Mathematics : Introduction To Three Dimensional Geometry

The perpendicular distance from the point $\left(1, \mathrm{\pi }\right)$ to the line joining (1, 0⁰) and $\left(1, \frac{\mathrm{\pi }}{2}\right)$ (in polar coordinates) is

• 2

• $\sqrt{3}$

• 1

• $\sqrt{2}$

If D(2, 1, 0), E(2, 0, 0) and F(0, 1, 0) are mid-points of the sides BC, CA and AB of $∆$ABC, respectively. Then, the centroid of$∆$ABC is

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

• $\left(\frac{4}{3}, \frac{2}{3}, 0\right)$

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

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

If in a $∆$ABC, r₁ = 2, r₂ = 3 and 7a = 6, then ae quals to

• 4

• 1

• 2

• 3

$\mathrm{If} \frac{2{\mathrm{x}}^3 + {\mathrm{x}}^2 - 5}{{\mathrm{x}}^4 - 25} = \frac{\mathrm{Ax} + \mathrm{B}}{{\mathrm{x}}^2 - 5} + \frac{\mathrm{Cx} \hspace{0.17em}+ 1}{{\mathrm{x}}^2 +\hspace{0.17em}5}, \mathrm{then} \left(\mathrm{A}, \mathrm{B}, \mathrm{C}\right) = ?$

• (1, 1, 1)

• (1, 1, 0)

• (1, 0, 1)

• (1, 2, 1)

If in a $∆\mathrm{ABC}$, r₁ = 2r₂ = 3r₃, then the perimeter of the triangle is equal to

• 3a

• 3b

• 3c

• 3(a + b + c)

If the equation to the locus of points equidistant from the points (- 2, 3), (6, - 5) is ax + by + c = 0, where a > 0, then the ascending order of a, b, c is

• a, b, c

• c, b, a

• b, c, a

• a, c, b

Match the following columns.

A	The centroid of the triangle formed by (2, 3, - 1), (5, 6, 3), (2, -3 ,1) is	p	(2, 2, 2)
B	The circumcentre of the triangle formed by (1,2, 3) (2, 3, 1), (3, 1, 2) is	q	(3, 1, 4)
C	The orthocentre of the triangle formed by (2, 1, 5), (3, 2, 3), (4, 0, 4) is	r	(1, 1, 0)
D	The incentre of the triangle formed by (0, 0, 0), (3, 0, 0), (4, 0, 4) is	s	(3, 2, 1)
E	The incentre of the triangle formed by (0, 0, 0), (3, 0, 0), (4, 0, 4) is	t	(0, 0, 0)

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If $\angle$BAC = 90°, and ar($∆$ABC) = 55sq. units, then the abscissa of the vertex C is :

• $1 + \sqrt{5}$

• $2 + \sqrt{5}$

• $1 + 2\sqrt{5}$

• $2\sqrt{5} - 1$

If the point P on the curve, 4x² + 5y² = 20 is farthest from the point Q(0, – 4), then PQ² is equal to :

• 29

• 48

• 21

• 36

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that MD² + MC² is minimums is ___