Q.Match each item given under the column $C_1$ to its correct answer given under column $C_2$. Column $C_1$:

Concept: 3D Coordinate Octants and Planes

Reasoning:

1. In the $xy$-plane, the $z$-coordinate is always zero → (a) matches (iii).
2. Point $(2,3,4)$ has all coordinates positive → lies in the first octant → (b) matches (i).
3. $x=0$ is the $yz$-plane → (c) matches (ii).
4. A line parallel to the $x$-axis has constant $y$ and $z$ → (d) matches (vi).
5. $x=0,\ y=0$ together give the $z$-axis → (e) matches (iv).
6. $z=c$ is a plane parallel to the $xy$-plane → (f) matches (v).
7. Planes $x=a,\ y=b$ intersect in a line parallel to the $z$-axis → (g) matches (viii). …

Let’s build the intuition first. In 3D space, we have three mutually perpendicular axes: $x$, $y$, and $z$. The coordinate planes are $xy$-plane (where $z=0$), $yz$-plane (where $x=0$), and $zx$-plane (where $y=0$). The eight octants are the regions where the signs of $x$, $y$, $z$ are fixed — the first octant has all three coordinates positive. A line parallel to an axis has the other two coordinates constant. A plane of the form $z=c$ is parallel to the $xy$-plane. And a disc is the region inside a circle, while a sphere is the surface of a ball.

Now, match each item step by step.

1. (a) In $xy$-plane — On the $xy$-plane, the $z$-coordinate is always zero. So (a) matches with (iii) $z$-coordinate is zero.

2. (b) Point $(2,3,4)$ lies in the — All three coordinates are positive. In 3D, the region where $x>0$, $y>0$, $z>0$ is the first octant. So (b) matches with (i) Ist octant.

3. (c) Locus of the points having $x$ coordinate 0 is — If $x=0$ for all points, that’s exactly the $yz$-plane (since $y$ and $z$ can vary freely). So (c) matches with (ii) $yz$-plane.

4. (d) A line is parallel to $x$-axis if and only — A line parallel to the $x$-axis has its direction along $x$, so $y$ and $z$ coordinates are constant for all points on the line. That means “all the points on the line have equal $y$ and $z$-coordinates”. So (d) matches with (vi) if all the points on the line have equal $y$ and $z$-coordinates.

5. (e) If $x=0,\ y=0$ taken together will represent the — The set of points where both $x=0$ and $y=0$ is the $z$-axis (the line through the origin along $z$). So (e) matches with (iv) $z$-axis.

6. (f) $z=c$ represent the plane — The equation $z=c$ describes a plane parallel to the $xy$-plane (since $x$ and $y$ are free). So (f) matches with (v) plane parallel to $xy$-plane.

7. (g) Planes $x=a,\ y=b$ represent the line — The intersection of two planes $x=a$ and $y=b$ is a line parallel to the $z$-axis (since $z$ is free). So (g) matches with (viii) parallel to $z$-axis. …