x = my + n, z = py + q and x = m'y + n', z =p'y + q' are perpendicular to each other, if

• mm' + pp' = 1

• $\frac{\mathrm{m}}{\mathrm{m}\text{'}} + \frac{\mathrm{p}}{\mathrm{p}\text{'}} = - 1$

• $\frac{\mathrm{m}}{\mathrm{m}\text{'}} + \frac{\mathrm{p}}{\mathrm{p}\text{'}} = 1$

• mm' + pp' = - 1

If p, q, r are simple propositions with truth values T, F, T, then the truth value of $\left(~ \mathrm{p} \vee \mathrm{q}\right) \wedge ~ \mathrm{r} \Rightarrow \mathrm{p}$ is

• true

• false
  
  

• true, if r is false

• true, if q is true

4.

If $\left|\mathrm{z}\right| \ge 3$, then the least value of $\left|\mathrm{z} + \frac{1}{4}\right|$

• $\frac{11}{2}$

• $\frac{11}{4}$

• 3

• $\frac{1}{4}$

The normal at the point (${{\mathrm{at}}_1}^2$, 2at₁) on the parabola meets the parabola again in the point (${{\mathrm{at}}_2}^2, 2{\mathrm{at}}_2$), then

• ${\mathrm{t}}_2 = - {\mathrm{t}}_1 + \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = - {\mathrm{t}}_1 - \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = {\mathrm{t}}_1 - \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = {\mathrm{t}}_1 + \frac{2}{{\mathrm{t}}_1}$

If a = $\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$, then the angle $\mathrm{\theta }$ between a and b is given by

• ${\tan }^{-1}\left(1\right)$

• ${\sin }^{-1}\left(\frac{1}{2}\right)$

• ${\sec }^{-1}\left(1\right)$

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

If the rectangular hyperbola is x² - y² = 64. Then, which of the following is not correct?

• The length of latusrectum is 16

• The eccentricity is $\sqrt{2}$

• The asymptotes are parallel to each other

• The directrices are x = $\pm 4\sqrt{2}$

The equation of tangents to the hyperbola 3x² - 2y² = 6, which is perpendicular to the line x - 3y = 3, are

• $\mathrm{y} = - 3\mathrm{x} \pm \sqrt{15}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{6}$

• $\mathrm{y} = - 3\mathrm{x} \pm \sqrt{6}$

• $\mathrm{y} = 2\mathrm{x} \pm \sqrt{15}$

$\underset{\mathrm{x}\to \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\lim }\frac{\tan \left(\mathrm{x}\right) - 1}{\cos \left(2\mathrm{x}\right)}$ is equal to

• 1

• 0

• - 2

• - 1

If the mean and variance of a binomial distribution are 4 and 2, respectively. Then, the probability of atleast 7 successes is

• $\frac{3}{214}$

• $\frac{4}{173}$

• $\frac{9}{256}$

• $\frac{7}{231}$