Consider the following statements in respect of the quadratic equation 4(x - p)(x - q) - r² = 0, where p, q and r are real numbers:

1) The roots are real

2) The roots are equal if p = q and r = 0

Where of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

How many real roots does the equation x² + 3|x| + 2 = 0 have ?

• 1 only

• 2 Only

• Both 1 and 2

• Neither 1 nor 2

Under which one of the following conditions will the quadratic equation x² + mx + 2 = 0 always have real roots ?

• $2\sqrt{3} \le {\mathrm{m}}^2 < 8$

• $\sqrt{3} \le {\mathrm{m}}^2 < 4$

• m² $\ge$8

• m² $\le \sqrt{3}$

If x = 1 + i, then what is the value of x⁶ + x⁴ + x² + 1 ?

• 6i - 3

• - 6i + 3

• - 6i - 3

• 6i + 3

If α and β are the roots of x² + x + 1 = 0, then what is   $\sum _{\mathrm{j} = 0}^2\left({\mathrm{\alpha }}^1 + {\mathrm{\beta }}^1\right) = ?$

• 8

• 6

• 4

• 2

What is the value of ${\left[\frac{\mathrm{i} + \sqrt{3}}{2}\right]}^{2019} + {\left[\frac{\mathrm{i} - \sqrt{3}}{2}\right]}^{2019}$ ?

• 1

• 2

• 2i

• - 2i

If both p and q belong to the set {1, 2, 3, 4}, then how many equations of the form px² + qx + 1 = 0 will have real roots ?

• 12

• 10

• 7

• 6

If p and q are the roots of the equation x² – 30x + 221 = 0, what is the value of p³ + q³ ?

• 7010

• 7110

• 7210

• 7240

69.

Let f(x) = x² + 2x – 5 and g(x) = 5x + 30

Consider the following statements:

1) f[g(x)] is a polynomial of degree 3.

2) g[g(x)] is a polynomial of degree 2.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1and 2

• Neither 1 nor 2

What is the argument of the complex number $\frac{1 - \mathrm{i}\sqrt{3}}{1 + \mathrm{i}\sqrt{3}}$

where $\mathrm{i} = \sqrt{ - 1}$ ?

• $240°$

• $210°$

• $120°$

• $60°$