Exercise 15.1

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

i. There are 27 days in a month._ (i) , a month at least has 28 days usually either 30 or 31 days.

ii. Makarasankranthi falls on a Friday._ {.reveal(when="blank-0")}(ii) , In a given year Makarasankranti may or may not fall on Friday.

iii. The temperature in Hyderabad is 2°C._ {.reveal(when="blank-1")}(iii) , Sometimes in winter there can be temperature at 2°C but not always.

iv. The earth is the only planet where life exists._ {.reveal(when="blank-2")}(iv) , As of now there exists no other planet except for earth which has life.

v. Dogs can fly._ {.reveal(when="blank-3")}(v) , Dogs cannot fly.

vi. February has only 28 days._ {.reveal(when="blank-4")}(vi) , In a leap year February has 29 days usually has 28 days.

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°. (i) , sum of interior angles of quadrilateral is always 360°.

ii. For any real number x, x² ≥ 0._ {.reveal(when="blank-0")}(ii) , when any real number is squared, the result is always greater than or equal to zero.

iii. A rhombus is a parallelogram._ {.reveal(when="blank-1")}(iii) , a rhombus has opposite sides parallel to each other, which is a defining property of parallelograms.

iv. The sum of two even numbers is even._ {.reveal(when="blank-2")}(iv) , if you add any two even numbers, the result will always be even.

v. Square numbers can be written as the sum of two odd numbers._ {.reveal(when="blank-3")}(v) , not all square numbers can be written as the sum of two odd numbers [For example: 9 cannot be written as sum of two odd numbers]

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

i. All numbers can be represented as the product of prime factors._ (i) Only numbers greater than can be represented as products of their prime factors.

ii. Two times a real number is always even._ {.reveal(when="blank-0")}(ii) Two times an integer is always

iii. For any x, 3x + 1 > 4._ {.reveal(when="blank-1")}(iii) For x > , 3x + 1 > 4 [∵ When x = 0, 3(0) + 1 = 1 < 4; When x = 1, 3(1) + 1 = 4]

iv. For any x, x³ ≥ 0._ {.reveal(when="blank-2")}(iv) For x ≥ , x³ ≥ 0 [∵ Cube of a non-negative number is non-negative]

v. In every triangle, a median is also an angle bisector._ {.reveal(when="blank-3")}(v) Only in an triangle, the median is also an angle bisector [∵ Due to equal sides and angles]

4. Disprove, by finding a suitable counter example, the statement x² > y² for all x > y.

Sol:

Given, Statement to disprove: x² > y² for all x > y

Let's find a counter example:

Take x = and y =

Here x > y is satisfied because -2 >

Now, let's calculate squares:

x² = (-2)² =

y² = (-4)² =

Therefore, x² < y² because 4 <

∴ We have found a counter example where:

x > y is true (-2 > -4)

But x² > y² is false (4 < 16)

Hence, the statement "x² > y² for all x > y" is [∵ We found a case where x > y but x² < y²]