1.

Suppose $\mathrm{\omega }$ is a cube root unity with $\mathrm{\omega } \ne 1$. Suppose P and Q are the points on the complex plane defined by $\mathrm{\omega } \mathrm{and} {\mathrm{\omega }}^2$. If O is the origin,then what is the angle between OP and OQ ?

• $60°$

• $90°$

• $120°$

• $150°$

If ${\mathrm{x}}^2 - \mathrm{px} + 4 > 0 \mathrm{for} \mathrm{all} \mathrm{real} \mathrm{values} \mathrm{of} \mathrm{x}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\left|\mathrm{p}\right| < 4$

• $\left|\mathrm{p}\right| \le 4$

• $\mathrm{p} > 4$

• $\mathrm{p} \ge 4$

If z = x + iy = ${\left(\frac{1}{\sqrt{2}} - \frac{\mathrm{i}}{\sqrt{2}}\right)}^{ - 25}, \mathrm{where} \mathrm{i} = \sqrt{ - 1}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{fundamental} \mathrm{amplitude} \mathrm{of} \frac{\mathrm{z} - \sqrt{2}}{\mathrm{z} - \mathrm{i}\sqrt{2}} ?$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

What is the range of the function y = $\frac{x^2}{1 + x^2}$

where, x $\in \mathrm{ℝ}$

• [0, 1)

• [0, 1]

• (0, 1)

• (0, 1]

A straight line intersects x and y axes at P and Q respectively. If (3, 5) is the middle point of PQ, then what is the area of the triangle OPQ ?

• 12 square units

• 15 square units

• 20 square units

• 30  square units

If a circle of radius b units with centre at (0, b) touches the line y = x $- \sqrt{2}$, then what is the value of b ?

• $2 + \sqrt{2}$

• $2 - \sqrt{2}$

• $2\sqrt{2}$

• $\sqrt{2}$

Consider the function

$\mathrm{f}\left(\mathrm{\theta }\right) = 4\left({\sin }^2\left(\mathrm{\theta }\right) + {\cos }^4\left(\mathrm{\theta }\right)\right)$

What is the maximum value of the function f($\mathrm{\theta }$)?

• 1

• 2

• 3

• 4

Consider the function

$\mathrm{f}\left(\mathrm{\theta }\right) = 4\left({\sin }^2\left(\mathrm{\theta }\right) + {\cos }^4\left(\mathrm{\theta }\right)\right)$

What is the minimum value of the function f($\mathrm{\theta }$)?

• 0

• 1

• 2

• 3

Consider the function

$\mathrm{f}\left(\mathrm{\theta }\right) = 4\left({\sin }^2\left(\mathrm{\theta }\right) + {\cos }^4\left(\mathrm{\theta }\right)\right)$

Consider the following statements :

$$\begin{gathered} 1. \mathrm{f}\left(\mathrm{\theta }\right) = 2 \mathrm{has} \mathrm{no} \mathrm{solution} \\ 2. \mathrm{f}\left(\mathrm{\theta }\right) = \frac{7}{2} \mathrm{has} \mathrm{a} \mathrm{solution} \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{statements} \mathrm{is}/\mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

Consider the curves

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}\left|\mathrm{x}\right| - 1 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \left\{\frac{3\mathrm{x}}{2}, \mathrm{x} > 0 \\ 2\mathrm{x}, \mathrm{x} \le 0\right\} \end{gathered}$$

Where do the curves intersect ?

• At (2, 3) only

• At ( - 1, - 2) only

• At (2, 3) and ( - 1, 2)

• Neither at (2, 3) nor at ( - 1,  - 2)