If $\frac{\sin \left(\mathrm{A}\right)}{\sin \left(\mathrm{C}\right)} = \frac{\sin \left(\mathrm{A} - \mathrm{B}\right)}{\sin \left(\mathrm{B} - \mathrm{C}\right)}$, then a², b², c² are in

• AP

• GP

• HP

• None of these

If A = $60°$, a = 5, b = 4$\sqrt{3}$ in $∆\mathrm{ABC}$, then B is equal to

• 30°

• 60°

• 90°

• None of these

If a + b + c = 0, the straight line 2ax + 3by + 4c = 0 passes through the fixed point

• $\left(2, \frac{4}{3}\right)$

• (2, 2)

• $\left(\frac{4}{3}, \frac{4}{3}\right)$

• No such fixed point

The orthocentre of the triangle formed by tie lines xy = 0 and x + y = 1 is

• $\left(\frac{1}{2}, \frac{1}{2}\right)$

• $\left(\frac{1}{3}, \frac{1}{3}\right)$

• (0, 0)

• $\left(\frac{1}{4}, \frac{1}{4}\right)$

15.

Equation of the circle which passes through the origin and cuts intercepts of lengths a and b on axes is

• x² + y² + ax + by = 0

• x² + y² + ax - by = 0

• x² + y² + bx + ay = 0

• None of the above

If focus ofa parabolais at (3, 3) and its directrix is 3x - 4y = 2, then its latusrectum is

• 2

• 3

• 4

• 5

If the straight line y = 4x + c is a tangent to the ellipse $\frac{{\mathrm{x}}^2}{8} + \frac{{\mathrm{y}}^2}{4} = 1$, then c will be equal to

• $\pm 4$

• $\pm 6$

• $\pm 1$

• $\pm \sqrt{132}$

The distance between the foci of a hyperbola is 16 and its eccentricity is - 2. Its equation will be

• x² - y² = 32

• y² - x² = 32

• x² - y² = 16

• y² - x² = 16

The locus of the point, which moves such that its distance from (1, - 2, 2) is unity, will be

• x² + y² + z² - 2x + 4y - 4z + 8 = 0

• x² + y² + z² - 2x - 4y - 4z + 8 = 0

• x² + y² + z² + 2x + 4y - 4z + 8 = 0

• x² + y² + z² - 2x + 4y + 4z + 8 = 0

The ratio in which the sphere x² + y² + z² = 504 divides the line segment AB joining the points A(12, - 4, 8) and B(27, - 9, 18) is given by

• 2 : 3 externally

• 2 : 3 internally

• 1 : 2 externally

• None of these