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

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

What is ${\mathrm{\omega }}^{100} + {\mathrm{\omega }}^{200} + {\mathrm{\omega }}^{300} \mathrm{equal} \mathrm{to}, \mathrm{where} \mathrm{\omega } \mathrm{is} \mathrm{the} \mathrm{cube} \mathrm{root} \mathrm{of} \mathrm{unity} ?$

• 1

• 3$\mathrm{\omega }$

• $3{\mathrm{\omega }}^2$

• 0

If c > 0 and 4a + c < 2b, then ax² - bx + c = 0 has a root in which one of the following intervals ?

• (0, 2)

• (2, 3)

• (3, 4)

• ( - 2, 0)

If $\mathrm{Re}\left(\frac{\mathrm{z} - 1}{\mathrm{z} +\hspace{0.17em}1}\right) = 0,$,where z = x + iy is a complex number, then which one of the following is corrcet ?

• z = 1 + i

• $\left|\mathrm{z}\right| = 2$

• z = 1 - i

• $\left|\mathrm{z}\right| = 1$

57.

If $\mathrm{z} = {\left(\frac{\sqrt{3}}{2} + \frac{\mathrm{i}}{2}\right)}^{107} + {\left(\frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\right)}^{107}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{imaginary} \mathrm{part} \mathrm{of} \mathrm{z} \mathrm{equal} \mathrm{to} ?$

• 0

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• 1

If both the roots of the equation x² - 2kx + k² - 4 = 0 lie between - 3 and 5, then which one of the following is correct ?

• $- 2 < \mathrm{k} < 2$

• $- 5 < \mathrm{k} < 3$

• $- 3 < \mathrm{k} < 5$

• $- 1 < \mathrm{k} < 3$

What is the number of distinct solutions of the equation  ${\mathrm{z}}^2 + \left|\mathrm{z}\right| = 0$ (where z is a complex number) ?

• One

• Two

• Three

• Five

Which of the following are correct in respect of the system of equations x + y + z = 8, x - y + 2z = 6 and 3x - y + 5z = k ?

1. They have no solution, if k = 15.

2. They have infinitely many solutions, if k = 20

3. They have unique solution, if k = 25.

Select the correct answer using the code given below :

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3