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वे आकृतियाँ जिनका आकार समान हो किन्तु विस्तार या माप या परिमाप अलग-अलग हो, समरूप कहलाती है। तथा उनका यह प्रगुण समरूपता कहलाता है।

Given expression = cos 80° cos20° cos40°

= (1/2) 2cos 80° cos20° cos40°

Use 2 cosAcosB = cos(A+B) + cos(A-B) formula.

= (1/2){ cos (80°+20°) + cos(80°–20°)}cos40°

= (1/2)( cos 100° + cos 60°)cos40°

= (1/2) cos 100° cos 40° + (1/2) cos60°cos40°

= (1/2)(1/2)2cos100°cos40° +(1/2)(1/2)cos40°

= (1/4){cos140° + cos60°}+ (1/4)cos40°

= (1/4)(cos140° + cos40°) + (1/4)cos 60°

= (1/4){cos(180°-40°) + cos 40) +(1/4) (1/2)

But cos (180° -x) = -cos x. So,

= (1/4)(-cos40°+cos40°) + 1/8

= 1/8

Irfan was brought up in a village in Bihar. At a very small age he entered into stock trading business and made up for his father’s losses in stock market. Later, Irfan set up his own portfolio management firm.

Phrasal Verbs: A phrasal verb is a verb followed by a preposition or an adverb; the combination creates a meaning different from the original verb alone.

To get – to obtain 

To get together – to meet

DearAnand,

How are you? Sorry that I didn’t mail you all these days. I needed time to settle down in this new set up. Moreover, we are allowed to use the internet facility only on Sundays. My new college is nice and I have no problem at all. Of course, I do feel homesick and miss my friends once in a while, but it’s not the fault of the college. The infrastructure is fantastic and the teachers are of a good standard. We have a lot of scope for extracurricular activities as we have many ceils and associations.

I have joined the Waste Management cell and NSS because I sincerely want to contribute my bit towards social welfare. We had an orientation programme which made us get acquainted with our teachers, seniors, and classmates. The Cabinet Members took us around the campus and gave us a thorough knowledge of what is where. Since I’m a good eater, I am happy that our college has a full-fledged canteen with quite a few food items, reasonably priced. When you visit me here, I will take you to the canteen. Do visit me as soon as possible. I miss you a lot.

Yours,

Venkatesh

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^77,232,917-1, having 23,249,425 digits.

Secretary
Cultural Committee
Mountain View P.U. College
Chikmagalur
19 January 2019
The Principal
Sarvodaya Vidyalaya
Birur Road
Koppa
Dear Sir/Madam,

It gives me immense pleasure to announce that our college is celebrating its Golden Jubilee this year. We intend to take this auspicious occasion to organize a Cultural Fest (Aayaas’19) on the 12th of February 2019 from 6 p.m. to9 p.m. in our college auditorium. The Fest reflects the spirit of youth and provides a platform for the students to show case their talents In various spheres that cover the many versatile aspects of our culture. Dance, classical music, singing, a fashion show, a play, and stand up comedy are some of the programmes lined up for the day. I am writing to request your honourable presence along with your staff in gracing the event. It will be our pleasure to have you as a part of Aayaas19 so that we can make the Fest bigger and brighter.

Earnestly waiting for a favourable response

Thanks and regards

Sincerely yours
Secretary
Cultural Committee
M.V.P.U. College
Chikmagalur

(a) Rickshaw Pullar 

 The life of rickshaw pullar is very hard. His life is, infact, very miserable. He lives by the sweat of his brow. He has to work like a beast of burden. He pulls heavy loads, men, women and children. He perspires from head to foot. It is very painful to look at him. A rickshaw pullar hardly gets as much as he deserves. People try to give him as little as possible. He has to work from morning to night but he gets very little to eat. It is sad that in the age of science, a man has to work like beast to earn his bread. 

 (b) Your Aim in Life 

My Aim in life is to become a doctor. A doctor can earn money as well as honour. He can serve the sick. He can serve his nation in a big way. India needs a large number of doctors. Our villagers suffer from many diseases. But there are few doctors in villages. I want to be a doctor and serve in the villages. My aim will be not to earn money but bring hope to the poor and the suffering people. Thus by serving people, I will be serving God also. 

(c) Hostel Life 

Hostel are places which provide board and lodging to school and college students who need it. Here a student comes leaving behind his home; sweet home, the tender love and care of his parents, brothers and sisters. But he soon makes new friends and gradually gets used to the new way of life. Hostel life is the surest way of eliciting the best in young man. 

(d) A Picnic 

There is a beautiful canal outside our town. Last Sunday, my friends and I decided to go there for a Picnic. We bought some sweets and fruits, cake and chocolates etc. We reached the canal and camped under a tree on its bank. We enjoyed there by playing badminton and cricket. Then we played Antakshari. After that we ate fruits and sweets with cold drinks. There were many people who came there for holiday. In evening we came back to our home. It was a lovely picnic.

Tenali. 

10-3-xx. 

Dear Ravi, 

It has been a long time since I wrote to you. How are your parents? How are you? How is your little sister? I hope you are studying well. We are all fine here. I am also doing well here. In this letter I would like to describe you about our school Annual Day which we celebrate next month. 

On 20th of April our school is going to celebrate the Annual Day. The local M.L.A. will be the Chief Guest of the function. Our Head Master will preside over the function. Many important people of the village will attend the programme. All the teachers are working hard to make it a big success. Competitions in games and sports are being conducted for the occasion, Many cultural activities are going to be held. I am going to dance for two songs. It is going to be a great fun and enjoyment. Do you want to grab the fun and entertain yourself by attending our Annual Day? I would like you to be at the function. I am looking forward to seeing you here. 

Convey my regards to your parents. Hoping you will turn up, I remain. 

Yours lovingly, 

x x x x x

Address on the Envelope: 

To

G. Suresh, 

S/o G. Satish, 

H.No: 8-95/A, 

Sai Baba Temple Street, 

Brahmin Street, 

Nizam Patnam. 

Guntur (Dt.)

(i) Mass copying in the examination has made the education to a joke. 

(ii) Lengthy courses of study and the system of annual examination forced the students to cram and copy. 

(iii) An examination should be entrusted with the difficult task. He will have to rise above petty worldly consideration and must be brave and impartial. 

 (iv) The word is 'Cram'.. 

To, 

The Headmaster 

Patna High School 

Patna 

Sub. :- Application to provide some book from library. 

R/Sir, 

I am a student of class X/A of your school. This year I will appear in Board Examination. I have completed my all course books and also done revision. Sir I want to do best preparation, so I need some more books of Maths and Science. 

Sir this is my highly request to you to please allow me to provide some books of Maths and Science from library. I will be thankful to you for this concern. 

Thanks. 

Yours obedient pupil 

 Rajesh 

 X/A

a. Naveena

b. Four

c. iii) Kavya

d. ii) cousins

e. ii) Anuhya and Rani are sisters.

Dt 2.4.20 xx 

Ravi Kumar, 

Class VII ZPHS 

Ramavarappadu 

VIJAYAWADA

The Headmaster 

ZP High School 

Ramavarappadu 

Vijayawada 

Sir

I bring to your kind notice that last night we received a call from our uncle’s place. He met with an accident and admitted in the hospital. Our family has to visit his place to be helpful to his family. 

So kindly grant me leave for three days. I promise that I shall attend the Exams without fail. 

Thanking you

Yours sincerely 

Ravi Kumar 

Class VII 

(Parent’s Signature)

13 December, 

2018 Patna

Respected father, 

 I hope you will receive this letter in the best of your mood and spirits. Father I received your letter yesterday. You have asked about my preparation for board examination.

I am doing all my preparations for the board examination. I have completed all subjects especially Maths and Science. Now my revisions are going on. Now I need only your bless for my future. 

 My best regard to my elders and love to my youngers. 

Your's loving son 

 Saroj  

Sudarsan Buildings 

Vinukonda 

2.4.20xx

Dear Anu

How are you ? Very pleased to learn about your success in Kabaddi competitions.

I am fine and finished my Summative Exams. I am regularly going for volleyball practice to the stadium. When Mum, Dad and I knew you won in Inter School Kabaddi competition, we felt very excited and happy. Congratulations

We shall celebrate when you come here. When are you coming ? I am eagerly waiting for your arrival. 

With love 

Kanchana 

Address on the Envelope: 

To 

R. Anusha, 

1-134, Girls Hostel, 

ZPHS, 

Guntur.

30, Gandhi Nagar,

Jhansi Raod,

Gwalior – 474011

25 November, 2012

Dear Mohit,

Yesterday I received your letter, I was very happy to read that you have passed your examination with flying colours. May I offer my heartiest congratulations on your brilliant success? Really my joy knew no bounds when I read that you have secured fifth rank in a state and also won a merit scholarship. May God bless you always in the same way. It is a great achievement of which not only you but your parents also ought to be proud. Undoubtedly you deserved the rank you have secured.

My parents also congratulate you on this success. When are you planning to give a party to celebrate such occasion? I’m excited enough to come to see you.

Please pay my regards and congratulations to Uncle and Aunty and love to Shobhit.

Please do reply. More when we meet.

Yours ever

Pratap

19 December, 2016 

 PATNA

  My dear friend Raja, 

 Hello,

I am fine. How are you? I came to know that you have got high percentage in the Board Examination. My joy knew no bound when I got this news. Please accept my heartiest congratulations on your brilliant success. Really it is a splendid success. You have brought honour to your family and to us also. Your hard work comes forward as your success. 

Please convey my congratulations to your dear parents. 

Your's loving friend 

 Ankur 

544, Gandhi Road,

New Delhi – 110023

25th November, 2012

My dear Father,

 I am writing this letter to find you in good health. I received your letter a few days ago. I am sorry I could not reply in time because I was busy in my examination. I did well in it. I hope to get through it with good marks.

Now I have firmly decided to make my career in the Banking Sectors. A number of vacancies are advertised in Banks. There is a great opportunity to get selected in this competitive examination. I know it is a difficult job but I’ll leave no stone unturned to make it up. I’ll work hard. I have decided to start my studies right now. A Career Institute gives coaching to students appearing at this examination. If I have problem, I’ll join it. I know your blessings are with me.

Rest is O.K.

          Kindly convey my best regards to mother and love to Ranu.

Yours loving daughter

Meena

28, Cantt. Road
Bareilly
Dated : June 30th, 20….

Dear Mohan,
Please, accept my heartiest congratulations on our success in the examination. I am very happy to know that you got first division in High School Examination. Really, you have fulfilled the expectation of your parents by getting first division. I hope, you will maintain the same division in Intermediate Examination also. My good wishes are always with you. I am sure, you will earn a good name there. With regards to elders and best wishes to youngers.

Yours sincerely,
Ramesh Kumar

Dear Sister,

     Mahua Mukundpur

      20/12/2019

I am fine here and hope that you are also will be the same to me. I am waiting for you beause it is good fine to visit at my place in holidays. We shall go to Nalanda and Rajgir. And will enjoy we shall spend for five day at Nalanda. So you come to me.

Please convay my best complements to uncle and aunty.

Kishunpur Telaur  

Vaishali 

XYZ

11B Lane, Road No-14, Rannch

205, Civil Lines,
Bareilly-243001,
Dated : October 10th, 20….

Dear Mummy,
Thanks for your kind letter that I received yesterday. I am working hard, there is no cause to worry. Winter has set in. So I need a new coat. The old one will not do this year. I, therefore, request you to persuade respected Papa to arrange a new one this year. I am coming to see you. I hope to be there on 10th Nov., as our college will be celebrating the founder’s week. With a lot of regards.

Yours obedient son,
Mohan

Bara Bazaar
Bareilly-243003
Dated : April 15th, 20….

Dear Raju,
You will be pleased to know that my examinations are over today. I have done my papers well. Now it is time for us to enjoy and recover our health. I hope you have not made any arrangement as yet for these holidays. So I propose a trip to Musoori. Will it not be a good change? If you agree, I may arrange for it and you should reach here on April 28th, 20….. Bring your camera with you, if possible.

 Yours sincerely,
Rakesh

30, Gandhi Nagar,

Jhansi Raod,

Gwalior – 474011

25 November, 2012

Dear Sister,

Thanks very much for your sweet letter. I was very happy to read it. At present I am working hard not only to get good marks in the Senior Secondary Examination but also to get through the most prestigious examination known as IIT. It’s my dream to study in Roorkee Engineering College. I hope you will also be working hard

 If I qualify the IIT exams and get a good division, I will be selected for the Engineering division. Nowadays our country needs qualified technical hands. There is very little scope for art students. I want to be a good engineer and wish to work in the automobile industry.

Will you kindly persuade dear father to send me to Roorkee in July this year?

I’ll earnestly wait for your reply.

Convey my regards to mummy and papa and love to Shreya.

Your loving brother

Jai Kumar

Room No. 21,Shanta Ayatan,

Banasthali Vidyapeeth,

Bansthali – 304022

08 July, 2012

My dear Father,

All is well here and hope you will be enjoying the same. Since long I was eager for your kind letter which I received yesterday evening and I felt overjoyed. There is nothing to worry about my accommodation in the hostel. I am glad to inform you that I am living in a double seated room. My room mate, Meenu, is my class fellow. She is a good student. Her father is a doctor. She comes from a well educated family.

In the beginning I felt shyness. I did not mix with other girls. They made fun of me. But now they have become my good friends.

Our hostel is quite good. We get healthy food to eat. Special food is also served time to time. On Thursday fruit are distributed. There is a prayer hall and a good library with a number of good books. We are allowed to play football, volley-ball, bat minton and other games of our own interest in the playground in the evening. There is a good healthy atmosphere for competitive examination.

I am also trying at my best to prove myself. You should not worry about me.

Please convey may respect to my loving mother and sweet love to Jai.

Rest is o. k.

Yours loving daughter

Shreya

33-Rajendra Nagar
Bareilly-243002
Dated : May 17th, 20….

Dear Sister Reeta,
You will be glad to know that yesterday, I came back from Agra. I went there with my friends. Our History teacher Mrs. Usha Gupta was also with us. We reached Agra in late night. We passed the night in a hotel. Next day, we went to see the Taj. We found it very grand and beautiful. We liked it very much. Our teacher told us that Taj Mahal, one of the seven wonders of the world, was built by Shahjahan, a Mughal emperor, in the memory of his beloved wife Mumtaz Mahal, after her death. As it is built of white marble. Shahjahan used to call it ‘a dream in white marble’. Luckily we saw it in the moon light too. It looked silver white. Its beauty cannot be described in words. Of course, it is beyond description. The rest is O.K. There is nothing more to write. With love to Kartik.

Yours loving brother,
Varun

Room No. 21, Shanta Ayatan,

Banasthali Vidyapeeth,

Banasthali – 304022

01 August, 2012

My dear Mother,

Thanks you very much for your kind letter. It was delivered to me yesterday evening. All is well here and hope you will be enjoying the same. I am working hard to get through my examination with good marks. Mummy, I need some reference books to consult the topics of my syllabus as suggested by my teachers.

Please send me Rs. 500/- for buying books by money order as earliest as possible.

I wish you to convey my regards to father and love to Jai.

Please do send the money.

More when we see.

Yours loving daughter

Shreya

J-8, Sarojini Nagar
Delhi, 100004,
Dated : June 10th, 20….

Respected Father,
Thanks for your kind letter that I received today. I am glad to know that you are quite well now. We, too, are happy here. Rani misses you very much. She becomes very sad in your absence. Mummy is also not well. She has been suffering from fever for a week. But there is no cause to worry. She is under the treatment of Dr. Keshav Agarwal. The doctor says that she will be well within two or three days. Papa, I have gone short of money. Our college has opened. We have to deposit our college fees. Rani has been admitted in a new school. I have to buy some books. I, therefore, request you to send me four hundred rupees as soon as possible.

With regards.

Yours obediently,
Bobbey

Amount (P) = 2000

Compound interest (r %) Year (n)

\(P(1+\frac{r}{100})^n=A;\) \(2000(1+\frac{r}{100})^2=2205\)

\((1+\frac{r}{100})^2=\frac{2205}{2000}=\frac{441}{400}\)

\((\frac{100+r}{100})=\frac{21}{20};\) \(100+r=\frac{21\times100}{20}=\frac{2100}{20}\)

r = 105 - 100 = 5

Interest rate = 5%

a. Let the number of students = n

Number of sweets given to one student = \(\frac{180}{n}\)

Since 9 students were absent = \(\frac{180}{n-9}\)

∴ \(\frac{180}{n-9}=\frac{180}{n}+1\)

180n = 180(n - 9) + n(n - 9)

180n - 180n + 1620 - n² + 9n = 0

n² - 9n - 1620 = 0

\(n=\frac{9\pm \sqrt{81+4\times1620}}{2}\) = \(\frac{9\pm81}{2}\) or \(\frac{9-18}{2}\)

= 45 or -36

∴ Total number of students = 45

b. Total present = 45 – 9 = 36

CD = 16 centimetre MC = 8 – x, MD = 8 + x

then, MA x MB = MC x MD

6 x 8 = (8 – x)(8 + x)

48 = 64 – x² , x² = 16, x = 4

MC = 8 – 4 = 4 centimetre

MD = 8 + 4 = 12 centimetre

Let one of the perpendicular side = x

Second side = 2x + 4, Area = 80

1/2 × (2x + 4) = 80

2x² + 4x = 160

x² + 2x = 80

x² + 2x + 12 = 80 + 12

(x + 1)² = 81

x +1 = ± 9

x + 1 = 9

x = 8

x = – 10 cannot be accepted as – ve sign cannot exist for the length of the side

∴ One side = 8cm

Second side = 2 × 8 + 4 = 20 cm

(-2)^-3 x (-2)^-2

= \(\frac{1}{{-2}^3} \times \frac{1}{{-2}^2}\)

= \(\frac{1}{-8} \times \frac{1}{4}\)

= \(\frac{-1}{32}\)

Let x be the length

Perimeter 34 cm, therefore width will be 17- x cm

Perimeter = 60 cm, therefore width be \(\frac{60}{x}\) cm

\(\frac{60}{x}=17-x,\)

60 = 17x - x²

x² - 17x + 60 = 0

\(x=\frac{17\pm \sqrt{289-240}}{2}=\frac{17\pm 7}{2}=12,5\)

x = 5 cm

length = 12 cm therefore width = 17 – 12 = 5 cm

length = 12 cm ,

width = 5 cm

Perimeter = 100 m .

Length + Breadth = 50 m.

Length = 25 + x.

Breadth = 25 – x

Area = (25 + x)(25 – x) = 600

25² – x² = 600

x² = 625 – 600 = 25, x = 5

Sides, 25 + 5 = 30 m

25 – 5 = 20 m

Let x be the length of side.

other side = x + 5

x (x + 5) = 150

x² + 5x – 150 = 0

Let the side of square = x

Sides of rectangle = x, x – 8

Area = x(x – 8) = 84

x² – 8x = 84

x² – 8x + 16 = 84 + 16

(x – 4)² = 100,

x – 4 = 10, x = 14

Side of square = 14 centimetre

Given:

The Least natural number leaves no remainder when divided by all the digits from 1 to 9 which are = ( 1, 2, 3, 4, 5, 6, 7, 8, 9 )

Calculation:

Now, here we have to find the LCM of all the numbers from 1 to 9

⇒ LCM of ( 1, 2, 3, 4, 5, 6, 7, 8, 9 ) = 8× 7× 9× 5 = 2520

∴ 2520 (The least natural number which leaves no remainder when divided by all the digits from 1 to 9)

Ans: 3

Given: 

Antony and Vikash together can complete a piece of work in 20 days.

Vikash alone can complete it in 25 days.

Formula used:

one day work = 1/total days taken to complete the work

Calculation:

Antony's and Vikash's one day work = 1/20

Vikash's one day work = 1/25

Antony's one day work = 1/20 - 1/25

⇒ (5-4)/100 = 1/100

∴ Antony will complete the work in 100 days.

Given:

The price of each Geography textbook is x

The price of each History textbook is (x + 2)

The price of each Mathematics textbook is (x - 2)

Calculation:

⇒ Total expenditure on Geography textbook = ax

⇒ Total expenditure on History textbook = (a + 2) × (x + 2) = ax + 2x + 2a + 4

⇒ Total expenditure on Mathematics textbook = (a - 2) × (x - 2) = ax - 2x - 2a + 4

⇒ The total sale = ax + ax + 2x + 2a + 4 + ax - 2x - 2a + 4 = 3ax + 8

∴ The required result will be "3ax + 8".

Oil refinery located near the of field is Nunamati.

Railway passenger coaches are manufactured at Perambur.

The correct option: (c) III IV II I