**Telangana**** SCERT Solution Class IX (9) Math Chapter 15 Proofs in Mathematics Exercise 15.4**

**1.)**

(i) No, humans have more that only blue color of eyes

(ii) x+7 = 18, Yes

X = 18 -7

=11

X can only be 11 in this case.

(iii) No, today might be Sunday or 6 other days of the week.

(iv) Yes, it is true for all numbers.

(v) No, It be any time of the day or night.

**2.)**

(i) False, all side of square are equal but opposite sides of rectangular are equal.

(iii) For, n= 11, 2n^{2} +11= 253 which is not a prime number.

(iv) True.

(v) False , A rhombus has equal side but it is not a square.

**3.)**

Let,

2k be an even number

∴ 2k +1 is an odd number, where k is an integer.

Now,

Let,

2a +1 & 2b +1 are two odd numbers

a, b are integers

∴ (2a + 1) +( 2b +1) = 2a +2b +2

= 2 (a+b+1).

Now,

a+b+1 is in itself an integer since sum of two integers and constant gives an integers.

: (a+b+1) = k

: (2a+1) + (2b +1) = 2k

Now,

2k is an even number hence proved as shown above.

**4.)**

Let,

2k be an even number.

When k is an integer.

Let,

2a , 2b, be two even number

Where a, b an integers.

Now,

2a * 2b = 4 ab

= 4 (a*b)

=2* 2(a*b)

Now,

Multiplication of two integer is where one integer so we can write,

2a * 2b = 2 * 2k

When 2k is even number.

Now, twice of any even number is bound to be even.

Hence,

Product of two even numbers is even.

**5.)**

Let,

2k be an even number.

2k +1 be an odd number where k is any integer.

Let,

2a +1 be an odd numbers.

∴ (2a +1)^{2} = 4a^{2} + 2 * 2a * 1 + 1

= 4a^{2} +4a + 1

= 4a x a + 4a +1

∴ Now,

Product of two integer is always an integer.

∴ ( 2a +1)^{2} = 4a + 4a +1

= 8a +1

= 4 * 2a +1

= 4 * 2k +1

Here 2a = 2k

Sine a, k are both integers.

We know,

When an even number is multiplied with any number the result is even.

∴ 4 * 2k is an even number.

∴ ( 2a +1)^{2} = 4 *2k +1

= 2k +1 [∵ 2k is an even number]

∴ 2k +1 is an odd number.

∴ Square of any odd No. x is odd proved.

(ii)

Now,

7 x 11 x 13 = 1001

abc be a 3 digit number

∴ abc x 1001 = abc abc

∴ The six digit number abc abc is divisible by 7, 11, 213.