Work And Energy

Work And Energy

This video lecture consists of 

1. Definition of Work

2. Dimension of work

3. Unit of work

4. Definition of energy

5. Types of mechanical energy

6. Conservation of energy 

Number Systems

1. Number Systems

Class : IX

Subject: Mathematics

NCERT Text book Solution.

EXERCISE 1.1

1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

Solution:

Yes zero is a rational number.

It can be written in the form p/q which is 0/1.

2. Find six rational numbers between 3 and 4.

Solution:

Both 3 and 4 are multiplied and divided by (6 + 1 = 7)

3 = 3×7/7 = 21/7

4 = 4×7/7 = 28/7

Required rational numbers are

22/7, 23/7, 24/7, 25/7, 26/7 and 27/7 Ans.

3. Find five rational numbers between ⅗ and ⅘.

Solution:

Both ⅗ and ⅘ multiplied and divided by (5+1=6)

⅗ = 18/30

⅘ = 24/30

Required rational numbers are

19/30, 20/30, 21/30, 22/30 and 23/30 Ans.

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

(ii) Every integer is a whole number.

(iii) Every rational number is a whole number.

Solution:

(i)

True, Natural numbers are a part of whole numbers.

(ii)

False, Negative integers are not whole numbers.

(iii)

False, ½ is a rational number but not a whole number.

प्रश्नावली 1.1

1. क्या शून्य एक परिमेय संख्या है? क्या इसे आप p/q के रूप में लिख सकते हैं, जहाँ p और q पूर्णांक हैं और q ≠ 0 है?

हल:

हां, शून्य एक परिमेय संख्या है। इसे p/q के रूप में लिखा जा सकता है जो 0/1 है।

2. 3 और 4 के बीच में छः परिमेय संख्याएँ ज्ञात कीजिए।

हल:

3 और 4 दोनों के अंश एवं हर को 7 से गुना करने पर।

3 = 21/7

4 = 28/7

अतः 6 परिमेय संख्या है

22/7, 23/7, 24/7, 25/7, 26/7 एवं 27/7।

3. ⅗ और ⅘ के बीच पाँच परिमेय संख्याएँ ज्ञात कीजिये ।

हल:

⅗ एवं ⅘ दोनों के अंश एवं हर को 6 से गुना करने पर।

⅗ = 18/30

⅘ = 24/30

अतः 5 परिमेय संख्या है

19/30, 20/30, 21/30, 22/30 एवं 23/30

4. नीचे दिए गए कथन सत्य हैं या असत्य? कारण के साथ अपने उत्तर दीजिए।

(i) प्रत्येक प्राकृत संख्या एक पूर्ण संख्या होती है।

(ii) प्रत्येक पूर्णांक एक पूर्ण संख्या होती है।

(iii) प्रत्येक परिमेय संख्या एक पूर्ण संख्या होती है।

हल:

(i)

सत्य है, क्योंकि पूर्ण संख्या के समूह में प्राकृत संख्या शामिल है।

(ii)

असत्य है, क्योंकि ऋणात्मक पूर्णांक पूर्ण संख्या नही होता है।

(iii)

असत्य है, जैसे ½ परिमेय संख्या है लेकिन पूर्ण संख्या नही है।

EXERCISE 1.2

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

Solution:

True, every irrational numbers are represented on a number line.

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

Solution:

False, every point on the number is of the form of real number.

(iii) Every real number is an irrational number.

Solution:

False, every real number is either rational or irrational.

2. 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.

Solution:

The square root of all positive integers are not irrational.

Example

√4 is a rational number.

3. Show how √5 can be represented on the number line.

Solution:

Following steps are used to show √5 on number line.

Step 1.

Apply Pythagoras theorem to get base for unit perpendicular and for hypotenuse of √5.

b² = (√5)² - 1² = 4 = 2² 

Therefore, 

b = 2.

Step 1.

Draw perpendicular on 2 of unit that is P.

Step 2.

Join O to P.

Step 3.

Draw an arch of radius OP and centre O. The arch intersects the number line at Q. This represent √5 on number line.

Figure is given below.

7. Triangles Class IX Mathematics

Exercise 7.1

Q.No. 1. In a quadrilateral ABCD, AC = AD and AB bisect <A (see Fig. 7.16). Show that △ABC⩭△ABD. What can you say about BC and BD.

Solution:

Consider about the △ABC and △ABD.

AB = AB {Common}

AC = AD {Given}

<CAB = <DAB {AB bisect <A}

△ABC ⩭ △ABD {Under SAS}

BC = BD.

Q.No. 2. ABCD is a quadrilateral in which AD = BC and <DAB = <CBA (see Fig. 7.17). Prove that

(i) △ABD ⩭ △BAC

(ii) BD = AC

(iii) <ABD = <BAC.

Solution:

(i)

Consider in △ABD and △BAC.

AB = AB {Common}

AD = BC {Given}

<DAB = <CBA {Given}

△ABD ⩭ △BAC. {Under SAS}Proved.

(ii)

BD = AC {△ABD ⩭ △BAC} Proved.

(iii)

<ABD = <BAC {△ABD ⩭ △BAC}Proved.

Q.No. 3. AD and BC are equal perpendicular to a line segment AB(see Fig. 7.18). Show that CD bisect AB.

Solution:

Consider about the △BOC and △AOD.

BC = AD {Given}

<BOC = <AOD {Vertically opposite angles}

<OBC = <OAD { Each right angles}

△BOC ⩭ △AOD {Under ASA}

BO = AO

∴ CD bisect AB Proved.

Q.No. 4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that △ ABC ⩭ △CDA.

Solution:
Consider about the △ABC and △CDA.
<BAC = <DCA {Alternate pair}
AC = AC {Common}
<ABC = <ADC {Opposite angles of a parallelogram}
∴△ABC⩭△CDA. Proved.

Q.No. 5. Line l is the bisector of an angle <A and B is any point on l. BP and BQ are perpendiculars from B to the arms of <A (see Fig. 7.20).
Show that:
(i) △APB⩭△AQB
(ii) BP = BQ or B is equidistant from the arms <A.

Solution:
(i)
Consider about the △APB and △AQB.
AB = AB {Common}
<APB = <AQB {Each right angles}
<PAB = <QAB {l is the bisector of <A}
∴△APB⩭△AQB {Under ASA} Proved.

(ii)
BP = BQ {△APB⩭△AQB} Proved.

Continuous ............................