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SSC - CGL
QUANTITATIVE APTITUDE
SURDS
t
e
n
.
a
!"
Conjugate Surd: If sum and product of two surds is a rational number then the two surds are called
Conjugate surds to each other.
"
"
Ex: 5 + ! 2, 5 - ! 2
!"
Rationalizing factor: If the product of two surds is a rational number then two surds are called
Rationalizing factors to each other.
" "
"
"
Ex: ! 7 + ! 5, ! 7 - ! 5
!"
Note: Every conjugate surd must be a rationalizing factor, but every rationalizing factor need not be a
conjugate surd.
1
#
n"
If a is a positive rational number, ‘n’ is positive integer and a is an irrational number. Then a n is
!"
h
b
i
t
a
r
a
n
called a surd of degree ‘n’.
"
! 7,
Ex:
e
e
.
w
w
Quadratic surd
p
u
d
3
"
! 7,
4"
!7
Cubic surd
Standard forms of a surd:
!
Bi-quadratic surd
t
e
n
Square root:
"
"
"
w
"
"
= !x +!y
! a+2.! b
(where x + y = a,
""""""""""
"
"
"
"
" "
a + 2. ! b + 2. ! c + 2. ! d = ! x + ! y + ! z
!
#
b
i
t
(where x + y + z = a, x.y = b, y.z = c, z.x = d)
Cube root:
"
3 "
"
a
r
p)
"
= x ±!y
! a ± b.! c
.
a
h
x . y = b)
"""
b#c
where x = "" and y = c $ x3 ± 3xy = a
3
(
1.
2.
3.
4.
u
d
a
!
n
e
!
e
.
!ww
w!
""""""""
""
""""
""""
""
a + a + a + ! a + ...... %
!
!
!
""""""""
""""""
""""
""
a . a . a . ! a . ...... %
!
!
!
!
"
! 1 + 4a - 1
= """"
2
= a
" """"""
"""""""
""""
""
a . a . a . ! a . .... n times
!
"
! 1 + 4a + 1
= """"
2
""""""""
""""""
""""
""
a - a - a - ! a - ...... %
!
Important formulae
( )
= a
2n -1
""
2n
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Surd
Rationalizing factor
"
a-!b
"
a + !b
3
" 3"
!a + !b
3
"
3
"
3
"
" 3"
!a - !b
3
"
3
"
3
"
3
! a2 + ! ab + ! b2
" 4"
!a + !b
(4!"a - 4!"b) (!"a +
(4!"a + 4!"b) (!"a +
4
4
"a
"
- 4! b
!
QUESTIONS
1.
"
"
"
2.
"""""""""
"
! 12 + 2.! 15 + 4. !"5 + 4.!"3
"
"
"
a) ! 3 + ! 5 + ! 15
3.
3 """"
"
! 72 # 32.! 5
e
e
.
w
w
!
w!
""""""""
""""""
""""
""
5 .! 5. ! 5 .! 5 .....%
""""""""
""""""
""""
!7.! 7.! 7 .....10 times
"
"
"
b) ! 2
"
"
d) 3 # ! 6
c) #1
d) # ! 2
t
e
n
"
.
a
h
"
c) ! 5
b
i
t
(
)
ra
= a
!20 + ! 20 + ! 20 + ! 20. ....%
= ?
d
a
p
u
b) 4
n
e
e
.
ww
"
"
c) 5 # ! 6
= ?
""""""""
""""""
""""
""
d) Can’t estimate
1
1# "
210
c) 7
1
3# "
210
d) 7
c) 5
d) 6
(
)
"
The conjugate surd of ! 2 # 1 is
"
"
b) ! 2 # 1
"
c) 1 + ! 3
The greatest among the following is ..........
"
" " " "
"
! 22 + ! 21 , ! 23 + ! 20 , ! 24 + ! 19
"
"
"
"
"
"
a) ! 22 + ! 21
b) ! 23 + ! 20
c) ! 24 + ! 19
"
" " " "
" "
"
The greatest among ! 21 # ! 13 , ! 19 # ! 11 , ! 13 # ! 5, ! 23 # ! 15
" "
"
"
"
"
a) ! 21 # ! 13
b) ! 19 # ! 11
c) ! 13 # ! 5
w
10.
"
d) 8 + ! 6
d) ! 2 # ! 3 # ! 5
"
b) 6 # ! 5
"
b) 10. ! 7
a) # ! 2 # 1
9.
"
c) 2 + ! 3 + ! 5
b) 1
a) 3
8.
"
h
b
i
!
!
a) 7
7.
u
d
"
"b)
6 ""
3 ""
"
2"
!2 + 3 . 1 # ! 2
a) 5
6.
"
a
n
= ?
a) 1
5.
r
p
"
c) 2! 3 + 6
b) ! 2 + ! 3 + ! 5
"
a) 3 # ! 5
4.
"
b) 2! 2 # ! 6
n
.
a
"
! b)
at
"
Square root of 14 # 8! 3 is .......
a) 2! 2 + ! 6
t
e
! a2 - ! ab + ! b2
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"
d) 2 + ! 3
d) Can’t estimate
"
"
d) ! 23 # ! 15
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11.
Arrange in decending order
3
"
3"
" 6
"
4
"
! 4 , !2 , ! 3 , ! 5
4
"
"
6
"
a) ! 4 > ! 5 > ! 2 > ! 3
"
12.
13.
6
"
3
"
"
4
b) 11
16.
(
)
e
e
.
w
w (
w
b) 9602
2
"
3
-2
"
3
"
"
"
4
"
c) 2.! 15
#1
"
"
6"
4"
!2 ÷ !6 = ?
b)
du
""
""
27
12 2
!
c) 10104
d) 11132
a
n
"
3
t
e
n
.
a
h
-1
b
i
t
"
c) 2 + ! 3
c)
)
"
d) 3.! 2
a
r
p
"
b) ! 3 + ! 2
"
d) 2(! 5 + ! 3
"
c) 2 + 2
3 """""""
! 9. !"3 + 11. !"2 = ?
d) 16
c) 2.! 3
1
"
3
t
e
"
n
.
a
h
b
i
)
b) 2 # 2
a) 1
""
""
2
12 27
!
1
"
3
d) 2 + 1
"
d) 11 + ! 2
""
6 2
d) "
9
!
1
1
1
"""
" + """"
"
" + """"
"
" + ........ 99 terms.
!2 + !3
!3 + !4
1 + !2
e
e
.
ww
a) 9
w
1
2
" "
3
3
If x = 3 + 3 + 3
b) 10
"
c) 1 + ! 99
"
d) ! 99 # 1
c) 14
d) 8
then x3 # 9x2 + 18x is
a) 10
22.
1
"
3
2 -2
" "
3
3
"
3
"
21.
6
Rationalizing factor of 2 # 2 + 1
a) 3 + ! 2
20.
p
u
d
a
n
a) 2 + 2 + 1
19.
"
t
a
r
"
"
"
"
!3 + !2
!3 # !2
"""
"
" # ""
"
" simplify
!3 # !2
!3 + !2
"
"
a) 2.! 6
b) 4.! 16
"
"
!3 # !2
1
4
If x = ""
=?
"
" then x + ""
!3 + !2
x4
2
"
3
18.
"
c) 14
b) 16
a) 8642
17.
6
3
"
"
"
"
!5 + !3
!5 # !3
If x = """
"
" and y = """
"
" then x + y = ?
!5 # !3
!5 + !3
a) 8
15.
"
c) ! 2 > ! 3 > ! 4 > ! 5
d) ! 3 > ! 4 > ! 2 > ! 5
"
"
The geometric mean of (! "
100 + ! 104 ) , ( ! 26 # 5) is
"
"
4 "
4 "
a) ! 2
b) 2. ! 6
c) 2.! 12
d) 2 . ! 12
"
If x = 4 # ! 15 then x2 # 8x + 15 =?
a) 10
14.
4
b) ! 5 > ! 4 > ! 3 > ! 2
b) 12
" x2 # 8
" x2# 8
+ 7 # 4.! 3
= 14 then x = ?
If 7 + 4. ! 3
(
a) ± 1, ± 3
)
(
)
b) ± 3, ± 7
"
c) ± 3, ± ! 7
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"
"
d) ± ! 3, ± ! 7
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23.
" ""
""
""
""
""
""
"
""""
"
" ""
3 + 3 + 3 + ! 3 + ..... %
! !
If x =
!
a) x2 # x + 3 = 0
24.
b) x2 # x # 3 = 0
"""
"
""
""""""
""
""
""
"
" ""
....
30
30
30 ! 30
%
!
x =
!
!
a) 3
25.
3
c) x2 + x # 3 = 0
"
n
.
a
c) 5
"
a) ! 20
d) 6
"
b) ! 20
t
e
d) x2 + x + 3 = 0
=?
b) 4
"
Pure surd of 2 . ! 5 is
3
then
c) ! 40
bh
i
t
EXPLANATIONS
a
r
3
"
d) ! 40
1.
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
""
"
"
"
"
"
"
14 # 8 ! 3 = 14 # 2 ! 3 × 16 = 14 # 2 ! 6 × 8 = (6 + 8) # 2 ! 6 × 8 = 2! 2 # ! 6
2.
! (3 + 4 + 5) + 2. ! 3.5
3.
From the concept of standard form of cube root of a surd (as mentioned above)
"
"
"
"
"
"
"
"
32 # 5
"
"
x = """ = 3 y = 5 & x # ! y = 3 # ! 5
3
a
n
e
e
! w.
w
w
!
4.
"""""
3 2 """"
""
! 2! 2 + 3
=
=
5.
p
u
d
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
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"
"
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"
"
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"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
" "
+ 2. ! 3.4 + 2. ! 4.5 = ! 3 + ! 4 + ! 5
3
t
e
n
"""""
! 1 # !"2
.
.
a
h
3 ""
3 ""
"
"
1 + !2 . 1 # !2
!
!
"
"
(1 + ! 2 ) (1 # ! 2
{
)}1/ 3 = (#1)
b
i
( a
)t
r
p
u
From important formulae No.3, answer is 5
6.
From Important formulae No.4 answer is = 7
7.
From Important formulae no.1, answer is 5.
1
1# "
210
d
a
Shortcut: If any number which can be written as a product of two consecutive numbers then, sum of
!n
e
e
.
ww
its square roots will be the greatest number of the two consecutive numbers.
Here, 20 = 4 × 5 &
8.
""""""""
""""""
""""
""
20 +
! 20 + ! 20 + ! 20. ....%
= 5
"
"
# ! 2 - 1 is a conjugate surd of ! 2 - 1
(from synopsis).
9.
w
The sum of two numbers of the surds are equal.
Hence, calculate the difference.
The surd with greater difference is greater.
"
"
& ! 24 + ! 19 is greater.
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10. When the differences of the numbers of the surds given are equal then, the surd with smaller numbers is
greater.
"
"
& ! 13 # ! 5 is greater.
11.
t
e
LCM of powers of surds given is 12.
1
" × 12
43
1
" × 12
22
1
" × 12
36
44, 26, 32, 53
3
"
4
"
h
b
i
6
"
"
12.
In descending order, 44 > 53 > 26 > 32 & ! 4 > ! 5 > ! 2 > ! 3
"
The geometric mean of a & b is ! ab.
13.
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
Therefore, ! (! 100 + ! 104 ) ( ! 26 - 5) = ! 2
"
x # 4 = # ! 15 then, squaring on both sides
a
n
From the formulae,
"
"
"
"
!a + !b
!a # !b
"""
"
" + """
"
"
!a # !b
!a + !b
t
a
r
p
u
d
x2 # 8x + 16 = 15 & x2 # 8x + 15 = 14
14.
n
.
a
1
" × 12
54
e
e
.
w
w
( ) ( )
a+b
5+3
= 2 "" & 2 "" = 8.
a#b
5#3
15.
t
e
n
From the formulae,
" "
" "
!a + ! b
!a #!b
"""
" " # """
" "
!a # ! b
!a+!b
w
.
a
h
"
""
4 ! ab
4.!3×2
"
& "" & """" = 4 . ! 6
a#b
3#2
16.
1
1 2
x4 + " = x2 + " # 2
x4
x2
(
)
1
1 2
& x2 + " = x + " # 2
x
x2
(
)
18.
a
r
p
u
d
a
1
Hence, x2 + " = (10)2 # 2 = 98,
x2
17.
b
i
t
n
e
e
.
w
w
w!!
1
= 982 # 2 = 9604 # 2 = 9602
x4 + "
4
x
"
"
The Rationalising factor of 3 ! a + 3 ! b is
( 3!"
a2 #
3"
! ab
+
3"
2
!b
)
From the formulae,
"""""""""""
3
"
"
"
"
(x + 3y) . ! x ± (3x + y) . ! y
= !x ± !y
"""""""""""
3
"
"
"
"
(3 + 3 × 2) ! 3 + (3 + 3 × 2) ! 2 = ! 3 ± ! 2
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19.
6"
2
1
"
3
!6
1 "
1
"
4 4
1
"
12
2
2
! 2 = """
"""
= ""
=
3
4"
"
12
2 .3
20.
!
12
3
"
2
"
27
t
e
1
1
1
""
" + """""
"
" + ........ + """"""
"
"
"
1 +! 2
!2 + !3
! 99 + !100
n
.
a
(rationalising each term)
h
b
i
"
"
"
"
"
"
"
"
"
!2 # 1
!3 # !2
!4 # !3
"
"
! 100 - ! 99
""" + """" + """" + ........ + """" = ! 100 - 1 = 9
1
1
1
1
21.
1
"
3
3
x#3 = 3 +3
x3 # 27 # 9x (x # 3) = 3 + 9 + 9 (x # 3)
x3 # 9x2 + 27x # 27 = 9x # 15
x3 # 9x2 + 18x = 12
22.
t
a
r
2"
"
p
u
d
a
n
2
2
" x #8
" x #8
+ (7 # 4! 3)
(7 + 4! 3)
"
"
= (7 + 4! 3 )1 + (7 # 4! 3 )1 (or)
"
"
= (7 + 4! 3 )#1 + (7 # 4! 3 )#1
x2 # 8 = 1 or x2 # 8 = #1
e
e
.
w
w
t
e
n
w
x2 = 9
x2 = 7
"
& x = ± 3 or ± ! 7
23.
24.
.
a
h
Squaring on both sides, x2 = 3 + x
x2 # x # 3 = 0
b
i
t
From Important formula No.2. answer is 5.
a
r
p
Shortcut: If any number which can be written as a product of two consecutive numbers then,
difference of the suqare roots will be the smallest number of the two consecutive numbers.
"
"
"
"
"
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"
"
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"
"
"
"
"
"
"
"
"
"
Here, 30 = 5 × 6 = 30 # !"
30 # ......% = 5
25.
3"
3"
!
u
d
a
3"
2 . ! 5 = ! 2 3 . 5 = . ! 40
n
e
e
.
ww
w
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Writer : Ch. Sudheer