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Mathematics Class 9 Real Numbers Topic 1 and 2
Engineering Mathematics I (2K6EN101)
Kannur University
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a Quick ,. Rt1bon wntl..-71 Review TOPIC-1 Rational tn Sumber the form : A p. number q, Numbers where ,. 1' r and ,s calle q ar d e a r co-p ational rime integers num be~ , .. 1f an JI d can q -t 1,e 0,
denoted For C\ amplc Symbolrcallv, by (l 2'. 1 4' 4 5' Q 2 3 "" clc. {1?. q' are q all O raand tiona. p, q I numberE ,J s.
roPlC -• • • ~,. ~.) r na 1 ;rr.,. r..
,. Decimal Expansion of Real Numbers : The dee imal r expansion of real roPIC -P liz 5 ciu:J,. .,r r r' V• _ .,• ;,.-,:-::
1, Cases in Rational Number: numbe r It. the dcall>d eau--ed a. mal rational number. ex to pansion of a real numbens either repre, ent a number on the number.. 1 emunating me. or non-t be p emunating (q -t 0) can recurribe expreng, ssed than as a dea..,, the real;_
e. Case di\i ding : (i) 1 : .!. 2 'When Remainder becomes Zero - p by = 0 q, when the remainder becomes zero, then the ecun Every · rational al num d. so r al q' 1 is called is a terminating a terminatidng eamal. decimal.
On (ii) dividing 100 52 = 0. 52 1 by 2, we get value 0 i., remamder equ lo zero, 2 52.. tJ d. I
On Case 2 by terminating recurring q, dividing when remainder never becomes zero : When remainder never becomes zero - A rational 52 by 100, or we get repeating decimal. value 0 i., and It remainder equal is denoted by the bar set of digits repeats periodicall y thnumber to zero, so -over expressed in 100 it. JS a term the an form ma the deamal ng of p. ea q or ma. is d1\ cafld 1510!
e. On dividing (ii) : (1). 1 = - 3 0 = 0 1 by 3, ...... .. we get 3 again and again = = 0. 0 - 3 i., remainder never becomes zero, 50 113 is a repeating decim
, There are infinitely many rational numbers between ,. On dividing 3 by Every integer is a rational number. 11 11, we get 27 again and again i., remainder neve any two ·. gnen r comes zero. be rational. numbersSo,. J 111 is a repeatin g de
, (ili)x If (i) (ii) x and x x - x + y y y y is is are is a a a rational number any rational number rational number two rational numbers, then :
(iv) x + y is a rational number, (y # OJ.
Q. l. L, ---"F.: ,Ris -J2 a rational number or not? Q. So, 2. Write ,~ 177 413 is a rational number. the simplest form of a rational number 1 R Sol. Q. 177 - U3 3. Find = 59x3 59x7 the decimal expansion of =~- 7 1000. 1 Sol. places 1000 58 = to 0 the left) (Decimal [Board Term point I, 2015, is shifted Set 20UIYN] three 1 R Q. 4. Calculate fraction ?_ 8 the. decimal [CBSE Marking which Scheme, represents 2015] the R Sol. Q. 7 5. = Write a real deci mal expansion. J. number which has terminating 1 [Q] Sol. QfG)nsert - 125 J1 = 0. three rational numbers between - ~ and 1 --. 2 3 [Board Term I, 2016, Set JQ22L5C]
Sol. and -- 1 3 2 3 - - ---- 12 4 12 8 1 So three rational --12' 5 numbers [CBSE -- 12 6 and--. Marking are 12 7 Scheme, 2016} [Ql Sol. Q@ Find two rational numbers between [Board Term 4 = - 4 5 x 5 I, 2016, and 5 4 Set and = - JQ22L5C] 5 5 5. x 5 i., The 1\umbers are 21 5 4 and = - 20 5 22 5 and 5= - 25 lQl Q. 8. Express where p the and rational q are [CBSE integers number Marking an4 0 q:;:. Scheme, in 0. the form 20161 p q , Sol. Let, lOx lOx - x x = = = 0 9 .... (9 ..... ) -.... (0 [Board .999 ....... ) Term I, 20141 ½ lUJ Q. 9. Calculate 9x x=l the =9 value of 2 in the form of P ½ q where p and q are integers and q :;:. O.
REAL NUMBERS Sol. ·nvo r.1tion num1'1'f'ii
t,i,1wc-rn n I 2221 (md O 1414414 At' ll ilnd 0 " 14 and l!\l le.) ,, imd ll\l 50 (CBSU .\1.. Schrmc, 201h) Q. t,. ln~11 thin• rahon.
3 numbn~ bctWl'cn. ,ind 5 IBfl Tetm I, 2012, Set
&O] Sol. lCM 0f Sand i i~ 35
3 7 21 3 - = -x-i::-s 7 3S ¼ and
5 2$ 5 = -x-=-¼ i 3:i 7 5 11. 23 24 25 ~,, -"-
¼ ~-<-<-<- 35 35 35 35 35 The required three ratw
nal numbers are , 35 35
and 35
½ (CBSE Marking Scheme,
2012] - p U Q. 7. Express 0 in the form of - , where
p and q are -
q integers and q:;. 0.
[Board Term I, 2013] Sol Let,
x = 0 X = 0.................
.. .. (i) multiplying 10 on both the sides, we get, 10 X = 6. .... (ii) 1 From (ii)-(i), we get 9x =
6 6 2 X = -=-9 3 A Q. 8. Represent 0.
in the form of E.., where
p and q q are integers and q :;. 0. [Board
Sol. Term I, 2012, Set 38] I 000 c .)
[ 5 217 237 X = m [CBSE Marking Scheme, 2012I
Alternative Method : Let r - 0.
= 0 ...... > lOOOx :.-237. ........
1/z > ]()(){)x X "'(237 7237
. ... ) - (Q .... )
1/z 999x = 237
1/z 237 x=-
¼ 999 fA! Q. 9. Express the decimal number 2.
in the form of E., where p and q are
integers and q:;. 0. q [Board Term I, 2012, Set 41] Sol. Let X
= 2 = 2. ........ lOx = 22. ........
½ lO00x = 2218.... . 1/z 1000x-10x = (2218.
..... J-(22 .... J 990x = 2196 ¼ x = 2196 2x3x3x122 122 1/z 990 2x3x3x55 55 [CBSE Marking Scheme, 2012] [BJ Q. is not a rational number as AA
and .ffs -. are not rational. State whether
it is true or false. Justify your answer [Board
Term I, 2012, Set-39] Sol. False Justification: AA=
fM7" = @"_ = .ffs V75 vzs s which is a rational number. (CBSE Marking Scheme, 2012]
Short Answer
Type Questions-I
I
3 marks each
-u Q. Find three rational
numbers between and
.2... 7 11 [Board Term I, 2014] Sol. Since LCM of 7 and
11 is 77, ==!.!.=~
1 771177 and 5=5=11=55 771177 Hence, three rational numbers between and .2.. are : 7
56 57 58 77'77'77 IQ] Q. 2. Find six rational numbers between 3 and
- [Board Term I, 2014] Sol. Let a = 3 and b = 4 Here, we find six rational
numbers, i., n = 6 b-a 4-3 1 Sod=-=-=- n+l 6+1 7 51 1 rational number= a+
d = 3 + .!_ = ½ 7 7
"""
6 l '
BSE Que~
"1A,.
Oswaal_ -"
o. ~- , 2 ii 1 2"'' '••li, 11 ,,11111111 111 ,, [!" ,. ~,-:i r •
6 6 5 30 I I = x - = - "' 5 x c, 30 5 ISO 5 7 I 1 l,I ½ :11\1 r,1ti.,,1 11 111111>,,r •
4 ,li/ , , J t ; c 7 11 2.'i I
½ I'" 1,1ti1111,1l n 11111 1'i,, ,,
., ,I,/ c: l l 7 • 7 21 ,
½ ;ih lllliu11,1l 11 11 111h1•1 ' ,
I· :;,1 • .l I 7 • 7 1111 ' 1,1 11 ,1 11 ,1l 11111nh1•1 ,1 I
r, 27 (1d ,1 I ½ 7 7 '2 21 24 25 2/i S11, ~,, 1,1tin 11 ,,1 1111111hl
•r~ ,111• ; , ' 7' 7' 7
7 Hild ~ i
I I U Q, ,l hnd lour r, 1110
11,11 nurnbcrs between 5
and Sol.
[Board Tenn I, 2012, Set-
781 S 111 r ,• I .CM of 5 ,llld 6 is
30 I I S 5525 • :: X - :: > - ½ 6 65 3 1Js Long Answer
Type Questions R Q lJGive two rational numbers
whose : (i) difference is a ration al number, (ii) sum is a rational numbe
r, (iii) produ ct is a rational number, (iv) division is a rational numbe
r. /usli/y also. (Board
Term I, 2015, Set 2]
Sol. Any example and
verification of example: Let 111 4 /5, 11 = 9/2 Difforcnrr 9 _ 4 ,, 37 (Rational Number) 2 5 10 4 9 53 Sum + (Rational Number) 5 2 IO 4 9 36 (Raticmal Number) X l'rmluct I() 5 2 945 4 Diviion. + (R,1tion,1l Number) 2 5 8
l}, Q12Arr,111g1• in ch 1
·rcndini; ordt•r lfi, ~'
'ff and '". jBuard
Term I, 2012, Set 57]
Hi•ncc, lour rational A/ numbers ~ 61 "%~ I ,1r1?: 6 26 27 28 29 ,so' IS{)' 150' ISO ½,.½~ [CBSE Marking S chelti - p e Q Express 0. 328 in the 4
fonn of - , Wh· '.. q IChire and q -t 0. [Board Ternq , Sol. Let
X = 0 28 _ O ' --32828 10x = 3, 282828 I. 1000x = 328 .282828 ... JOOOOx- 10 x = 328. 2828. • 3)-SGJ 990 x = 325 325 65 x=,,__ 990 198 [CBSE Marking Sche1t1e, 4 rnarkse~ Sol. LCM of 3, 4, 6, and
--... 12 is 12. ,,,i 'ti, "2-' 2 'm I 3 ,, 5 12 I~ vs "5' I l 712 ifj "7' 'ffl I ';/3 "31? "';/3 Descending order is 'ms, 'm, 'm, ,;13 1 i., 1/5, ifi, ifj., ;/3 [CBSE Marking Scheme,llll:
Alternative Method : Since, LCM of 3, 4, 6, 12 is 12 1 ·. V2=2l•: =2~ ='fr' ='m 13 J vs= z,·, = 2" = '½' = 'ifuf, I l I ifi = 7' ·, = 7" = 'fi' = 'ffl ';/3 " ';/3 In Descending order '~, 'ffl, 'l/16, '(3 i., 1/5, 'ff, fi., ';/3.
§l __
05waal CBSE Question B
1 ank, MATHEMATICS
I Cl~
Very Short
Answer Type
Questions
1 markeq, 1
F&#039; Q, l. ldcnhly an
mahon,11 number ,1mong
the followin& number~
o o, orns, 0:1001300013..
[Board Term I, 2014] Sol. 0. D i, J lcmun,1lmg
number So, 1l is not
an urati(lnal number. Cl~ = 0.
15" repeating continuou
sly, '('I 11 ts not an ITTJlion l number. ODis = 0. 13151315. , 1s repeating continuously, so 11 i, not ,111 1rrahonal
number. O.'l0130(l\300013. ., non-terminating
and non- recurring deon1al. Henc
e, it is an irrational number. So, 0 i, an irrational number.
1 R Q. l. ls lhe product
of two irrational numbers
always an irrational number? Sol. No, it may be rational or irrational.
R Q. 3. Write the sum
of 2,/s and Sol. Sum of 2,/s and = 2,/s + U Q, 4 :C:alculate the irrational number
1 between 2 'li
- Sol. Since, Js = 2 Hence, the irrational
number between 2 and
2 Js. 8- Q. 5. Simplify the number (
- Jsj2 · Sol. (fi.+Jsj2 = (J'i.)
+(Jsj2 = 2+s+2,Jw. = 7 + 2M (Irrational Number).
Short Answer
Type Questions-I
2marks eacn
B Q. 1. Find any two
irrational numbers between
- and 0.
[Board Term I, 2014] Sol. Required two irrational number
are : (i). ..... .... (ti). ..... .. . S: Q. 2. Find any two
irrational numbers between
- and 0. [Board
Term I, 2012, Set 14] Sol. 0....
and 0 ..... [CBSE Marking
[i] Q. Scheme, 2012] 3. Find an irrational number
between and 7 when it is given that!
= 0. 7 [Board Term I, 2012 , Set 5~ Sol,
- = 0.142.........
1 / 2 7 = 0. ..... Hence, required number
can be 0 ... [CBSE Marking Scheme
, 20121
Short Answer
Type Questions
-I I
3 marks each
JI Q. 1. Find three irrational
numbers between i and
ii. [Board Term I, 2016, Set
7AEDLQR, NCERT] Sol.
5 -- 7 = 0·714285 9 - 11 = 0·81 Hence three inrrational numbers between
and 7 ! can be: 11
0·n7'227222 ........... 0·737337333 ........... Q·74 7447444 ........... [CBSE
Marking Scheme, 20161 IAl Q. 2. Represent ./
on the number line. [Board Term Sol. ...,.,.; I -2 -1 Let
I, 20131
,ff,,, ... LU 0 1 P 02 A B AB = BC = 1 unit length
t-+ .. 3
####### ) ,.
REAL NUMSERS
[ 9
l ,in~ l'yth,~'r,1, th,--.,n·m
, ,, l' ~,•t• th
A Q lh•prc,cn l f~
5 on the number line.
/5. ex·• ,- 1 i.
!
) [Board Term I, 2014
l\m,tn1ct CD • I unit
kn~h r,•rr1.•ndkul,1r to
OC th,•n u,m~ l) th,1~,,r,1, th,·,,rcm ,
w,• ,l'c th,1t ()I) ,(,'/~1i::.J3 ,,, ll, ing ,.1 .:omp,1,, with centre
O ,ind r On, Jr,.1"· .m .l'1: which mh:r,.t',t~ the numl:,l'r line
.it the r-,int (l, th,m (l ''-'l'TC~~"'"d~
to .J A Q. J Kcr~nt , 5
0n thl' num~•r lim·. [Board Torm L 201 4,
BQS6IZK) Sol. \'e know th,.1t, ,/s
2 O = ,':f+l , ·2~ + 1 0 '"' - 1 -2 0 3 Dra,,· a right angll?d __
OBA., such that C'B = 2 units. :.B = 1
= 90 ° unit and LOBA ::S:ow b,· using P•,thago ras theorem ,
we have 2.. OA" = 0B 1 2 2 2 +AB= 2 +1 OA = ,M=. 1 :&#039;ow, take O as centre ,
OA = __ as radius, draw an arc which intersects the line at point
C. Hence, the point C represents
1 .. [CBSE Marking Scheme, 2014] A Q. -.. Represent .J9
on the number line. 2016, Set 20CNJE9] [Board Term I, Sol
_D ./······ .......... ../95 •v:s • I BC E A O +--9.~ Marks the distance 9.
units from a fixed point A on a given line to obtain a point B such that 9 units from AB =
B, marks a distance of 1 unit and mark the new
point as C. Find the mid point of AC and mark
that point as 0. Draw a centre O semi-circle with and radius OC. Draw a line perpendicular to
AC pas sing through B
and intersecting the semi-circle
at D. 1¼ then BD
= .J9 ../9 on To represent the number line, let us treat the line BC as the number
zero (c) line, with Bas . as 1 and so on
½ Draw an arc with centre B
and radius BD which intersect the number line at E E represents ../9 :.
1 Scheme, [CBSE Marking Sol. 2016]
·,.,Ji;s p;g A O BC E -":S-- distance 4 Mark the units from a fixed point
A on ,, given line to obtain a poin t
B such that AB = 4. units. From B, mark a d istance of 1 unit and mark the new point as C. Find the mid-point of
and mark AC that point as 0. Draw a
semi-circle with centre O and radius OC. Draw a line
perpendicular to AC passing throu gh Band intersecting the se
mi-circle a t D. Then , BD
= To represent on the
number line, let us treat th e line BC as th e numb
er lin e, with B as zero,
C as 1, and so on.
1 Draw an arc with centr e
B an d radiu s BD, which intersect s th e number line at
E. 1 ents :. E repre s
1 18] Q. 6 Repre sent ,/9.
on the number line.
[Delhi Board Sol.
2013 Board Term I, 20
11, Set-12; 2010 , Set Bl 2012 , Set-50] ':-: i ....... ····•., .... ./9 _l./9._
A 0; BCE
1 +-- 9 .3---+++ ) , 1 Mark the distance 9 unit s from a fixed point A on a given line to obtain a point B such that AB = 9 units from B, mark a distance of 1 unit C. Find and mark the new point as
the mid -point of AC and mark that point as
- Draw a semi- circle with centre O and
Draw a line radiu s OC. perpendicular to AC
passin g through B and intersecting the semi-circle at D.
1 Then,BD= J9.
1 ent J93 on To repre s the numbe r line us tre at the line BC as the numb
er line, with B as zero, C and so on. as 1, Draw an arc with centre B and radius BD, which intersect s the number line at
E. 1 ,/9 :. E represents [CBSE Marking Scheme, 2013] ~] ~Examine whether
is rational or irrati ona
l. [Board Term I, 2016, Set
QGL21FS]
Mathematics Class 9 Real Numbers Topic 1 and 2
Course: Engineering Mathematics I (2K6EN101)
University: Kannur University
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