The number of integral points (integral points means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is

• 133

• 190

• 233

• 105

162.

The sides AB, BC and CA of a $∆\mathrm{ABC}$ have respectively 3, 4 and 5 points lying on them. number of triangles that can be constructed using these points as vertices is

• 205

• 220

• 210

• None of these

P is a fixed point (a, a, a) on a line through the origin is equally inclined to the axes, then any plane through P perpendicular to Op, makes intercepts on the axes, the sum of whose reciprocal is equal to

• a

• $\frac{3}{2\mathrm{a}}$

• $\frac{3\mathrm{a}}{2}$

• None of these

The tangent at (1, 7) to the curve x² = y - 6 touches the circle x² + y2 + 16x + 12y + c = 0 at

• (6, 7)

• (- 6, 7)

• (6, - 7)

• (- 6, - 7)

In an equilateral triangle, the inradius, circumradius and one of the exradii are in the ratio

• 2 : 3 : 5

• 1 : 2 : 3

• 1 : 3 : 7

• 3 : 7 : 9

The perimeter of the triangle with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1) is

• 3

• 2

• 2$\sqrt{2}$

• 3$\sqrt{2}$

If a line in the space makes angle $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ with the coordinate axes, then

$\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ equals

• - 1

• 0

• 1

• 2

Y-axis cuts the line joining the points (- 3, - 4) and (1, - 2) in the ratio

• 1 : 3

• 2 : 3

• 3 : 1

• 3 : 2

If the points (1, 1), (- 1, - 1), $\left(- \sqrt{3}, \sqrt{3}\right)$ are the vertices of a triangle, then this triangle is

• a right-angled triangle

• an isosceles triangle

• an equilateral triangle

• None of the above

The point of intersection of the lines $\frac{\mathrm{x} - 5}{3} = \frac{\mathrm{y} - 7}{- 1} = \frac{\mathrm{z} + 2}{1}$ and $\frac{\mathrm{x} + 3}{- 36} = \frac{\mathrm{y} - 3}{2} = \frac{\mathrm{z} - 6}{4}$ is

• (2, 10, 4)

• (21, 5/3, 10/3)

• (5, 7, - 2)

• (- 3, 3, 6)