First of all, we construct a line \(\bar{PQ}\) of length 7 cm.
Step I Place the zero mark of the ruler at P.
Step II Mark a point Q at a distance 7 ern from P.
                    
Step III Mark another point A between P and Q at distance 3.2 cm from P such that PA = 3.2 cm.
Step IV Measure line segment \(\bar{QA}\). We fin4 that \(\bar{QA}\) = 7.0 - 3.2 = 3.8 cm.

To construct a line segment of length 5 cm using a ruler and compass, we use the following steps:
                   
Step I Draw AB of length 5 cm.
Step II Draw a line I and mark a point C on it.
                   
Step III Open the compasses equal to \(\bar{AB}\) .
Step IV Keeping the same opening. Place the pointer on C and mark a point D on I. This \(\bar{CD}\) is a equal to \(\bar{AB}\) .

No, they are parallel to each other.

Longest chord is diameter. Yes, it will pass through the centre.

CD \(\bot \) AB

Diameter

30^o

8 cm

We know that longest chord of a circle is a diameter. So, radius of circle is 4 cm.
Steps of construction are as follows:
Step I Draw a circle of radius 4 cm. The longest chords is AB = 8 cm.
Step II Now, draw the perpendicular bisector of AB, which intersect AB at O.
                           
Step III 'O' is the centre of the circle.

Steps of construction are as follows:
Step I Draw \(\bar{PQ}\) of length 4 cm.
                     
Step II Draw a line l and mark a point R on it.
Step III Open the compasses equal to \(\bar{PQ}\).
Step IV Keeping the same opening. Place the pointer on R and mark a point S on I. Thus, \(\bar{RS}\) is a equal to \(\bar{PQ}\).
                    

Steps of construction are as follows:
Step I Draw a line segment \(\bar{AB}\) of length 5 cm.
Step II Take a point P and place the pointer of compasses at P draw an arc such that it intersects \(\bar{AB}\) at X and Y.
                       
Step III With X and Y as centres and radius greater than \(\bar{PX}\), draw two arcs such that they intersect at C.
Step IV Join CP Thus, \(\bar{CP}\bot\bar{AB}\)

We know that, perpendicular bisector of a line segment is its axis of symmetry.
Step I Draw a line segment \(\bar{PQ}\) = 7 cm.
Step II With P and 0 as centres and radius more than half of \(\bar{PQ}\), draw two arcs which intersect each other at A and B.
                        
Step III Join A and B.
Thus, AB is the axis of symmetry of \(\bar{PQ}\).

15 = 3 x 5, 21= 3 x 7,
36 = 2 x 2 x 3 x 3,
42 = 2 x 3 x 7,
48 = 2 x 2 x 2 x 2 x 3
∴ HCF=3
Hence, the largest length of the rod is 3 m.

The conditions remaining the same, they will cross each other again after 45 + 45 = 90 minutes.

Ratio of income to expenditure = 7 : 5
Let the income and expenditure per month be Rs 7x and 5x respectively.
Then, saving per month = Rs 7x - Rs 5x = Rs 2x
According to the question,
2x= 2000
⇒ x =\(\frac{2000}{2}\) =1000
⇒ 7x = 7 x 1000 = 7000
Hence, his income is Rs 7000 per month.