St. Xavier’s Sr. Sec. School, Chandigarh

Class 12 English Worksheet-4 (Literature)

Crossing The Bar Tennyson wrote “Crossing the Bar” in 1889, three years before he died. The poem describes his placid and accepting attitude toward death. Tennyson uses the metaphor of a sand bar to describe the barrier between life and death. A sandbar is a ridge of sand built up by currents along a shore. Shortly before he died, Tennyson told his son Hallam to "put 'Crossing the Bar' at the end of all editions of my poems". The extended metaphor of "crossing the bar" represents travelling serenely and securely from life through death. The Pilot is a metaphor for God, whom the speaker hopes to meet face to face. Alfred Lord Tennyson, 1809 - 1892. Sunset and evening star, And one clear call for me! And may there be no moaning of the bar, When I put out to sea, But such a tide as moving seems asleep, Too full for sound and foam, When that which drew from out the boundless deep Turns again home. If the sandbar is Tennyson's metaphor for the boundary between life and death, then "Crossing the Bar" is all about crossing from life to death. The poem is composed of four rhyming stanzas of four lines each - quatrains. Each of those quatrains has alternating rhymes, giving the entire poem a rhyme scheme of ABAB CDCD EFEF. Themes and Meanings

The enduring popularity of this short, meditative lyric lies in its ability to appeal to many people on many levels, despite attempts to limit it to one interpretation. The poem‟s themes of death and dying have made it a popular selection for memorial services over the years, including Tennyson‟s own. The poem, however, has significance for all readers. Daily life, a journey in itself, requires individuals to travel regularly from the safety of home, across a threshold, and into the unknown. Like the world of Tennyson‟s traveller, the world beyond the safe region is dark and mysterious, yet at day‟s end, people return home.

In this way, “Crossing the Bar” draws parallels between familiar and repeated patterns of ordinary, daily routine with nature‟s daily cycles, such as night and day and the flow and ebb of the tide. Similarly, Tennyson includes the “evening star” and the Pilot as reminders of sources that guide individuals. These elements eloquently diminish the horror of death by drawing attention to the fact that the journey into death is merely part of a cycle: The going out is also a return home to “the boundless deep,” from which this traveller, like all people, came.

The themes of sea and death recur frequently in Tennyson‟s life and work. As a child, Tennyson first saw the sea on a family vacation at Mablethorpe on the Lincolnshire coast, a place he revisited often for comfort and solitude when an adult. One...

Crossing the Bar: About the poem Crossing the Bar, an elegy written by the British poet Alfred, Lord Tennyson, is a poem focusing on the transience of life and the finality of death. Lord Tennyson was a poet of the Victorian period and remained the poet laureate of Great Britain and Ireland during his lifetime. He is well celebrated to this day for his short lyrics. „Crossing the bar‟ was written in 1889 when the poet was visiting the Isle of Wight and published in a volume Demeter and Other Poems (1889). He was eighty years old at the time and was down with a severe illness, from which he eventually recovered. The illness, however, made the poet ponder on Death as he himself was very old and nearing his time. He uses the metaphor of crossing a sand bar to represent death in this poem. He died three years later, and although he wrote a few more poems, he requested that all of his poetry volumes be ended with this poem. Thus, the poem is an important one and can be seen as Alfred, Lord Tennyson‟s choice of his final words. Crossing the Bar: Form and Structure The poem consists of four stanzas, and each of them are quatrains. The poet uses the classical rhyme scheme of abab. The structure of the poem is akin to that of a ballad verse but it falls short of the metre. There is no apparent metre to the poem. The length of the lines is a feature of interest in the poem. The poet varies the length of the lines between ten, six, and four syllables per line randomly throughout the poem. The entire poem is connected, both in theme and conceit. The stanzas do not stand individually on their own. They are tightly knit and carry the meaning forward to the next ones. Crossing the Bar: Line by Line Explanation First stanza: “Sunset and evening star, And one clear call for me! The poem begins with the speaker describing the atmosphere. He says it is sunset and the evening star can be seen in the sky. Someone is calling the speaker. It is a clear, unmistakable call. It is the call of death. The speaker believes that his death is close. It is interesting to note here the imagery the poet presents before us at the start of the poem. „Sunset‟ and „Evening star‟ represent the end of the day. Just as the day is about to end, the speaker says that his life is drawing to an end as well. And may there be no moaning of the bar, When I put out to sea, Here the poet uses his famous metaphor of „Crossing the bar‟, describing death as an act of passing beyond life. The word „bar‟ here means a sandbar. A sandbar is a geographical structure which forms around the mouth of a river, or extends from a „Spit‟ by slow deposition of sediments carried by the current over millions of years. The structure forms a kind of barrier between the water inside (the river water) and outside it (the open sea). The poet uses this sandbar as a symbol of death, with the water inside representing his life, and the water beyond representing the afterlife. He wants to „put out to sea‟ without the „moaning of the bar‟. The poet wishes his death to be without pain and without mourning.

Second stanza: But such a tide as moving seems asleep, Too full for sound and foam, Through the poem, the poet Alfred, Lord Tennyson compares his impending death to crossing a bar. In the stanza, the speaker of the poem talks about the inevitability of death. The poet wishes that when he „put(s) out to sea‟, that is when he dies, let it be like a ride which seems asleep as it moves. The speaker wants his death to be smooth. Like a calm sea wave, which is „too full for sound and foam‟ the speaker hopes that his death will be silent, smooth and quick, making no fuss. When that which drew from out the boundless deep Turns again home. In the subsequent lines, the poet uses the example of the river and the sea to express the kind of death he wishes for himself. The water from the sea evaporates and turns into clouds; these clouds bring rain, entering that water into the river, and these rivers too flow, carrying their water and eventually pouring it into the sea. They, thus complete a cycle, and the water returns from where it came. Just so, the speaker, considering himself like the water, says that he is returning where he came from. „The boundless deep‟ here apparently stands for the sea, and in an allegorical sense to the place the poet believes he will go to after his death. Here, we should notice that this stanza is a strict continuation of the idea introduced in the first stanza. The last lines of the first stanza together with this one makes up the meaning of the verse. Third Stanza: Twilight and evening bell, And after that the dark! In the third stanza, the poet again resorts to describing the atmosphere to convey his inner feelings. It was sunset when the speaker started the poem, but now it is twilight. The sun has already gone down the horizon and dusk is settling. The speaker can hear the evening bell tolling. It is the indication that night is approaching. Then after a while it gets dark. It is night. The poet here uses twilight to show us the state of his life. Just as the day has ended, his life too is about to end. Here twilight stands for sadness, darkness and grief portray the speaker‟s miserable state before his death. And may there be no sadness of farewell, When I embark; The speaker expresses his hope that there will be no „sadness of farewell‟ upon his death. The „sadness of farewell‟ is ambiguous and can mean both the speaker‟s own sadness as he departs from life, or the sadness of the people whom he leaves behind and who are saying farewell to him. However, we think, the former is more relevant. Again, Lord Tennyson writes „When I embark‟ to convey the idea of the speaker‟s death. Thus, it is evident from the word ‟embark‟ that death is not seen as a final destination by the poet, but rather as a new beginning.

Fourth stanza: For tho‟ from out our bourne of Time and Place The flood may bear me far, In the previous stanza of the poem, we see the speaker‟s positive attitude towards death. It is seen to be exemplified in this final stanza of the poem. We understand that the speaker has accepted his reality – inevitability of death. He appears to have made his peace with the idea of his fast approaching death. He says that he will be beyond the boundaries of time and place and the flood of death will carry him far away. This is going beyond the reach of this world. The speaker suggests that there is a place beyond our time and space where he hopes to go after his death. We are, thus, acquainted with the poet‟s belief in afterlife. I hope to see my Pilot face to face When I have crost the bar. These final lines of the poem are shrouded in allusions and hidden meanings. Firstly, we are told that the speaker hopes to see his pilot face to face when he will have crossed the bar. Here, the word pilot is a direct reference to God. Lord Tennyson had peculiar views on religion. On one hand, he disapproved of Christianity, while on the other; we see wide use of religious connotations and ideas in his works. Since God is considered to drive the world and all living things, we see the pilot reference of the divine world in the poem. Also, the use of the word „crost‟ is interesting. While it might simply be a word to suggest „Crossing‟ the bar, it is speculated that it might be a reference to Christ, as crost is similar in sound to both Christ and Cross. If so, then we find another allusion from the poet to region and afterlife. The poem thus ends on a positive note with the poet both accepting the finality of death and hoping to meet God in the afterlife. Further Thinking How (if at all) might the following facts affect the way you read the poem? First: it was written three years before Tennyson died, when the poet was 80. Second: the story goes that he wrote it in twenty minutes on the Isle of Wight ferry. Third: he asked future editors to place it last in collections of his work. Crossing the Bar is interpreted as Tennyson’s elegy for himself. Comment. „Crossing the Bar‟ can be analysed as Tennyson‟s elegy, for himself. It describes his anticipation of the „crossing‟ he must make from life to death. The poem focuses on the transience of life and the finality of death. Lord Tennyson was a poet of the Victorian period and remained the poet laureate of Great Britain and Ireland during his lifetime. He was eighty years old at the time he wrote this poem and was down with a severe illness, from which he eventually recovered. The illness, however, made the poet ponder on Death as he was nearing his end. He uses the metaphor of crossing a sand bar to represent death in this poem. He died three years later, and although he wrote a few more poems, he requested that all of his poetry volumes be ended with this poem. Thus, the poem is an important one and can be seen as Alfred, Lord Tennyson‟s choice of his final words.

The imagery the poet presents before us at the start of the poem. „Sunset‟ and „Evening star‟ represent the end of the day. Just as the day is about to end, the speaker says that his life is drawing to an end as well. „Crossing the bar‟, describes death as an act of passing beyond life. The sandbar is used as a symbol of death, with the water inland representing his life, and the water beyond representing the afterlife. He wants to „put out to sea‟ without the „moaning of the bar‟. The poet wishes his death to be without pain and without mourning. The poet wishes that when he „put(s) out to sea‟, that is when he dies, it should be like a ride which seems as though one is asleep as it moves. The speaker wants his death to be smooth. Like a calm sea wave, which is „too full for sound and foam‟ the speaker hopes that his death will be silent, smooth and quick, making no fuss. When that which drew from out the boundless deep Turns again home. As water returns from where it came, the speaker, considers himself like water, and says that he is returning to where he came from. „The boundless deep‟ here apparently stands for the sea, and in an allegorical sense to the place the poet believes he will go to after his death. The poet uses twilight to show us the state of his life. Just as the day has ended, his life too is about to end. Here twilight stands for sadness, darkness and grief portray the speaker‟s miserable state before his death. The speaker expresses the hope that there will be no „sadness of farewell‟ upon his death. This „farewell‟ is ambiguous and can mean both the speaker‟s own sadness as he departs from life or the sadness of the people whom he leaves behind and who are saying farewell to him. Again, Lord Tennyson writes „When I embark‟ to convey the idea of the speaker‟s death. Therefore it is evident from the word ‟embark‟ that death is not seen as a final destination by the poet, but rather as a new beginning. He says that he will be beyond the boundaries of time and place and the flood of death will carry him far away. This is going beyond the reach of this world. The speaker suggests that there is a place beyond our time and space where he hopes to go after his death. We are, thus, acquainted with the poet‟s belief in afterlife. On one hand, he disapproved of Christianity, while on the other; we see wide use of religious ideas in his works. Since God is considered to drive the world and all living things, we see the pilot reference of the divine world in the poem. Also, the use of the word „crost‟ is interesting. While it might simply be a word to suggest „Crossing‟ the bar, it is speculated that it might be a reference to Christ, as crost is similar in sound to both Christ and Cross. The poem thus ends on a positive note with the poet both accepting the finality of death and hoping to meet God in the afterlife.

Class 12 Commerce Worksheet-4

Chapter - 2 (FINANCING)

Business is an economic activity, which is related to continuous and regular production

and distribution of goods and services for satisfying human wants. Finance is the

management of money and other valuables. Finance is the lifeline of any business.

Money required for carrying out business activities is called business finance. Thus

finance is required for acquiring not only fixed assets like plant and machinery, building ,

etc., but also carrying out day to day operations like buying raw material, salaries and

wages etc.

Characteristics of business finance:

It is needed in all types of organisation, i.e. large, medium or small, trading,

manufacturing or service, profit–oriented or non-profit- oriented.

It involves estimation of funds and identification of the sources of funds based on

the estimation.

Business Finance includes all types of funds, i.e., long-term, short-term or for

tangible or intangible assets.

Financing includes raising the funds from different sources in an economical

manner as well as investment of funds for different purposes in a judicious manner

so as to bring down the overall cost of funds.

Importance of Finance for business:

Finance is the lifeline of business. It is required

For establishing an enterprise

For purchase of fixed and current assets for carrying on present operations and ;

For expansion, growth and modernisation of business

Adequate finance provides the following benefits to a business concern:

1. The firm can meet its liabilities in time. Prompt payment of debts in raising the

credit-worthiness and enables the firm to borrow funds as and when required.

2. It can take advantages of business opportunities e.g. it can buy materials in bulk at

a low price.

3. Firm can carry on business smoothly and without any interruptions.

4. Finance can bridge the gap between production and sales of goods and services.

5. A business enterprise with adequate funds only can pay the employees and other

related costs on time. This helps not only to retain and motivate the employees but

also in the smooth functioning of the entire enterprise,

6. Adequate finance ensures the capacity to purchase fixed assets and

implementation of advance technology.

7. Sound financial position enables an enterprise to enjoy goodwill in the business

world.

Sources of finance for different types of business enterprises

The terms “capital “refers to the investments made in an enterprise for the purpose of

earning profits. In business capital is required for both production as well as distribution

of goods and services. Capital requirement differ according to the nature and size of

business. Requirements of capital and sources of capital for different types of business

firm are given below:

1. Sole Proprietorship: A sole proprietor is an individual, who runs a business in his

own name, provides all the funds and bears all risks. The capability of a sole trader

to raise funds is limited as compared to a joint stock company because he operates

on a small scale. The sole trader may utilise his personal capital, retained profits,

etc., for raising funds. He can also take loan from banks and financial institutions.

Sources may include capital contributed by sole proprietor, loan from family

members and friends, loan from banks, financial institutions, etc., ploughing back

of profits.

2. Partnership: Partnership is a relationship between persons who have agreed to

share their profits and losses of a business carried on by all or any of them acting

for all. The capital base of partnership is generally more than that of a sole trader.

Partners introduce capital in agreed ratio as per partnership deed. In case of

partnership, the capability to raise funds from different sources as stated in case of

sole proprietor goes up substantially higher.

3. Joint Stock Company: A joint stock company is voluntary association of persons

to carry on a business having separate legal entity distinct from its shareholders. A

joint stock company generally requires a large amount of capital. A public

company can raise capital by issuing shares. Like proprietorship and partnerships,

companies can borrow money from banks and other lenders. It can also raise

capital by issuing debentures (Loaned capital) or by issuing share (Share capital or

owner`s capital). Thus, a joint stock company can raise very huge funds from

diverse sources.

Financial Planning:

Financial planning is the process of

Estimating the financial requirement of an organisation,

Choosing the sources of funds,

Efficient utilisation of funds.

Features of Financial Planning:

Financial planning involves linkage between organisational objectives and

financial structure.

Like any other planning it is future oriented and involves forecasting, it is dynamic

in nature.

Wider in scope as it involves estimating the financial requirements, selecting the

appropriate sources of funds and their ratio and formulating the policies for us of

funds and disposal of earnings.

Importance of Financial Planning:

Financial planning is the key to success for any business organisation. Financial

planning is almost as important as setting up a business. The importance of financial

planning is discussed below:

(i) Usage of funds: The benefit of financial planning is judiciously utilising all the

funds that are available with the business organisation. Financial plans made

will in advance help in the utilisation of scarce funds as well as in planning the

financials goals of the business organisation.

(ii) Long-term planning: Financial planning works by preparing specific plan for

future. A financial plan is not static but dynamic in nature. So, detailed analysis

is required before setting financial goals which help in growth of business in

the long-run.

(iii) Mitigation of risk: Financial planning helps a business organisation prepare for

risk. There can be contingencies like major unforeseen expenditures, natural

calamities causing major loss etc., but financial plans will help the business

organisation prepare for such contingencies.

(iv) Evaluation of performance: The finance team while planning prepares

numerous reports which help the business organisation to analyse and evaluate

the profits and losses. This analysis helps in cost control and strategic planning

to maximise profits.

(v) Developing Capital Structure: Financial planning helps in developing a sound

capital structure for the organisation. It ensures parity between debt and equity.

Different asset classes have different risk return characteristics. Financial

Planning ensure proper allocation of funds so as to maximise returns.

Assignment:

Short Answer:

(i) Define business finance.

(ii) What is meant by financial planning?

(iii) Give any two characteristics of business finance.

Long Answer:

(i) Explain the nature and importance of business finance.

(ii) Discuss finance for different types of business firms.

(iii) Why is financial planning necessary?

Class 12 Accounts Worksheet – 4

CHAPTER – 1 (Practical Questions) Before you actually start solving practical sums, one more thing need to be understood and that is how to calculate and treat Manager‟s Commission in the accounts of a partnership firm. MANAGER’S COMMISSION ON NET PROFITS : Sometimes the manager is to be allowed a certain percentage of net profits as his commission. Such commission is an item of Profit & Loss A/c and NOT an item of Profit & Loss Appropriation A/c. It is calculated as below:

(1) On profits before charging such commission:- For example if the profits before charging his commission is Rs 22,000 and the manager is to be allowed a commission of 10% on the net profits before charging such commission, the commission will be:

22,000 X 10/100 = Rs 2,200 (2) On profits after charging such commission:- For example, if the profit before

charging his commission is Rs 22,000 and the manager is to be allowed a commission of 10% on the net profit after charging such commission, the commission will be:-

22,000 X 10/110 = Rs 2,000. On the basis of all previous worksheets understand the following illustrations before solving unsolved questions.

On the basis of above worksheet, solve following questions neatly in a notebook.

Following unsolved questions are exactly similar to the above solved questions in

sequence. For example to solve question number 6. refer to illustration number 6.

And you need the knowledge of previous worksheets too.

Class 12 Economics Worksheet-4

TOPIC- BANKS: COMMERCIAL BANKS AND CENTRAL BANKS

Read the text given below and answer the following questions:

Very Short Answer Questions (2 marks)

Question 1. What is a commercial bank or a bank? .

Question 2. Define bank money.

Question 3. Explain discounting bills of exchange as one of the functions of the bank.

Question 4. Distinguish between demand deposits and time deposits

Question 5. What do you mean by overdraft facility ?

Short Answer Questions (3 marks)

Question 1.Explain various types of deposits with the commercial banks.

Question 2Highlight the various agency functions provided by commercial

banks.

Question 3.Explain the ‘acceptance of deposits’ function of a commercial bank .

Long Answer Questions (6 marks)

Question-Explain the various functions performed by commercial banks in India .

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Class XII Maths Worksheet-4

MATRICES AND DETERMINANTS (CONTD…..)

QUICK RECAP:

In the last worksheet we understood Minors and Co-factors and we also learnt how to find the

determinant of a 3x3 matrix.

Let‟s take an example (Q. 41) A = 1 𝑎 𝑏−𝑎 1 𝑐−𝑏 −𝑐 1

To find |A|, let us expand it along Row 1

Therefore |A| = 1 C11 + a C12 + b C13

= 1 1 𝑐−𝑐 1

- a −𝑎 𝑐−𝑏 1

+ b −𝑎 1−𝑏 −𝑐

[Remember sign convention: C11 = +ve, C12= -ve, C13= +ve]

= 1 (1+c2 ) –a(-a+bc) +b(ac +b) = 1 +c2 +a2 – abc + bac + b2

= 1 + a2 + b2 + c2

ADJOINT OF A MATRIX

Adjoint of a matrix: Adjoint of a matrix is transpose of co-factor matrix. In the previous lessons , we

have already understood co- factor matrix.

For ex: if A = 2 4 1

8 5 2−1 3 7

In order to find co factor matrix of A, C11 = + 5 2

3 7 , C12 = -

8 2−1 7 , C13 = +

8 5−1 3

= + (29) = -( 58) = + (29)

C21 = - 4 1

3 7 , C22 = +

2 1−1 7 , C23 = -

2 4−1 3

= - (25) = + 15 = - (10)

C31 = + 4 1

5 2 , C32 = -

2 1

8 2 , C33 = +

2 4

8 5

= + (3) = - (-4) = + (-22)

Therefore co-factor matrix of A = 29 −58 29−25 15 −10

3 4 −22

Adjoint of matrix A will be transpose of it , written as adj(A) = 29 −25 3−58 15 4

29 −10 −22

Important properties of adjoint of a matrix

1. A (adj.A) = (adj. A) A = |A| In

2. |Adj. A| = |𝑨|𝒏−𝟏

Where n is order of the matrix.

3. Adj.(AB)= (Adj.B) (Adj.A)

ILLUSTRATION 8:

Given matrix A = 3 −1 1−15 6 −5

5 −2 1

, Prove that (i) A (adj. A ) = |A| I3 (ii) |Adj. A| = |𝐴|𝑛−1

Solution : First of all we will find cofactors of matrix A ⇒ C11 = + ( 6 -10) = -4, C12 = - ( -15+25) = -10 C13 = + ( 30-30) = 0

C21 = - ( -1+2) = -1 C22 = + (3-5) = -2 C23 = - (-6+5) = +1

C31 = + ( 5-6) = -1 C32= - ( -15+15) = 0 C33 = + (18-15) = +3

Therefore co-factor matrix of A = −4 −10 0−1 −2 1−1 0 3

Hence adj. A = −4 −1 −1−10 −2 0

0 1 3

Now we will find |A| = 3 C11 -1 C12 + 1C13 = 3 (-4) -1 (-10) + 1(0) = -2

Therefore A (Adj. A) = 3 −1 1−15 6 −5

5 −2 1

−4 −1 −1−10 −2 0

0 1 3

|A| I3

= −12 + 10 + 0 −3 + 2 + 1 −3 + 0 + 3

60 − 60 + 0 15 − 12 − 5 15 + 0 − 15−20 + 20 + 0 −5 + 4 + 1 −5 + 0 + 3

= -2 1 0 0

0 1 0

0 0 1

= −2 0 0

0 −2 0

0 0 −2

= −2 0 0

0 −2 0

0 0 −2

(ii)Adj. A = −4 −1 −1−10 −2 0

0 1 3

therefore |Adj. A| = -4 C11 + 1 C12 -1 C13 [ Remember sign conventions]

|Adj.A| = -4 (-6-0) +1(-30-0) -1 (-10 -0 ) = 24-30+10 = +4

On the other hand, |𝐴|𝑛−1 = (-2) 3-1 = 4 [ |A| = -2, n =3 order of matrix]

Therefore |Adj. A| = |𝐴|𝑛−1

ILLUSTRATION 9: For the matrix A = 5 2−3 −1

, verify that adj A/ = (adj. A) /

Solution: L.H.S. R.H.S.

A/ = 5 −3

2 −1 co factor matrix of A =

−1 3−2 5

Co factor matrix of A/ = −1 −2

3 5 Adj. A =

−1 −2

3 5

Adj. A/ = −1 3−2 5

(Adj.A)/ = −1 3−2 5

36. Given that A = cos 𝑥 sin𝑥 − 𝑠𝑖𝑛𝑥 cos𝑥

and A (Adj. A) = k 1 0

0 1 , Find k. Ans: 1

37. Without computing adj. A , find |Adj. A|

A = −2 0 0

3 4 0

10 −7 3

[Hint: |Adj. A| = |𝐴|𝑛−1 ] Ans: 576

38. Find the adjoint of matrix A = −1 −2 −2

2 1 −2

2 −2 1

, [Check step: cofactor matrix= −3 −6 −6

6 3 −6

6 −6 3

hence show that A(adj. A) = |A| I3

39. If A = 𝑎 0 0

0 𝑎 0

0 0 𝑎 . find | A . Adj A| Ans: a 9

[ Find Adj. A first , then find product of A.Adj A, then find its determinant. ]

40. For the matrix A = 2 −1 3

4 2 5

0 4 −1

, verify that A (Adj. A ) = O

41. If A = 2 −3

5 1 , B =

−1 3

4 0 , verify that adj.(AB) = (adj.B) (adj.A)

INVERSE OF A MATRIX

We have learnt operations of addition, subtraction and multiplication of matrices. But there is no division

of matrices defined. For ex; for matrices A, B and C if A B= C then we can not say B = 𝐶𝐴

Matrices can not be divided this way. Yes there is a different way called inverse of a matrix.

In such a case if AB = C , then B = 𝐴−1C --------------------------------------------(1)

So in order to understand inverse of a matrix let us understand one more term first.

Singular Matrix: A square matrix is said to be a singular matrix if its determinant (value ) is zero.

e.g. A = 2 5

4 10 , then |A| = (2x10) – (5x4) = 0 therefore A is a Singular Matrix.

Therefore if a matrix is singular , A-1 does not exist. As is clear from (1)

In other words for a matrix to be invertible , matrix A should be Non Singular.

Inverse of a matrix A-1 = 𝑨𝒅𝒋.𝑨

|𝑨| , therefore |A| should be non zero.

Properties of Inverse of a matrix:

(1) A A-1 = A-1 A = In where n is the order of a matrix.

(2) (𝐴/)−1 = (𝐴−1)/

(3) ( AB)-1 = B-1 A-1 -Reversal Law

ILLUSTRATION 10: If A= 2 −3−4 7

, Show that 2A-1 = 9I –A

Solution: |A| = (2x7) – (-4x-3) = 2 ≠ 0 therefore A-1 exists.

Step 1: Co-factor matrix of A = 7 −4

3 2 Remember sign conventions

+ −− +

Step 2: Adj. A = transpose of co-factor matrix i.e. = 7 3−4 2

Step 3: A-1 = 𝐴𝑑𝑗 .𝐴

|𝐴| =

1

2

7 3−4 2

Step 4: Putting in the required equation: 2A-1 = 9I –A ⇒ 2x 1

2

7 3−4 2 =

9 0

0 9 -

2 −3

4 7

⇒ 7 3−4 2

= 7 3−4 2

ILLUSTRATION 11: If A = 1 𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑥 1

, show that A/ A-1 = cos 2𝑥 − sin 2𝑥sin 2𝑥 cos 2𝑥

Solution: |A| = 1 + tan2x = sec2x

Co-factor matrix of A = 1 tan 𝑥− tan𝑥 1

, therefore Adj. A = 1 − tan𝑥

tan𝑥 1

A-1 = 𝐴𝑑𝑗 𝐴

|𝐴| =

1𝑠𝑒𝑐 2𝑥 1 − tan𝑥

tan𝑥 1 =

1/𝑠𝑒𝑐2𝑥 −𝑡𝑎𝑛𝑥/𝑠𝑒𝑐2𝑥𝑡𝑎𝑛𝑥/𝑠𝑒𝑐2𝑥 1/𝑠𝑒𝑐2𝑥 = 𝑐𝑜𝑠2𝑥 −𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥

sin𝑥 cos 𝑥 𝑐𝑜𝑠2𝑥

Now A/ A-1 = 1 −𝑡𝑎𝑛𝑥

tan𝑥 1 𝑐𝑜𝑠2𝑥 −𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥

sin𝑥 cos𝑥 𝑐𝑜𝑠2𝑥

⇒ 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥 − sin𝑥 cos𝑥 − sin𝑥 cos𝑥

sin𝑥 cos𝑥 + sin𝑥 cos𝑥 −𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 = cos 2𝑥 – sin 2𝑥sin 2𝑥 cos 2𝑥

Hence Proved.

42. If A = 4 5

2 1 , show that A -3I = 2 ( I + 3A-1 )

43. Find A-1 for given matrix A = 𝑎 𝑏𝑐 1+𝑏𝑐𝑎 and show that a A-1 = ( a2 + bc+1) I – aA

[ check step: A-1 = 1+𝑏𝑐𝑎 −𝑏−𝑐 𝑎

44. Computer A-1 for A = 1 0 2−2 1 0

0 −1 2

Ans: 1

6

2 −2 −2

4 2 −4

2 1 1

45. Given A = 5 0 4

2 3 2

1 2 1

and B-1 = 1 3 3

1 4 3

1 3 4

, Computer (AB )-1

Hint: Since ( AB)-1 = B-1 A-1 , therefore find A-1 Ans: −2 19 −27−2 18 −25−3 29 −42

and then multiply B-1 with A-1

To be continued in Worksheet-5……..

Class 12 Physical Education Worksheet-3

Intervals related test match in the game of cricket:

Interval between the innings Interval for lunch Interval for tea Interval for drinks (water or juice).

Follow on: the team which bats first leads up by 200 runs in a five day match, or four days (National matches or practice matches), 150 runs in three day match, 100 runs in two day match has the option to ask the other team to follow their innings.

How’s that: it is a mode of appealing which covers all the ways of getting out.

Batting order: the order which the batsman, bat from the openers, through the middle order till the lower order.

Batsman’s retirement: A batsman may retire at any time during his innings (in case of injury or other reasons), after informing the umpires. He may resume his innings only at the fall of a wicket or the retirement of another batsman.

Straight drive: A form of drive, played to a good length ball pitching around middle or off stump, by which the ball is sent back down the pitch (or over the bowler‟s head) into the area directly behind the bowler‟s wicket.

Pull shot: A shot played t the leg side to a short pitched delivery, between mid-wicket and backward square leg.

Reverse sweep: a right handed batsman sweeping the ball like a left hander batsman and vice versa.

Upper cut: A typical shot played against a short ball or bouncer. Here the batsman makes a cut above his head and the ball usually goes to the third man area.

Late cut: A cut made with a delayed action so as to send the ball to the off side behind the wicket.

Leg glance: A delicate shot played at a ball aimed slightly on the leg side, using the bat to flick the ball as it passes the batsman, deflecting towards the square leg or fine leg area.

Hook shot: a shot played on a ball pitched on leg side or outside leg stump above the shoulder height in the fine leg region.

Square cut: an attacking stroke played to a delivery itched outside the off stump (ball is played square on the off side) in the region between point and third man.

Cover drive: this is played to a half volley that pitches wider of the off stump. The left foot is moved further across to allow the head to be in line with the ball. The full face of the bat hits through the ball and points in the direction that you wish the ball to go.

Back foot drive: played to a short ball, slightly wide of the off stump. This again, is an extension of the defensive stroke. The back lift is higher and the bat comes through more quickly. The weight of body is transferred forward into the stroke. The head is still kept in lie. The left arm still controls the shot and the full face of the bat hits through the ball in the intended direction of the shot.

Variations of pace bowling: I. in swing

II. out swing III. reverse swing IV. slower one V. beamer

VI. bouncer VII. cutters (leg cutter, off cutter)

VIII. full toss IX. Yorker

Variations of spin bowling: I. Leg spin

II. Off spin III. Googly IV. Doosra V. Top spin

The stance: place feet either either side of, and parallel to, the crease. The left side of the batsman points towards the bowler. Bend the knees slightly to allow for easy movement. Keep the head upright and the eyes level. Place your feet comfortably, but not too far apart. Bat rests behind back right foot. Left shoulder towards bowler and head turned to face bowler with eyes level.

Bowling:

The fundamental of good bowling are: A correct grip A smooth, economical run up An easy, rhythmical and well balanced delivery, making full use of your height

and body. A deliberate and fluent follow through

The grip:

The grip varies according to the type of delivery, but for all types of the ball is held in the fingers and not the palm.

The run up:

The run up should start slowly, gradually increase speed, and over the strides, full pace should be reached.

Fielding:

Fielding is very important in the modern game and even more so if everyone tries. The fielder‟s first job is to stop the ball and there are two methods of fielding.

The orthodox position: get in line with the ball early. Form a “v” with the heels touching when facing the ball. Bend the knees and keep the head above you8r hands. Point the fingers down and stop the ball in front of your feet. The following will apply to right handed throwers:

The long barrier position: get into the line of the ball as quickly as possible. Turn sideways to the line of the ball. Drop onto the left knee, so that the knee overlaps, with right foot. Head is above the hands, with fingers pointing down, when the ball is stropped.

Wicket keeping:

Of all the positions in the field, the wicket keeper is the most important and the most demanding. A wicket keeper should make sure that his stance.

Is comfortable and not strained. Allows him the best possible sight of the ball. Enables him to take the ball with the minimum of movement. Is close enough to the wicket, so that he can stump the batsman without having to

reach for the stumps.

The following feet movements allow wicket keepers to take balls on the offside and down the leg side while keeping the body facing and close to the stumps.

The body and head must be kept still. Stay down for as long as possible, only rising to meet the rise of the ball off the

pitch. The feet should move as little as possible, just making sure that the body is behind

the ball. Always take the ball with the fingers pointing down, never at the ball.

Board of control for cricket in India. (BCCI): BCCI is the national governing body for cricket in India. The board was formed December 1928 as a society, registered under the Tamil Nadu societies Registration Act. It is a consortium of state cricket associations and the state associations select their representatives who in turn elect the BCCI officials.

The following office bearers of the BCCI shall be elected by the full members of the BCCI from amongst their representatives at an Annual General Meeting:

The president The vice president The secretary The joint secretary The Treasurer

International cricket council (ICC): the international cricket council is the international governing body of cricket. It was founded as the imperial cricket conference in 1909 by representatives from England, Australia and South Africa, renamed the international cricket conference in 1965, and too up its current name in 1989.

It organizes world championship events such as cricket world cup, women‟s cricket world cup, ICC T20 world cup, ICC women‟s T20 world cup, ICC championships trophy and under 19 cricket world cup.

The ICC has instituted the ICC Awards to recognize the honors the best international cricket players of the previous 12 months. The ICC player rankings are a widely followed system of ranking for international cricketers based on their recent performances.

Important tournaments:

National Tournaments: Ranji trophy, Irani trophy, C. K. Naidu Trophy, Duleep Trophy, Deodhar trophy, wills trophy etc.

International Tournaments: ICC world cup, (50 over‟s, ODI), T20, Asia cup, etc.