Questions covered in Hard

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 ratio in which the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is divided by the xz-plane is
2. The locus of a point for which $x=0$ is
3. If the distance of the point $(a, b, c)$ from the z-axis is $d$, then
4. Three vertices of a parallelogram ABCD are $A(3, -1, 2)$, $B(1, 2, -4)$ and $C(-1, 1, 2)$. The coordinates of the fourth vertex D are
5. The points $(0, 7, 10)$, $(-1, 6, 6)$ and $(-4, 9, 6)$ form
6. Find the coordinates of the point which divides the line segment joining the points $P(2, -1, 4)$ and $Q(4, 3, 2)$ in the ratio $2:3$ externally.
7. Find the ratio in which the line segment joining the points $(2, -1, 3)$ and $(-1, 2, 1)$ is divided by the plane $x + y + z = 5$.
8. Using section formula, show that the points $A(2, -3, 4)$, $B(-1, 2, 1)$ and $C(0, 1/3, 2)$ are collinear.
9. Find the equation of the set of points $P$ such that the sum of the squares of its distances from the points $A(3, 4, 5)$ and $B(-1, 3, -7)$ is equal to $2k^2$.
10. Find the lengths of the medians of the triangle with vertices $A(0, 0, 6)$, $B(0, 4, 0)$ and $C(6, 0, 0)$.
11. Find the equation of the set of points P such that its distance from $A(4, 0, 0)$ is twice its distance from $B(-4, 0, 0)$.
12. Case Study: Solar panels are placed at vertices of a triangle $A(2, 3, 4)$, $B(-1, 2, 1)$, $C(5, 0, -2)$.
    (i) Find the coordinates of the centroid of the triangle.
    (ii) Find the length of the median drawn from vertex A.
    (iii) Find the coordinates of a point D such that ABCD is a parallelogram.