1.

Let L₁ be a tangent to the parabola y² = 4(x + 1) and L₂ be a tangent to the parabola y² = 8(x + 2) such that L₁ and L₂ intersect at right angles. Then L₁ and L₂ meet on the straight line :

• x + 3 = 0

• 2x + 1 = 0

• x + 2y = 0

• x + 2 = 0

$$\begin{gathered} \mathrm{If} \sum _{\mathrm{i} = 1}^{\mathrm{n}}\left({\mathrm{x}}_{\mathrm{i}} - \mathrm{a}\right) = \mathrm{n} \mathrm{and} \sum _{\mathrm{i} = 1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}} - \mathrm{a}\right)}^2 = \mathrm{na}, \left(\mathrm{n}, {\mathrm{a}}_1 > 1\right) \mathrm{then} \mathrm{the} \mathrm{standard} \mathrm{deviation} \mathrm{of} \mathrm{n} \\ \mathrm{observations} {\mathrm{x}}_1, {\mathrm{x}}_2, ... {\mathrm{x}}_{\mathrm{n}} \mathrm{is} : \end{gathered}$$

• a - 1

• $\sqrt{\mathrm{n}\left(\mathrm{a} - 1\right)}$

• $\mathrm{n}\sqrt{\left(\mathrm{a} - 1\right)}$

• $\sqrt{\left(\mathrm{a} - 1\right)}$

$\mathrm{A} \underset{\mathrm{x}\to 1}{\lim }\left(\frac{{\int }_0^{{\left(\mathrm{x} - 1\right)}^2}\mathrm{tcos}\left({\mathrm{t}}^2\right)d\mathrm{t}}{\left(\mathrm{x} - 1\right)\sin \left(\mathrm{x} - 1\right)}\right) = ?$

• 1

• does not exist

• $\frac{1}{2}$

• $- \frac{1}{2}$

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,  $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{2} = 1$ from any of its foci?

• $\left( - 2, \sqrt{3}\right)$

• $\left( - 1, \sqrt{2}\right)$

• (1, 2)

• $\left( - 1, \sqrt{3}\right)$

Out of 11 consecutive natural number if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is :

• $\frac{10}{99}$

• $\frac{15}{101}$

• $\frac{5}{33}$

• $\frac{5}{101}$

Let a , b, c, d and ay non zero distinct real numbers such that (a² + b² + c²)p² – 2(ab + bc + cd)p + (b² + c² + d²) = 0. Then :

• a, b, c, d are in A.P.

• a, c, p are in G.P.

• a, b, c, d are in G.P.

• a, c, p, are in A.P.

$\mathrm{If} \left\{\mathrm{p}\right\} \mathrm{denotes} \mathrm{the} \mathrm{fractional} \mathrm{part} \mathrm{pf} \mathrm{the} \mathrm{number} \mathrm{p}, \mathrm{then} \left\{\frac{3^{200}}{8}\right\} = ?$

• $\frac{1}{8}$

• $\frac{3}{8}$

• $\frac{7}{8}$

• $\frac{5}{8}$

The region represented by $\left\{\mathrm{z} = \mathrm{x} + \mathrm{iy} \in \mathrm{C} : \left|\mathrm{z}\right| - \mathrm{Re}\left(\mathrm{z}\right) \le 1\right\}$ is also given by the inequality

• ${\mathrm{y}}^2 \ge 2\left(\mathrm{x} + 1\right)$

• ${\mathrm{y}}^2 \le \mathrm{x} + \frac{1}{2}$

• ${\mathrm{y}}^2 \le 2\mathrm{x} +\hspace{0.17em}1$

• ${\mathrm{y}}^2 \ge \mathrm{x} + 1$

The negation of the Boolean expression p v (~ p $\wedge$ q) is equivalent to :

• $\mathrm{p} \wedge ~ \mathrm{q}$

• $~ \mathrm{p} \vee \mathrm{q}$

• $~ \mathrm{p} \wedge ~ \mathrm{q}$

• $~ \mathrm{p} \vee ~ \mathrm{q}$

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ?

• 2! 3! 4!

• 3!(4!)³

• 3! . 2 . (4!)

• (3!)³ . (4!)