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EXERCISE 1
Write the correct answer in each of the following:
1. Every rational number is
(A) a natural number
(B)
(C) a real number
(D)
2. Between two rational numbers
(A) there is no rational number
(B) there is exactly one rational number
(C) there are infinitely many rational numbers
an integer
a whole number
(D)
4. The product of any two irrational numbers is
(A) always an irrational number
(B) always a rational number
(C) always an integer
3. Decimal representation of a rational number cannot be
(A) terminating
(B) non-terminating
(C) non-terminating repeating
(D)
(D)
non-terminating non-repeating
5. The decimal expansion of the number
(A) a finite decimal
(B) 1.41421
(C) non-terminating recurring
(D) non-terminating non-recurring
2 is
(B) 0.1416
8. A rational number between
(A)
(A)
2+ 3
2
(C) 0.1416
2 and
2⋅ 3
2
(B)
(D) 0.4014001400014...
3 is
(C)
1.5
(D)
1.8
p
9. The value of 1.999... in the form q , where p and q are integers and q ≠ 0 , is
(A)
19
10
(B)
1999
1000
(C)
2
(D)
1
9
6
(C)
3 3
(D)
4 6
5 6
(C)
25
(D)
10 5
10. 2 3 + 3 is equal to
(A)
11.
2 6
(B)
10 × 15 is equal to
(A)
6 5
(B)
sometimes rational, sometimes irrational
6. Which of the following is irrational?
7. Which of the following is irrational?
(A) 0.14
there are only rational numbers and no irrational numbers
4
9
(B)
12
3
(C)
7
(D)
81
1
12. The number obtained on rationalising the denominator of
7+2
3
(A)
7–2
3
(B)
7+2
5
(B)
1
3+2 2
(D)
3+2 2
is
7+2
45
(D)
1
is equal to
9– 8
13.
(A)
1
3– 2 2
2
(C)
3– 2 2
(
)
14. After rationalising the denominator of
(A)
13
(B)
(A)
16. If
4 3
(A)
(B)
2
2 = 1.4142, then
(C)
5
(D)
35
2
(C)
4
(D)
8
2 –1
is equal to
2 +1
(A) 2.4142
(B) 5.8282
(C)
(D) 0.1718
0.4142
2 2 equals
−
2
1
6
(B)
18. The product
(A)
2
19. Value of
4
(A)
19
7
, we get the denominator as
3 3 –2 2
32 + 48
is equal to
8 + 12
15. The value of
17.
(C)
7–2
1
9
3
2– 6
(C)
1
26
(D)
26
(D)
12
(D)
1
81
2 ⋅ 4 2 ⋅12 32 equals
( 81) −2
(B)
2
(C)
12
(B)
1
3
(C)
9
2
32
is
20. Value of (256)0.16 × (256)0.09 is
(A) 4
(B) 16
(C)
64
(D)
256.25
(D)
x 7 × x 12
21. Which of the following is equal to x?
12
7
(A)
x –x
5
7
12
(B)
1
4 3
(x )
2
3 3
12
(x)
(C)
7
EXERCISE 2
1. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an
irrational number? Give an example in support of your answer.
2. Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer
by an example.
3. State whether the following statements are true or false? Justify your answer.
(i)
2
is a rational number.
3
There are infinitely many integers between any two integers.
(ii)
(iii) Number of rational numbers between 15 and 18 is finite.
p
(iv) There are numbers which cannot be written in the form q , q ≠ 0 , p, q both
are integers.
(v) The square of an irrational number is always rational.
(vi)
12
is not a rational number as 12 and
3
(vii)
15
p
is written in the form , q ≠ 0 and so it is a rational number.
q
3
3 are not integers.
4. Classify the following numbers as rational or irrational with justification :
(i)
196
(ii)
3 18
(iii)
12
75
9
27
(v)
– 0.4
(vi)
(viii)
(1 + 5) – (4 + 5 )
(ix) 10.124124... (x)
(iv)
(vii) 0.5918
1.010010001...
28
343
EXERCISE 3
1. Find which of the variables x, y, z and u represent rational numbers and which
irrational numbers:
x2 = 5
(i)
(ii)
y2 = 9
(iii)
z2 = .04
(ii)
0.1 and 0.11
(iv)
1
1
and
4
5
(iv)
u2 =
17
4
2. Find three rational numbers between
(i) –1 and –2
(iii)
5
6
and
7
7
3. Insert a rational number and an irrational number between the following :
(i)
2 and 3
(ii)
0 and 0.1
(iii)
(iv)
–2
1
and
5
2
(v) 0.15 and 0.16
(vi)
(vii)
(x)
2.357 and 3.121 (viii) .0001 and .001
6.375289 and 6.375738
1
1
and
3
2
2 and 3
(ix) 3.623623 and 0.484848
p
4. Express the following in the form q , where p and q are integers and q ≠ 0 :
(i)
0.2
(ii)
0.888...
(iii)
5.2
(iv)
(v)
0.2555...
(vi)
0.134
(vii)
.00323232...
(viii) .404040...
0.001
5. Simplify the following:
(i)
(ii)
45 – 3 20 + 4 5
3 3 + 2 27 +
(v)
7
(vi)
3
2 3
(ix)
3
–
(
3– 2
)
4
(iii)
24
54
+
8
9
2
(vii)
4
4 28 ÷ 3 7 ÷ 3 7
(iv)
12 × 7 6
81 – 8 3 216 + 15 5 32 + 225
(viii)
3
1
+
8
2
3
6
6. Rationalise the denominator of the following:
2
(i)
3 3
3+ 2
3– 2
(vii)
(ii)
40
3
(viii)
3 5+ 3
5– 3
(iii)
(ix)
3+ 2
4 2
(iv)
16
41 – 5
2+ 3
2– 3
(v)
6
2+ 3
(vi)
4 3+5 2
48 + 18
7. Find the values of a and b in each of the following:
(i)
5+ 2 3
=a−6 3
7 +4 3
3– 5
19
=a 5–
3+2 5
11
(ii)
8. If a = 2 + 3 , then find the value of a –
(iii)
2+ 3
= 2– b 6
3 2 –2 3
(iv)
7+ 5 7 – 5
7
–
=a+
5b
7 – 5 7+ 5
11
1
.
a
3 = 1.732 and 5 = 2.236
9. Rationalise the denominator in each of the following and hence evaluate by taking 2 = 1.414 ,
upto three places of decimal.
4
3
(i)
6
6
(ii)
(iii)
10 – 5
2
(iv)
2
2+ 2
10. Simplify :
(i)
1
13 + 2 3 + 33 2
(
)
−
(vi)
64
1
3
1
64 3
(ii)
–
2
64 3
3
5
4
8
5
1
83
−12
×
(vii)
32
1
16 3
−
1
3
32
5
6
(iii)
1
27
−2
3
(iv)
1
3+ 2
(v)
(625)
1
−
2
−
1
4
2
1
93
(v)
1
36
× 27
×3
−
−
1
2
2
3
EXERCISE 4
1. Express 0.6 + 0.7 + 0.47 in the form
2. Simplify :
3. If
p
, where p and q are integers and q ≠ 0 .
q
7 3
2 5
3 2
–
–
.
10 + 3
6+ 5
15 + 3 2
2 = 1.414, 3 = 1.732 , then find the value of
4. If a =
5. If x =
3+ 5
2
2
, then find the value of a +
4
3
+
.
3 3–2 2 3 3+2 2
1
.
a2
3+ 2
3– 2
and y =
, then find the value of x2 + y2.
3– 2
3+ 2
( )
6. Simplify : (256 )
− 4
7. Find the value of
−3
2
4
(216 )
2
−
3
+
1
( 256)
3
−
4
+
2
1
(243 )− 5
ANSWERS
EXERCISE 1
1.
(C)
2.
9.
(C)
10.
16.
(C)
3.
(D)
4.
(D)
(C)
11.
(B)
12.
(A)
(C)
18.
(C)
17.
(B)
19.
5.
13.
(A)
(D)
(D)
20.
(A)
EXERCISE 2
1. Yes. Let x = 21, y =
2 be a rational number.
2 = 21 + 1.4142 ... = 22.4142 ...
Which is non-terminating and non-recurring. Hence x + y is irrational.
Now x + y = 21 +
2. No. 0 × 2 = 0 which is not irrational .
3.
(i)
False. Although
p
2
is of the form
but here p, i.e., 2 is not an integer.
q
3
(ii)
(iii)
False. Between 2 and 3, there is no integer.
False, because between any two rational numbers we can find infinitely
many rational numbers.
(iv)
True.
(v)
False, as
(vi)
False, because
2
3
is of the form
(4 2)
2
p
but p and q here are not integers.
q
= 2 which is not a rational number.
12
= 4 = 2 which is a rational number.
3
(vii)
False, because
15
3
= 5=
5
which is p, i.e.,
1
5 is not an integer.
6.
(C)
14.
(B)
21.
(C)
7.
8.
(D)
15.
(B)
(C)
4.
(i)
Rational, as
(ii)
3 18 = 9 2 , which is the product of a rational and an irrational number and so an irrational number
.
9
1
=
, which is the quotient of a rational and an irrational number and so an irrational number
27
3
.
(iii)
28
(iv)
(v)
196 = 14
=
343
2
, which is a rational number.
7
Irrational, − 0.4 = −
12
(vi)
=
75
2
10
, which is the quotient of a rational and an irrational.
2
, which is a rational number.
7
(vii)
Rational, as decimal expansion is terminating.
(viii)
(1 + 5) − (4 + 5 ) = –3 , which is a rational number.
(ix)
Rational, as decimal expansion is non-terminating recurring.
(x)
Irrational, as decimal expansion is non-terminating non-recurring.
EXERCISE 3
1. Rational numbers: (ii), (iii)
Irrational numbers: (i), (iv)
2.
3.
4.
(i)
–1.1, –1.2, –1.3
(ii)
0.101, 0.102, 0.103
(i)
2.1, 2.040040004 ...
(ii)
0.03, 0.007000700007, ...
(v)
0.151, 0.151551555 ...
(vi)
1.5, 1.585585558 ...
(ix)
1, 1.909009000 ...
(x)
6.3753, 6.375414114111 ...
(i)
(vi)
1
5
133
990
(ii)
(vii)
8
9
8
2475
(iii)
47
9
(viii)
40
99
(iv)
1
999
(iii)
51 52 53
, ,
70 70 70
(iii)
5
, 0.414114111 ...
12
(vii)
(v)
23
90
3, 3.101101110 ...
(iv)
9 17 19
, ,
40 80 80
(iv)
(viii)
0, 0.151151115 ...
0.00011, .0001131331333 ...
5.
(i)
5
(ii)
7 6
12
(iii) 168 2
6
(i)
2
3
9
(ii)
2
30
3
(iii)
(ix)
(iv)
2+3 2
8
8
3
(iv)
(v)
34 3
3
(vi) 5 − 2 6
(v) 7 + 4 3
41 + 5
(vii) 0
(vi) 3 2 − 2 3
9+4 6
15
(ii) a =
7. (i) a = 11
8.
2 3
9.
(i) 2.309
9
11
(ii) 2.449
10. (i) 6
(ii)
2025
64
(iii) b =
−5
6
(iv) a = 0, b = 1
(iii) 0.463
(iv) 0.414
(v)
0.318
(iii) 9
(iv) 5
(v) 3– 3
1
(vi) –3
(vii) 16
EXERCISE 4
1.
167
90
(vii) 5 + 2 6
2. 1
3. 2.063
4. 7
5. 98
6.
1
2
7. 214
(viii)
5
2
4
(viii) 9 + 2 15
(ix)
3
2