NCERT Solutions For Class 12 Maths Chapter 7 Integrals can be checked from here. The NCERT maths book of class 12 includes the concept of integrals in chapter 7. In this chapter students learn about integral calculus (definite and indefinite), its formulas and their properties. This topics is extremely important for both CBSE board exam and for competitive exams like JEE and NEET.
NCERT class 12 maths solutions for chapter 7 integrals given here are detailed and easy to understand. These simple class 12 maths NCERT solutions for integrals are very simple and can help the students to understand them very easily. The NCERT Solutions For Class 12 Maths Chapter 7 Integrals PDF is also available for download.
The NCERT solutions is one of the best tool to prepare physics for class 12 Class 12 Maths Integrals Rational Numbers is a crucial chapter in CBSE class 12. Students must prepare this chapter well to score well in their examination. The NCERT Solutions Class 12 Maths Integrals Rational Numbers is given below so that students can understand the concepts of this chapter in depth.
NCERT Solutions For Class 6 Maths Chapter 8 Exercises
- NCERT Solutions For Class 6 Maths Chapter 8 Decimals Exercise 8.1
- NCERT Solutions For Class 6 Maths Chapter 8 Decimals Exercise 8.2
- NCERT Solutions For Class 6 Maths Chapter 8 Decimals Exercise 8.3
- NCERT Solutions For Class 6 Maths Chapter 8 Decimals Exercise 8.4
Exercise 8.1
1. Using the given diagram write the data in form of numbers in the table given below:
Hundreds (100) | Tens
(10) |
Ones
(1) |
Tenths
(\(\frac{1}{10}\)) |
Answer:
Hundreds (100) | Tens
(10) |
Ones
(1) |
Tenths
(\(\frac{1}{10}\)) |
Number |
0 | 3 | 1 | 2 | 31.2 |
1 | 1 | 0 | 4 | 110.4 |
2. Write the given decimals in the place value table.
(a) 20.4
(b) 0.4
(c) 18.5
(d) 302.1
Answer:
Hundreds (100) | Tens
(10) |
Ones
(1) |
Tenths
(\(\frac{1}{10}\)) |
Number | |
(a) | 0 | 2 | 0 | 4 | 20.4 |
(b) | 0 | 0 | 0 | 4 | 0.4 |
(c) | 0 | 1 | 8 | 5 | 18.5 |
(d) | 3 | 0 | 2 | 1 | 302.1 |
3. Write the given values in the form of decimals:
(a) five-tenths
(b) Four tens and seven-tenths
(c) Sixteen point three
(d) Six hundred and eight-ones
(e) Nine hundred point one
Answer:
(a) five-tenths = 5 tenths
= \(\frac{5}{10}\)
= 0.5
(b) Four tens and seven-tenths = 4 tens and 7 tenths
= \((4 \times 10) + \frac{7}{10}\)
= 40 + 0.7
= 40.7
(c) Sixteen point three = 16.3
(d) Six hundred and eight-ones = 6 hundreds + 8 ones
= 600 + 8
= 608
(e) Nine hundred point one = 900.1
4. Write the fractions given below in the form of decimals:
(a) \(\frac{5}{10}\)
(b) \(3 + \frac{7}{10}\)
(c) \(200 + 60 + 5 + \frac{1}{10}\)
(d) \(70 + \frac{8}{10}\)
(e) \(\frac{88}{10}\)
(f) \(4\frac{2}{10}\)
(g) \(\frac{3}{2}\)
(h) \(\frac{2}{5}\)
(i) \(\frac{12}{5}\)
(j) \(3\frac{3}{5}\)
(k) \(4\frac{1}{2}\)
Answer:
(a) \(\frac{5}{10}\) = 0.5
(b) \(3 + \frac{7}{10}\) = 3.7
(c) \(200 + 60 + 5 + \frac{1}{10}\) = 200 + 60 + 5 + 0.1
= 265.1
(d) \(70 + \frac{8}{10}\) = 70 + 0.8
= 70.8
(e) \(\frac{88}{10}\) = \(\frac{80 + 8}{10}\)
= \(\frac{80}{10}\) + \(\frac{8}{10}\)
= 8 + 0.8
= 8.8
(f) \(4\frac{2}{10}\) = 4 + \(\frac{2}{10}\)
= 4 + 0.2
= 4.2
(g) \(\frac{3}{2}\) = \(\frac{3 \times 5}{2 \times 5}\)
= \(\frac{15}{10}\)
= \(\frac{10 + 5}{10}\)
= \(\frac{10}{10} + \frac{5}{10}\)
= 1 + 0.5
= 1.5
(h) \(\frac{2}{5}\) = \(\frac{2 \times 2}{5 \times 2}\)
= \(\frac{4}{10}\)
= 0.4
(i) \(\frac{12}{5}\) = \(\frac{12 \times 2}{5 \times 2}\)
= \(\frac{24}{10}\)
= \(\frac{20 + 4}{10}\)
= \(\frac{20}{10} + \frac{4}{10}\)
= 2 + 0.4
= 2.4
(j) \(3\frac{3}{5}\) = 3 + \(\frac{3}{5}\)
= 3 + \(\frac{3 \times 2}{5 \times 2}\)
= 3 + \(\frac{6}{10}\)
= 3 + 0.6
= 3.6
(k) \(4\frac{1}{2}\) = 4 + \(\frac{1}{2}\)
= 4 + \(\frac{1 \times 5}{2 \times 5}\)
= 4 + \(\frac{5}{10}\)
= 4 + 0.5
= 4.5
5. Write the given decimals in the form of fraction. Reduce them to the lowest terms:
(a) 0.6
(b) 2.5
(c) 1.0
(d) 3.8
(e) 13.7
(f) 21.2
(g) 6.4
Answer:
(a) 0.6 = \(\frac{6}{10}\)
= \(\frac{3}{5}\)
(b) 2.5 = \(\frac{25}{10}\)
= \(\frac{5}{2}\)
(c) 1.0 = \(\frac{10}{10}\)
= 1
(d) 3.8 = \(\frac{38}{10}\)
= \(\frac{19}{5}\)
(e) 13.7 = \(\frac{137}{10}\)
(f) 21.2 = \(\frac{212}{10}\)
= \(\frac{106}{5}\)
(g) 6.4 = \(\frac{64}{10}\)
= \(\frac{32}{5}\)
6. Express the following as cm using decimals:
(a) 30 mm
(b) 2 mm
(c) 116 mm
(d) 4cm 2mm
(e) 162 mm
(f) 83 mm
Answer:
(a) Since, 10mm = 1cm
Therefore, 1mm = \(\frac{1}{10}\) cm
Therefore, 30mm = \(\frac{1}{10}\) x 30 = 3.0cm
(b) Since, 10 mm =1cm
Therefore, 1mm = \(\frac{1}{10}\) cm
Therefore, 2mm = \(\frac{1}{10}\) x 2 = 0.2 cm
(c) Therefore, 116 mm = \(\frac{1}{10}\) x 116 = 11.6 cm
(d) Since, 10mm = 1cm
\(4cm+\frac{2}{10}cm\)4 + 0.2 = 4.2 cm
(e) 1mm = \(\frac{1}{10}\) cm
162 mm = \(\frac{1}{10}\times 162 =16.2cm\)
(f) Since, 10 mm = 1cm
Therefore, 1mm = \(\frac{1}{10}\) cm
Therefore, 83 mm = \(\frac{1}{10}\times 83 =8.3cm\)
7. Between which two whole numbers do the given numbers lie? Then find the closest whole number to the given numbers.
(a) 5.1
(b) 0.8
(c) 6.4
(d) 2.6
(e) 4.9
(f) 9.1
Answer:
(a) From 5 to 6, 5.1 is nearest to 5.
(b) From 0 to 1, 0.8 is nearest to 1.
(c) From 6 to 7, 6.4 is nearest to 6.
(d) From 2 to 3, 2.6 is nearest to 3.
(e) From 4 to 5, 4.9 is nearest to 5.
(f) From 9 to 10, 9.1 is nearest to 9
8. Show the given numbers on number line:
(a) 0.2
(b)1.9
(c) 1.1
(d) 2.5
Answer:
9. From the given number line, write down decimal number represented by the points P, Q, R and S.
Answer:
P = 0 + \(\frac{8}{10}\)
= 0.8
Q = 1 + \(\frac{3}{10}\)
= 1 + 0.3
= 1.3
R = 2 + \(\frac{2}{10}\)
= 2 + 0.2
= 2.2
S = 2 + \(\frac{9}{10}\)
= 2 + 0.9
= 2.9
10. (a) The length of Mahesh’s textbook is 10 cm and 6 mm. What will be the length of textbook in cm?
(b) The length of a rose plant is 75 mm. Write the length in cm form.
Answer:
(a) 10 cm 6 mm = 9 cm + 6 mm
= 9 + \(\frac{6}{10}\)
= 9.6 cm
(b) 75 mm = \(\frac{75}{10}\) cm
= 7.5 cm
Exercise 8.2
1. Write numbers according to the information given in the form of table.
Ones | Tenths | Hundredths | Number | |
( a ) | – | – | – | – |
( b ) | – | – | – | – |
( c ) | – | – | – | – |
Answer
Ones | Tenths | Hundredths | Number | |
( a ) | 0 | 5 | 6 | 0.56 |
( b ) | 1 | 0 | 8 | 1.08 |
( c ) | 1 | 2 | 7 | 1.27 |
2. Write the numbers from the table given below showing place value of numbers.
Hundreds (100) | Tens (10) | Ones (1) | Tenths (\(\frac{1}{10}\)) | Hundredths (\(\frac{1}{100}\)) | Thousandths (\(\frac{1}{1000}\)) | |
(a) | 0 | 0 | 2 | 5 | 7 | 0 |
(b) | 1 | 0 | 6 | 4 | 9 | 1 |
(c) | 0 | 1 | 8 | 9 | 2 | 0 |
(d) | 2 | 1 | 3 | 5 | 0 | 3 |
(e) | 0 | 2 | 3 | 0 | 1 | 5 |
Answer
(a) \(2 + \frac{5}{10} + \frac{7}{100} = 2 + 0.5 + 0.07 = 2.57\)
(b) \(100 + 6 + \frac{4}{10} + \frac{9}{100} + \frac{1}{1000} = 100 + 6 + 0.4 + 0.09 + 0.001 = 106.491\)
(c) \(10 + 8 + \frac{9}{10} + \frac{2}{100} = 10 + 8 + 0.9 + 0.02 = 18.92\)
(d) \(200 + 10 + 3 + \frac{5}{10} + \frac{3}{1000} = 200 + 10 + 3 + 0.5 + 0.003 = 213.503\)
(e) \(20 + 3 + \frac{1}{100} + \frac{5}{1000} = 20 + 3 + 0.01 + 0.005 = 23.015\)
3. Write the decimal values given below in place value table.
(a) 0.46
(b) 1.73
(c) 3.08
(d) 12.69
(e) 452.103
Answer
(a) \(0.46 = 0.4 + 0.06 = \frac{4}{10} + \frac{6}{100}\)
(b) \(1.73 = 1 + 0.7 + 0.03 = 1 + \frac{7}{10} + \frac{3}{100}\)
(c) \(3.08 = 3 + 0.08 = 3 + \frac{8}{100}\)
(d) \(12.69 = 10 + 2 + 0.6 + 0.09 = 12 + \frac{6}{10} + \frac{9}{100}\)
(e) \(452.103 = 400 + 50 + 2 + 0.1 + 0.003 = 452 + \frac{1}{10} + \frac{3}{1000}\)
Hundreds (100) | Tens (10) | Ones (1) | Tenths (\(\frac{1}{10}\)) | Hundredths (\(\frac{1}{100}\)) | Thousandths (\(\frac{1}{1000}\)) | |
(a) | 0 | 0 | 0 | 4 | 6 | 0 |
(b) | 0 | 0 | 1 | 7 | 3 | 0 |
(c) | 0 | 0 | 3 | 0 | 8 | 0 |
(d) | 0 | 1 | 2 | 6 | 9 | 0 |
(e) | 4 | 5 | 2 | 1 | 0 | 3 |
4. Write the following in the form of decimals.
(a) \(30 + 8 + \frac{5}{10} + \frac{2}{100}\)
(b) \(200 + 50 + 3 + \frac{4}{100}\)
(c) \(\frac{8}{10} + \frac{2}{100} + \frac{7}{1000}\)
(d) \(25 + \frac{6}{10} + \frac{9}{1000}\)
(e) \(600 + 37 + \frac{7}{100}\)
(f) \(1 + \frac{2}{1000}\)
Answer
(a) \(30 + 8 + \frac{5}{10} + \frac{2}{100}\) = 30 + 8 + 0.5 + 0.02 = 38.52
(b) \(200 + 50 + 3 + \frac{4}{100}\) = 200 + 50 + 3 + 0.04 = 253.04
(c) \(\frac{8}{10} + \frac{2}{100} + \frac{7}{1000}\) = 0.8 + 0.02 + 0.007 = 0.827
(d) \(25 + \frac{6}{10} + \frac{9}{1000}\) = 25 + 0.6 + 0.009 = 25.609
(e) \(600 + 37 + \frac{7}{100}\) = 600 + 37 + 0.07 = 637.07
(f) \(1 + \frac{2}{1000}\) = 1 + 0.002 = 1.002
5. Write in words the given below decimals.
(a) 0.05
(b) 1.35
(c) 106.54
(d) 12.09
(e) 0.058
(f) 2.007
(g) 102.107
Answer
(a) 0.05 = zero point zero five
(b) 1.35= one point three five
(c) 106.54 = one zero six point five four
(d) 12.09 = one two point three zero nine
(e) 0.058 = zero point zero five eight
(f) 2.007 = two point zero zero seven
(g) 102.107 = one zero two point one zero seven
6. Decimals are given below. Between which numbers in 10^{th} places will the number will lie on number line?
(a) 0.08
(b) 0.23
(c) 0.97
(d) 0.15
(e) 0.46
(f) 0.68
(g) 0.89
Answer
(a) 0.08 – 0 and 0.1
(b) 0.23 – 0.2 and 0.3
(c) 0.97 – 0.9 and 1.0
(d) 0.15 – 0.1 and 0.2
(e) 0.46 – 0.4 and 0.5
(f) 0.68 – 0.6 and 0.7
(g) 0.89 – 0.8 and 0.9
7. Write the following decimals in the form of fractions in the lowest terms.
(a) 0.04
(b) 0.65
(c) 0.50
(d) 0.24
(e) 0.15
(f) 0.115
(g) 0.095
Answer
(a) 0.04 = \(\frac{40}{100} = \frac{4}{10} = \frac{2}{5}\)
(b) 0.65 = \(\frac{65}{100} = \frac{13}{20}\)
(c) 0.50 = \(\frac{50}{100} = \frac{5}{10} = \frac{1}{2}\)
(d) 0.24 = \(\frac{24}{100} = \frac{6}{25}\)
(e) 0.15 = \(\frac{15}{100} = \frac{3}{20}\)
(f) 0.115 = \(\frac{115}{1000} = \frac{23}{200}\)
(g) 0.095 = \(\frac{95}{1000} = \frac{19}{200}\)
Exercise 8.3
Q1.Which number is greater?
- 0.3 or 0.5
- 0.08 or 0.03
- 5 or 0.9
- 0.3 or 0.03
- 2.6 or 2.61
- 0.09 or 0.019
- 1.7 or 1.70
- 5.36 or 5.26
- 2.6 or 2.6000
- 4.89 or 4.9
Answer
1. 0.3 or 0.5
In these numbers, whole part is same but the tenth part of 0.3 is smaller than 0.5.
Therefore, 0.5 > 0.3
2. 0.08 or 0.03
In these numbers the hundredth place of 0.08 is greater than 0.03.
Therefore, 0.08 > 0.03
3. 5 or 0.9
In these numbers, whole part 5 is greater than 0 in 0.9.
Therefore, 5 > 0.9
4. 0.3 or 0.03
In these numbers the tenth place of 0.3 is greater than 0.03.
Therefore, 0.3 > 0.03
5. 2.6 or 2.61
In these numbers the hundredth place of 2.6 is smaller than 2.61.
Therefore, 2.61 > 2.6
6. 0.09 or 0.019
In these numbers, hundredth place of 0.019 is smaller than 0.09.
Therefore, 0.09 > 0.019
7. 1.7 or 1.70
In these numbers, whole part is same. Even tenth place is equal. But in first number there is no hundredth place that means that there is zero. So both the numbers are equal.
Therefore, 1.7 = 1.70
8. 5.26 or 5.36
In these numbers, whole part 5 is same. Now tenth place of 5.26 is smaller than that of 5.36.
Therefore, 5.36 > 5.26
9. 2.6 or 2.6000
In these numbers, whole part 2 is equal. Tenth place is also same. But there are three zeros in case of second number. In first number it is till tenth place but it implies that there are zeros present. So they are equal.
Therefore, 2.6 = 2.6000
10. 4.89 or 4.9
In these numbers, the whole part is same but tenth place of 4.89 is smaller than that of 4.9.
Therefore, 4.9 > 4.89
Exercise 8.4
Q1. Express in the form of rupees using decimal places.
- 6 paise
- 85 paise
- 30 paise
- 60 rupees 40 paise
- 625 paise
- 450 paise
Answer
We know that, 100 paise is equal to 1 rupee.
- 6 paise = \(\frac{6}{100}\) rupees = Re 0.06
- 85 paise = \(\frac{85}{100}\) rupees = Re 0.85
- 30 paise = \(\frac{30}{100}\) rupees = Re 0.30
- 60 rupees 40 paise = \(60 + \frac{40}{100}\) = Re 60.40
- 625 paise = (\(\frac{625}{100}\)) rupees = Re 6.25
- 450 paise = \(\frac{450}{100}\) rupees = Re 4.50
Q2. Express the following in the form of metres using decimal places.
- 16 cm
- 5 cm
- 3 m 25 cm
- 5 m 8 cm
- 625 cm
- 213 cm
Answer
We know that, 100 cm is equal to 1 m.
- 16 cm = \(\frac{16}{100}\) m = 0.16 m
- 5 cm = \(\frac{5}{100}\) m = 0.05 m
- 3 m 25 cm = (\(3 + \frac{25}{100}\)) m = 3.25 m
- 5 m 8 cm = (\(5 + \frac{8}{100}\)) m = 5.08 m
- 625 cm = \(\frac{625}{100}\) m = 6.25 m
- 213 cm = \(\frac{213}{100}\) m = 2.13 m
Q3. Express the following in the form of centimetres using decimal places.
- 6 mm
- 50 mm
- 155 mm
- 10 cm 8 mm
- 98 mm
- 289 mm
Answer
We know that, 10 mm is equal to 1 cm.
- 6 mm = \(\frac{6}{10}\) cm = 0.6 cm
- 50 mm = \(\frac{50}{10}\) cm = 5.0 cm
- 155 mm = \(\frac{155}{10}\) cm = 15.5 cm
- 10 cm 8 mm = \(10 + \frac{8}{10}\) cm = 10.8 cm
- 98 mm = \(\frac{98}{10}\) cm = 9.8 cm
- 289 mm = \(\frac{289}{10}\) cm = 28.9 cm
Q4. Express the following in the form of kilometres using decimal places.
- 9 m
- 78 m
- 825 m
- 9456 m
- 25 km 726 m
Answer
We know that, 1000 m is equal to 1 km.
- 9 m = \(\frac{9}{1000}\) km = 0.009 km
- 78 m = \(\frac{78}{1000}\) km = 0.078 km
- 825 m = \(\frac{825}{1000}\) km = 0.825 km
- 9456 m = \(\frac{9456}{1000}\) km = 9.456 km
- 25 km 726 m = (25 + \(\frac{726}{1000}\)) km = (25 + 0.726) km = 25.726 km
Q5. Express the following in the form of kilograms using decimal places.
- 5 g
- 56 g
- 654 g
- 5452 g
- 15 kg 348 g
- 54681 g
Answer
We know that, 1000 g is equal to 1 kg.
- 5 g = \(\frac{5}{1000}\) kg = 0.005 kg
- 56 g = \(\frac{56}{1000}\) kg = 0.056 kg
- 654 g = \(\frac{654}{1000}\) kg = 0.654 kg
- 5452 g = \(\frac{5452}{1000}\) kg = 5.452 kg
- 15 kg 348 g = (15 + \(\frac{348}{1000}\)) kg = (15 + 0.348) kg = 15.348 kg
- 54681 g = \(\frac{54681}{1000}\) kg = 54.681 kg
CBSE is one of the popular educational board in India. The Central Board of Secondary Education follow the NCERT syllabus to conducts its class 10th and class 12th board examinations. The NCERT Solutions Class 12 Maths Integrals Rational Numbers is given so that students can understand the concepts of this chapter in depth.
Differential Calculus is centred on the concept of the derivative. Some key points to remember from the chapter are given below.
Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation.
From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis
A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution.