NCERT Exemplar Solutions for Class 9 Math’s Chapter 1 Number Systems https://www.swiflearn.com/ Exercise No. 1.3 Short Answer Questions: Question: 1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers: (i) 2 5 x (ii) 2 9 y (iii) 2 0.04 z (iv) 2 17 4 u Solution: (i) 2 5 5 x x Which is an irrational number. (ii) 2 9 9 3 y y y Which is a rational number. (iii) 2 0.04 0.04 0.2 z z z Which is a rational number. (iv) 2 17 4 17 4 17 2 u u u Where, is not an integer. Which is an irrational number. 17
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NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
Exercise No. 1.3 Short Answer Questions: Question: 1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers: (i) 2 5x
(ii) 2 9y
(iii) 2 0.04z
(iv) 2 17
4u
Solution: (i)
2 5
5
x
x
Which is an irrational number.
(ii)
2 9
93
y
yy
Which is a rational number.
(iii)
2 0.04
0.040.2
z
zz
Which is a rational number. (iv)
2 174
174172
u
u
u
Where, is not an integer. Which is an irrational number. 17
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
(x) and 6.375289 6.375738 Solution:
(i) A rational number between 2 and 3 is: . 2 3 5 2.5
2 2
(ii) 0.04 is rational number which lies between 0 and 0.1.
(iii) 1 1 4 4 1 1 6 6 and 3 3 4 12 2 2 6 12
is a rational number between and . Which is also lying between and . 5
124
126
1213
12
Now, and . 1 0.333333
1 0.52
Now, 0.414114111…. is a non-terminating and non-recurring decimal.
Hence, 0.414114111… is an irrational number lying between and . 13
12
(iv) and 2 0.4
5
1 0.52
0 is rational number between -0.4 and 0.5 i.e., 0 is a rational number between and2
5
. 12Again, 0.131131113… is a non-terminating and non-recurring decimal which lies between -0.4 and 0.5.
Hence, 0.131131113… is an irrational number lying between and . 2
5 1
2
(v) 0.151 is a rational number between 0.15 and 0.16. Similarly, 0.153, 0.157, etc, are rational number lying between 0.15 and 0.16. 0.151151115 is an irrational number between 0.15 and 0.16.
(vi) … and 2 1.4142135 3 1.732050807Now, 1.5 which lies between 1.4142135…. and 1.732050807…
Since, 1.5 is a rational number between and . 2 3
Now, 1.5755755557… is an irrational number lying between and . 2 3
(vii) 3 is a rational number between 2.357 and 3.121. Again, 3.101101110 is an irrational number between 2.357 and 3.121.
(viii) 0.00011 is a rational number between 0.0001 and 0.001 Again, 0.0001131331333 is an irrational number between 0.0001 and 0.001.
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
By Pythagoras theorem, 5OB Using a compass with center O and radius OB, draw an arc which intersects the number line at a point C. Then, C corresponds to . 5
Presentation of on number line: 10We can write 10 as the sum of the square of two natural numbers:
2 2
10 1 91 3
On the number line, take OA = 3 units. Draw BA = 1 unit, perpendicular to OA join OB. By Pythagoras theorem, 10OB Using a compass with center O and radius OB, draw an arc which intersects the number line at a point C. Then, C corresponds to . 10
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
Presentation of on number line: 17We can write 17 as the sum of the square of two natural numbers:
2 2
10 1 161 4
On the number line, take OA = 4 units. Draw BA = 1 unit, perpendicular to OA join OB. By Pythagoras theorem, 17OB Using a compass with center O and radius OB, draw an arc which intersects the number line at a point C. Then, C corresponds to . 17
6. Represent geometrically the following numbers on the number line: (i) 4.5
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
(ii) 5.6(iii) 8.1(iv) 2.3 Solution: (i) Presentation of on number line: 4.5
Mark the distance 4.5 units from a fixed point A on a given line to obtain a point B such that AB = 4.5 units. From B, mark a distance of 1 units and mark the new points as C. Find the mid-point of AC and mark that points as O. Draw a semicircle with center O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = . 4.5Now, draw an arc with center B and B radius BD, which intersects the number line in E. Thus, E represent . 4.5 (ii) Presentation of on number line: 5.6
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
Mark the distance 5.6 units from a fixed point A on a given line to obtain a point B such that AB = 5.6 units. From B, mark a distance of 1 units and mark the new points as C. Find the mid-point of AC and mark that points as O. Draw a semicircle with center O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = . 5.6Now, draw an arc with center B and B radius BD, which intersects the number line in E. Thus, E represent . 5.6 (iii) Presentation of on number line: 8.1
Mark the distance 8.1 units from a fixed point A on a given line to obtain a point B such that AB = 8.1 units. From B, mark a distance of 1 units and mark the new points as C. Find the mid-point of AC and mark that points as O. Draw a semicircle with center O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = . 8.1 Now, draw an arc with center B and B radius BD, which intersects the number line in E. Thus, E represent . 8.1
NCERT Exemplar Solutions for Class 9 Math’s Chapter 1
Number Systems
https://www.swiflearn.com/
Mark the distance 2.3 units from a fixed point A on a given line to obtain a point B such that AB = 2.3 units. From B, mark a distance of 1 units and mark the new points as C. Find the mid-point of AC and mark that points as O. Draw a semicircle with center O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = . 2.3Now, draw an arc with center B and B radius BD, which intersects the number line in E. Thus, E represent . 2.3
7. Express the following in the form , where and are integers and pq