AP State Syllabus AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.1 Textbook Questions and Answers.

## AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.1

Question 1.

State whether the following sentences are always true, always false or ambiguous. Justify your answer.

i) There are 27 days in a month.

Solution:

Always false. Generally 30 days or 31 days make a month except February.

ii) Makarasankranthi fells on a Friday.

Solution:

Ambiguous. Makarasankranthi may fall on any day of the week.

iii) The temperature in Hyderabad is 2°C.

Solution:

Ambiguous. Sometimes the temperature may go down to 2°C in winter.

iv) The earth is the only planet where life exist.

Solution:

We can’t say always true. To the known fact, so far we can say this.

v) Dogs can fly.

Solution:

Always false,’as dogs can never fly.

vi) February has only 28 days.

Solution:

Ambiguous. A leap year has 29 days for February.

Question 2.

State whether the following statements are true or false. Give reasons for your answers.

i) The sum of the interior angles of a quadrilateral is 350°.

Solution:

False. Sum of the interior angles of a quadrilateral is 360°.

ii) For any real number x, x^{2} > 0

Solution:

True. This is true for all real numbers,

iii) A rhombus is a parallelogram.

Solution:

True. In a rhombus, both pairs of opposite sides are parallel and hence every rhombus is a parallelogram.

iv) The spm of two even numbers is even.

Solution:

True. This is true for any two even numbers.

v) Square numbers can be written as the sum of two odd numbers.

Solution:

Ambiguous. Since square of an odd number can’t be written as sum of two odd numbers.

Question 3.

Restate the following statements with appropriate conditions, so that they become true statements.

i) All numbers can be represented in prime factorization.

Solution:

Any natural number greater than 1 can be represented in prime factorization.

ii) Two times a real number is always even.

Solution:

Two times a natural number is always even.

iii) For any x, 3x + 1 > 4.

Solution:

For any x > 1; 3x + 1 > 4.

iv) For any x, x^{3} ≥ 0.

Solution:

For any x > 0; x^{3} ≥ 0.

v) In every triangle, a median is also an angle bisector.

Solution:

In an equilateral triangle, a median is also an angle bisector.

Question 4.

Disprove, by finding a suitable counter example, the statement

x^{2} > y^{2} for all x > y.

Solution:

If x = – 8 and y = – 10

Here x > y

x^{2} = (- 8)^{2} = 64 and y^{2} = (- 10)^{2} = 100

But x^{2} > y^{2} is false here. [ ∵ 64 < 100]

(This can be proved for any set of nega-tive numbers or a negative number and a positive number)