NCERT Solutions Class 9 Mathematics

Exercise 1.5

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

i 2 - 5      ii 3 + 23- 23

iii 2777        iv 12     v 2π

Q.2. Simplify each of the following expressions:

i 3 + 32 + 2  ii 3 + 33 - 3

iii 5 + 22             iv 5 - 25 + 2

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

Q.4. Represent 9.3 on the number line.

Q.5. Rationalise the denominators of the following:

i17          ii17-6

iii15+2         iv17-2

NCERT Solutions Class 9 Mathematics

Exercise 1.5

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

i 2 - 5      ii 3 + 23- 23

iii 2777        iv 12     v 2π

Solution:

(i) 2 - 5 = 2 - 2.2360679… = - 0.2360679…

2 is rational and 5 is irrational.

Since the number (2 - 5) involves subtraction of a irrational number from a rational number, it is an irrational number.

(ii) 3 + 23- 23=3 + 23- 23

= 3 = 31

Since the number can be represented in pq form, it is a rational number.

iii2777=27

Since the number can be represented in pq  form, it is a rational number.

iv 12 = 22= 0.7071067811

Since the number involves, division of an irrational number by a rational number, it is an irrational number.

(v) 2π = 2 × 3.1415… = 6.2830…

π is an irrational number. Since the number involves, multiplication of an irrational number with a rational number, it is an irrational number.

Q.2. Simplify each of the following expressions:

i 3 + 32 + 2  ii 3 + 33 - 3

iii 5 + 22             iv 5 - 25 + 2

Solution:

(i) 3 + 32 + 2

= 3×2 + 3×2 + 3×2+ 3×2

= 6 + 23+32+ 6

(ii) Using (a + b)(a - b) = a2 - b2, we have,

3 + 33 - 3

= 32 - 32

= 9 - 3 = 6

(iii) Using (a + b)2 = a2 + b2 + 2ab, we have,

5 + 22

=52+22 + 2 ×5 × 2

= 5 + 2 + 2 × 5× 2 

= 7 +210

(iv) Using (a + b) (a - b) = a2 - b2, we have,

5- 25+ 2

= 52 - 22 

= 5 - 2 = 3

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

Solution:

There is no contradiction in here. When we measure a value with a scale, we only obtain an approximate value. We never obtain an exact value. Therefore, we may not realise that either c or d is irrational. Hence their ratio will also be irrational.

Q.4. Represent 9.3 on the number line.

Solution:

  1. Draw a line segment of unit 9.3. Extend it to C so that BC = 1 unit.

  2. Now, AC = 10.3 units. Find the midpoint of AC. Let it be O.

  3. Draw a semi circle with radius OA = OC and centre O.

  4. Draw a perpendicular to AC at point B which intersects the semicircle at D. Join OD.

  5. Now, OBD is a right angled triangle. The length of BD is 9.3.

  6. Taking BD as radius and B as centre draw an arc which cuts the line segment AC at E. Now BE is equal to 9.3.

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Q.5. Rationalise the denominators of the following:

i17          ii17-6

iii15+2         iv17-2

Solution:

i 17=17×77=77

ii17-6=17-6×7+67+6

=7+67-6=7+6 

iii15+2=15+2×5-25-2

=5-25-2=5-23 

iv 17-2=17-2×7+27+2

=7+27-4=7+23