Ncert – class 10- Real Numbers – Hots Question

Question -1

In a classroom activity on real numbers, the students have to pick a
number card from a pile and frame question on it if it is not a rational
number for the rest of the class. The number cards picked up by first 5
students and their questions on the numbers for the rest of the class are as
shown below. Answer them.
(i) Suraj picked up 8√8 and his question was – Which of the following is
true about 8√8?
(a) It is a natural number
(b) It is an irrational number
(c) It is a rational number
(d) None of these

(ii) Shreya picked up ‘BONUS’ and her question was – Which of the
following is not irrational?
(a) 3-4√5 (b) √7 -6
(c) 2+2√9 (d) 4√11-6
(iii) Ananya picked up -√15 , -√10 and her question was – √15 , -√10
_________is number.
(a) a natural (b) an irrational
(c) a whole (d) a rational

Solution

(i) (b):
Here √8 = 2√2 = product of rational and irrational numbers = irrational number
(ii) (c):
Here, √9 = 3
So, 2 + 2√9= 2 + 6 = 8 , which is not irrational.
(iii) (b):
Here.-√15 and -√10 are both irrational and difference of two irrational numbers is also
irrational.

Question – 2

There are 7 friends playing with marbles. When they divide the marbles
evenly between three friends, there are two marbles left over. When the same
marbles are divided evenly among five friends, there are two marbles left
over. When the marbles are divided among seven friends there are two
marbles left over. What is the least possible number of marbles in the bag. Is
there any other possible number of marbles in the bag?

Solution

x ÷ 3 = 2
x ÷ 5 = 2
x ÷ 7 = 2
LCM(3,5,7) = 105
Least possible number of marbles in the bag = 102 + 2 = 104
Another possible number of marbles in the bag = 102 x 2 = 204 + 2 = 206

Question -3

The Muscle Gym has bought 63 treadmills and 108 elliptical machines. The
gym divides them into several identical sets of treadmills and elliptical
machines for its branches located throughout the city, with no exercise
equipment left over. What is the greatest number of branches the gym can
have in the city?

Solution

Let the greatest number of branches in the city be ‘n’.
Let the number of treadmills sent to each branch be ‘t’ and the number
of ellipticals sent to each branch be ‘e’.
63 treadmills were distributed to ‘n’ branches with each branch getting ‘t’
treadmills.
i.e., nt = 63
108 elliptical machines were distributed to ‘n’ branches with each branch
getting ‘e’ elliptical machines.
i.e., ne = 108.
HCF of 108 and 63 is 9. So ,The greatest number of branches the gym can
have in the city is 9.

Question -4

Show that 5 + √3 is irrational.

Solution

Let us assume that 5 + √3 is a rational number.
We can find coprime a and b (b ≠ 0) such that 5 + √3 = a/b.
So,
√3 = (a/b) – 5
√3 = (a – 5b)/b
Since a and b are integers, we get (a/b) – 5 is rational, and so √3 is
rational.
But this contradicts the fact that √3 is irrational.
This contradiction has arisen because of our incorrect assumption that 5+√3 is rational.
Therefore, we conclude that 5 + √3 is irrational

Question -5

Find the value of a, if 6/(3√2 – 2√3) = 3√2 – a√3 .

Solution

Given,
6/(3√2 – 2√3) = 3√2 – a√3….(i)
Consider LHS,
6/(3√2 – 2√3)
Rationalizing the denominator, we get;
[6/(3√2 – 2√3)] [(3√2 + 2√3)/ (3√2 + 2√3)]
= [6(3√2 + 2√3)]/ [(3√2)2 – (2√3)2]
= [6(3√2 + 2√3)]/ (18 – 12)
= 6(3√2 + 2√3)/6
= 3√2 + 2√3….(ii)
From (i) and (ii),
-a = 2
a = -2

Question -6

The product of the LcM and HcF of two natural numbers is 24. The difference of two
numbers is 2. Find the numbers.

Solution

Let the numbers be p and (p + 2).
∴ Product of numbers = HCF × LCM
⇒ p (p + 2) = 24
⇒ p2 + 2p – 24 = 0
⇒ p2 + 6p – 4p – 24 = 0
⇒ p (p + 6) – 4 (p + 6) = 0
⇒ (p – 4) (p + 6) = 0
⇒ p = 4 or p = – 6
Numbers = p = 4 and p + 2 = 4 + 2 = 6.
∴ Numbers are 4 and 6.

Question -7

Dinesh says that if √4×2 is irrational then √4 + √2 is also irrational. Therefore,
if √ab, √bc and √ca is an irrational number, then show that √a + √b + √c is
also irrational.

Solution

Let (√a + √b + √c) be a rational number
√a + √b + √c = p / q
Squaring both sides
(√a + √b + √c)² = (p/q)²
a + b + c + 2√ab + 2√bc + 2√ac = p²/q²
2(√ab + √bc + √ac) = p²/q² – (a + b + c)
√ab + √bc + √ac = p²/2q² – (a + b + c)/2
But is given that √ab, √bc, √ac are irrational, therefore our assumption is wrong
√a + √b + √c is an irrational number

Question – 8

Shreya takes 35 seconds to pack a gift and label it. For Ravi the same job takes 42 seconds and for Priya, it takes 28 seconds. If they all start using the labeling machine at the same time, after how many seconds will they be using the labeling machine together and by then what is the total number of gifts they all would have packed and labeled together?

Solution

35 = 7 x 5
42 = 7 x 2 x 3
28 = 7 x 2 x 2
LCM(35, 42, 28) = 7 x 5 x 2 x 3 x 2 = 420
Ans – All of them will be using labelling machine together after 420 seconds
In 420 seconds Shreya will pack 420 ÷ 35 = 12 gifts
Similarly, in 420 seconds, Ravi and Priya will pack 420 ÷ 42 = 10 gifts
and 420 ÷ 28 = 15gifts respectively
Therefore, total gifts packed 12 + 10 + 15 = 37

Question – 9

An army contingent of 612 members is to march behind an army band of 48
members in a parade. The two are to march in the same number of columns.
What is the maximum number of columns in which they can march?

Solution

Clearly, the maximum number of columns = HCF(612, 48)
Now, 612 = (2² × 3² × 17 and 48 = 2⁴ × 3
HCF(612, 48) = 2² × 3 = 12

Question : 10

In a school there are two sections, namely A and B, of class X. there are 30
students in section A and 28 students in section B. find the minimum number
of books required for their class library so that they can be distributed equally
among students of section A or section B.

Solution

Clearly, the required number of books are to be distributed equally among
the students of section A or section B,
So the numbers of these books must be a multiple of 30 as well as that of
28.
The required number is LCM(30, 28) = 420
Hence the required number of books = 420