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NCERT Class 9 Maths Solutions Chapter - 1 Number Systems, Ex 1.5

Ex - 1.5

Question 1.  Classify the following numbers as rational or irrational:



Solution

  (i) 
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
 (ii) 
It can be represented in  form so it is a rational number.
(iii)  
As it can be represented in  form, so it is a rational number.
(iv)  
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
(v)  
As decimal expansion is non terminating non recurring, so it is an irrational number.
Concept Insight: Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating.  Remember Sum/difference/Product of a rational and irrational number may or may not be  irrational.

Question 2.  Simplify each of the following expressions:

Solution

Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2,(a+b)(a-b) etc to simplify the given expressions.
Equivalent Identities used here are

Question 3.  Recall,  is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is,  . This seems to contradict the fact that  is irrational. How will you resolve this contradiction?

Solution

There is no contradiction. Since    here circumference or diameter are not given to be integers . When we measure a length with scale or any other instrument, we only get an approximate rational value. We never get an exact value. c or d  may be irrational. So, the fraction   is irrational. Therefore,  is irrational.

Concept Insight: A rational number is the number of the form   where p and q are
integers. In    c and d are not integers. Also remember that no measurement is exact.

Question 4.  Represent   on the number line.

Solution

(i)    Mark a line segment OB = 9.3 on number line.
(ii)    Take BC of 1 unit.
(iii)    Find mid point D of OC and draw a semicircle on OC while taking D as its centre.
(iv)     Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E. Length of perpendicular BE =  .
(v)    Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is  i.e point F represents   on number line
Verification: In EDB
ED2=EB2+DB2      Using Pythagoras theorem
Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.
The key idea to represent  is to create a length of   units.
In ODB
DB = 

Question 5.  Rationalise the denominators of the following:

Solution

Concept Insight: Rationalisation of denominator means converting the irrational  denominator  to rational  i.e .  removing the radical sign from  denominator.A number of the form   can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities



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Ex - 1.5

Question 1.  Classify the following numbers as rational or irrational:



Solution

  (i) 
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
 (ii) 
It can be represented in  form so it is a rational number.
(iii)  
As it can be represented in  form, so it is a rational number.
(iv)  
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
(v)  
As decimal expansion is non terminating non recurring, so it is an irrational number.
Concept Insight: Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating.  Remember Sum/difference/Product of a rational and irrational number may or may not be  irrational.

Question 2.  Simplify each of the following expressions:

Solution

Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2,(a+b)(a-b) etc to simplify the given expressions.
Equivalent Identities used here are

Question 3.  Recall,  is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is,  . This seems to contradict the fact that  is irrational. How will you resolve this contradiction?

Solution

There is no contradiction. Since    here circumference or diameter are not given to be integers . When we measure a length with scale or any other instrument, we only get an approximate rational value. We never get an exact value. c or d  may be irrational. So, the fraction   is irrational. Therefore,  is irrational.

Concept Insight: A rational number is the number of the form   where p and q are
integers. In    c and d are not integers. Also remember that no measurement is exact.

Question 4.  Represent   on the number line.

Solution

(i)    Mark a line segment OB = 9.3 on number line.
(ii)    Take BC of 1 unit.
(iii)    Find mid point D of OC and draw a semicircle on OC while taking D as its centre.
(iv)     Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E. Length of perpendicular BE =  .
(v)    Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is  i.e point F represents   on number line
Verification: In EDB
ED2=EB2+DB2      Using Pythagoras theorem
Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.
The key idea to represent  is to create a length of   units.
In ODB
DB = 

Question 5.  Rationalise the denominators of the following:

Solution

Concept Insight: Rationalisation of denominator means converting the irrational  denominator  to rational  i.e .  removing the radical sign from  denominator.A number of the form   can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities



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