Here, we have providing chapter 1 real numbers NCERT solutions for class 10th math which is beneficial for students. This solutions are updated according to 2020-21. Syllabus.
Exercise : 1.1
Exercise : 1.1
Use Euclid's division algorithm to find the HCF of
( 1 ) 135 and 225. ( 2 ) 196 and 38220.
( 3 ) 867 and 255.
Answer :
( 1 ) 135 and 225.
since 225 > 135 . We apply the division lemma to obtain
225 = 135* 1 + 90
135 = 90* 1 +45
90 = 45*2 + 0
Scince the reminder is zero. The process stop.
Lastly our answers is 45.
( 2 ) 196 and 38220
Here , 38220 > 196 .we apply the division lemma ,to obtain
38220 = 196*195 +0
Since the remainder is zero, The process stop.
So, lastly our answer is. 196.
( 3 ) 867 and 225.
Here, 867 >225. We apply the division lemma , to obtain.
867 = 225*3 + 102.
225 = 102*2 + 51.
102 = 51*2 + 0
Scince, the remaider is zero.,the process stop.
So, lastly our answer is , 51
Q,No.2..Show that any positive odd integer is the form 6q+1,or 6q+3,or 6q+5, where q is some integer.
Answer :
Let a, be any positive integer and b=6. Then,by Euclid's algorithm a= 6q+r where r= 0,1,2,3,4,5
Therefore
a=6q+0 or 6q ,or 6q+1'or 6q+2,or 6q+3,or 6q+4,or 6q+5.
also,
6q+1= 2(3q) +1. =2k+1
6q+3= 2(3q+1)+1. =2k+1
6q+5= 2(3q+2)+1. =2k+1
Here all the integers have 1 remainder.( if they are even number then it is exactly divisible by 2 and no remainder leave.)
Therefore any odd positive integers are is the form of 6q+1,or 6q+3,or 6q+5. Proved.
Q. No.3.. An army contingent of 616 members is to march behind an army band of 32 members in a parade.The two groups are to march in the same number of columns.What is the maximum number of columns in which they can march.
Answer :
Hcf( 616,32) will give the maximum number of columns in which they can march.
We can use Euclid's algorithm to find the HCF.
616= 32*19+8.
32=8*4+0
The HCF ( 616,32 ) is 8.
Therefore, they can march in 8 columns each.
Q. No. 4. Use Euclid's division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m.
Answer :
Let, a be any positive integer and b=3.
Then a =3q+r for some integer q >_0
and r =0,1,2 because 0<_ r<3
therefore, a =3q or 3q+1 or 3q+2
Or
a^2= (3q)^2 or (3q +1)^2 or (3q+2)^2
a^2= (9q^2) or 9q^2 +6q+1 or 9q^2 +12q +4
= 3(3q^2) or 3(3q^2 +2q) +1 or 3(3q^2. +4q+1) +1
=3k1 or 3k2+1 or 3k3 +1
where k1, k2 and k3 are some positive integers
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.
Q. No 5.
Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
Answer :
Let a be any positive integer and b= 3q+r, where q>_0 and 0 <_ r< 3.
Therefore
a=3q or 3q+1 or 3q+2
Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a=3q,
a^3 = (3q)^3 =27q^3 = 9(3q^3) = 9m,
where m is an integer such that m=3q^3.
Case 2 : When a = 3q+1,
a^3= (3q+1)^3
a^3 = 27q^3 +27q^2 + 9q+1
a^3 = 9(3q^3+ 3q^2 + q) +1.
a^3 = 9m+1
Where, m is an integer such that m= (3q^3 + 3q^2 +q).
Case 3 : when a=3q+2 ,
a^3 = (3q+2)^3
a^3 = 27q^3 + 54q^2 + 36q + 8
a^3 = 9(3q^3 + 6q^2 + 4q) + 8
a^3 = 9m+8
Where m is integer such that m= (3q^3 +6q^2 + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
Q,No.2..Show that any positive odd integer is the form 6q+1,or 6q+3,or 6q+5, where q is some integer.
Answer :
Let a, be any positive integer and b=6. Then,by Euclid's algorithm a= 6q+r where r= 0,1,2,3,4,5
Therefore
a=6q+0 or 6q ,or 6q+1'or 6q+2,or 6q+3,or 6q+4,or 6q+5.
also,
6q+1= 2(3q) +1. =2k+1
6q+3= 2(3q+1)+1. =2k+1
6q+5= 2(3q+2)+1. =2k+1
Here all the integers have 1 remainder.( if they are even number then it is exactly divisible by 2 and no remainder leave.)
Therefore any odd positive integers are is the form of 6q+1,or 6q+3,or 6q+5. Proved.
Q. No.3.. An army contingent of 616 members is to march behind an army band of 32 members in a parade.The two groups are to march in the same number of columns.What is the maximum number of columns in which they can march.
Answer :
Hcf( 616,32) will give the maximum number of columns in which they can march.
We can use Euclid's algorithm to find the HCF.
616= 32*19+8.
32=8*4+0
The HCF ( 616,32 ) is 8.
Therefore, they can march in 8 columns each.
Q. No. 4. Use Euclid's division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m.
Answer :
Let, a be any positive integer and b=3.
Then a =3q+r for some integer q >_0
and r =0,1,2 because 0<_ r<3
therefore, a =3q or 3q+1 or 3q+2
Or
a^2= (3q)^2 or (3q +1)^2 or (3q+2)^2
a^2= (9q^2) or 9q^2 +6q+1 or 9q^2 +12q +4
= 3(3q^2) or 3(3q^2 +2q) +1 or 3(3q^2. +4q+1) +1
=3k1 or 3k2+1 or 3k3 +1
where k1, k2 and k3 are some positive integers
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.
Q. No 5.
Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
Answer :
Let a be any positive integer and b= 3q+r, where q>_0 and 0 <_ r< 3.
Therefore
a=3q or 3q+1 or 3q+2
Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a=3q,
a^3 = (3q)^3 =27q^3 = 9(3q^3) = 9m,
where m is an integer such that m=3q^3.
Case 2 : When a = 3q+1,
a^3= (3q+1)^3
a^3 = 27q^3 +27q^2 + 9q+1
a^3 = 9(3q^3+ 3q^2 + q) +1.
a^3 = 9m+1
Where, m is an integer such that m= (3q^3 + 3q^2 +q).
Case 3 : when a=3q+2 ,
a^3 = (3q+2)^3
a^3 = 27q^3 + 54q^2 + 36q + 8
a^3 = 9(3q^3 + 6q^2 + 4q) + 8
a^3 = 9m+8
Where m is integer such that m= (3q^3 +6q^2 + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
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