AP State Syllabus AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.4 Textbook Questions and Answers.

## AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.4

Question 1.

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

i) She has blue eyes.

Solution:

This is not a mathematical statement. no mathematics is involved in it.

ii) x + 7 = 18

Solution:

This is not a statement, as its truthness cant be determined.

iii) Today is not Sunday.

Solution:

This is not a statement. This is an am biguous open sentence.

iv) For each counting number x, x + 0 = x.

Solution:

This is a mathematical statement.

v) What tune is it?

Solution:

This is not a riathematical statement.

Question 2.

Find counter examples to disprove the following statements.

i) Every rectangle is a square.

Solution:

A rectangle and square are equiangular i.e., all the four angles are right angles. This doesn’t mean that they have equal sides.

ii) For any integers x and y,

\(\sqrt{x^{2}+y^{2}}\) = x + y

Solution:

Let x = 3; y = 8

\(\sqrt{x^{2}+y^{2}}=\sqrt{3^{2}+8^{2}}\)

= \(\sqrt{9+64}=\sqrt{73}\)

x + y = 3 + 8 = 11

Here, √73 ≠ 11

i.e., \(\sqrt{x^{2}+y^{2}}\) ≠ x + y

iii) If n is a whole number then 2n^{2} +11 is a prime.

Solution:

If n = 11 then 2n^{2}+ 11 = 2 (11)^{2} + 11

= 11 (2 × 11 + 1) = 11 × (22 + 1)

= 11 × 23 is not a prime.

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

Solution:

If the corresponding angles are equal then the triangles are only similar.

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

Solution:

A rhombus is not a square, but all its sides are equal.

Question 3.

Prove that the sum of two odd numbers is even.

Solution:

Steps | Reasons |

1) (2m + 1); (2n + 1) be the two odd numbers | General form of an odd number. |

2) (2m + 1) + (2n + 1) = (2m + 2n + 2) = 2 (m + n + 1) = 2K Hence proved. |
Adding the two numbers General form of an even number. |

Question 4.

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

Solution:

Steps | Reasons |

1) Let 2m and 2n be two even numbers. | General form of an even number. |

2) 2m.2n = 4mn = 2(2mn) = 2K | Taking the product Rearranging the numbers. |

3) 2K where K = 2mn | K=2mn |

4) Even number Hence proved. |
General form of an even number. |

Question 5.

Prove that if x is odd, then x^{2} is also odd.

Solution:

Let x be an odd number.

Then x = 2m + 1

(general form of ah odd number) Squaring on both sides,

x^{2} = (2m + 1)^{2}

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

= 2 (2m^{2} + 2m) + 1

= 2K + 1 where K = 2m^{2}+ 2

Hence x^{2} is also odd.

Question 6.

Examine why they work ?

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

Solution:

Choose a number = x say

Double it = 2x

Add nine = 2x + 9

Add your original number

= 2x + 9 + x = 3x + 9

Divide by 3 = (3x + 9) ÷ 3

= \(\frac{3 x}{3}+\frac{9}{3}\) = x + 3

Add 4 ⇒ x + 3 + 4 = x + 7

Subtract your original number =

x + 7 – x = 7

Your result is 7 – True.

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.

Solution:

Let a three digit number be xyz.

Repeat the digit = xyzxyz

= xyz × (1001)

= xyz × (7 × 11 × 13)

Hence the given conjecture is true.