2
SQUARE ROOT & CUBE ROOTS
CHAPTER Important Facts And Formulae :
Next, we bring down the next period which is 13. Now, trial divisor is 12 × 2 = 24 and trial dividend is 313. So, we take 241 as dividend and 1 as quotient. The remainder is 72. Bring down the next period i.e., 69. Now, the trial divisor is 121 × 2 = 242 and the trial dividend is 7269. So, we take 3 as quatient and 2423 as dividsor. The remainder is then zero. Hence, 1471369 = 1213.
Square Root : If x2 = y, We say that the square root of y is x and we write,
y x . Thus,
4 =2,
9 =3, 196 =14.
Cube Root : The cube root of a given number x is the number whose cube is x. We denote the cube root of x by 3 x . Thus,
3
8 = 3 2 2 2 =2,
3
7 7 7 =7 etc.
1 1471369 (1213 1 22 47 44 313 241 241 7269 2423 7269 ×
Note : 1.
xy = x × y
2.
y xy x x x y= y= y× y= y .
EXAMPLES Ex.1 Sol.
Evaluate 6084 by factorization method, Method : Express the given number as the product of prime factors. Now, take the product of these prime factors choosing one out of every pair of the same primes. This product gives the squeare root of the given number. Thus, resolving 6084 into prime factors, we get:
2 6084 2 3042 3 1521 3 507 13 169 13
Ex.2 Sol.
Ex.3 Sol.
248 51 169 . Given expression Evaluate :
248 51 13 = 248 64 = 248 8 =
=
256 =16. Ex.4
If a*b*c =
a 2 b 3 c 1
, then find the value
of 6*15*3. Sol.
6*15*3 =
6084 = 22 × 32 × 132 6084 = (2 × 3 × 13) = 78.
=
Find the square root of 1471369. Explanation : In the given number, mark off the digits in pairs starting from the unit’s digit. Each pair and the remaining one digit is called a period. Now, 1 2 = 1. One subtracting, we get 0 as remainder. Now, bring down the next period i.e., 47. Now, trial divisor is 1 × 2 = 2 and triall dividend is 47. So, we take 22 as dvsor and put 2 as quotient. The remainder is 3.
Ex.5 Sol. Ex.6 Sol.
6 215 3 3 1
818 144 = 4 4
12 = 3. 4
Find the value of 1
=
1
9 . 16
25 25 5 1 9 1 . = 16 4 16 16 4
What is the square root of 0.0009?
9 9 3 = = 0.03 0.0009 = 10000 = 10000 100
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Ex.7
Evaluate
175.2976 .
Ex.11
Sol.
Method : We make even number of decimal places by affixing a zero, If necessary. Now we mark off periods and extract the square root as shown.
3 upto three places of
decimal. Sol.
1 3.000000 (1.732 1 27 200 189 343 1100 1029 7100 3462 6924
1 175.2976 (13.24 1 23 75 69 262 629 524 2644 10576 10576 × Ex.12
Find the value of
3 = 1.732.
If
3 =1.732,
175.2976 = 13.24.
192
find
the
value
of
1 48 75 correct to 3 places of 2
decimal. Ex.8
What will come in place of question mark in each of the following questions?
32.4 2 ?
(i) Sol.
(i) Let
(ii)
Sol.
86.49 5 (?) 2 12.3
32.4 32.4 = 2. Then, = 4 4x x x
3 3 2 3 3 1.732
= 32.4 x = 8.1 (ii) Let
86.49 5 x 2 =12.3.
Then, 9.3 +
5 x =12.3 2
5 x
2
= 12.3 - 9.3 = 3
Ex.9
5+x2 = 9 x2— = 9 - 5 = 4 x= 4 =2.
Find the value of
Evaluate :
Sol.
Given exp. =
0.289 . 0.00121
If
1
x 13 169 x 13 = 1 1 144 144 12 144 12
x 25 x = 25. 144 144
95 85 18900 = 5 5 900 17 19 21
= 5 × 30 = 150. Simplify :
x 13 1 = , then find the value of x. 144 12 2
Sol.
9.5 .0085 18.900 .0017 1.9 0.021 Now, since the sum of decimal places in the numerator and denominator and under the radial sign is the same, we remove the decimal.
Given exp. =
Ex.14 Ex.10
9.5 .0085 18.9 . .0017 1.9 0.021
Ex.13
0.289 0.28900 28900 170 = = = 0.00121 0.00121 121 11
Sol.
1 48 75 2 1 64 3 16 3 25 3 2 1 8 3 4 3 5 3 2 192
[(12.1) 2 (8.1) 2 ] [(0.25) 2 (0.25)(19.95)] Sol.
Given exp. = =
(12.1 8.1)(12.1 8.1)
0.250.25 19.95
4 0.25
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400 16 4 . 25
20.2 4 0.25 20.2
Ex.15 Sol.
If x = 1+ 2 and y = 1- 2 , find the value of (x2+y2).
Ex.21
x2 + y2 = (1 + 2 )2 =2(1)2 + ( 2 ) 2 ] = 2 × 3 =6.
2 Sol.
Ex.16 Sol.
2 2 = 1.4142, find the value of (2 2 ) .
If
Evaluate 0.9 upto three place of decimal.
(2 2 )
Ex.22
(2 2 )
×
=
(2 2 )
2 2 2 (4 2)
2( 2 1) ( 2 1) = (1.4142–1) = 0.4142. 2
=
9 0.900000 (.948 81 900 184 736 16400 1888 15104
(2 2 )
2 =
5 3 If x = and y = 5 3
5 3 5 3 , find the
value of (x2 + y2). Ex.17 Sol.
If
0.9 = 0.948 15 = 3.88, find the value of 5 53 15 3.88 = 3 3 3 3 3
5 . 3
Sol.
x=
=
( 5 3) ( 5 3) ( 5 3)2 (5 3) ( 5 3) ( 5 3)
5 3 2 15 4 15 . 2
= 1.2933 .......... = 1.29 3 . Ex.18 Sol.
Ex.19 Sol.
Find the least square number which is exactly divisible by 10, 12, 15 and 18. L.C.M. of 10, 12, 15, 18 = 180. Now, 180 = 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5. To make it a perfect square, It must be multiplied by 5. Required number = (22 × 32 ×52)= 900. Find the greatest number of five digits which is a perfect square. Greatest number of 5 digits is 99999.
3 99999 (316 9 61 99 61 626 6899 3756 143
( 5 3)
y=
=
( 5 3)2 (5 3)
x2 + y2 = (4 + ( 15 ) 2 =2 [(4) 2 ( 15 ) 2 ] = 2 × 31 = 62. Ex.23
Find the cube of root of 2744.
Sol.
Method : Resolve the given number as the product of prime factors and take the product of prime factor. Resolving 2744 as the product of prime factors, we get :
2744 = 23 × 73.
2 2744 2 1327 2 686 7 343 49 7 7
Find the smallest number that must be added to 1780 to make it a perfect square.
Sol.
Required to be added = (43)2 – 1780 = 1849 – 1780 = 69.
5 3 2 15 4 15 . 2
3
4 1780 (42 16 82 180 164 16
( 5 3)
( 5 3) ( 5 3)
Required number = (99999 – 143) = 99856. Ex.20
3
2744 = 2 × 7 = 14.EXERCICE
Ex.24
By what least number 4320 be multiplied to obtain a number which is a perfect cube ?
Sol.
Clearly, 4320 = 23 × 33 × 22 × 5. To make it a perfect cube, it must be multiplied by 2 × 52 i.e., 50.
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EXERCISE 1.
2.
53824 ? (A) 202 (B) 232
(B) 347
The value of
(C) 363
The digit in the unit’s place in the square root of 15876 is (A) 2 (B) 4 (C) 3 (D) None
14.
How many two digit-numbers satisfy this property : The last digit (unit’s digit) of the square of the two-digit number is 8 ? (A) 1 (B) 2 (C) 3 (D) None
15.
What is the square root of 0.16 ? (A) 0.004 (B) 0.04 (C) 0.4
(D) 332
The square root of 64009 is (A) 253
3.
(C) 242
13.
(D) 803
10 25 108 154 225
is (A) 4
(B) 6
(C) 8
(D) 10
16.
0.0004761 is -
The value of
(A) 0.00021 (B) 0.0021 4.
5.
Evaluate :
41 21 19 – 9 -
(A) 3
(B) 5
17. (D) 6.9
176 2401 is equal to (A) 14
6.
(C) 6
(B) 15
(C) 18
(D) 24
625 14 11 is equal to 11 25 196 (A) 5 (B) 6 (C) 8 (D) 11
7.
8.
225 25 16 ? 729 144 81 1 5 5 (A) (B) (C) 48 48 16
(C) 0.0609
(D) 0.069
0.0225 = ? (A) 0.0375 (B) 0.3375
(C) 3.275
(D) 32.75
(C) 0.42
(D) None
0.01 0.0064 = ? (A) 0.03
(B) 0.3
The value of
0.01 + 0.81 + 1.21 + 0.0009
is (A) 2.03 (D) None
(B) 2.1
(C) 2.11
.0025 2.25 .0001 = ?
21.
(A) .000075 (B) .0075
(C) .075
(D) None
(B) 200
(C) 240
(D) 256
If x * y x y xy , the value of 6 * 24 is (A) 41 (B) 42 (C) 43 (D) 44
22.
(C) 1.45
(D) 1.55
1.5625 = ? (A) 1.05
23.
If
(B) 1.25
.00000676 =.0026, the square root of
10.
If y = 5, then what is the value of 10y y 3 y 2 ?
67,60,000 is (A)
1 26
11.
(A) 50 2 1 110 =? 4 (A) 10.25
If
18225 =135, then the value of
12.
(D) 2.13
The square root of (2722– 1282) is (A) 144
9.
(D) 0.21
1.52 ×
19.
20.
(C) 0.021
0.00004761 equals (A) 0.00069 (B) 0.0069
18.
(D) 4
(B) 100
(C) 200 5
(D) 500 24.
(B) 10.5
25 1 – =? 81 9 2 4 (A) (B) 3 9
(C) 11.15
(D) 19.5
(B) 26
(C) 260
( 182.25 1.8225 0.018225 0.00018225 is :
(C)
16 81
(D)
25 81
(D) 2600
(A) 1.499985
(B) 14.9985
(C) 149.985
(D) 1499.85
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25.
Given that value of (A) 36.164
26.
13 =3.605 and 130 =11.40, find the 1.3 1300 0.013 (B) 36.304
36.
(C) 37.164 (D) 37.304
169 52 = , the value of x is 289 x (A) 52 (B) 38 (C) 62
37.
If
* * =1 For what value of * the statement 15 135 is true ?
(A) 15 28.
(B) 25
(C) 35
(A) 12 30.
(B) 0.1
162 . ? 128 (C) 144 (D) 196
32.
(B) 15, 12
42.
0.25 to
(C) 50
43.
(D) 125
3 n =729, then the value of n is : (A) 6 (B) 8 (C) 10 (D) 12
If
If
(D) None
Three-fifth of the square of a certain number is 126.15. What is the number ? (A) 14.5 (B) 75.69 (C) 145 (D) 210.25 0.361 ? 0.00169 1 .9 19 (A) (B) 13 13
34.
(B) 24
(C) 28
5 17 16 (C) 12 17
(D) 32
3 of 2872. 4 (B) 676 (C) 1396
28 ? + 1426 = (A) 576
.45. (D) 1444
? 54 169 39 (A) 108 (B) 324
46. (C) 2916
1 .9 130
(D)
190 13
1 17 7 (D) 129 17
(B) 12
x 14 , then x is equal to 169 13 (A) 1 (B) 13
If
1
(C) 27 35.
(C)
48.4 is equal to 0.289
44.
18 14 x =84, then x equals :
(A) 22
(D) None
a .04 .4 a = .004 × .4 × b , then b is (A) 16×10-3 (B) 16×10-4
(A) 1 33.
(C) 22, 19
If
(C) 16×10-5
give the result as 25 ? (B) 25
1369 .0615 x =37.25, then x is equal to(A) 10-1 (B) 10-2 (C) 10-3 (D) None
(D) 100
What number should be divide by (A) 12.5
(D) None
(A) 8, 5
(C) 10
(C) 1000
If ( x 1)( y 2) 7, x and y being positive whole numbers, then the values of x and y respectively are -
If 0.13 ÷ p2 = 13, then p equals : (A) 0.01
31.
(B) 14
0.0169 ? 1.3 (A) 10 (B) 100
40.
What should come in place of both the question
?
(D) None
If
(D) 45
Which number can replace both the question 1 4 marks in the equation 2 ? . ? 32 1 (A) 1 (B) 7 (C) 7 (D) None 2
marks in the equation
(C) 4.9
39.
41. 29.
0.196 0 .2 ? (A) 0.49 (B) 0.7
(D) 68 38.
27.
x 441 = 0.02, then the value of x is (A) 0.1764 (B) 1.764 (C) 1.64 (D) 2.64
If
If
1
(D) 4800 (A) 1
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(D) None of these 55 x 1 , then the value of x is 729 27
(B) 3
(C) 5
(D) 7
47.
The value of (A) 1.410
48.
3
(B) 2 3
24 216
The value of
3 (A) 4 52.
53.
54.
(C) 3 3
(D) 4 3
59.
9.5 .085 equals .0017 .19 (A) .05 (B) 5
(C) 50
(D) 500
(0.03) 2 (0.21) 2 (0.065) 2
(D) 3( 3 2 )
60.
The value of
(0.003) 2 (0.021) 2 (0.0064) 2
is -
?
96 (A) 2 6
51.
0.081 0.324 4.624 is equal to 1.5625 0.0289 72.9 64 (A) 0.024 (B) 0.24 (C) 2.4 (D) 24
By how much does 12 18 exceed 3 2 ? (A) 2 4 3 (B) 3 2 2 (C) 2( 3 2 )
50.
58.
(2 27 75 12 ) is equal to (A)
49.
2 upto three place of decimal is(B) 1.412 (C) 1.413 (D) 1.414
(A) 0.1
2 (B) 2
(C) 6 2
80 112 1 3
6
61.
(C) 1
(C) 102
(D) 103
The square root of (7+ 3 5 )(7- 3 5 ) is (A)
is -
45 63 (B) 1
(D)
(B) 10
(B) 2
5
(C) 4
(D) 3 5
2
7 9
(D) 1
3 4
62.
1 3 simplifies to 3 4
If 3 5 + 125 =17.88, then what will be the value of 80 6 5 ? (A) 13.41 (B) 20.46 (C) 21.66 (D) 22.36
(A)
3 4
(B)
(C)
4 3
(D) None of these
50 98 is equal to (A) 63.75 (B) 65.95 (C) 70
1 2 is equal to 2 1 1 1 (A) 2 (B) 3 (C) 4 2 2 2
3
2
(D) 70.25
63.
Given 2 =1.414. The value of
(D) 5
1 2
8 2 32 3 128 4 50 is (A) 8.246
(B) 8.484
(C) 8.526
(D) 8.876
64.
If a = 0.1039, then the value of
4a 2 4a 1 3a
is 3 12
55.
The approximate value of (A) 1.0605
(B) 1.0727
2 21
2 28
(C) 1.6007
98
(A) 0.1039
(D) 1.6026
.081 .484 is equal to .0064 6.25
(A) 0.9
57.
(B) 0.99
65.
66.
(C) 9
(D) 99
0.204 42 is equal to 0.07 304 1 (A) (B) 0.06 (C) 0.6 6
(C) 1.1039
(D) 2.1039
is -
(D) 6
67.
(0.75) 3 [0.75+(0.75)2+1] is1 0.75 (B) 2 (C) 3 (D) 4
The square root of (A) 1
56.
(B) 0.2078
If 3a = 4b = 6a and a + b + c = 27 29 , then a 2 b 2 c 2 is (A) 3 29
(B) 81
(C) 87
(D) None of these
The square root of 0. 4 is (A) 0. 6 (B) 0. 7 (C) 0. 8
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(D) 0. 9
68.
Which one of the following numbers has rational square root ? (A) 0.4
(B) 0.09
(C) 0.9
The value of
0.4 is (B) 0.2 (C) 0.51
(D) 0.63
0.121 is (B) 0.11 (C) 0.347
(D) 1.1
80.
(D) 0.025
The least perfect square, which is divisible by each of 21, 36 and 66 is (A) 213444 (B) 214344 (C) 214434
69.
(A) 0.02 70.
The value of (A) 0.011
81.
The value of (A) 0.008
The least number by which 294 must be multiplied to make it a perfect square is (A) 2
82. 71.
(D) 231444
0.064 is (B) 0.08 (C) 0.252
(D) 0.8
(B) 3
72.
73.
The value of
(C) 0.63
(D) None of these
(A) 5
If
1 0.01 1 0.1
(B) 1.1
84.
is clese to (C) 1.6
(D) 1.7
5 = 2.236, then the value of
(B) 0.000004
(C) 0.02
(D) 0.04
If
8 is 3 (C) 1.633 (D) 2.666
24 =4.899, the value of
(B) 1.333
6 =2.449, then the value of (B) 0.8163
3 2
If 5 =2.236, then the value of
(B) 7.826
86.
The smallest number added to 680621 to make the sum a perfect square is (A) 4 (B) 5 (C) 6 (D) 8
87.
The greates four-digit perfect square number, is(A) 9000 (B) 9801 (C) 9900 (D) 9981
88.
The least number of a digits which is a perfect square is (A) 1000 (B) 1016 (C) 1024 (D) 1036
89.
Given
is -
2 3 (C) 1.223 (D) 1.2245 5 10 125 2 5
(C) 8.944
1
(D) 10.062
17
(B)
29
(C) 12
(D) 13
The least perfect square number divisible by 3, 4, 5, 6 and 8 is (A) 900
(B) 1200
(C) 2500
(D) 3600
5 =2.2361,
equal to (A) 1.89
If 2 * 3 = 3 and 3 * 4 =5, then the value of 5 *12 is (A)
(D) 81
What is the least which should be subtracted from 0.000326 to make it a perfect square ? (A) 0.000002
(A) 5.59
79.
(C) 62
(D) None of these
is equal to -
78.
(B) 36
(C) .745 If
(D) 30
What is the smallest number to be subtracted form 549162 in order to make it a perfect square? (A) 28
85.
(C) 15
(B) .447
(A) 0.6122
77.
5
is -
(B) 6
(A) .367
(A) 0.544
76.
(D) 11
(B) 0.2
1
75.
(C) 7
(A) 0.02
(A) 0.6 74.
(B) 3
The least number by which 1470 must be divided to get a number which is a perfect square is -
The value of
83.
(D) 24
Find the smallest number by which 5808 should be multiplied so that the product becomes a perfect square (A) 2
0.16 is 0.4
(C) 6
1
90.
3 =1.7321, then
(B) 1.984
1
(C) 1.9841
5 3 (D) 2
1
( 9 8) ( 8 7) ( 7 6) 1 1 is equal to ( 6 5) ( 5 4) 1 (A) 0 (B) (C) 1 (D) 5 3
manishkumarphysics.in
is
91.
1 1 2 2 simplifies to 2 2 2 2 (A) 2- 2 (B) 2 (C) 2+ 2 (D) 2 2
92.
If
nearest to (A) 0.172 (B) 0.414
93.
94.
95.
96.
98.
99.
is
(D) 1.414
is equal to 7 5 (A) 1 (C) 6 35 If
52 3
74 3 (A) a = –11, b= – 6 (C) a = 11, b= – 6 If
2 1 2 1
is
=? 3 5 2 12 32 50 (A) 3 (B) 3 2 (C) 6 (D) None of these
2 3 2 3 3 1 simplifies to 2 3 2 3 3 1 (A) 16– 3 (B) 4– 3 (C) 2– 3 (D) 2+ 3 1 If x = (7 – 4 3 ), then the value of x isx (A) 3 3 (B) 8 3
If x =
and y =
3 1 (x2 + y2) is (A) 10 (B) 13
3 1 3 1
, then the value of
(C) 14
(D) 15
5 1
(C)
3 5
, the value of
(D)
5 3
104.
A group of student decided to collect s many paise from each member of the group as is the number of members. If the total collection amounts to Rs. 59.29, The number of members in the group is (A) 54 (B) 67 (C) 77 (D) 87
105.
The cube root of .000216 is (A) .6 (B) .06 (C) .006 (D) None of these
106.
107.
12 =? 125 2 3 (A) 1 (B) 1 5 5 3
4
(C) 1
4 5
(D) 2
2 5
3
.000064 = ? (A) .02 (C) 2
(B) .2 (D) None of these
108.
The largest four-digit number which is a perfect cube, is (A) 8000 (B) 9261 (C) 9999 (D) None of these
109.
By what least number 675 be multiplied to obtain a number which is a perfect cube ? (A) 5 (B) 6 (C) 7 (D) 8
110.
What is the smallest number by which 3600 be divided to make it a perfect cube ? (A) 9 (B) 50 (C) 300 (D) 450
(D) 14 + 8 3
3 1
5 1 2 a ab b 2 a 2 ab b 2 is 4 3 (A) (B) 3 4
5 1
A General wishes to draw up his 56581 soldiers in the form of a solid square. After arranging them, he found that some of them are left over. How many are left ? (A) 65 (B) 81 (C) 100 (D) None of these
(B) 0.414 (D) 1.414
3 6
and b =
103.
(B) a = –11, b= 6 (D) a = 6, b= 11
2 =1.414, the square root of
5 1
A man plants 15376 apple trees in this garden and arranged them so that there are as many rows as there are apples trees in each row. The number of row is (A) 124 (B) 126 (C) 134 (D) 144
(B) 2 (D) 6 35
a b 3 , then -
If a =
102.
7 5
(C) 14
100.
(C) 0.586
2 1
3 2 4 3 6 =? 6 2 8 12 6 3 (A) 3 2 (B) 3 2 (C) 5 3 (D) 1
nearest to (A) 0.172 (C) 0.586 97.
2 1
2 =1.4142, the square root of
101.
manishkumarphysics.in
111.
The square root of 73.96 is (A) 8.6 (B) 86 (C) 0.86 (D) None of these
122.
2 2 2 2 2 is equal to 31
(A) 0 x 4 = then the value of x is 7 49 (A) 9 (B) 25 (C) 16 (D) 8
(B) 2
(C) 1
(D) 2 32
112.
If
123.
113.
Which of the following can not be a digit in the unit place of a perfect square (A) 1 (B) 5 (C) 7 (D) 0
The greatest six digit number which is perfect square is (A) 998004 (B) 998006 (C) 998049 (D) 998001
124.
Divide the number 26244 by the smallest number so that the quolient is a pertect cube, so the smallest number is (A) 4 (B) 6 (C) 36 (D) 16
125.
The pertect cube cube nearest to 2750 is (A) 2749 (B) 2747 (C) 2744 (D) 2754
126.
The least number which must be added to 4931 to make it perfect square is (A) 110 (B) 120 (C) 130 (D) None
127.
The least number which must be added to 4931 to make it perfect square is (A) 20 (B) 30 (C) 40 (D) 50
128.
Least pertect square of 6 digits is (A) 998000 (B) 998001 (C) 998002 (D) None of these
129.
A 8 × 6 × 4 cm3 metalic cube is melted. Find the minimum volume of molten metal which should be added to mould it into a cube whose edge is an x, where x is an interger (A) 20 (B) 21 (C) 23 (D) 24
130.
Value of (A) 50
114.
If
35 = 5.9160, then value of
7– 5 (B) 10.9160 (D) 12
(A) 9.1060 (C) 11.9160 115.
7 5
is -
3018 36 169 = 0.9 × 0.09 × 2
If x =
x is 2 (B) 44
then value of (A) 55 116.
117.
118.
(D) 42
0.9 0.09 x =0.9 × 0.09 × 2 then value of x is 2 (A) 0.081 (B) 0.810 (C) 0.81 (D) 8.09 If
The value of (A)
6 1
(C)
5 2
If x = 2 5 4 (C) 3 5
(A) 1
119.
(C) 63
3
2
7 2 10 is -
4 3 (D) 2 + 5 (B)
93 , then value of x is 125 1 (B) 2 5 1 (D) 4 5
A number is 64 time of the square of its reciprocal. The number is (A) 10 (B) 4 (C) 2 (D) 16
120.
The smallest perfect square number exactly divisible by 4, 5, 6, 15, 18, is (A) 1800 (B) 225 (C) 361 (D) 900
121.
A gardner plants 17956 trees in such a way that there are as many rows as there are trees in each row. The number of trees in a row are (A) 136 (B) 164 (C) 134 (D) 166
3
392 3 448 is (B) 52 (C) 54
(D) 56
131.
The smallest number by which 137592 should be multiply to make it perfect cube (A) 1183 (B) 1180 (C) 1181 (D) None of these
132.
The volume of two cubes are in the ratio of 343 : 1331, the ratio of their edges is (A) 7 : 10 (B) 7 : 11 (C) 7 : 12 (D) None of these
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ANSWER KEY Q.No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A A C B A D C B D B B D D C B B B B D Ans. B Q.No 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 B D B D B D D C B A D C B B A A B C A Ans. D Q.No 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A D C A A D C B B B D C B A B D A C B Ans. C Q.No 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 C C C B C A B D C C C C B C D B D D A Ans. B Q.No 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 B D D A A B C C D B B C D C B D A C C Ans. C Q.No 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 A C C B B B B A D A C C C A A C A B D Ans. B Q.No 121 122 123 124 125 126 127 128 129 130 131 132 D D C C A C B D D A B Ans. C
Hints & Solution 1.
3.
2 53824 (232 4 43 138 129 462 924 924 ×
7.
9.
Given exp.
5.
225 25 – = 144 729
10 25 108 154 15
=
10 25 108 169
=
10 25 108 13 = 10 25 121
=
10 25 11 = 10 36 = 10 6
6*24 = 6 + 24 +
10.
13.
=
41 – 21 19 – 3 = 41 – 21 16
=
41 – 21 4 =
176 49 = 225 = 15.
144
= 50 × 125 – 25 = 50 × 100 = 50 × 10 = 500
Given exp.
=
6 24 = 30 +
10 y y3 – y 2 = 10 × 5 53 – 52
1 15876 (126 1 22 58 44 246 1476 1476 ×
Given exp.
81
= 30 + 12 = 42.
=
41 – 5 =
16
15 9 5 = . 108 4 16
53824 232 .
= 16 = 4. 4.
Given exp.
36 = 6.
4 2401 (49 16 89 801 801 ×
14.
15876 126
A number ending in 8 can never be a perfect square.
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4761
0.00004761
17.
8
4761
=
8
10 10 69 = = 0.0069 10000
69
25.
Given exp. = 1.30 1300 0.0130
10 4
130 130 13 100 100 10000
=
130 130 13 10 10 100 11.40 11.40 3.605 10 = 10 100 =
18.
225 10000
1.52 × 0.0225 = 1.52 × = 2.225 ×
15 = 2.25 × 0.15 = 0.3375 100
0.01 0.0064 1.5 2
19.
225 10000
26.
64 10000
= 2 0.01 0.0064 0.01
= 1.14 + 36.05 + 0.114 = 37.304.
27.
8 0.01 0.08 0.09 0.3. = 0.01 100
20.
Given exp. =
1 81 121 9 100 100 100 10000
1 9 11 3 10 10 10 100 = 0.1 + 0.9 + 1.1 + 0.03 =2.13.
Given Exp. =
15 2 3 2 15 3 45.
1 2 x . x 32
4 28.
Let
9 Then, x2 = 32 × = 144 x = 2
25 255 1 10000 100 10000
29.
5 15 1 75 0.00075. = 100 10 100 100000
Let
x
128
Then, x2 =
144 = 12
162 . x
128 162 64 2 18 9
8 2 6 2 3 2 = 8 × 6 × 3 = 144
22.
1 1.5625(1.25 1 22 56 44 245 1225 1225 ×
0.13
30.
= 0.00000676 10
p2
13 p2 0.13 1 p 13 100
6760000 0.00000676 1012
23.
24.
Let the missing number be x Then, x 2 = 15 × 136 x = 15 135 =
=
21.
52 169 52 13 52 17 x 38 x 289 x 17 13
12
31.
.0026 10 2600. 6
1 1 1 0.1. 100 100 10
Let the required number be x. Then, x x 25 25 x 25 0.5 12.5 0 .5 0.25
Given exp. =
18225 10 2
18225 10 4
18225 10 6
18225
32.
108
18225 18225 18225 18225 10 10 2 10 3 10 4 135 135 135 135 = 10 100 1000 10000 =
6 2 6 2 n 3 n = 729 = 3 ( 3 n ) = (3 ) 3
= 1312 n = 12 33.
18 14 x 84 18 14 x 84 84 x
= 13.5 + 1.35 + 0.135 + 0.0135 = 14.9985
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84 84 = 28 18 14
34.
35.
Let 28 x + 1426 = 3 × 718 x 54 x 54 . Then, x 169 39 13 39
Let
44.
48.4 0.289
45.
1
59 2 = 13 18 x 18 324 39
36.
37.
54 0.02 x 13 441 39 = 18 x = (0.42)2 = 0.1764. x
.0196 .0169 0.2. Then, x x .0196 1.96 = =.49. 0.04 x = .04 4
Let
39.
37 +
1
= 40.
10 3
10
–3
a b
.004 .4 =
.04 .4
2 2 – 75 12 2 9 3 – 25 3 4 3 = 6 3 5 3 2 3 3 3.
49.
( 12 18 ) ( 3 2 )
= ( 43 9 2) – ( 3 2) = (2 3 3 2 ) ( 3 2 )
42.
= (2 3 3 ) (3 2 2 ) 3 2 2 .
a .004 .4 .004 .4 b .04 .4
.0000064 = .04
Let the number be x. Then, 3 2 x 126.15 x 2 5 5 = 126.15 210.25 x 210.25 14.5 3 0.36100 0.00169
36100 190 . 169 13
4 6 36 6 16 6
96 =
2 6 6 6
45 163 =
4 6
80 112
51.
52. 43.
24 216
50.
a .00064 16 .00016 5 = 16 × 10–5 b 4 10
0.361 0.00169
2 = 1.414.
48.
(x – 1) = 7 and (y + 2) = 7
41.
27 x 27 x 28 x 1 27
.
( x 1)( y 2) 7 (x – 1)(y + 2) = (7)2
784 27 x 28 729 27 27
55 x 1 729 27
1 2.000000 (1.414 1 24 100 96 281 400 281 11900 2824 11296
.0615 x 37.25 .0615 x 0.25
.615 + x = (0.25)2 = 0.0625 x = 0.001
784 27 x 28 729 27 27
x x 169 27
47.
0.0169 x 1.3. Then, 0.0169x = (1.3)2 1.69 100 . = 1.69 x = 0.0169
38.
1
=
Let
48400 220 16 12 . 289 17 17
27 x 27 x 28 x 1. 27
46.
48.400 0.289
4 5 4 7 3 5 3 7
8 6
2.
4 6
16 5 16 7 95 97 4( 5 7 ) 3( 5 7 )
=
4 1 1 3 3
3 5 125 17.88 3 5 25 5 17.88 3 5 5 5 17.88 8 5 17.88 5 2.235 80 6 5 16 5 6 5 = 4 5 6 5 10 5 (10 2.235) 22.35
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50 98 50 98 4900 70.
53. 54.
3
Given exp. 63.
= 2 2 8 2 24 2 20 2 = 6 2 6 1.414 8.484. 55.
3 2
6 3
4 7 56.
98
2 28
2 21 7 3 4 7
21 6 4 7 21
=2
2 43 3 47
49 2
2 21
28 3
= ( 1 + a) = (1 + 0.1039) = 1.1039.
3 2 4
Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal.
Given exp. =
58.
Given exp. = =
59.
81 484 9 22 = 0.99. 64 625 8 25
204 42 36 6. 7 34
66.
81 324 4624 9 18 68 15625 289 729 64 125 17 27 8
9.5 08500 .19 .0017
95 82500 19 17
2
2
100[(0.03) 2 (0.21) 2 (0.065) 2 ]
0.032 0.212 00652
100 4 2. 25
4a = 6c b =
(7 3 5 )(7 3 5 )
72 (3
2
2 4 c and 3a = 4b a = b 3 3
43 c 2c . 32
=
9 c 27 29 c 6 29 . 2
a 2 b2 c2
3 (729 29) 2 3c 2 c 2 2c 2 2
729 29 2 13 c 2 2
5 )2 =
2
3 cc 2
3 3 2 = (27 29 ) 2 2c c c c c 2c 2 2
=
1 1 1 3 ( 3 ) 2 2 3 3 3 3
1 0.25
(a b c ) 2 2(ab bc ca )
100 10.
49 45 4 2. 62.
(0.75) 3 [(1) 3 (0.75) 3 ] 1 0.75
Given exp. = 0.03 0.21 0.065 10 10 10
2
27 29
(0.03) 2 (0.21) 2 (0.065) 2
(0.75) 3 (1 0.75)[(1) 2 (0.75) 2 1 0.75] 1 0.75
a + b + c = 27 29 2c
5 500 2500 50
60.
3 0.024. 125
Given exp. =
(0.75) 3 [0.75 (0.75) 2 1] (1 0.75)
65.
3 1.414 3 0.3535 1.0605 4
57.
12 2a 2 2 1 2a 3a
(1 2a ) 2 3a (1 a ) 3a
2 21 21 6
1 1 1 2 4 4 . 2 2 2
4a 2 4a 1 3a
64.
7 2
Given exp. =
61.
1 1 1 2 2 2 ( 2 ) 2 2 2 3
Given exp.
2
2
4 2 2 16 2 3 64 2 4 25 2
=
1 1 4 2 1 . 3 3 3
(729 29) 13 (6 29 ) 2
(729 29) 13 (6 29 ) 2 =
29(729 468)
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29 261 29 29 9 29 × 3 =87.
= 67.
68.
77.
4 2 = = 0.666..... = 0. 6 . 9 3
0. 4
9 3 0.3 , Which is rational. 100 10 0.09 has rational square root. 0.09
5 10 ( 5 ) 2 20 2 5 5 5 125 2 5 2 5 2 20 50 35 5 = = 2 5 2 5 5 =
78.
69.
35 5 7 2.236 7 × 1.118 = 7.826. 10 2
Clearly, a + b =
6 0.400000 ( .63 36 400 123 369
5 * 12 5 2 12 2 25 144 169 13 .
79.
L.C.M. of 3, 4, 5, 6, 8 is 120. Now, 120 = 2 × 2 × 2 × 3 × 5. To make it a perfect square, it must be multiplied by 2 × 3 × 5. So, required number = 22 × 22 × 32 × 52 = 3600.
80.
L.C.M. of 21, 36, 66 = 2772. Now, 2772 = 2 × 2 × 3 × 3 × 7 × 11. To make it a perfect square, It must be multiplied by 7 × 11. So required number = 22 × 32 × 72 × 112 = 213444.
81.
294 = 7 × 7 × 2 × 3 To make it perfect square, It must be multiplied by 2 × 3 i.e., 6. Required number is = 6.
82.
5808 = 2 × 2 × 2 × 2 × 3 × 11 × 11 = 22 × 22 × 3 × 112 = 213444. To make it a perfect square, It must be multiplied by 3.
83.
1470 = 7 × 7 × 5 × 6. To make it a perfect square, it must be a divided by 5 × 6, i.e., 30.
84.
7 549162 ( 741 49 144 591 576 1481 1562 1481 81 Required number to be subtracted = 81.
85.
0.000326 =
70.
3 0.121000 ( .347 9 310 64 256 5400 687 4809
a 2 b2
71.
2 0.064000 ( .252 4 45 240 225 502 1500 1006
72.
73.
0.16 0.4
1 0.01 1 0.1
0.16 16 4 0.4 0.63 . 0 .4 40 10
1 0.1 1.1 1 0.316 0.684
3 0.100000 ( .316 9 61 100 60 62 3900 1100 1.6. = 3750 684 74.
75.
76.
1
5
1
5
5 5
8 83 3 3 3
3 2 2 3
3 2 2 3
5 2.236 =0.447. 5 5
24 4.899 1.633. 3 3
3 3
6 2.449 =1.2245. 2 2
326 10 6
1 326 ( 18 1 28 226 224 2
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Required number to be subtracted = =
2
10 6
0.000002. =
86.
8 680621 ( 824 64 162 406 324 1644 8221 6576 1645
( 5 4)
( 9 8) ( 8 7) ( 7 6) (9 8) (8 7) (7 6) ( 6 5) ( 5 4 ) (6 5) (5 4)
= ( 9 8) ( 8 7) ( 7 6)
( 6 5) ( 5 4) ( 9 4) = 3 + 2 = 5.
91.
1 (2 2 ) 2 ) + (2 2 ) (2 2 )
Given exp. = ( 2 +
1
Greates number of foud digits is 9999.
–
9 9999 ( 99 81 189 1899 1701 198 Required number = ( 9999 – 198 ) = 9801. 88.
( 5 4) ( 5 4)
–
Number to be added = (825)2 – 680621 = 680625 – 680621 = 4. 87.
1
Least number of four digits is 1000.
(2 2 ) (2 2 )
= (2 2 )
(2 2 ) ( 2 2 ) ( 4 2) ( 4 2)
= (2 2 )
1 1 ( 2 2 ) ( 2 2 ) 2. 2 2
7 92.
(2 2 )
(3 2 )
7
(3 2 )
(3 2 ) (3 2 )
7 (3 2 ) (3 2 ) (9 2) = ( 3 – 1.4142) = 1.5858. =
3 1000 ( 31 9 61 100 61 39
93.
Given exp.
3 2 =
( 6 3)
(31)2 < 1000 < (32)2. Hence, required number = (32)2 = 1024.
1 89.
( 5 3)
1 ( 5 3)
( 5 3)
=
( 5 3)
=
Given exp.
1 =
( 9 8)
( 9 8) ( 9 8)
( 8 7) ( 8 7) 1
( 8 7)
( 7 6)
( 7 6) ( 7 6)
( 6 3) ( 6 2) ( 6 2)
4 3 ( 6 2) 6 2( 2 3 )
3
( 3 2)
( 3 2) ( 3 2)
2 ( 6 3 ) 3 ( 6 2 ) 3( 3 2 )
= 12 6 18 6 3 3 3 2 = 2 3 3 2 3 3 3 2 5 3.
1
1
( 6 3)
3 2 ( 6 3) 4 3( 6 2 ) (6 3) (6 2)
(2.2361 1.7321) 3.9682 1.9841 = 2 2 90.
7 5 94.
7 5
( 6 5)
( 6 5) ( 6 5)
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7 5 7 5
( 7 5) ( 7 5)
( 7 5 ) 2 7 5 2 35 = (7 5) 2
100.
12 2 35 6 35. 2 (5 2 3 ) (7 4 3 ) ab 3 (7 4 3 ) (7 4 3 ) 35 20 3 14 3 24
7
=
(4 3 ) a = 11, b = – 6. 2
2 1 96.
2 1
11 6 3 11 6 3 49 48
2 1
( 2 1) (1.414 1) = 0.414.
3 6 5 3 4 3 4 2 4 2
x 2 y 2 (2 3 ) 2 (2 3 ) 2 = 2 (4 + 3) = 2 × 7 = 14.
101.
( 5 1) ( 5 1) ( 5 1) 2 a= (5 1) ( 5 1) ( 5 1)
b=
( 3 2)
( 3 2)
( 3 2) ( 3 2)
=
3 3 3 2 3 2 2 3 3. (3 2)
( 3 1) ( 3 1) ( 3 1) ( 3 1) =
(2 3 ) 2 (2 3 ) 2 ( 3 1) ( 3 1) (4 3) (4 3) ( 3 1) ( 3 1)
=
(2 3 ) 2 (2 3 ) 2 ( 3 1) 2 (4 3) (4 3) (3 1)
42 3 = [(2 3 ) (2 3 ) ] 2
( 5 1) ( 5 1)
( 5 1) 2 (5 1)
(3 5 ) 2 (3 5 ) 2 2(9 5) 7. = 4 4
102.
1 15376 ( 124 1 22 53 44 976 244 976 × Number of row = 124.
2
= 2 (4 + 3) + 2 –
3 = 16 –
3.
1 1 (7 4 3 ) (7 4 3 ) x (7 4 3 ) ( 7 4 3 )
= (7 4 3 )
( 5 1)
5 1 2 5 3 5 2 . 4 2 ( 3 5 ) (3 5 ) 2 a2 + b 2 = 4 4
(2 3 ) (2 3 ) (2 3 ) (2 3 ) (2 3 ) (2 3 ) (2 3 ) (2 3 )
x
=
Given exp.
2
5 2 1 5 3 5 2 . 4 ( 5 1)
(3 6 )
=
=
3 1 2 3 2 3 . 2
=
(3 6 )
99.
=
Givne exp. =
98.
=
( 2 1) ( 2 1) ( 2 1) 2 . ( 2 1) ( 2 1)
2 1
97.
2
( 3 1) ( 3 1) ( 3 1) 2 (3 1) ( 3 1) ( 3 1)
3 1 2 3 2 3 . 2 ( 3 1) ( 3 1) ( 3 1) 2 y= (3 1) ( 3 1) ( 3 1)
= 95.
x
(7 4 3 ) (49 48)
= (7 – 4 3 ) + (7 + 4 3 ) = 14.
103.
1 36581 ( 191 1 29 265 261 481 381 381 100 Number of men left = 100.
manishkumarphysics.in
104.
Money collected = (59.29 × 100) paise = 5929 paise Number of members = 1/ 3
105.
(
.000216)1/3
6 10
2
216 = 6 10
5929 = 77.
666 2 2 2 10 10 10
107.
3
Clearly, 9261 is a perfect cube satisfying the given property.
109.
675 = 5 × 5 × 3 × 3 × 3. To make it a perfect cube, it must be multiplied by 5.
110.
3600 = 23 × 52 × 32 × 2. To make it a perfect cube, it must be divided by 52 × 32 × 2 i.e., 450. (A) Do your self.
112.
(C)
113.
(C) Do your self.
114.
(C)
116.
117.
x 4 or 49 7
7 5
7 5
( 7 5 )2 75
3018 49
=
3018 7 3025 55
(C)
1 x2
x3
120.
(D) LCM of 4, 5, 6, 15, 18 = 180 required perfect square = 180 × 5 = 900
121.
(C)
122.
(D)
123.
(D) Greatest number of siz digits = 999999 number nearest to it is 998001 which is perfect square.
124.
(C) 26244 = 2 × 2 × 3 × 3 × 32 × 32. Hence required no. to divide is 2 × 2 × 3 × 3 = 36.
125.
(C) 143 = 2744 & 153 = 3375
126.
(A)
x 0.9 0.09 0.9 0.09 0.081 5 0.9 0.09
2 2 2 2 2 2
31 32
71 7 4931 49 141 31 141 110
127.
(C)
128.
(B)
129.
(D) Volume = 8 × 6 × 4 = 192 Cube of 5 = 53 = 125 Cube of 6 = 63 = 216 So 216 – 192 = 24
130.
(D)
131.
(A)
132.
(B)
318 36 169 3018 36 13
=
(A)
(B) Let number be x so x = 64
x 4 x 4 x 16 7 7
7 5 7 5 7 5 2 35 12 2 35 6 35 = 2 2 = 6 + 5.9160 = 11.9160 (A)
2
8
108.
115.
119.
8 3 1 . 5 5
8 .000064 3 .008 . 6 1000 10 10 8 2 3 .000064 3 .008 3 0 .2 1000 10
111.
5 2 5 2
64 x 4
12 3 512 8 8 8 4 4 125 125 5 5 5 64
2
= 5 2
6 .06 . 100 1/ 3
106.
5 2 2 2
=
3
392 448 do prime factorization & find.
x3 y
3
7 2 10 5 2 2 5 2
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343 x 7 . 1331 y 11
