NUMBER SYSTEMS
Directions for questions 1 – 15:
State true or false.
1. There are infinitely many natural numbers between 1 and 2.
2. The difference between any two consecutive natural numbers is one.
3. There are infinitely many rational numbers.
4. All real numbers are rational numbers.
5. Every point on the real number line can be written in the from n , where ‘n’ is a natural number.
6. Every real number is either rational or irrational.
7. The sum of two irrational numbers is always irrational.
8. Every irrational number can be converted into a rational number.
9. 5 3 and 10 3 are rationalizing factors of each other.
10. Every irrational number has a unique rationalizing factor.
11. The product of a
and a
is a
.
m
n
mn
12. If (–a)
= a
, then ‘n’ is an even number.
n
n
13. All prime numbers are odd numbers.
14. All decimal numbers with non–terminating decimal parts are irrational numbers.
15. If ‘a’ and ‘b’ are two consecutive integers and ‘a’ lies to the right of ‘b’, then a – b > 0.
16. Write two integers which are not whole numbers.
17. Insert one rational number between 3 and 4.
18. Insert 3 rational numbers between 4 and 5.
2 and 5
3 .
19. Insert 4 rational numbers between 5
p , where p and q are integers and q 0.
20. Express 3 and –2 in the f orm of q
21. Give two irrational numbers whose sum is rational.
22. Give two irrational numbers whose sum is irrational.
23. Give two irrational numbers whose difference is (i) rational (ii) irrational.
24. Give two irrational numbers whose product is (i) rational (ii) irrational.
25. Give two irrational numbers whose quotient is (i) rational (ii) irrational.
26. Locate 2 and 3 on the number line.
27. Locate the points representing 2+ 1 and 2 –1 on the number line.
____________________________________________________________
1
Get free notes for Class X and IX on
www.cbsekey.com
…
_________________________________________________________________________________
28. Construct a square root spiral starting from 2 upto 5 .
29. Classify the following rational number into numbers with
(i) terminating decimal expansion
(ii) non–terminating decimal expansion.
1 (b) 3
1 (c) 5
3 (d) 7
3
(a) 4
3 and 35
3
30. Find the decimal representation of 24
31. State which of the following fractions are convertible into terminating decimals, doing actual division.
33 (b) 25
4 (c) 15
5
5
(a) 18
(d) 13
32. Write two numbers whose decimal expansion are non–terminating and non repeating.
p .
33. Convert the following decimal numbers into rational number in the form q
(i) 0.08 (ii) 1.205 (iii) 0.001 (iv) 3.2125
34. Convert the following decimal numbers into rational numbers in the form p/q.
(i) 0. 312 (ii) 0. 092 (iii) 0.12345 (iv) 0.4 (v) 2.21347
1 and 3
2 .
35. Find four irrational number between 3
36. Find four irrational numbers between 4 and 5.
37. Classify the following numbers as rational or irrational
(i) 26 (ii) 324 (iii) 0.125 (iv) 0.934984….. (v) 4.02121121112………..
38. Visualize 2.34 on the number line using successive magnification.
39. Visualize 4. 3 on the number line upto three decimal places using successive magnification.
40. Find the sum of (3 + 4 3 + 5 2 )
41. Simplif y:
(i) (5 2 + 7 3) – (3 2 + 2 3 ) (ii) 5 2 × 2 10 × 5 15
6
3
(iii) 5 3 × 7 3 × 2 2 × 3 2 (iv) 3
2
42. Find the simplest rationalizing factors of the following irrationals.
(i) 8
(ii) 18
3 (iii) 5 12 (iv) 7 7
43. Rationalize the denominators of the following.
1 (ii)
1
2
(i) 3
(iii) 1
3
-
-
3
2
____________________________________________________________
2
_________________________________________________________________________________
44. If a = 3 12 + 2 18 and b = 2 + 3 , find the value of
(i) a + b (ii) a – b (iii) 2a + 3b (iv) 5a – 4b
45. Rationalize the denominators
5
3
3
2
(i)
- (ii)
-
10
3
6
2
46. Evaluate
(i) 4
× 4
÷ 4
(ii) 5
× 5
÷ 5
–5
8
7
7
3
–2
× 1
56
0
1 × 16
2
2
9
(iii) (3
)
× (3
)
(iv)
×
2
3
–2
2
28
5
1
-
3
/
5
(v)
(
625
)
(vi)
-
3
/
4
32
47. Evaluate
8
1
/
2
(i) (243)
÷ (64)
(iii) 18
× 8
(iv)
3/ 5
–1/ 4
1/ 2
1/ 2
8
1
/
3
48. Find the point representing 5
3 on the number line geometrically.
.
49. Find the point representing 42 on the number line geometrically.
50. Find the point representing 35 on the number line geometrically.
51. In the following figure, AB = 7 units, BC = 5 units and AC is the diameter of the semicircle. What is the
value of ‘x’?
D
a
x b
A B
C
7
5
52. In the figure, PQ = 12, QS = 6 and PR is the diameter of the semicircle. Find PR.
S
x
6 y
P Q
R
12
____________________________________________________________
3
_________________________________________________________________________________
53. In the figure, ‘O’ is the origin of the number line ‘l ’. AB = 1 unit and AB l . The semi circle with
centre origin and radius OB intersects the line at ‘C’ and ‘D’. Find the numbers represented by ‘C’ and
‘D’.
B
x
1
A
D 1
O C
2 3
54. Look at the square root spiral shown below. Given that P
P
= P
P
= P
P
= P
P
= 1 unit. Find the
1
2
2
3
3
4
4
5
length of the segment OP
.
5
P
5
P
4
P
3
P
2
O P
1
2
3 4 5
55. In the figure, ‘O’ is the centre of the number line ‘ l ’. AB l and CD l . AB = 1 unit, the arc with
centre ‘O’ and radius OB intersects the line at ‘C’. CD = 2 units, the arc with centre ‘O’ and radius OD
intersects the line at ‘E’. Find the number that represents the point E. Check whether E represents a
natural number.
D
B
A
O E
1
2 C
56. In the figure, ‘O’ is the centre of the number line ‘l ’, OA = 6 and OC = 7. An arc with centre ‘O’ and
radius OC intersects the perpendicular on A and ‘B’. Find AB.
B
l
O
1
2
3 4 5 6 7
3
C
A
____________________________________________________________
4
_________________________________________________________________________________
57. In the figure, ‘O’ is the centre of the number line ‘ l ’. A semi circle is drawn with centre ‘O’ and radius
10 units. The semi circle intersects the perpendiculars at the points A and B, respectively represent –6
and 5, at D and C. Calculate the area of the quadrilateral ABCD.
(Take 3 =1.732)
D C
l
5
–12 –10 –8 –6 –4 –2 O 2 4 6 8 10 12
A B
5
NUMBER SYSTEMS
ANSWER KEY
1. False 2. True 3. True
40. 5
4. False 5. False 6. True
41. (i) 2
+ 5
(ii) 500
2
3
3
7. False 8. True 9. True
(iv) 1260 (iv) 3
10. True 11. True 12. True
42. (i)
(ii)
2
2
13. False 14. False 15. True
(iii)
(iv)
3
7
-
6
4
16. (–2, –3) 20.
,
3
43. (i)
(ii) 2(
+
)
2
2
3
2
3
21. (3–
, 4 +
)
5
5
3 +
1
(iii)
22. (
,
)
2
3
2
44. (i) 7
+ 7
23. (i) 3 +
, 2 +
3
2
5
5
(ii) 5
+ 5
(ii) 3 +
, 2 + 2
3
2
5
5
(iii) 15
+ 15
24. (i) 2
,
(ii) 2
,
3
2
5
5
3
2
(iv) 26
+ 26
25. (i)
, –
,
,
3
2
2
2
2
3
29. (i) (a), (c) (ii) (b), (d)
5 + (ii) 2
(
30
3
)
3 +
(
3
1
)
45. (i) 7
30. [0.125, 0.
]
0857143
46. (i) 4
(ii) 5
–4
12
31. (b)
9
2
241
(iii) 3
(iv)
2
33. (i)
(ii)
100
25
200
1
1
257
(v)
(vi) 8
(iii)
(iv)
125
1000
80
47. (i) 216 (ii) 12 (iii)
2
104 (ii) 999
92
34. (i) 333
4115 (iv) 5
2
51.
52. 15
35
(iii) 3333
53.
, –
54. 2
10
10
5
110563
(v) 49950
55.
, No 56.
7
13
37. (i) Irrational (ii) Rational
(iii) Rational (iv) Irrational
(v) Irrational
Directions for questions 1 – 15:
State true or false.
1. There are infinitely many natural numbers between 1 and 2.
2. The difference between any two consecutive natural numbers is one.
3. There are infinitely many rational numbers.
4. All real numbers are rational numbers.
5. Every point on the real number line can be written in the from n , where ‘n’ is a natural number.
6. Every real number is either rational or irrational.
7. The sum of two irrational numbers is always irrational.
8. Every irrational number can be converted into a rational number.
9. 5 3 and 10 3 are rationalizing factors of each other.
10. Every irrational number has a unique rationalizing factor.
11. The product of a
and a
is a
.
m
n
mn
12. If (–a)
= a
, then ‘n’ is an even number.
n
n
13. All prime numbers are odd numbers.
14. All decimal numbers with non–terminating decimal parts are irrational numbers.
15. If ‘a’ and ‘b’ are two consecutive integers and ‘a’ lies to the right of ‘b’, then a – b > 0.
16. Write two integers which are not whole numbers.
17. Insert one rational number between 3 and 4.
18. Insert 3 rational numbers between 4 and 5.
2 and 5
3 .
19. Insert 4 rational numbers between 5
p , where p and q are integers and q 0.
20. Express 3 and –2 in the f orm of q
21. Give two irrational numbers whose sum is rational.
22. Give two irrational numbers whose sum is irrational.
23. Give two irrational numbers whose difference is (i) rational (ii) irrational.
24. Give two irrational numbers whose product is (i) rational (ii) irrational.
25. Give two irrational numbers whose quotient is (i) rational (ii) irrational.
26. Locate 2 and 3 on the number line.
27. Locate the points representing 2+ 1 and 2 –1 on the number line.
____________________________________________________________
1
Get free notes for Class X and IX on
www.cbsekey.com
…
_________________________________________________________________________________
28. Construct a square root spiral starting from 2 upto 5 .
29. Classify the following rational number into numbers with
(i) terminating decimal expansion
(ii) non–terminating decimal expansion.
1 (b) 3
1 (c) 5
3 (d) 7
3
(a) 4
3 and 35
3
30. Find the decimal representation of 24
31. State which of the following fractions are convertible into terminating decimals, doing actual division.
33 (b) 25
4 (c) 15
5
5
(a) 18
(d) 13
32. Write two numbers whose decimal expansion are non–terminating and non repeating.
p .
33. Convert the following decimal numbers into rational number in the form q
(i) 0.08 (ii) 1.205 (iii) 0.001 (iv) 3.2125
34. Convert the following decimal numbers into rational numbers in the form p/q.
(i) 0. 312 (ii) 0. 092 (iii) 0.12345 (iv) 0.4 (v) 2.21347
1 and 3
2 .
35. Find four irrational number between 3
36. Find four irrational numbers between 4 and 5.
37. Classify the following numbers as rational or irrational
(i) 26 (ii) 324 (iii) 0.125 (iv) 0.934984….. (v) 4.02121121112………..
38. Visualize 2.34 on the number line using successive magnification.
39. Visualize 4. 3 on the number line upto three decimal places using successive magnification.
40. Find the sum of (3 + 4 3 + 5 2 )
41. Simplif y:
(i) (5 2 + 7 3) – (3 2 + 2 3 ) (ii) 5 2 × 2 10 × 5 15
6
3
(iii) 5 3 × 7 3 × 2 2 × 3 2 (iv) 3
2
42. Find the simplest rationalizing factors of the following irrationals.
(i) 8
(ii) 18
3 (iii) 5 12 (iv) 7 7
43. Rationalize the denominators of the following.
1 (ii)
1
2
(i) 3
(iii) 1
3
-
-
3
2
____________________________________________________________
2
_________________________________________________________________________________
44. If a = 3 12 + 2 18 and b = 2 + 3 , find the value of
(i) a + b (ii) a – b (iii) 2a + 3b (iv) 5a – 4b
45. Rationalize the denominators
5
3
3
2
(i)
- (ii)
-
10
3
6
2
46. Evaluate
(i) 4
× 4
÷ 4
(ii) 5
× 5
÷ 5
–5
8
7
7
3
–2
× 1
56
0
1 × 16
2
2
9
(iii) (3
)
× (3
)
(iv)
×
2
3
–2
2
28
5
1
-
3
/
5
(v)
(
625
)
(vi)
-
3
/
4
32
47. Evaluate
8
1
/
2
(i) (243)
÷ (64)
(iii) 18
× 8
(iv)
3/ 5
–1/ 4
1/ 2
1/ 2
8
1
/
3
48. Find the point representing 5
3 on the number line geometrically.
.
49. Find the point representing 42 on the number line geometrically.
50. Find the point representing 35 on the number line geometrically.
51. In the following figure, AB = 7 units, BC = 5 units and AC is the diameter of the semicircle. What is the
value of ‘x’?
D
a
x b
A B
C
7
5
52. In the figure, PQ = 12, QS = 6 and PR is the diameter of the semicircle. Find PR.
S
x
6 y
P Q
R
12
____________________________________________________________
3
_________________________________________________________________________________
53. In the figure, ‘O’ is the origin of the number line ‘l ’. AB = 1 unit and AB l . The semi circle with
centre origin and radius OB intersects the line at ‘C’ and ‘D’. Find the numbers represented by ‘C’ and
‘D’.
B
x
1
A
D 1
O C
2 3
54. Look at the square root spiral shown below. Given that P
P
= P
P
= P
P
= P
P
= 1 unit. Find the
1
2
2
3
3
4
4
5
length of the segment OP
.
5
P
5
P
4
P
3
P
2
O P
1
2
3 4 5
55. In the figure, ‘O’ is the centre of the number line ‘ l ’. AB l and CD l . AB = 1 unit, the arc with
centre ‘O’ and radius OB intersects the line at ‘C’. CD = 2 units, the arc with centre ‘O’ and radius OD
intersects the line at ‘E’. Find the number that represents the point E. Check whether E represents a
natural number.
D
B
A
O E
1
2 C
56. In the figure, ‘O’ is the centre of the number line ‘l ’, OA = 6 and OC = 7. An arc with centre ‘O’ and
radius OC intersects the perpendicular on A and ‘B’. Find AB.
B
l
O
1
2
3 4 5 6 7
3
C
A
____________________________________________________________
4
_________________________________________________________________________________
57. In the figure, ‘O’ is the centre of the number line ‘ l ’. A semi circle is drawn with centre ‘O’ and radius
10 units. The semi circle intersects the perpendiculars at the points A and B, respectively represent –6
and 5, at D and C. Calculate the area of the quadrilateral ABCD.
(Take 3 =1.732)
D C
l
5
–12 –10 –8 –6 –4 –2 O 2 4 6 8 10 12
A B
5
NUMBER SYSTEMS
ANSWER KEY
1. False 2. True 3. True
40. 5
4. False 5. False 6. True
41. (i) 2
+ 5
(ii) 500
2
3
3
7. False 8. True 9. True
(iv) 1260 (iv) 3
10. True 11. True 12. True
42. (i)
(ii)
2
2
13. False 14. False 15. True
(iii)
(iv)
3
7
-
6
4
16. (–2, –3) 20.
,
3
43. (i)
(ii) 2(
+
)
2
2
3
2
3
21. (3–
, 4 +
)
5
5
3 +
1
(iii)
22. (
,
)
2
3
2
44. (i) 7
+ 7
23. (i) 3 +
, 2 +
3
2
5
5
(ii) 5
+ 5
(ii) 3 +
, 2 + 2
3
2
5
5
(iii) 15
+ 15
24. (i) 2
,
(ii) 2
,
3
2
5
5
3
2
(iv) 26
+ 26
25. (i)
, –
,
,
3
2
2
2
2
3
29. (i) (a), (c) (ii) (b), (d)
5 + (ii) 2
(
30
3
)
3 +
(
3
1
)
45. (i) 7
30. [0.125, 0.
]
0857143
46. (i) 4
(ii) 5
–4
12
31. (b)
9
2
241
(iii) 3
(iv)
2
33. (i)
(ii)
100
25
200
1
1
257
(v)
(vi) 8
(iii)
(iv)
125
1000
80
47. (i) 216 (ii) 12 (iii)
2
104 (ii) 999
92
34. (i) 333
4115 (iv) 5
2
51.
52. 15
35
(iii) 3333
53.
, –
54. 2
10
10
5
110563
(v) 49950
55.
, No 56.
7
13
37. (i) Irrational (ii) Rational
(iii) Rational (iv) Irrational
(v) Irrational
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