9th Maths 1.2

NCERT Class 9th solution of Exercise 1.1

NCERT Class 9th solution of Exercise 1.3

NCERT Class 9th solution of Exercise 1.4

NCERT Class 9th solution of Exercise 1.5

NCERT Class 9th solution of Exercise 1.6

NCERT Class 9th Maths Projects

Exercise 1.2

Q1. State whether the following statements are true or false. Justify your answer.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form √m, where m is a natural number.

(iii) Every real number is an irrational number.

Answer :

(i) True, since the collection of real numbers is made up of rational and irrational numbers.

(ii) False, no negative number can be the square root of any natural number.

(iii) False, for example, 2 is a rational number.

Q2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Answer :

No. For example, `sqrt4` = 2 is a rational number. 

Q3. Show how `sqrt5` can be represented on the number line.

Sol. :

In ◺OAB

By Pythagoras theorem

`(OB)^2`=`(OA)^2`+ `(AB)^2`

`(OB)^2`=`(1)^2`+ `(1)^2`

`(OB)^2`=`1`+ `1`

`(OB)^2`=`2`

`(OB)`=`sqrt2`

In ◺OBC

By Pythagoras theorem

`(OC)^2`=`(OB)^2`+ `(BC)^2`

`(OC)^2`=`(sqrt2)^2`+ `(1)^2`

`(OC)^2`=`2`+ `1`

`(OC)^2`=`3`

`(OC)`=`sqrt3`

In ◺OCD

By Pythagoras theorem

`(OD)^2`=`(OC)^2`+ `(CD)^2`

`(OD)^2`=`(sqrt3)^2`+ `(1)^2`

`(OD)^2`=`3`+ `1`

`(OD)^2`=`4`

`(OD)`=`sqrt4`=`2`

In ◺ODE

By Pythagoras theorem

`(OE)^2`=`(OD)^2`+ `(DE)^2`

`(OE)^2`=`(2)^2`+ `(1)^2`

`(OE)^2`=`4`+ `1`

`(OE)^2`=`5`

`(OE)`=`sqrt5`

Take center O, draw an arc with radius OE it intersects the number line at a point. 

Q4. Classroom activity 

(Constructing the 'square root spiral').

Answer :

Constructing Square root spiral

Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment OP₁ of unit length. Draw a line segment P₁P₂ perpendicular to OP₁ of unit length. Now draw a line segment P₂P₃ perpendicular to OP₂. Then draw a line segment P₃P₄ perpendicular to OP₃. Continuing in this manner, you can get the line segment Pn-₁Pn by drawing a line segment of unit length perpendicular to OPn-₁. In this manner, you will have created the points P₂, P₃,....Pn...., and joined them to create a beautiful spiral depicting `sqrt2`, `sqrt3`, `sqrt4`, .....