Mathematics : Introduction To Three Dimensional Geometry

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

• $\frac{11\sqrt{3}}{4}$

• $\frac{5\sqrt{3}}{4}$

• $\frac{5}{4}$

• $\frac{11}{4}$

In the triangle with vertices at A(6, 3), B(- 6, 3) and C(- 6, - 3), the median through A meets BC at P, the line AC meets the x-axis at Q, while R and S respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is

      List-I                        List-II

(i)    P                      (A)    (0, 0)

(ii)   Q                      (B)    (6, 0)

(iii)   R                     (C)    (- 2, 1)

(iv)   S                     (D)    (- 6, 0)

                               (E)    (- 6, - 3)

                               (F)    (- 6, 3)

In $∆$ABC the mid points of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, $\frac{{\mathrm{AB}}^2 + {\mathrm{BC}}^2 + {\mathrm{CA}}^2}{{\mathrm{l}}^2 + {\mathrm{m}}^2 + {\mathrm{n}}^2} = ?$

• 2

• 4

• 8

• 16

$\mathrm{The} \mathrm{perimeter} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{with} \mathrm{vertices} \mathrm{at} \left(1, 0, 0\right), \left(0, 1, 0\right) \mathrm{and} \left(0, 0, 1\right) \mathrm{is}$

• 3

• 2

• $2\sqrt{2}$

• $3\sqrt{2}$

$$\begin{gathered} \mathrm{If} \mathrm{a} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{space} \mathrm{makes} \mathrm{angle} \mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma } \mathrm{with} \mathrm{the} \mathrm{coordinate} \mathrm{axes}, \mathrm{then} \\ \cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) +\hspace{0.17em}{\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right) =? \end{gathered}$$

• - 1

• 0

• 1

• 2