51.

The centroid of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and 4x - y = - 4 lies in the quadrant

• I

• II

• III

• IV

All the points on the X-axis have

• x = 0

• y = 0

• x = 0, y = 0

• y = 0, z = 0

For all values of a and b the line (a + 2b)x + (a - by + (a + 5b) = 0 passes through the point.

• (- 1, 2)

• (2, - 1)

• (- 2, 1)

• (1, - 2)

The incentre of triangle formed by the lines x + y = 1, x =1, y = 1 is

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

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

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

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

The orthocentre of triangle formed by the lines x + 3y = 10 and 6x² + xy - y² = 0 is

• (1, 3)

• (3, 1)

• (- 1, 3)

• (1, - 3)

The point P is equidistant from A(1, 3), B(- 3, 5)and C(5, - 1), then PA is equal to

• 5

• $5\sqrt{5}$

• 25

• $5\sqrt{10}$

A particle moves along the curve y = x² + 2x. Then, the point on the curve such that x and y coordinates of the particle change with same rate is

• (1, 3)

• $\left(\frac{1}{2}, \frac{5}{2}\right)$

• $\left(- \frac{1}{2}, - \frac{3}{4}\right)$

• (- 1, - 1)

If PM is the perpendicular from P(2, 3) onto the line x + y = 3, then the coordinates of M are

• (2, 1)

• (- 1, 4)

• (1, 2)

• (4, - 1)

If OA is equally inclined to OX, OY and OZ and if A is $\sqrt{3}$units from the origin, then A is :

• (3, 3, 3)

• (- 1, 1, - 1)

• (- 1, 1, 1)

• (1, 1, 1)

The area (in square unit) of the triangle formed by the points with polar coordinates $\left(1, 0\right), \left(2, \frac{\mathrm{\pi }}{3}\right) \mathrm{and} \left(3, \frac{2\mathrm{\pi }}{3}\right)$ is

• $\frac{11\sqrt{3}}{4}$

• $\frac{5\sqrt{3}}{4}$

• $\frac{5}{4}$

• $\frac{11}{4}$