### Real Numbers

Four Marks Questions

1. Prove that  is an irrational number.

Sol:

TWO MARKS QUESTIONS

1. 12n can end with the digit 0 or 5 for any Natural Number 'n'. Do you agree with the statement? Why?/ Why not?

Sol: 12n = (22 × 3)n  = 22n × 3n

i.e., The prime factorisation of 12n contains the Prime Numbers 2 and 3. As for any number to end with '0' or '5', It must be divisible by 5.

∵ 5 is not present in the prime factorisation of 12n (n ∈ N), it follows that 12n cannot end with the digit 0 or 5 for any natural number n. Hence, the given statement is wrong.

2. The formula for calculating pH is pH = − log10 [H+] where pH is the acidity or basicity of the solution and [H+] is the hydrogen ion concentration.

If Shankar's Grandma's Lux soap has a hydrogen ion concentration of 9.2 × 10−12 what is its pH?

Sol: Hydrogen ion concentration in Lux soap is 9.2 × 10−12

pH = −log10[H+] = −log10 (9.2 × 10−12)

= − [log109.2 + log10 10−12]

= −[0.9638 + (−12)]     [ ∵ From Logarithm table log10 9.2 = 0.9638]

= −0.9638 + 12

= 11.0362

4. Shriya says that 24 = 16 and 42 = 16. Hence it can be concluded that m= nm for any natural number m and n. Do you agree?

Sol: It is true in the case of m = 2 and n = 4 but not always.

for m = 3 and n = 2

mn = 32 = 9

nm = 23 = 8

∴ It cannot be concluded that mn = nm for any natural numbers m and n.

6. Draw the Venn diagram of Real Numbers.

Sol:

N ⊂ W ⊂ Z ⊂ Q

Q ∪ Q' = R

7. Find the L.C.M. and H.C.F. of 72 and 108 by prime factorisation method. How these are related to the product of the given numbers.

Sol: 72 = 2 × 2 × 2 × 3 × 3

= 23 × 32

108 = 2 × 2 × 3 × 3 × 3

= 22 × 33

L.C.M. 23 × 33 = 8 × 27 = 216

H.C.F. 22 × 32 = 4 × 9 = 36

L.C.M. × H.C.F. = 216 × 36 = 7776

Product of the numbers = 72 × 108 = 7776

It is observed that

The Product of the numbers = Product of their L.C.M. and H.C.F.

One Mark Questions

4. How will you show that (17 × 11 × 2) + (17 × 11 × 5) is a Composite Number? Explain.

Sol. (17 × 11 × 2) + (17 × 11 × 5) = 17 × 11 × (2 + 5)

= 17 × 11 × 7 (The product of primes)

By the Fundamental Theorem of Arithmetic, Every Composite number can be expressed as product of primes uniquely.

∴ (17 × 11 × 2) + (17 × 11 × 5) is a composite number.

5. Write 3825 as a product of its prime factors.

3825 = 3 × 3 × 5 × 5 × 17

= 32 × 52 × 17

6. 'Every rational number is a terminal decimal'. Do you agree? Why/ Why not?

Sol. Every rational number need not be a terminating decimal, a non - terminating recurring decimal is also a rational number.

e.g.:  is a rational number which is non - terminating but recurring decimal.

=

8. Which of the following are rational?

(i) 43.123456789 (ii) 0.120120012000120000....

Sol: (i) 43.123456789 is a terminating decimal which can be written in  form where p, q are integers and q ≠ 0.

∴ 43.123456789 is a rational number

(ii) 0.120120012000120000 .... is a non - terminating, non - recurring decimal.

It cannot be written in  form where q ≠ 0 and p, q ∈ Z.

∴ 0.120120012000120000.... is not a rational number.

9. Write 64 = 82 and 64 = 43 in logarithmic form. What do you Infer?

Sol:  Logarithmic form of

(i) 64 = 82  log8 64 = 2

(ii) 64 = 43  log4 64 = 3

∴ The logarithms of the same number to different bases are different.

10. Write the exponential form of (i) log22 = 1  (ii) log55 = 1. What can you conclude?

Sol: (i) Exponential form of log22 = 1 is 21 = 2

(ii) Exponential form of log55 = 1 is 51 = 5

As a1 = a, where 'a' is a positive number.

We can conclude that the logarithmic form of a1 = a    i.e. logaa = 1

11. How do you prove that loga1 = 0?

Sol: We know that a0 = 1 for any positive number 'a'.

Writing a0 = 1 in logarithmic form we get loga1 = 0.

14. ''The sum of two irrational numbers need not be irrational". Justify your answer.

Sol: Consider two irrational numbers 2 +  and 2 -

Their sum = 2 +  + 2 -  = 4 Which is a rational number.

∴ It is concluded that the sum of two irrational numbers need not be irrational.

15. "The product of two irrational numbers need not be irrational". Justify.

Sol: Consider two irrational numbers  and

Their product  ×   = ± 9 is a rational number.

∴ The product of two irrational numbers need not be irrational.

16. Write the second law of Logarithms.

Sol: For x, y and a > 0, a ≠ 1

Second law of Logarithms states that

17. What are the uses of Laws of Logarithms?

Sol: Logarithms are used for all sorts of calculations in Engineering, Science, Business, Economics and includes calculating compound interest, exponential growth and decay, pH value in chemistry, measurement of the magnitude of earthquakes etc.

Eight Mark Questions

18. Prove that √3 - √5  is an Irrational number?

rational number and hence √15 is a rational number. This contradicts with the fact that √15 is an irrational number.

∴ Our assumption is false and Hence √3 - √5  is an irrational number.

Try these: F Prove that √2 + 3√5  is an irrational number. F Show that 2√3 - 3√2 is an irrational number.

18. Use division algorithm to show that the square of any positive integer is of the form 4p or 4p + 1.

Sol: Let ‘a’ be any positive integer and b = 4 By division algorithm,

a = 4q + r for some q 0 and 0 r < 4  i.e. The possible remainders are 0, 1, 2 and 3.

We have, a =  4q + r

a2 = (4q + r)2

= 16q2 + r2 + 8qr

= 4 (4q2 + 2qr) + r2

= 4k + r2 [for some integer k = 4q2 + 2qr]

a2= 4k + r2 …. (1)

If  r = 0,  a2 = 4k

If r = 1,  a2 = 4k + 12

= 4k + 1

If r = 2, a2 = 4k + 22

= 4k + 4

= 4(k + 1)

= 4p  [for p = k + 1]

If r = 3,   a2 = 4k +32

= 4k + 9

= 4(k + 2) + 1

= 4p1 + 1 (for some integer p1= k + 2) ∴ The square of any positive integer is of the form 4p or 4p + 1

Try these:

* Use Euclid’s division algorithm to show that the square of any positive integer is of the form 5m, 5m + 1 or 5m + 4.

* Use Euclid’s division lemma to show that the cube of any positive integer is of the form 7k, 7k + 1 or 7K + 6.

19.

x2 + y2 - 14xy = 0

20. If x2 + y2 = 16xy, then prove that 2 log (x + y) = log2 + 2 log3 + logx + logy (4 marks)

Sol: Given x2 + y2 = 16xy

⇒ x2 + y2 + 2xy = 16xy + 2xy

⇒ (x + y)2 = 18xy

⇒ log (x + y)2 = log18xy

2 log (x + y) = log18 + logx + logy

[ log axn = n logax]

[logaxy = logax + logay]

⇒ 2 log (x + y) = log2 + log32 + logx + logy

= log2 + 2log3 + logx + logy

Try this:

* If x2 + y2 =10xy then prove that 2 log(x - y) = 3 log2 + logx + logy

21. Find the L.C.M. and H.C.F of 8, 15, 72 by the prime factorization method. (2 marks)

Sol: We have, 8 = 2 × 2 × 2 = 23

15 = 3 × 5 72 = 2 × 2 × 2 × 3 × 3

= 23 × 32

L.C.M. (8, 15, 72) = 23 × 32 × 5 = 360

H.C.F. (8, 15, 72) = 1

Try This: F Find the L.C.M and H.C.F. of 21, 27, 54 by prime factorization method

22. Express 5025 a product of prime factors. (2 marks)

Try this:

* Express a) 1024     b) 3685 as the product of prime factors

Posted Date : 30-10-2021

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