1 How will you describe the position of a table lamp on your study table to another person?
Solution :
Let us consider the given below figure of a study stable, on which a study lamp is placed.$\\$
From the Figure Above,$\\$ $\bullet $ Consider the lamp on the table as a point$\\$ $\bullet $ Consider the table as a plane.$\\$ $\bullet $ We can conclude that the table is rectangular in shape, when observed from the top.$\\$ $ \ \ \ \ \ \ \ \ \ \circ $The table has a short edge and a long edge$\\$ $ \ \ \ \ \ \ \ \ \ \circ $Let us measure the distance of the lamp from the shorter edge and the longer edge.$\\$ $ \ \ \ \ \ \ \ \ \ \circ $Let us assume$\\$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \diamond$ Distance of the lamp from the shorter edge is $15 cm$$\\$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \diamond$ Distance of the lamp from the longer edge, its $25 cm.$$\\$ Therefore, we can conclude that the position of the lamp on the table can be described in two ways depending on the order of the axes as $(15, 25)$ or $(25, 15).$
2 (Street Plan): A city has two main roads which cross each other at the center of the city. These two roads are along the North–South direction and East-West direction. All the other streets of the city run parallel to these roads and are $200 m$ apart. There are $5$ streets in each direction. Using $1 cm = 200 m,$ draw a model of the city on your notebook. Represent the roads/streets by single lines. There are many cross- streets in your model. A particular cross- street is made by two streets, one running in the North-South direction and another in the East- West direction. Each cross street is referred to in the following manner: If the $2^{ nd}$ street running in the North-South direction and $5^{ th}$ in the East-West direction meet at some crossing, then we will call this cross-street $(2, 5)$. Using this convention, find:$\\$ (i) How many cross - streets can be referred to as $(4, 3).$$\\$ (ii) How many cross - streets can be referred to as $(3, 4).$$\\$
Solution :
$\bullet $ Draw two perpendicular lines as the two main roads of the city that cross each other at the center$\\$ $\bullet $ Mark it as N-S and E-W.$\\$ $\bullet $ Let us take the scale as $1 cm = 200 m.$$\\$ $\bullet $ Draw five streets that are parallel to both the main roads, to get the given below figure.$\\$
Street plan is as shown in the figure:$\\$ (i) There is only one cross street, which can be referred as $(4, 3)$.$\\$ (ii) There is only one cross street, which can be referred as $(3, 4).$$\\$
3 Write the answer of each of the following questions:$\\$ (i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?$\\$ (ii) What is the name of each part of the plane formed by these two lines?$\\$ (iii) Write the name of the point where these two lines intersect.
Solution :
(i) The horizontal line that is drawn to determine the position of any point in the Cartesian plane is called as x-axis.$\\$ The vertical line that is drawn to determine the position of any point in the Cartesian plane is called as y-axis.$\\$
(ii) The name of each part of the plane that is formed by x-axis and y-axis is called as quadrant.$\\$
(iii) The point, where the x-axis and the y-axis intersect is called as origin (O)$\\$
4 See the figure, and write the following:$\\$ (i) The coordinates of $B.$$\\$ (ii) The coordinates of $C.$$\\$ (iii) The point identified by the coordinates $(-3, -5).$$\\$ (iv) The point identified by the coordinates $(2, -4).$$\\$ (v) The abscissa of the point $D.$$\\$ (vi) The ordinate of the point $H.$$\\$ (vii) The coordinates of the point $L.$$\\$ (viii) The coordinates of the point $M.$$\\$
Solution :
From the Figure above,$\\$ (i)The coordinates of point B is the distance of point B from x-axis and y-axis.$\\$ Therefore, the coordinates of point B are $(-5, 2).$$\\$ (ii)The coordinates of point C is the distance of point C from x-axis and y-axis.$\\$ Therefore, the coordinates of point C are $(5, -5).$$\\$ (iii)The point that represents the coordinates $(-3, -5)$ is E.$\\$ (iv)The point that represents the coordinates $(2, -4)$ is G.$\\$ (v)The abscissa of point D is the distance of point D from the y-axis. Therefore, the abscissa of point D is $6$.$\\$ (vi)The ordinate of point H is the distance of point H from the x-axis. Therefore, the abscissa of point H is $-3$.$\\$ (vii)The coordinates of point L in the above figure is the distance of point L from x-axis and y-axis. $\\$Therefore, the coordinates of point L are $(0, 5).$$\\$ (viii)The coordinates of point M in the above figure is the distance of point M from x-axis and y-axis.$\\$ Therefore the coordinates of point M are $ (-3, 0).$
5 In which quadrant or on which axis do each of the points $(-2, 4), (3, -1), (-1, 0), (1, 2)$ and $(-3, -5)$ lie? Verify your answer by locating them on the Cartesian plane.
Solution :
To determine the quadrant or axis of the points $(-2, 4), (3, -1), (-1, 0), (1, 2)$ and $(-3, -5).$$\\$ Plot the plot the points $(-2, 4), (3, -1), (-1, 0), (1, 2)$ and $(-3, -5)$ on the graph, to get $\\$
From the figure above, we can conclude that the points$\\$ $\bullet $ Point $(-2, 4)$ lie in II nd quadrant.$\\$ $\bullet $ Point $(3, -1)$ lie in IV th quadrant.$\\$ $\bullet $ Point $(-1, 0)$ lie on the negative x-axis.$\\$ $\bullet $ Point $(1, 2)$ lie in I st quadrant.$\\$ $\bullet $ Point $(-3, -5)$ lie in III rd quadrant.
6 Plot the points $(x, y)$ given in the following table on the plane, choosing suitable units of distance on the axes.$\\$ $\begin{array}{|c|c|} \hline x& -2 & -1 & 0 & 1 & 3 \\ \hline y & 8 & 7 &-1.25 & 3 & -1 \\ \hline \end{array}$
Solution :
Given,$\\$ $\begin{array}{|c|c|} \hline x& -2 & -1 & 0 & 1 & 3 \\ \hline y & 8 & 7 &-1.25 & 3 & -1 \\ \hline \end{array}$
Draw $X'OX $ and $Y'OY$ as the coordinate axes and mark their point of intersection $O$ as the Origin $ (0, 0)$$\\$ In order to plot the table provided above,$\\$ $\bullet $ Point $A(-2, 8)$$\\$ $\quad \circ $ Take$ 2$ units on $OX'$ and then $8$ units parallel to $OY$$\\$ $\bullet $ Point $B (-1, 7)$$\\$ $\quad \circ $ Take $-1$ unit on $OX'$ and then $7$ units parallel to $OY$$\\$ $\bullet $Point $C(0, -1.25)$,$\\$ $\quad \circ $ Take $1.25$ units below x-axis on $OY'$ on the y-axis$\\$ $\bullet $ Point $D(1, 3)$$\\$ $\quad \circ $ Take $1$ unit on $OX$ and then $3$ units parallel to $OY$$\\$ $\bullet $Point $E(3, -1),$$\\$ $\quad \circ $Take $3$ units on $OX$ and then move $1$ unit parallel to $OY'$