The shortest distance from the plane 12x + y + 3z = 327 to the sphere x² + y² + z² + 4x - 2y - 6z = 155 is

• 13

• 26

• 39

• $41\frac{4}{13}$

If the vertices of a· triangle are A(0, 4, 1), B(2, 3, - 1) and C(4, 5, 0), then the orthocentre of $∆$ABC, is

• (4, 5, 0)

• (2, 3, - 1)

• (- 2, 3, - 1)

• (2, 0, 2)

The normals at three points· P,Q and R of the parabola y² = 4ax meet at (h, k). The centroid of the $∆$PQR lies on

• x = 0

• y = 0

• x = - a

• y = a

A tower AB leans towards West making an angle $\mathrm{\alpha }$ with the vertical. The angular elevation of B, the top most point of the tower is $\mathrm{\beta }$ as observed from a point C due East of A at a distance 'd' from A. If the angular elevation of B from a point D due East of C at a distance 2d from C is r, then 2$\tan \left(\mathrm{\alpha }\right)$ can be given as

• $3\cot \left(\mathrm{\beta }\right) - 2\cot \left(\mathrm{\gamma }\right)$

• $3\cot \left(\mathrm{\gamma }\right) - 2\cot \left(\mathrm{\beta }\right)$

• $3\cot \left(\mathrm{\beta }\right) - \cot \left(\mathrm{\gamma }\right)$

• $\cot \left(\mathrm{\beta }\right) - 3\cot \left(\mathrm{\gamma }\right)$

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

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$\sqrt{2}$