Solutions: Real Numbers - Vardaan Learning Institute

1. The HCF of 135 and 225 is:

Ans: (b) 45
Detailed Solution:
$135 = 3^3 \times 5$
$225 = 3^2 \times 5^2$
HCF = $3^2 \times 5$ $= 9 \times 5$ $= 45$.

2. Which of the following is an irrational number?

Ans: (c) $\pi$

3. The product of a non-zero rational and an irrational number is:

Ans: (a) Always irrational

4. The LCM of the smallest prime number and the smallest composite number is:

Ans: (b) 4
Detailed Solution:
Smallest prime = 2. Smallest composite = 4. LCM(2, 4) = 4.

5. If HCF(a, b) = 12 and $a \times b = 1800$, then LCM(a, b) is:

Ans: (c) 150
Detailed Solution:
HCF $\times$ LCM = $a \times b$.
$12 \times \text{LCM} = 1800$ $\Rightarrow$ $\text{LCM} = 150$.

6. The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after:

Ans: (d) Four decimal places
Detailed Solution:
Denominator $1250 = 2 \times 5^4$. Highest power is 4.

7. If LCM(x, 18) = 36 and HCF(x, 18) = 2, then x is:

Ans: (c) 4
Detailed Solution:
LCM $\times$ HCF = Product of numbers.
$36 \times 2 = x \times 18 \Rightarrow 72 = 18x \Rightarrow x = 4$.

8. The ratio of LCM and HCF of the least composite and the least prime numbers is:

Ans: (b) 2:1
Detailed Solution:
Least composite = 4, Least prime = 2.
LCM(4, 2) = 4, HCF(4, 2) = 2.
Ratio = 4:2 = 2:1.

9. $\sqrt{p}$ is an irrational number if p is:

Ans: (b) A prime number

10. Assertion (A): The number $5^n$ cannot end with the digit 0, where n is a natural number.
Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.

Ans: (b) Both A and R are true but R is not the correct explanation of A.
Detailed Solution:
$5^n$ ends in 0 if it has factors 2 and 5. It lacks factor 2. A is correct. R is a true statement facts about 5, but doesn't explicitly mention the absence of 2 as the cause (though implied). Best fit is (b).

11. Using prime factorization, find the HCF and LCM of 72 and 120.

Ans: HCF = 24, LCM = 360
Detailed Solution:
$72 = 2^3 \times 3^2$
$120 = 2^3 \times 3 \times 5$
HCF = $2^3 \times 3$ $= 24$.
LCM = $2^3 \times 3^2 \times 5$ $= 360$.

12. Check whether $6^n$ can end with the digit 0 for any natural number $n$.

Ans: No
Detailed Solution:
For a number to end with 0, its prime factorization must contain 2 and 5.
$6^n = (2 \times 3)^n$ $= 2^n \times 3^n$. Does not contain 5.

13. Explain why $7 \times 11 \times 13 + 13$ is a composite number.

Ans: Composite
Detailed Solution:
Expression = $13(7 \times 11 + 1)$ $= 13(78)$ $= 13 \times 13 \times 6$.
Has factors other than 1 and itself.

14. Prove that $3 + \sqrt{2}$ is irrational, given that $\sqrt{2}$ is irrational.

Ans: Proof
Detailed Solution:
Let $3+\sqrt{2} = a/b$ (rational).
$\sqrt{2} = a/b - 3$.
RHS is rational, LHS is irrational. Contradiction.

15. Prove that $\sqrt{5}$ is an irrational number.

Ans: Standard Proof
Detailed Solution:
Assumption of rational $a/b$, showing $a$ and $b$ divisible by 5, contradiction.

16. Find the largest number that divides 245 and 1029 leaving remainder 5 in each case.

Ans: 16
Detailed Solution:
Find HCF of $(245-5)$ and $(1029-5)$, i.e., HCF(240, 1024).
$240 = 16 \times 15$,
$1024 = 16 \times 64$.
HCF is 16.

17. An army contingent of 616 members...

Ans: 8 columns
Detailed Solution:
HCF(616, 32).
$616 = 32 \times 19 + 8$.
$32 = 8 \times 4 + 0$.
HCF is 8.

18. Three bells ring at intervals of 4, 7 and 14 minutes...

Ans: 6:28 AM
Detailed Solution:
LCM(4, 7, 14) = 28 minutes.
6:00 + 28 mins = 6:28 AM.

19. Find HCF and LCM of 404 and 96 and verify that HCF $\times$ LCM = Product of the two numbers.

Ans: HCF = 4, LCM = 9696
Detailed Solution:
$404 = 2^2 \times 101$. $96 = 2^5 \times 3$.
HCF = $2^2 = 4$. LCM = $2^5 \times 3 \times 101 = 9696$.
Verification: $4 \times 9696 = 38784$. $404 \times 96 = 38784$. LHS = RHS.

20. Prove that $5 - \sqrt{3}$ is an irrational number.

Ans: Proof
Detailed Solution:
Let $5-\sqrt{3} = a/b$. Then $\sqrt{3} = 5 - a/b = (5b-a)/b$.
Since $a,b$ are integers, RHS is rational. But $\sqrt{3}$ is irrational. Contradiction.

21. Prove that $\sqrt{2} + \sqrt{5}$ is irrational.

Ans: Proof
Detailed Solution:
Let $x = \sqrt{2} + \sqrt{5}$. Squaring both sides: $x^2 = 2 + 5 + 2\sqrt{10}$.
$\Rightarrow \sqrt{10} = (x^2 - 7)/2$.
If $x$ is rational, RHS is rational. But $\sqrt{10}$ is irrational. Contradiction.

22. A sweets seller has 420 kaju barfis and 130 badam barfis...

Ans: 10
Detailed Solution:
We need to find HCF(420, 130).
Using Euclid's algorithm: $420 = 130 \times 3 + 30$.
$130 = 30 \times 4 + 10$.
$30 = 10 \times 3 + 0$.
HCF is 10. So, 10 barfis per stack.

23. Case Study: Seminar Hall

Detailed Solution:
(i) Max participants per room = HCF(60, 84, 108).
$60 = 12 \times 5$, $84 = 12 \times 7$, $108 = 12 \times 9$. HCF = 12.

(ii) LCM of 60, 84, 108.
$60=2^2 \times 3 \times 5$, $84=2^2 \times 3 \times 7$, $108=2^2 \times 3^3$.
LCM = $2^2 \times 3^3 \times 5 \times 7$ $= 4 \times 27 \times 35$ $= 3780$.

(iii) Total Rooms = Total Participants / HCF.
$(60 + 84 + 108) / 12$ $= 252 / 12$ $= 21$.