Let  $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation :

${\mathrm{x}}^2 - \left(1 - 2{\mathrm{a}}^2\right)\mathrm{x} + \left(1 - 2{\mathrm{a}}^2\right) = 0$

Under what condition does the above equation have real roots ?

• ${\mathrm{a}}^2 < \frac{1}{2}$

• ${\mathrm{a}}^2 > \frac{1}{2}$

• ${\mathrm{a}}^2 \le \frac{1}{2}$

• ${\mathrm{a}}^2 \ge \frac{1}{2}$

Let  $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation :

${\mathrm{x}}^2 - \left(1 - 2{\mathrm{a}}^2\right)\mathrm{x} + \left(1 - 2{\mathrm{a}}^2\right) = 0$

Under what condition $\frac{1}{{\mathrm{\alpha }}^2} + \frac{1}{{\mathrm{\beta }}^2} < 1 ?$

• ${\mathrm{a}}^2 < \frac{1}{2}$

• ${\mathrm{a}}^2 > \frac{1}{2}$

• ${\mathrm{a}}^2 > 1$

• ${\mathrm{a}}^2 \in \left(\frac{1}{3}, \frac{1}{2}\right) \mathrm{only}$

What is $\sqrt{\frac{1 + {\mathrm{\omega }}^2}{1 + \mathrm{\omega }}}$ = ?, where $\mathrm{\omega }$ is cube root of unity

• 1

• $\mathrm{\omega }$

• $\mathrm{\omega }$²

• None of the above

4.

In an examination, 70% students passed in Physics, 80% students passed in Chemistry, 75% students passed in Mathematics and 85% students passed in Biology, and x% students failed in all the four subjects. What is the minimum value of x ?

• 10

• 12

• 15

• None of the above

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

Under what condition does the above system of equations have infinitely many solutions ?

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } \ne 9$

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } = 9$

• $\mathrm{\lambda } = 0 \mathrm{and} \mathrm{\mu } = 5$

• $\mathrm{\lambda } = 9 \mathrm{and} \mathrm{\mu } \ne 5$

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

Under what condition does the above system of equations have unique solution ?

• $\mathrm{\lambda } = 5, \mathrm{\mu } = 9$

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } = 7 \mathrm{only}$

• $\mathrm{\lambda } \ne 5 \mathrm{and} \mathrm{\mu } \mathrm{has} \mathrm{any} \mathrm{real} \mathrm{value}$

• $\mathrm{\lambda }$ has any real value and $\mathrm{\mu } \ne 9$

What is the number of odd integers between 1000 and 9999 with no digit repeated ?

• 2100

• 2120

• 2240

• None of the above

What is the greatest value of the positive integer n satisfying the condition :

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... + \frac{1}{2^{\mathrm{n} - 1}} < 2 - \frac{1}{1000} ?$

• 8

• 9

• 10

• 11

2x² + 3x - $\mathrm{\alpha }$ = 0 has roots - 2 and $\mathrm{\beta }$ while the equation x² - 3mx + 2m² = 0 has both roots positive, where a > 0 and $\mathrm{\beta }$ > 0. What is the value of $\mathrm{\alpha }$?

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• 1

• 2

• 4

2x² + 3x - $\mathrm{\alpha }$ = 0 has roots - 2 and $\mathrm{\beta }$ while the equation x² - 3mx + 2m² = 0 has both roots positive, where a > 0 and $\mathrm{\beta }$ > 0. $\mathrm{If} \mathrm{\beta }, 2, 2\mathrm{m} \mathrm{are} \mathrm{in} \mathrm{GP}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\beta }\sqrt{\mathrm{m}} ?$

• 1

• 2

• 4

• 6