CBSE NCERT Solutions for Class 10 Mathematics Chapter 1 Back of Chapter Questions class10
CBSE NCERT Solutions for Class 10 Mathematics Chapter 1
Back of Chapter Questions
1. Use Euclid’s division algorithm to find the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255
Solution:
(i) 135 and 225
Step 1:
Since 225 is greater than 135, we can apply Euclid's division lemma to a = 225 and b = 135 to find q and r such that
225 = 135q + r, 0 ≤ r < 135
So, dividing 225 by 135 we get 1 as the quotient and 90 as remainder.
i. e 225 = (135 × 1) + 90
Step 2:
Remainder r is 90 and is not equal to 0, we apply Euclid's division lemma to b = 135 and r = 90 to find whole numbers q and r such that
135 = 90 × q + r, 0 ≤ r < 90
So, dividing 135 by 90 we get 1 as the quotient and 45 as remainder.
𝑖𝑖. 𝑒𝑒 135 = (90 × 1) + 45
Step 3:
Again, remainder r is 45 and is not equal to 0, so we apply Euclid's division lemma to b = 90 and r = 45 to find q and r such that
90 = 45 × q + r, 0 ≤ r < 45
So, dividing 90 by 45 we get 2 as the quotientand 0 as remainder.
𝑖𝑖. 𝑒𝑒. 90 = (2 × 45) + 0
Step 4:
Since the remainder is zero, the divisor at this stage will be HCF of (135, 225)
Since the divisor at this stage is 45, therefore, the HCF of 135 and 225 is 45.
(ii) 196 and 38220
Step 1:
Since 38220 is greater than 196, we can apply Euclid's division lemma to a = 38220 and b = 196 to find whole numbers q and r such that
38220 = 196 q + r, 0 ≤ r < 196
So dividing 38220 by 196, we get 195 as the quotient and 0 as remainder r
𝑖𝑖. 𝑒𝑒 38220 = (196 × 195) + 0
Because the remainder is zero, divisor at this stage will be HCF
Since divisor at this stage is 196, therefore, HCF of 196 and 38220 is 196.
(iii) 867 and 255
Step 1:
Since 867 is greater than 255, we can apply Euclid's division lemma, to
a = 867 and b = 255 to find q and r such that 867 = 255q + r, 0 ≤ r <
255. So, dividing 867 by 255 we get 3 as the quotient and 102 as remainder.𝑖𝑖. 𝑒𝑒 867 = 255 × 3 + 102
Step 2:
Since remainder is 102 and is not equal to 0, we can apply the division lemma to a = 255 and b = 102 to find whole numbers q and r such that 255 = 102q + r where 0 ≤ r < 102. So, dividing 255 by 102 we get 2 as the quotient and 51 as remainder.𝑖𝑖. 𝑒𝑒 255 = 102 × 2 + 51
Step 3:
Again remainder 51 is not equal to zero, so we apply the division lemma to a = 102 and b = 51 to find whole numbers q and r such that 102 = 51 q + r where 0 ≤ r < 51. So, dividing 102 by 51 we get 2 as the quotient and 0 as remainder.𝑖𝑖. 𝑒𝑒 102 = 51 × 2 + 0. Since, the remainder is zero, the divisor at this stage is the HCF. Since the divisor at this stage is 51, therefore, HCF of 867 and 255 is 51
2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Solution:
Let a be any odd positive integer, we need to prove that a is in the form of 6q + 1 , or 6q + 3 , or 6q + 5, where q is some integer.
Because a is an integer, we can consider b to be 6 as another integer. Applying Euclid's division lemma, we get a = 6q + r for some integer q ≥ 0, and r = 0, 1, 2, 3, 4, 5 Where 0 ≤ r < 6.
Therefore, a can be any of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
However, since a is odd, a cannot take the values 6q, 6q + 2 and 6q + 4
(since all these are divisible by 2)
Also, 6q + 1 = 2 × 3q + 1 = 2k1 + 1, where k1 is a positive integer
6q + 3 = (6q + 2) + 1 = 2 (3q + 1) + 1 = 2k2 + 1, where k2 is an integer
6q + 5 = (6q + 4) + 1 = 2 (3q + 2) + 1 = 2k3 + 1, where k3 is an integer
Clearly, 6q + 1, 6q + 3, 6q + 5 are in the form of 2k + 1, where k is an integer.
Therefore, 6q + 1, 6q + 3 and 6q + 5 are odd numbers.
Therefore, any odd integer can be expressed in the form of
6q + 1, or 6q + 3, or 6q + 5 where q is some integer
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? Solution:
Maximum number of columns in which the Army contingent and the band can march is equal to the HCF of 616 and 32. Euclid's algorithm can be used here to find the HCF.
Step 1:
Since 616 is greater than 32, so by applying Euclid's division lemma to a = 616 and b = 32 we get integers q and r as 19 and 8
𝑖𝑖. 𝑒𝑒 616 = 32 × 19 + 8
Step 2:
Since remainder r is 8 and is not equal to 0, we can again apply Euclid's lemma to 32 and 8 to get integers 4 and 0 as the quotient and remainder respectively.
𝑖𝑖. 𝑒𝑒 32 = 8 × 4 + 0
Step 3:
Since remainder is zero so divisor at this stage will be the HCF. The HCF(616, 32) is 8. Therefore, they can march in 8 columns each.
4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
[Hint: Let 𝑥𝑥 be any positive integer then it is of the form 3q, 3q + 1or3q + 2.
Now square each of these and show that they can be rewritten in the form 3m or
3m + 1.]
Solution:
Let 𝑥𝑥 be any positive integer we need to prove that 𝑥𝑥2 is in the form of 3m or 3m + 1 where m is an integer.
Let b = 3 be the other integer, so by applying Euclid's division lemma to 𝑥𝑥 and b = 3
Then 𝑥𝑥 = 3q + r for another integer q ≥ 0 and r = 0, 1, 2 because 0 ≤ r < 3
Therefore, 𝑥𝑥 = 3q, for r = 0 or 3q + 1, for r=1 or 3q + 2, for r=2
Now Consider 𝑥𝑥2
𝑥𝑥2 = (3q)2 or (3q + 1)2 or (3q + 2)2
𝑥𝑥2 = (9q2) or 9q2 + 6q + 1 or 9q2 + 12q + 4
𝑥𝑥2 = 3 × (3q2) or 3(3q2 + 2q) + 1 or 3(3q2 + 4q + 1) + 1
𝑥𝑥2 = 3k1 or 3k2 + 1 or 3k3 + 1
Where k1 = 3q2, k2 = 3q2 + 2q and k3 = 3q2 + 4q + 1 since q, 2, 3, 1 etc are
all integers, so, their sum and product will be integer
So k1,k2,k3 are all integers.
Hence, it can be said that the square of any positive integer is either in the form of 3m or 3m + 1 for any integer m.
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.
Solution:
Let 𝑎𝑎 be any positive integer and b equals to 3, then 𝑎𝑎 = 3q + r, where q ≥ 0 and
0 ≤ r < 3, ∴ 𝑎𝑎 = 3q or 3q + 1 or 3q + 2
Therefore, every number can be represented in these three forms. There are three cases.
Case 1:
When 𝑎𝑎 = 3q (r=0)
𝑎𝑎3 = (3q)3 = 27q3 = 9(3q3) = 9 m
Where m is an integer such that m = 3q3
Case 2:
When 𝑎𝑎 = 3q + 1 (r=1), 𝑎𝑎3 = (3q + 1)3
𝑎𝑎3 = 27q3 + 27q2 + 9q + 1
𝑎𝑎3 = 9(3q3 + 3q2 + q) + 1
𝑎𝑎3 = 9m + 1
Where m is an integer such that m = (3q3 + 3q2 + q)
Case 3:
When 𝑎𝑎 = 3q + 2 (r=2)
𝑎𝑎3 = (3q + 2)3
𝑎𝑎3 = 27q3 + 54q2 + 36q + 8
𝑎𝑎3 = 9(3q3 + 6q2 + 4q) + 8
𝑎𝑎3 = 9m + 8
Where m is an integer such that m = (3q3 + 6q2 + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
♦ ♦ ♦
EXERCISE 1.2
1. Express each number as a product of its prime factors:
(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429
Solution:
(i) 140 = 2 × 2 × 5 × 7 = 22 × 5 × 7
(ii) 156 = 2 × 2 × 3 × 13 = 22 × 3 × 13
(iii) 3825 = 3 × 3 × 5 × 5 × 17 = 32 × 52 × 17
(iv) 5005 = 5 × 7 × 11 × 13
(v) 7429 = 17 × 19 × 23
2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 and 91
(ii) 510 and 92
(iii) 336 and 54
Solution:
(i) 26 and 91
26 = 2 × 13
91 = 7 × 13
13 is the largest number which divides both 26 and 91. So, HCF = 13.
LCM = 2 × 7 × 13 = 182
Product of the two numbers = 26 × 91 = 2366
HCF × LCM = 13 × 182 = 2366
Hence, product of two numbers = HCF × LCM
(ii) 510 and 92
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23
2 is the largest number which divides both 510 and 92. So, HCF = 2.
LCM = 2 × 2 × 3 × 5 × 17 × 23 = 23460
Product of the two numbers= 510 × 92 = 46920
HCF × LCM = 2 × 23460 = 46920
Hence, product of two numbers = HCF × LCM
(iii) 336 and 54
336 = 2 × 2 × 2 × 2 × 3 × 7 = 24 × 3 × 7 54 = 2 × 3 × 3 × 3 = 2 × 33
6 is the largest number which divides both 336 and 54. So, HCF = 6.
LCM = 24 × 33 × 7 = 3024
Product of the numbers = 336 × 54 = 18144
HCF × LCM = 6 × 3024 = 18144
Hence, product of two numbers = HCF × LCM
3. Find the LCM and HCF of the following integers by applying the prime factorization method.
(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25
Solution:
(i) 12, 15 and 21 12 = 22 × 3
15 = 3 × 5
21 = 3 × 7
3 is the largest number which divides 12, 15 and 21. So, HCF = 3.
LCM = 22 × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
17 = 1 × 17
23 = 1 × 23 29 = 1 × 29
1 is the largest number which divides 17, 23 and 29. So, HCF = 1.
LCM = 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
8 = 2 × 2 × 2
9 = 3 × 3
25 = 5 × 5
1 is the largest number which divides 8, 9 and 25. So, HCF = 1.
LCM = 2 × 2 × 2 × 3 × 3 × 5 × 5 = 1800
4. Given that HCF (306, 657) = 9, find LCM (306, 657).
Solution:
HCF (306, 657) = 9
We know that, Product of two numbers is equal to product of their LCM and HCF.
∴ LCM × HCF = 306 × 657
306 × 657 306 × 657
LCM = =
HCF 9
LCM = 22338
5. Check whether 6n can end with the digit 0 for any natural number n.
Solution:
If any number ends with the digit 0, it should be divisible by 10 or in other words the prime factorization of the number must include 2 and 5 both.
Prime factorization of 6n = (2 × 3)n
We can see that 5 is not a prime factor of 6n.
Hence, for any value of n, 6n will not be divisible by 5.
Therefore, 6n cannot end with the digit 0 for any natural number n.
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Solution:
There are two types of numbers, namely – prime and composite. Prime numbers have only two factors namely 1 and the number itself whereas composite numbers have factors other than 1 and itself.
It can be observed that
7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × (77 + 1)
= 13 × 78
= 13 × 13 × 6
The given expression has 6 and 13 as its factors. Therefore, it is a composite number.
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1)
= 5 × (1008 + 1)
= 5 × 1009
1009 cannot be factorised further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?
Solution:
It is given that Ravi and Sonia do not take same of time. Ravi takes lesser time than Sonia for completing 1 round of the circular path.
As they are going in the same direction, they will meet again when Ravi will complete 1 round of that circular path with respect to Sonia.
(i) e., When Sonia completes one round then Ravi completes 1.5 rounds. So they will meet first time at the time which is a common multiple of the time taken by them to complete 1 round
i.e.,LCM of 18 minutes and 12 minutes.
Now
18 = 2 × 3 × 3 = 2 × 32 and, 12 = 2 × 2 × 3 = 22 × 3
LCM of 12 and 18 = product of factors raised to highest exponent = 22 × 32 = 36
Therefore, Ravi and Sonia will meet together at the starting point after 36 minutes.
♦ ♦ ♦
EXERCISE 1.3
1. Prove that is irrational.
Solution:
Let us assume, on the contrary, that is a rational number.
𝑎𝑎
Therefore, we can find two co-prime integers 𝑎𝑎, b (b ≠ 0) such that b
𝑎𝑎
b
⇒ 𝑎𝑎 b
⇒ 𝑎𝑎2 = 5 b2
Therefore, 𝑎𝑎2 is divisible by 5, then 𝑎𝑎 is also divisible by 5.
So, 𝑎𝑎2 = (5k)2 = 5(5k2) = 5b2
⇒ b2 = 5k2
This means that b2 is divisible by 5 and hence, b is divisible by 5.
This implies that 𝑎𝑎 and b have 5 as a common factor.
And this is a contradiction to the fact that 𝑎𝑎 and b are co-prime.
So our assumption that is rational is wrong.
Hence, cannot be a rational number. Therefore, is irrational.
2. Prove that is irrational.
Solution:
Let us assume, on the contrary, that is rational.
Therefore, we can find two integers 𝑎𝑎, b (b ≠ 0) such that
𝑎𝑎
b
𝑎𝑎
b
1 𝑎𝑎
1 𝑎𝑎
Since 𝑎𝑎 and b are integers, − 3 will also be rational and therefore, is
2 b
rational.
This contradicts the fact that is irrational.
Hence, our assumption that is rational is false.
Therefore, is irrational.
3. Prove that the following are irrationals:
(i)
(ii)
(iii)
Solution:
(i)
Let us assume that is rational.
Therefore, we can find two integers 𝑎𝑎, b (b ≠ 0) such that
1 𝑎𝑎
= √2 b b
𝑎𝑎
b is rational as 𝑎𝑎 and b are integers.
𝑎𝑎
Therefore, is rational.
This contradicts the fact that is irrational.
Hence, our assumption is false and is irrational.
(ii)
Let us assume that is rational.
𝑎𝑎
Therefore, we can find two integers 𝑎𝑎, b (b ≠ 0) such that b
𝑎𝑎
∴ √5 =
7b
𝑎𝑎
is rational as a and b are integers.
7b
Therefore, should be rational.
This contradicts the fact that is irrational.
Therefore, our assumption that is rational is false.
Hence, is irrational.
(iii)
Let be rational. Therefore, we can find two integers 𝑎𝑎, b (b ≠ 0) such that 𝑎𝑎 b
𝑎𝑎
b
a
Since 𝑎𝑎 and b are integers, − 6 is a rational number and hence, b
should be rational.
This contradicts the fact that is irrational.
Therefore, our assumption is false and hence, is irrational.
♦ ♦ ♦
EXERCISE 1.4
1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Solution:
(i)
3125 = 55
The denominator is of the form 2m × 5n.
Here, m=0 and n=5
Hence, the decimal expansion of is terminating
(ii)
8 = 23
The denominator is of the form 2m × 5n.
Here m=3 and n=0.
Hence, the decimal of expansion of is terminating.
(iii)
455 = 5 × 7 × 13
Since the denominator is not in the form 2m × 5n, as it also contains 7 and 13 as its factors.
Hence, its decimal expansion will be non-terminating repeating.
(iv)
1600 = 26 × 52
The denominator is of the form 2m × 5n.
Here m=6 and n=2.
Hence, the decimal expansion of is terminating.
(v)
343 = 73
Since the denominator is not in the form 2m × 5n, as it has 7 as its factor.
Hence, decimal expansion of is non-terminating repeating.
(vi)
The denominator is of the form 2m × 5n.
Here m=3 and n=2.
Hence, the decimal expansion of is terminating.
(vii)
Since the denominator is not of the form 2m × 5n as it also has 7 as its factor.
The decimal expansion of is non-terminating repeating.
(viii) = =
The denominator is of the form 2m × 5n.
Here m=0 and n=1.
Hence, the decimal expansion of is terminating.
(ix)
The denominator is of the form 2m × 5n
Here m=1 and n=2
Hence, the decimal expansion of is terminating.
(x) = =
30 = 2 × 3 × 5
Since the denominator is not in the form 2m × 5n, as it also has 3 as its factor. Hence, the decimal expansion of is non-terminating repeating.
2. Write down the decimal expansions of those rational numbers in Question1 above which have terminating decimal expansions.
Solution:
(i) = 0.00416
18750
×
(ii) = 2.125
40
×
(iv) = 0.009375
8000
×
(vi) = = 0.115
1000
×
(viii) = = = 0.4
20
× (ix) = 0.7
350
×
3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p
, what can you say about the prime factors of q?
q
(i) 43.123456789
(ii) 0.120120012000120000. ..
(iii) 43. 123456789
Solution:
(i) 43.123456789
Since this has a terminating decimal expansion, it is a rational number of p m × 5n,
the form and q is of the form 2 q
i.e., the prime factors of q will be either 2 or 5 or both.
(ii) 0.120120012000120000 …
The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
(iii) 43. 123456789
Since the decimal expansion is non-terminating recurring, the given p number is a rational number of the form and q is not of the form q
2m × 5n i.e., the prime factors of q will also have a factors other than 2 or 5.
♦ ♦ ♦ By shikha kaushal
