The sum of the coefficients of all odd degree terms in the expansion of ${\left(x + \sqrt{x^3 -1}\right)}^5 + {\left(x - \sqrt{x^3-1}\right)}^5, (x>1) is$

• 2

• -1

• 0

• 1

The common chord of the circles x² + y² - 4x - 4y = 0 and 2x²+ 2y² = 32 subtends at the origin an angle equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

The locus of the mid-points of the chords of the circle x² + y² + 2x - 2y - 2= 0, which make an angle of 90° at the centre is

• x² + y² - 2x - 2y = 0

• x² + y² - 2x + 2y = 0

• x² + y² + 2x - 2y = 0

• x² + y² + 2x - 2y - 1 = 0

The expression $\frac{{\left(1 + \mathrm{i}\right)}^{\mathrm{n}}}{{\left(1 - \mathrm{i}\right)}^{\mathrm{n} - 2}}$

• $- {\mathrm{i}}^{\mathrm{n} + 1}$

• ${\mathrm{i}}^{\mathrm{n} + 1}$

• $- 2{\mathrm{i}}^{\mathrm{n} + 1}$

• 1

Let z = .x + iy, where x and y are real. The  points (x, y) in the X-Y plane for which $\frac{\mathrm{z} + \mathrm{i}}{\mathrm{z} - \mathrm{i}}$ is purely imaginary, lie on

• a straight line

• An ellipse

• a hyperbola

• a circle

If p, q are odd integers, then the roots of the equation 2px² + (2p + q) x + q= 0 are

• rational

• irrational

• non-real

• equal

If a, b $\in$ {1, 2, 3} and the equation ax² + bx + 1 = 0 has real roots, then

• a > b

• $\mathrm{a} \le \mathrm{b}$

• number of possible ordered pairs (a, b) is 3

• a<b

The complex number z satisfying the equation $\left|\mathrm{z} - \mathrm{i}\right| = \left|\mathrm{z} + 1\right| = 1$ is

• 0

• 1 + i

• - 1 + i

• 1 - i

129.

If ${\mathrm{z}}_{1 }, {\mathrm{z}}_2, {\mathrm{z}}_3$ are imaginary numbers such that $\left|{\mathrm{z}}_1\right| = \left|{\mathrm{z}}_2\right| = \left|{\mathrm{z}}_3\right| = \left|\frac{1}{{\mathrm{z}}_1} + \frac{1}{{\mathrm{z}}_2} + \frac{1}{{\mathrm{z}}_3}\right|$ = 1, then $\left|z_1 + z_2 + z_3\right|$ is

• equal to 1

• less than

• greater than 1

• equal to 3

If p.q are the · roots of the equation x² + px + q =0, then

• p = 1, q = - 2

• p = 0, q = 1

• p = - 2, q = 0

• p = - 2, q = 1