1.

The set of all real values of $\mathrm{\lambda }$ for which the quadratic equation ($\mathrm{\lambda }$² + 1) x² – 4$\mathrm{\lambda }$x + 2 = 0 always have exactly one root in the interval (0, 1) is :

• ( - 3, - 1)

• (0, 2)

• (1, 3)

• (2, 4)

Let p, q, r be three statements such that the truth value of $\left(\mathrm{p} \wedge \mathrm{q}\right) \to \left(~ \mathrm{q} \vee \mathrm{r}\right)$ is F. Then the truth values of p, q, r are respectively :

• T, T, F

• T, T, T

• T, F, T

• F, T, F

The Plane which bisects the line joining the points (4, – 2, 3) and (2, 4, – 1) at right angles also passes through the point :

• (0, - 1, 1)

• (4, 0, - 1)

• (4, 0, 1)

• (0, 1, - 1)

$\mathrm{If} \mathrm{a} ∆\mathrm{ABC} \mathrm{has} \mathrm{vertices} \mathrm{A}\left( - 1, 7\right), \mathrm{B}\left( - 7, 1\right) \mathrm{and} \mathrm{C}\left(5, - 5\right), \mathrm{then} \mathrm{its} \mathrm{orthocentre} \mathrm{has} \mathrm{coordinates}$

• ( - 3, 3)

• (3, - 3)

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

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

If the term independent of x in the expansion of ${\left(\frac{3}{2}{\mathrm{x}}^2 - \frac{1}{3\mathrm{x}}\right)}^9$, then 18k is equal to:

• 11

• 5

• 9

• 7

If z₁, z₂ are complex numbers such that Re(z₁) = |z₁ = 1| and Re(z₂) = |z₂ = 1| and arg (z₁ – z₂) = $\frac{\mathrm{\pi }}{6}$, then Im (z₁ + z₂) is equal to :

• $2\sqrt{3}$

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{\sqrt{3}}$

• $\frac{2}{\sqrt{3}}$

$$\begin{gathered} \mathrm{Let} {\mathrm{x}}_{\mathrm{i}}\left(1 \le \mathrm{i} \le 10\right) \mathrm{be} \mathrm{ten} \mathrm{observation} \mathrm{of} \mathrm{a} \mathrm{random} \mathrm{variable} \mathrm{X}. \mathrm{If} \sum _{\mathrm{i} = 1}^{10}\left({\mathrm{x}}_{\mathrm{i}} - \mathrm{p}\right) = 3 \\ \mathrm{and} \sum _{\mathrm{i} = 1}^{10}{\left({\mathrm{x}}_{\mathrm{i}} - \mathrm{p}\right)}^2 = 9 \mathrm{where} 0 \ne \mathrm{p} \in \mathrm{R}, \mathrm{then} \mathrm{the} \mathrm{standard} \mathrm{deviation} \mathrm{of} \mathrm{these} \mathrm{observations} \mathrm{is} \end{gathered}$$

• $\frac{4}{5}$

• $\sqrt{\frac{3}{5}}$

• $\frac{9}{10}$

• $\frac{7}{10}$

$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\left(\mathrm{a} + 2\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\left(3\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{{\left(3\mathrm{a} + \mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\left(4\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} \left(\mathrm{a} \ne 0\right) = ?$

• $\left(\frac{2}{9}\right){\left(\frac{2}{3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left(\frac{2}{3}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left(\frac{2}{9}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $\left(\frac{2}{3}\right){\left(\frac{2}{9}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Let e₁ and e₂ be the eccentricities of the ellipse, $\frac{{\mathrm{x}}^2}{25} + \frac{{\displaystyle {\mathrm{y}}^2}}{{{\displaystyle \mathrm{b}}}^2} = 1\left(\mathrm{b} < 5\right) \mathrm{and} \mathrm{the} \mathrm{hyperbola}, \frac{{\mathrm{x}}^2}{16} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$respectively satisfying e₁e₂ = 1. If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $\left(\mathrm{\alpha }, \mathrm{\beta }\right)$ is equal to:

• (8, 10)

• $\left(\frac{20}{3}, 12\right)$

• (8, 12)

• $\left(\frac{24}{5}, 10\right)$

$$\begin{gathered} \mathrm{Let} \mathrm{A} \mathrm{be} \mathrm{a} 3 \times 3 \mathrm{matrix} \mathrm{such} \mathrm{that} \\ \mathrm{adj} \mathrm{A} = \left[\begin{array}{ccc}2& - 1& 1\\ - 1& 0& 2\\ 1& - 2& - 1\end{array}\right] \mathrm{and} \mathrm{B} = \mathrm{adj}\left(\mathrm{adj} \mathrm{A}\right) \end{gathered}$$

$\mathrm{If} \left|\mathrm{A}\right| = \mathrm{\lambda } \mathrm{and} \left|{\left({\mathrm{B}}^{ - 1}\right)}^{\mathrm{T}}\right| = \mathrm{\mu }, \mathrm{Then} \mathrm{the} \mathrm{ordered} \mathrm{pair}, \left(\left|\mathrm{\lambda }\right|, \mathrm{\mu }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\left(3, \frac{1}{81}\right)$

• $\left(9, \frac{1}{9}\right)$

• (3, 81)

• $\left(9, \frac{1}{81}\right)$