SURDS MATHEMATICS

WORK SHEET - 1

Single Answer Type

1. The order of surd 6
7 is
1) 4 2) 5 3) 6 4) 7
2. The radicand of the surd 3
2012 is
1
1) 3 2) 1/3 3) 2012 4)
2012
3. Surd of order 3 is called
1) Quadratic surd 2) Cubic surd 3) Biquadratic surd 4) Non original
surd
4. 2 3
2
1) 6 2 2) 6 2 3) 3
2 4) 2
2
5. Which of the following is not a surd ?
1) 3
4 2) 2
3 3) 2
10 4) 3
27
6. 5
1024 simplified
1) 2 2) 4 3) 6 4) 8
7. Which of the following is true
1) 10
34255 2) 10
34255 3) 10
34255 4) 10
34255
8. If x  8 7, y  4 7, z  2 7 then the ascending order of the variable is
1) z,x,y 2) z,y,x 3) x,y,z 4) y,z,x
9. Pure surd among the following is
1) 3 6 2) 5 3 7 3) 4 3 15 4) 7
10. Which of the following is trinomial is
1) 5  6 1 2) 2
3  5 7  9 11 3) 7  11  13 4) 2  2012 2
11. Which of the following is an irrational number
1) 121 2) e 3) 22/7 4) 3.24317
12. The union of Rational number and irrational number is
1) Whole numbers 2) Complex numbers
3) Real numbers 4) Imaginary numbers
13. The value of 6
12 ÷ 3 3 2 is

1 1
1) 3 2) 3
2 3) 4) 3
2 3

IX Class - Maths 6
MATHEMATICS SURDS

1
14. The value of 294 - 3 -5 6 + 252 is
6

12 6  3 7 3 6  12 7
1) 2) 3) 6 4) 3
2 2

 4  
4x +7 2x +7
5 11
15. If = 64 , then the value of x is
1) 2 2) – 2 3) 3 4) – 3

Multi Answer Type

16. Which of the following is true
1) 2 3,5 3, 6 3 are similar surd 2) Every surd is an irrational

3) 3  7 is mixed surd
4) The product of two similar quadratic surds is a Rational number
17. 4
81  8 3 216  225  15 5 32
1 0
1) 0 2) 24 0 3) 4)
0 1
18. 4
2 x  4  a and 3
3 x  4  b where x=3 then a  b is

1) 4
235 2) 4
235 3) 3
245
4

4)   2  5
3

Reasoning Answer Type
19. Statement I : 8 and 5 2 are similar surds

Statement II : If a b  x  y then a b  x y
1) Both Statements are true 2) Both Statements are false
3) Statement I is true, Statement II is false.
4) Statement I is false, Statement II is true.

Comprehension Type
Writeup - 1
Find the simplified forms of the following using laws of surds
20. If 5  20  45
1) 2 5 (b) 4 5 (c) 6 5 4) 8 5

21. If 48  72  27  2 18
1) 2 2) 3 3) 5 4) 7

7 IX Class - Maths
SURDS MATHEMATICS

22. If A  2 3  5 3 and B  5 3  2 3 then the value of A2  B 2 

1) 256 2) 252 3) 120 4) 242
Writeup - 2
A surd having unity as its rational factor and the other factor being irrational
is called a pure surd
23. Express 2012 4 2011 as pure surd

1) 4
2011 2012 2) 4
 2012 
4
 2011 3) 4
 2012  4  2011 4) 4  2012 2   20112
2011
24. Express 5 5 as a pure surd
625

2011 2011 3 2011

1) 5 2) 5 3) 5
2011 5 4) 5
625 625 5

2010
25. Express 5 as a pure surd
125

2010 125 2010 402

1) 2) 3) (d) 10
25 2 5 25

Matrix Matching Type

26. Matrix match Type
Column-I Column-II
1) 2
3 3 4 (1) 5
5
2) 5
30  5 6 (2) 6
5

3) 2 3
5 (3) 6
27 / 4
2
3
4) 3 (4) 6
432
2
27. Column-I Column-II
1) The greater among 3
3, 4 5 is (1) 3 4

2) The greater among 3

3, 4 4 is (2) 4
4
3) The lesser among 4
3, 3 4 is (3) 4
3
4) The lesser among 4
3, 6 10 is (4) 4
5
(5) 6
10
IX Class - Maths 8
MATHEMATICS SURDS

28. Column-I Column-II

1) Simple surd (1) 2009  3 2010  13 2012
2) Mixed surd (2) 2009  2011

3) Compound surd (3) 2011

2012
4) Dissimilar surd (4) perfect square

(5) 2008, 3 2011, 4 4 2012

Integer Answer Type

29. Simplity 8 3  4 75  3 300  9  x 3 then x =

30. 2 3 5
32  6 x ; Then x =

31. The value of 3

54  33  22  15 3 4k then k 
32. Mixed surd form of 5400  30 k then k 

SYNOPSIS - 2
Conjugate surds: The binomial surds of the form a  b , a  b are called
conjugate surds.
Here, each surd is called the conjugate of the other.
The sum and the product of conjugate surds are rational numbers.
Example: 7  8, 7  8 are two conjugate surds.

   
Sum = 7  8  7  8  14 which is rational number.

Product =  7  8  7  8   41 which is rational number.

Rationalising Factor: If the product of two surds is a rational number, then

each of them is called a rationalising factor of the other. The rationalising
1
1
factor of n
a is given by a n .
Example: Rationalising factor of 8 is 2.
Rationalising factor of 2 is also 2.
Note: The R.F. of a given surd is not unique. A surd has infinite number of R.F.’s.
Example: 2 2, 3 2, 4 2 ........ are R.F.’s of 5 2

If m
a n is a surd then its R.F. m
a mn .

9 IX Class - Maths
MATHEMATICS SURDS

a b c d  x y z
By squaring both sides, and comparing rational and irrational parts on
either sides, we get, x  y  z  a.

1 bd 1 bc 1 cd
x ,y  andz 
2 c 2 d 2 b
Example
1. Find the square root of 7  4 3.

Let 74 3  x  y
Squaring both the sides,
7  4 3  x  y  2 xy
 x  y  7and xy  2 3  xy  12
By solving, we get x = 4 and y = 3
x  y  4  3 2 3
2. Find the square root of 10  24  60  40.

Let the given expression be equal to a  x  y  z.

As per the method discussed,
a  10, b  24, c  60andd  40
1 bd 1 24  40
x  2
2 c 2 60
1 bc 1 24  60
y  3
2 d 2 40
1 cd 1 60  40
z  5
2 b 2 24
Alternative method

 10  24  60  40

 10  2 6  2 15  2 10

  2  3  5   2 2  3  2 3  5   2 2  5 

 
2
 2 3 5

 2 3 5

11 IX Class - Maths
 1+ 2 1- 2
4. The value of + correct to three places of decimal is
5+ 3 5- 3

SURDS Q 5  2.236, 6  2.449  MATHEMATICS

1) 0.213 2) –1 3) –0.213 4) 2

7 3 2 5 3 2
5. The value of - - is
10 + 3 6+ 5 15 + 3 2

1) 0 2) 1 3) 3 4) 3

6. A rationalizing factor of 3  2 is

12  8
1) 3 2 2) 32 3) 4) none
2
1
7. A rationalizing factor of
3
25  3
 1 is
25
1 1 1 1 1 1
1) 5 3  5 3 2) 5 3  5 3 3) 25 3  25 3 4) none

3 1
8. x then 4 x 3  2 x 2  8 x  7 
2
1) 0 2) 8 3) 9 4) 10

26  15 3
9.
5 2  38  5 3

1 1
1) 2) 3 3) 2 3 4)
3 2 3

IX Class - Maths 12
of decimal is

MATHEMATICS SURDS

10. If 23  x 10  18  5 then x 
1) 2 2) 3 3) 5 4) 6
11. 3
2 5 3
1) 15
864 2) 5
24 3) 3
250 4) 15
648

3 6
12. 
75  48  32  50
1) 2 2) 3 3) 3 2 4) 3 2
13. If x  1  x  1  1 then x 
1) 5/2 2) 4/5 3) 5/4 4) 2

Multi Answer Type

14. A rationalizing factor of 3
16  3 4  1 is
1 1 1 1
1) 2) 3) 16 6  1 4) 16 6  1
43 1 43  1
15. A rationalizing factor of 4  5 is

64  80 16  20
1) 4 5 2) 3) 4) 4 5
4 2

16. 4
17  12 2
1) 2 1 2) 4
4 1 3) 2 1 4) 6
8 1

17. If 19  4 x  12  7 then x 
1) 21 2) 42/2 3) 441 4) 20

1 2 3
18.
  
15  4 14 12  2 35 13  4 10

1) 2 5 2) 0 3) 3 7 4) 1  20120

13 IX Class - Maths
SURDS MATHEMATICS

Matrix Matching Type

19. Column-I Column-II

27
1) The value of 4
28 4 (1) 12
2
2) The value of 6
46 4 (2) 5
177147

3) The value of 5
33  5 34  5 36 (3) 20
2592
4) The value of 4
2 5 3 (4) 5
1594323
(5) 1
20. Column-I Column-II
1
1) If x  5  2 6 then x 
2
(1) 2 7
x2

5  21 1
then x  2
2
2) If x  (2) 8
2 x

 1
3) If x  2 2  7 then  x   (3) 7
 x

x2  1
4) If x  4  15 then (4) 40 6
x
(5) 23
21. Column-I Column-II

1 2
1) If  a  b then (a,b) (1) (1,3/4,7/3)
3 2 2
4
2) If  a  b  c then (a,b,c) (2) 1
2 3  7

5
3) If  a  b then (a,b) (3) (5, 30)
6 5
1 1
4) If a then a  (4) (9,5)
2 2 2
(5) (1,4/3,7/3)

IX Class - Maths 14
MATHEMATICS SURDS

Reasoning Answer Type

22. Statement I : The rationalizing factor of 9
102 is 9
107

Statement II : If a n is a surd, then its rationalizing factor is m a m  n (m > n).

m

1) Both Statements are true 2) Both Statements are false

3) Statement I is true, Statement II is false.
4) Statement I is false, Statement II is true.
23. Statement I : The resultant after dividing 6
12 by 3  3 2 and simplified is

1
3 .
3
Statement II : The Rationalizing factor of 2 is 3 2 . 3

1) Both Statements are true 2) Both Statements are false

3) Statement I is true, Statement II is false.
4) Statement I is false, Statement II is true.
24. Statement-I: Rationalizing Factor of 4
2  4 3  4 8  4 12  4 18  4 27
Statement-II: Rationalizing Factor of 4
a  4 b  4 a 3  4 a 2b  4 ab 2  4 b3
1) Both Statements are true 2) Both Statements are false
3) Statement I is true, Statement II is false.
4) Statement I is false, Statement II is true.

Comprehension Type
WRITEUP-1
If m
a n is a surd, then its rationalizing factor is m
amn .

25. Rationalizing factor of 7

93 is

1
1) 9
73 2) 7 3) 7
94 4) 9
37
94

26. Rationalizing factor of 11

127  510 is

1
1) 12
117  5 2) 11
124  5 3) 12
711  5 4) 11 5
124

27. Rationalizing factor of 15

2311  210  513 is

1) 15
234  25  52 2) 15
235  24  52 3) 13
1511  25  55 4) 23
1510  25  53

15 IX Class - Maths
SURDS MATHEMATICS

WRITEUP-2
If a,b, a 2  b are positive rational numbers and b is a surd then

a  a2  b a  a2  b
a b  x  y , where x  ,y 
2 2

28. 52 6 
1) 3  2 2) 3 2 3) 3 2 4) 3 4

29. 13  120 
1) 10  3 2) 10  3 3) 10  3 4) 10  3

30. 3 8 
1] 2  1 2] 2  1 3] 1  2 4] none
WRITEUP-3
If ‘a’ is a rational number and b , c , d are surds, and if

a  b  c  d  x  y  z , then

1 bd 1 bc 1 cd
x ,y ,z  ,x y z  a
2 c 2 d 2 b

31. 10  2 6  2 10  2 15 

1) 3 5 2 2) 2 3 5 3) 2 3 5 4) 3 5 2
32. 21  4 5  8 3  4 15 
1) 5 22 3 2) 5  4  12 3)  5  4  12 4)  5  4  12

33. 
Square root of 25  2 35  2 91  2 65  is

1] 13  7  5 2] 13  7  5 3] 13  7  5 4]  13  7  5

IX Class - Maths 16
MATHEMATICS SURDS

KEY & HINTS

WORK SHEET – 1 (KEY)

1) 3 2) 1 3) 2 4) 2 5) 4

6) 2 7) 2 8) 3 9) 4 10) 4

11) 2 12) 3 13) 4 14) 2 15) 1

16) 1,2,3,4 17) 1 18) 1 19) 1 20) 3

21) 2 22) 3 23) 2 24) 3 25) 3

26) A-4 27) A-4 28) A-3 29) 2 30) 2

B-1 B-3 B-2
C-2 C-1 C-1
D-3 D-5 D-5

31) 5 32) 6

1. c
2. a
3. b
4. b
5. d
6. b
1 1 1
7. 310 , 2 4` ,55
L.C.M. of 10, 4, 5, is 20
20 20 20
 101   14   15 
3  , 2  , 5 
     
32 , 25 , 54
9 < 32 < 625
10 3 < 4
2 < 5
5
Ans: (b)

19 IX Class - Maths
MATHEMATICS SURDS

x  2
31. 5  3  3 584  15. 3 20
K  5
32. 5400  6  900  30 6
K  6

WORK SHEET – 2 (KEY)

1) 3 2) 2 3) 1 4) 3 5) 2

6) 3 7) 2 8) 4 9) 1 10) 4

11) 1 12) 2 13) 3 14) 3 15) 1,2,3

16) 1,2,4 17) 1,2,3 18) 2,4 19) A-5 20) A-4
B-5 B-5
C-4 C-1
D-3 D-2

21) A-3 22) 1 23) 3 24) 1 25) 3

B-5
C-4
D-2

26) 2 27) 1 28) 3 29) 2 30) 1

31) 4 32) 2 33) 1

1 1
1. 5 4 12  2 11  5 4  11  1
Ans: (3)
2. By using formula
Ans: (2)
3. Ans: (1)

4.
1
2

1 2  5  3  1 2    
5 3 


1
5  3  10  6  5  3  10  6 
2
 5 6

23 IX Class - Maths