Exercise 15.4

1. State which of the following are mathematical statements and which are not? Give reason.

Sol:

(i) , this is not a mathematical statement [∵ Humans can have eyes of different colors, not just blue] i. She has blue eyes

(ii) , this is a mathematical statement {.reveal(when="blank-0")}∵ x + 7 = 18 {.reveal(when="blank-0")}⇒ x = 18 - 7 {.reveal(when="blank-0")}⇒ x = ii. x + 7 = 18

(iii) , this is not a mathematical statement [∵ Today could be any day of the week, not just Sunday] iii. Today is not Sunday.

(iv) , this is a mathematical statement [∵ It is true for all counting numbers] iv. For each counting number x, x + 0 = x

(v) , this is not a mathematical statement [∵ It's a question about time which can vary] v. What time is it?

2. Find counter examples to prove the following statements:

Sol:

i. Every rectangle is a square. (i) , all sides of square are equal but opposite sides of rectangle are equal

ii. For any integers x and y, x2+y2 = x + y

(ii) , for x = 3 and y = 4:

32+42 = + = = 5

But 3 + 4 = 7 ≠ 5

iii. If n is a whole number then 2n2 + 11 is a prime.

(iii) , for n = 11:

2112 + 11 = 2(121) + 11 = 242 + 11 = 253

253 is not a prime number [∵ 253 = 11 × 23]

iv. Two triangles are congruent if all their corresponding angles are equal.

(iv) , two triangles with equal corresponding angles are similar but not necessarily congruent

v. A quadrilateral with all sides are equal is a square.

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

3. Prove that the sum of two odd numbers is even.

Sol:

Let 2k be an number {.reveal(when="blank-0")}∴ 2k+1 is an odd number, where k is an

Let 2a+1 and 2b+1 be two odd numbers

where a,b are

∴ 2a+1+2b+1 = 2a+2b+2

= 2a+b+1

Now, a+b+1 is an integer since sum of two integers and a constant gives an

Let a+b+1 =

∴ 2a+1+2b+1 = k

Since 2k is an even number, the sum of two odd numbers is

4. Prove that the product of two even numbers is an even number.

Sol:

Let 2k be an number

where k is an

Let 2a and 2b be two even numbers

where a,b are

Now, 2a×2b = 4ab

= 4a×b

= 2×2a×b

Since multiplication of two integers is an integer, we can write:

2a×2b = 2×2k

where 2k is an number

Now, twice of any even number is bound to be

Hence, the product of two even numbers is

5. Prove that if x is odd, then x² is also odd.

Sol:

Let 2k be an number

∴ 2k+1 is an odd number where k is any

Let 2a+1 be an odd number

∴ 2a+12 = 4a2+4a+1

= 4a×a+4a+1

Now, product of two integers is always an

∴ 2a+12 = 4a+4a+1

= 8a+1

= 4×2a+1

= 4×2k+1 [∵ 2a=2k]

Since a,k are both integers

We know that when an even number is multiplied with any number, the result is

∴ 4×2k is an even number

∴ 2a+12 = 4×2k+1

= 2k+1 [∵ 2k is an even number]

∴ 2k+1 is an odd number

Hence, square of any odd number x is

6. Examine why they work?

Sol:

i. Choose a number. Double it. Add nine. Add your original number. Divide by three. Add four. Subtract your original number. Your result is seven.

Let the chosen number be x

Double it: x

Add nine: 2x+9

Add original number: 2x+9+x = x + 9`

Divide by three: 3x+93 = `x +

Add four: x+3+4 = x +

Subtract original number: `x + 7 - x =

Hence, regardless of the number chosen, the final result is always

ii. Write down any three-digit number (for example, 425). Make a six-digit number by repeating these digits in the same order (425425). Your new number is divisible by 7, 11, and 13.

Let abc be a three-digit number

∴ The six-digit number becomes abcabc

Now, 7×11×13 =

∴ abc×1001 = abcabc

Hence, the six-digit number abcabc is divisible by , , and