Questions covered in Standard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The point on the y-axis which is at a distance of $\sqrt{10}$ from the point $(1, 2, 3)$ is
2. If the distance between the points $(a, 2, 1)$ and $(1, -1, 1)$ is 5, then $a$ is
3. The coordinates of the point which divides the line segment joining the points $(-2, 3, 5)$ and $(1, -4, 6)$ in the ratio $2:3$ externally are
4. The centroid of a triangle $ABC$ is at $(1, 1, 1)$. If the coordinates of $A$ and $B$ are $(3, -5, 7)$ and $(-1, 7, -6)$ respectively, then the coordinates of $C$ are
5. The perpendicular distance of the point $P(3, 3, 4)$ from the x-axis is
6. Find the ratio in which the line segment joining the points $(4, 8, 10)$ and $(6, 10, -8)$ is divided by the YZ-plane.
7. Show that the points $A(1, 2, 3)$, $B(-1, -2, -1)$, $C(2, 3, 2)$ and $D(4, 7, 6)$ are the vertices of a parallelogram.
8. Find the equation of the set of points $P$ such that its distances from the points $A(3, 4, -5)$ and $B(-2, 1, 4)$ are equal.
9. Find the coordinates of the point which divides the line segment joining the points $P(2, -4, 3)$ and $Q(-4, 5, -6)$ in the ratio $2:1$ internally. Also find the ratio in which the XY-plane divides the line segment PQ.
10. Three vertices of a parallelogram ABCD are $A(3, -1, 2)$, $B(1, 2, -4)$ and $C(-1, 1, 2)$. Find the coordinates of the fourth vertex D.
11. Find the coordinates of the point which is equidistant from the four points $O(0, 0, 0)$, $A(a, 0, 0)$, $B(0, b, 0)$ and $C(0, 0, c)$.
12. Case Study: Two points are given as $A(2, 3, 5)$ and $B(4, 3, 1)$.
    (i) Find the distance between points A and B.
    (ii) Find the coordinates of the point which divides AB in the ratio 1:3 internally.
    (iii) Find the coordinates of a point C on the x-axis such that $AC = BC$.