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com  22,  2019 -  7

The molarity of pure water is...

SOLUTIONS
Continued from Jan 20th

Krupakar Pendli
Centre Head
Urbane junior colleges
7893774888

JEE Main
Special
10  22,  2019 -  ntnews.com ntadilabad.com

Navodaya Entrance Model Paper-2

P kavitha
Director
MCC9 Navodaya &Sainik
Coaching Centre
Nalgonda
9393949497

1.3 2.2 3.2 4.3 5.4 6.3

7.2 8.1 9.2 10.3 11.1 12.3
13.3 14.2 15.1 16.4 17.4 18.4
key

19.1 20.3 21.2 22.2 23.2 24.3

25.3 26.1 27.1 28.4 29.4 30.2
31.2 32.3 33.2 34.4 35.3 36.3
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Who is the most followed person on twitter 2018?

General Awareness October 2015 with headquarters in 1) Uttar Pradesh 2) Maharashtra 3) Bihar of each deposits in
Mumbai that forms part of IDFC, an inte- 4) West Bengal 5) Madhya Pradesh a bank.
1. What is Indian economy expected grated infrastructure finance company. 16. Which of the following is the rate at C. DICGC is a sub-
growth rate for the financial year 2018- B. IDFC Bank and non-banking financial which the RBI lends money to the sidiary of RBI.
19, as per first advance estimates of company Capital First had announced banks for a short term? Which of the
National Income for 2018-19 released completion of their merger on December 1) Base Rate 2) Reserve Repo Rate above statements
by The Central Statistics Office (CSO) 18, 2018. 3) Bank Rate 4) Repo Rate 5) MSF Rate are correct?
under the Ministry of Statistics and C. Following the merger, the board of 17. Jair Bolsonaro is the president of which 1) Only A and B
Programme Implementation? IDFC Bank had approved the appoint- country? 2) Only B and C
1) 6.2 per cent 2) 7.2 per cent ment of V Vaidyanathan, founder and 1) Argentina 2) Brazil 3) Cuba 3) Only A and C A.Ravi Prakash Reddy
3) 8.2 per cent 4) 8.2 per cent chairman of Capital First Ltd, as manag- 4) Mexico 5) None of these 4) All A, B and C Senior Faculty
5) None of these ing director and chief executive officer of 18. Which of the following is Asia's first 5) None of the above
2. Pradhan Mantri Jan Dhan Yojana the merged entity. stock exchange? 25. Who is the CEO of Apple Inc?
(PMJDY) was launched on 28 August D. IDFC Bank name has been changed to 1) Bombay Stock Exchange 1) Jack Dorsey 2) Satya Nadella
2014. As apprised by banks, the num- IDFC First Bank Ltd with effect from 12 2) National Stock Exchange 3) Sundar Pichai' 4) Jeff Bezos
ber of accounts opened under PMJDY January 2019. 3) Shanghai Stock Exchange 5) Tim Cook
as on 26 December 2018? Which of the above statements are 4) Singapore Stock Exchange 26. Who is the most followed person on
1) 30.66 crore 2) 31.66 crore corect? 5) Tokyo Stock Exchange twitter 2018?
3) 32.66 crore 4) 33.66 crore 1) Only A and B 2) Only B and C 19. Among the following the correct share of 1) Katy Perry (An American singer)
5) None of these 3) Only A, C and D 4) A, B, C and D Central Government, State Government 2) Cristiano Ronaldo(Portugal Football
3. Match the following: 5) None of the above and Sponsor Bank Regional Rural player)
Pakistan Personalities Position 10. The headquarters of PFRDA are Banks (RRBs) in…. 3) Shakira(Colombian Singer)
A) Arif Alvi (i) Chief Justice located at? 1) 50%, 25%, 25% 2) 50%, 30%, 20% 4) Narendra Modi(Prime Minister of
of Pakistan India) 5) None of the above
B) Imran Khan (ii) President of
Pakistan
FOR ALL COMPETITIVE EXAMS 27. What is capital of Zimbabwe?
1) Pretoria 2) Abuja 3) Harare
C) Asif Saeed Khosa (iii) Foreign 4) Nairobi 5) Cairo
Minister of Pakistan 28. National Financial Switch an ATM net-
D) Shah Mahmood (iv) Prime work is regulated by…
Qureshi Minister of Pakistan 1) Reserve Bank of India(RBI)
A B C D 2) National Payments Corporation of
1) (i) (ii) (iii) (iv) India (NPCI)
2) (iv) (iii) (ii) (i) 3) Securities and Exchange Board of
3) (ii) (iv) (i) (iii) India (SEBI) 4) NITI Aayog
4) (i) (iii) (ii) (iv) 5) None of the above
5) (iii) (iv) (i) (ii) 29. World Economic outlook published by..
4. Prominent Indian-American economist 1) World Bank(IBRD & IDA)
Gita Gopinath has joined as the chief 2) International Monetary Fund(IMF)
economist of the … 3) International Finance Corporation(IFC)
1) United Nations Organization (UNO) 4) World Economic Forum(WEF)
2) World Bank (IBRD & IDA) 5) None of the above
3) International Monetary Fund (IMF) 30. Which of the following is the India
4) Bank for International Settlements (BIS) largest commodity exchange?
5) World Trade Organization (WTO) 1) Multi Commodity Exchange of India
5. Which of the following provides for 10 Ltd(MCX)
per cent reservation in jobs and educa- 1) New Delhi 2) Mumbai 3) Chennai 3) 25%, 25%, 50% 4) 50%, 15%, 35% 2) Securities and Exchange Board of
tional institutions to economically back- 4) Kolkata 5) None of the above 5) None of the above India(SEBI)
ward section in the general category? 11. Which act allows banks and other 20. The World Tourism Organization is the 3) National Stock Exchange of India
1) 121st Constitutional Amendment Bill financial institution to auction residen- United Nations agency responsible for Limited(NSE)
2) 122nd Constitutional Amendment Bill tial or commercial properties to recov- the promotion of responsible, sustain- 4) Bombay Stock Exchange(BSE)
3) 123rd Constitutional Amendment Bill er loans? able and universally accessible tourism. 5) None of the above
4) 124th Constitutional Amendment Bill 1) Industrial Disputes Act 1947 Where is the head quarters of World 31. The minimum paid-up equity capital
5) 125th Constitutional Amendment Bill 2) Banking Regulation Act, 1949 Tourism Organization? for small finance banks shall be…..
6. The Reserve Bank of India(RBI) has 3) Foreign Exchange Management Act, 1) Geneva, Switzerland 2) Paris, France 1) Rs.100 crore 2) Rs.200 crore
appointed a five member committee to 1999 3) Madrid, Spain 4) Rome, Italy 3) Rs.300 crore 4) Rs.400 crore
suggest ways to improve digital pay- 4) SARFAESI Act, 2002 5) None of the above 5) Rs.500 crore
ments. This committee is headed by.. 5) None of the above 21. Real is the currency of….. 32. Which of the following is known as
1) Nandan Nilekani 2) Urjit Patel 12. Which among the following bank has 1) Saudi Arabia Lender of Last Resort in India?
3) Y.V.Reddy 5) N.K.Singh its headquarters in Mumbai? 2) United Arab Emirates (UAE) 3) Russia 1) RBI 2) SBI 3) LIC 4) NABARD
5) None of these 1) Bank of Maharashtra 4) Switzerland 5) Brazil 5) None of these
7. The USA President Donald Trump has 2) Bank of India 22. Original(initial) headquarters of RBI 33. What is Marginal Standing Facility
appointed Zalmay Khalilzad as the 3) Corporation Bank 4) Indian Bank were located at which of the following (MSF) Rate?
spcial envoy of ... 5) Punjab National Bank place? 1) Long term lending rate of RBI.
1) Pakistan 2) China 3) North Korea 13. SIDBI related statements are given. 1) New Delhi 2) Bombay 3) Calcutta 2) Short term lending rate of RBI.
4) Afghanistan 5) Russia Pick the wrong Statement? 4) Madras 5) Bangalore 3) Overnight lending rate of RBI.
8. Who announced his resignation from 1) SIDBI stands for Small Industries 23. In which year was the Banking 4) Minimum Lending rate of Banks.
the post of President of World Bank on Development Bank of India. Regulation Act passed in India? 5) None of the above
7 January 2019 and he will leave the 2) It was established on April 2, 1990. 1) 1934 2) 1949 3) 1955 4) 1959 34. Sattriya is a classical dance is originat-
post by 1 February 2019, three and a 3) Its head office is in Mumbai. 5) None of the above ed in…
half years before the expiry of his term 4) It is the principal financial institution 24. Consider the following statements 1) Assam 2) Bihar 3) Punjab 4) Haryana
in 2022? for the promotion, financing and develop- regarding Deposit Insurance and Credit 5) None of the above
1) Antonio Guterres(Portugal) ment of industry in the small-scale sector. Guarantee Corporation (DICGC): Answers
2) Jim Yong Kim(Korean-American) 5) None of the above A. DICGC was established on 15 July
14. "Inglorious Empire: What the British 1978 under Deposit Insurance and Credit 1) 2 8) 2 15) 2 22) 3 29) 2
3) Christine Lagarde(France) 2) 4 9) 4 16) 4 23) 2 30) 1
4) Roberto Azevedo(Brazil) Did to India" is a book by… Guarantee Corporation Act, 1961 for the
1) Chetan Bhagat 2) Amitav Ghosh purpose of providing insurance of deposits 3) 3 10) 1 17) 2 24) 4 31) 1
5) Tedros Adhanom Ghebreyesus (Ethiopia) 4) 3 11) 4 18) 1 25) 5 32) 1
9. Consider the following statements 3) Shashi Tharoor 4) Arundhati Roy and guaranteeing of credit facilities.
5) None of the above B. DICGC insures all bank deposits, such 5) 4 12) 2 19) 4 26) 1 33) 3
regarding IDFC Bank: 6) 1 13) 3 20) 3 27) 3 34) 1
A. The bank started operations on 1 15. Name the second most populous state as saving, fixed, current, Recurring
in India? deposit for up to the limit of Rs 100,000 7) 4 14) 3 21) 5 28) 2
 .
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Define simple harmonic motion?

OSCILLATIONS
Very Short Answer Questions
N. SUNIL KUMAR
Senior Faculty

Jr. Inter - PHYSICS

Long Answer Questions

IÁ‹Δäˇ BÌ≥t-r*≈yä˛ sÒ˝ã‹t&2019

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The mean and median of Unimodal grouped data are

7) The mean of the following data is Graph: Sol:-
Prepared by
7.68 find the missing frequency (P) Class xi − a
of the following distribution Y Frequency f xi
4i =
4
fi4i Mr. K. Kumar
x 3 5 7 9 11 13 5-15 6 10 -3 -18 Senior Faculty
f 6 8 15 P 8 4 55
l.c.f curve 15-25 11 20 -2 -22 Hyderabad
50 (300,50)
Sol: 45 (250,46) 25-35 21 30 -1 -21 ph:9492956214
Scale 35-45 23 40(a) 0 0
x F xxf 40
(200,38) on-X-axis 1cm 45-55 14 50 1 14  20 − 13 
L.C.F

35
3 6 18 30 =50 units 45-55 5 60 2 10 M = 55 +  × 5
(150,28)  13 
5 8 40 25 on-Y-axis 1cm εfi=80 εfi4i=37
=5 units 7 35
7 15 105
20
M = 55 + × 5 = 55 +
15 (100,15)  ε fi 4i  13 13
9 8 9P x=a+ ×4
10
(50,6)  ε fi  M = 55 + 2.7 = 57.5
5
11 8 88 X
∴median = 57.7
Here a = 40, 4 = 10 ε fi 4i = −37, ε fi = 80
100
150
200
250
300
350

13 4 52 0
50

upper boundery 37 5) Find the missing frequencies f1, f2 if

x = 40 − x = 40 − 4.625 x = 35.375 mean of 50 observation given below
εfi=41+P εfixi=3030+09P 8
is 36.4
ε fixi 2) The marks of students of x class in a 4) Find the median for grouped data
x= Given x = 7.68 Class 0-10 10-20 20-30 30-40 40-50 50-60
mathematics exam are given here
ε fi Weight (in kg) 40-45 45-50 50-55 55-60
Frequency 3 5 F1 10 F2 8
find the median of the data by draw-
No.of students 3 4 6 13
303 + 98 ing Ogive curve class midvalue frequency
7.68
41 + 8 60-65 65-70 70-75 0-10 5 3 5x3=15
marks
20oct 15 5 15x5=75
(41 + P )7.68 = 303 + 9 P interval 5-10 10-15 15-20 20-25 25-30 7 5 2
20-30 25 f1 25xf1=25f1
⇒ 11.88 = 1.32 P No. of 30-40 35 10 35x10=350

⇒P=
11.88
=9
Students 2 4 6 7 10
10th Class special 40-50
50-60
45
55
f2
8
45x12=4512
55x8=440
1.32 60-70 65 5 65x5=325
∴value of P = 9
30-35
9
35-40
5
40-45
4
45-50
3
MATHS Σf1=31+f1 Σf1xi={1025
+f2 +25f1+45f2}
8) The mean and median of Uni modal sol :-
grouped data are 28.2 and 30.5 marks No. of U.B l.c.f Points l.B q.c.f points
respectively find the mode of the interval Students
data
5-10 2 10 2 (10,2) 5 48+2=50 (5,50)
()
Sol : Given mean x =28.5,median=30.5 10-15 4 15 2+4=6 (15,6) 10 44+4=48 (10,48)
15-20 6 20 6+6=12 (20,12) 15 38+6=44 (15,44)
mod e = 3 × median − 2 × mean 20-25 7 25 12+7=19 (25,19) 20 31+7=38 (20,38)
mode=3 × median − 2 × mean 25-30 10 30 19+10=29 (30,29) 25 21+10=31 (25,31)
mode=3 × 30.5 − 2 × 28.5 30-35 9 35 29+9=38 (35,38) 30 12+9=21 (30,21)
35-40 5 40 38+5=43 (40,43) 35 7+5=12 (35,12)
= 91.5 − 56.4 = 35.1
40-45 4 45 34+4=47 (45,47) 40 3+4=7 (40,7)
value of mode 35.1 45-50 3 50 47+3=50 (50,50) 45 3 (45,3)

4 Marks
Sol :-
1) The following distribution gives the
Weight No. of Cumulative
daily profits (in rupees) carried by ε fx
50 shops in a locality. in kg students(f) Frequency Mean =
40-45 3 3 ε fi
Daily profits 0-50 50-100 100-150
(in rupees) 45-50 4 3+4=7 Given π = 36.4
Frequency 6 9 13 50-55 6 7+6=13(cf)
55-60 13(f) 13+3=26 1025+25f1+45f2
150-200 200-250 250-300
36.4 =
60-65 7 26+7=33 50
10 8 4 65-70 5 33+5=38
H = 50 70-75 2 38+2=40 36.4 × 50 = 1025 + 25f1 +45f2
Convert the above distribution to a less
than type cumulative Frequency εfi=40 1820 = 1025 + 25f1 + 45f2
50
distribution and draw its Ogive. Median = 2.5 20
1820=1025 = 25f1 +45f2
20 4 40
Sol :- th
Here 4 = 40, = = (20 th observation ) 615 = 25f1 + 45f2 …………1
n 2 2
Daily No.of Upper L.C.F Points median =   observation A/1 31+f1 +f2 = 50
2 n 
profits Shop boundary  2 − cf  F1 +f2 = 50-31
From the above Median = l +   ×4
0-50 6 50 6 (50,6)  f  F1+f2 = 19 ……….2
graph We observe Median = 28  
Solving 1 and 2 we get
50-100 9 100 6+9=15 (100,15) Note: - Locate the point 25 on the Here 55 − 60 is the median class
F1 = 12 and f2 = 7
y-axis from that point draw a par- l = 55, f = 13, cf = 13, 4 = 5
100-150 13 150 15+13=28 (150,28) value of f1 =12 , f2 = 7
allel to the x- axis getting the curve  20 
 40 − 13 
150-200 10 200 28+10=38 (200,38) at point is called median.  2 
3) Find the mean for step devia-
Median = 55 +   ×5
B|æø£ ù|J\ô|’ MT n_ÛÁbÕj·÷\qT, dü\Vü‰\qT
200-250 8 250 38+8=46 (250,46) tion method. 13
e÷≈£î ‘Ó*j·TCÒj·T+&ç.
250-300 4 300 46+4=50 (300,50) Class 5-15 15-25 25-35 35-45 45-55 55-65
dipika.navatelangana21@gmail.com
Frequency 5 11 21 23 14 5
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>∑T]+∫ á XÊÁdüÔ+ n<Ûä´q+ #˚düTÔ+~. düe÷q‘·«+, Hê´j·T+, Vü≤≈£îÿ\ |ü≥¢ eT+∫
viii). Á|üuÛÑT‘·« $<ÛëHê\ n<Ûä´j·Tq+: Ä<ÛäTìø£ sê»˙‹
ne>±Vü≤q ÇdüTÔ+~.
XÊÁdüÔy˚‘·Ô˝…’q &˚$&é ádüºHé, #ê¬s¢dt *+&é u≤yéT \T iv).Vü≤≈£îÿ\T, u≤<Ûä´‘·\ |ü≥¢ ne>±Vü≤q: sê»˙‹ XÊÁdüÔ+
sê»˙‹ XÊÁkÕÔìï Á|üuÛÑT‘·« $<ÛëHê\ s¡÷|üø£\Œq\≈£î Á|ü»\˝À Vü≤≈£îÿ\T, u≤<Ûä´‘·\ |ü≥¢ #·ø£ÿì ne>±
dü+ã+~+∫q XÊÁdüÔeTì $e]+#ês¡T. sê»ø°j·T Vü≤qqT n+~düTÔ+~. Á|üC≤kÕ«eT´ $»j·T+
ì]à‘·T&ÉT, sê»ø°j·T esêZ\ bÕÁ‘·qT á XÊÁdüÔ+ Vü≤≈£îÿ\T, uÛ≤<ä´‘·\ô|’ Ä<Ûës¡|ü&ç ñ+≥T+~. Á|ü‹
n<Ûä´j·Tq+ #˚düTÔ+~. bÂs¡T&ÉT sê»´+, düe÷»+˝À u≤<Ûä´‘·j·TT‘·yÓTÆq
ô|’q ‘Ó*|æq $<Ûä+>± sê»˙‹ XÊÁdüÔ |ü]~Û Ç{Ïe\ bÕÁ‘· b˛wæ+#˚≥≥T¢ sê»˙‹ XÊÁdüÔ n<Ûä´j·Tq+
Á|ü‹ì~∏ n˙, Á|üuÛÑT‘·«+ ˝Ò≈£î+&Ü sê»´+ ñ+&É<äì ø±\+˝À uÛ≤>± $düÔ]+∫+~. <√Vü≤<ä|ü&ÉT‘·T+~.
eTTK´ Á|üX¯ï\T`»yêãT\T yês¡T uÛ≤$+#ês¡T. Á|üuÛÑT‘·«+ <ë«sêH˚ sê»´ v). eT+∫ bÂs¡dü‘·« >∑TD≤\ uÀ<Ûäq: sê»˙‹ n<Ûä´j·Tq+
1. sê»˙‹ XÊÁkÕÔìï ìs¡«∫+∫, Ä XÊÁdüÔ |ü]~Ûì ÄX¯j·÷\T H˚s¡y˚s¡‘êsTT. ø=+‘· eT+~ e´≈£îÔ\T Ç+≥sY Á|ü‘˚´ø£+ e\¢ e´øÏÔ eT+∫ bÂs¡T&ç>± r]Ã~<ä›ã&É‘ê&ÉT. eT+∫
$e]+#·+&ç? sê»´+ ‘·s¡|ü⁄q n~Ûø±sê\T #Ó˝≤sTT+#·&Üìï
i) sê»˙‹ XÊÁdüÔ+ sê»´+‘√ Äs¡+uÛÑyÓTÆ sê»´+‘√H˚ Á|üuÛÑT‘·«+ n+{≤s¡T. $$<ä Á|üuÛÑT‘ê«\ |üìrs¡TqT yÓTT<ä{Ï dü+e‘·‡s¡+ bÂs¡T&ÉT #·{≤º\T m˝≤ s¡÷bı+~+#·ã&ÉT‘êsTT.
Á|üuÛÑT‘·«$<ÛëHê\ s¡÷|üø£\Œq eT]j·TT |üìrs¡T
n+‘·eTe⁄‘·T+~.- >±s¡ïsY.
ii) sê»˙‹ XÊÁdüÔ eT+fÒ Á|üuÛÑT‘·«+ >∑T]+∫
>∑T]+∫ n<Ûä´j·Tq+ #˚düTÔ+~. bÂs¡XÊÁdüÔ+ >∑T]+∫ ‘Ó\TdüTø√e&ÜìøÏ, bÂs¡T\qT Ä<äs¡Ù bÂs¡T\T
iv).dü+|òü÷\T, dü+düú\ n<Ûä´j·Tq+: e´øÏÔ JeHêìï >± r]Ã~~› bÂs¡dü‘·« >∑TD≤\qT n+~düTÔ+~.
n<ä´j·Tq+ #˚ùd XÊÁdüÔ+.- d”ºô|òHé ©ø±ø˘. Á|üuÛ≤$‘·+ #˚ùd nH˚ø£ dü+|òü÷\T, dü+düú\T ø£\e⁄. 2. sê»˙‹ XÊÁdüÔ n<Ûä´j·Tq ÁbÕeTTK´‘·qT $e]+#·+&ç? vi). Á|ü|ü+#· e´eVü≤sê\ |ü]C≤„q+: sê»˙‹ XÊÁdüÔ
sê»˙‹ XÊÁdüÔ |ü]~Û: Á|ü‹ e´øÏÔ sê»´+˝Àì _Ûqï dü+|òü÷\T, dü+düú\˝À sê»˙‹ XÊÁdüÔ n<Ûä´j·Tq+ @+‘√ Á|üjÓ÷»qø£s¡yÓTÆ n<Ûä´j·Tq+ düeTø±©q Á|ü|ü+#·+˝À »]π> dü+|òüT
i) düe÷»+: sêC≤´\‘√ e÷qe⁄ìøÏ >∑\ dü+ã+<Ûë\ n<Ûä´ düuÛÑT´&ç>± ñ+{≤&ÉT. á dü+|òü÷\T dü+düú\T e´≈£îÔ\ q~ eT]j·TT $\TyÓ’q~. á XÊÁdüÔ |ü]C≤„q+ bÕ\ ≥q\ |ü]C≤„Hêìï n+~düTÔ+~. ‘·<ë«sê n+‘·sê®
j·Tq+: sê»˙‹ XÊÁdüÔ+ e÷qe⁄ìøÏ, düe÷C≤ìøÏ dü+|üPs¡í $ø±kÕìøÏ, n_Ûe�~ΔøÏ ‘√&ÉŒ&ÉT‘êsTT. ≈£î\T, bÕ*‘·T\T Çs¡Te⁄]øÏ m+‘·>±H√ neX¯´yÓTÆ+~. rj·T düeTdü´\ |ü≥¢ ne>±Vü≤q ô|s¡T>∑T‘·T+~.
eT<Ûä´ >∑\ dü+ã+<Ûëìï $e]düTÔ+~. düe÷»+ sê»˙‹ XÊÁdüÔ+ dü«#·Ã+<ä dü+düú\T, ñ<√´>∑ã�+<ä sê»˙‹ XÊÁdüÔ ÁbÕeTTK´‘·: vii). n+‘·sê®rj·T dü+düú\ |ü]C≤„q+: ◊ø£´sê»´dü$T‹
|ü⁄≥Tºø£, |ü]D≤eT+, ñ<˚ΔXÊ´\qT ‘Ó\T|ü⁄‘·T+~. e÷q dü+|òü÷\T eT]j·TT Á|üuÛÑT‘·« n+>±\qT >∑T]+∫ i). sê»´+ >∑T]+∫ düe÷#ês¡+: sêC≤´ìøÏ dü+ã+~+∫q <ëì nqTã+<ä dü+düú\ |ü]C≤„Hêìï n+~düTÔ+~.
e⁄&ÉT dü+|òüTJ$ nì n]kÕº{Ï˝Ÿ uÛ≤$+#ê&ÉT. düe÷ n<ä´j·Tq+ #˚düTÔ+~. |ü]C≤„Hêìï sê»˙‹ XÊÁdüÔ+ n+~düTÔ+~. sê»´+ Á|ü|ü+#· sêC≤´\ eT<Ûä´ >√&Ée\T eùdÔ yê{Ïì m˝≤
»+˝À e÷qe⁄&ÉT @$<ä+>± düs¡T›u≤≥T #˚düTø√ì v). Vü≤≈£îÿ\T-u≤<Ûä´‘·\ n<ä´j·Tq+: sê»´+˝À Á|ü‹ ne‘·s¡D <ëì dü«uÛ≤e+ eT]j·TT $<ÛäT\ >∑T]+∫ |ü]wüÿ]+#ê\ì, Äj·TT<ë\e\¢ @s¡Œ&˚ Á|üe÷<ë\T,
J$kÕÔ&√ sê»˙‹ XÊÁdüÔ+ $e]düTÔ+~. bÂs¡T&ÉT $$<ä s¡ø±˝…’q Vü≤≈£îÿ\T eT]j·TT sê»´˙‹ XÊÁdüÔ+ n<Ûä´j·Tq+ #˚düTÔ+~. Á|ü‹bÂs¡TìøÏ ìsêj·TTBÛø£s¡D neX¯´ø£‘·qT, Á|ü#·Ã¤qï j·TT<ä›+,
ii) sê»´ n<Ûä´j·Tq+: bÕ˝Ÿ C≤HÓ{Ÿ, ã¢+{Ÿ wæ©¢, >±s¡ïsY u≤<Ûä´‘·\qT ø£*– ñHêïs¡T. sê»ø°j·T eT]j·TT kÕe÷õø£ $wüj·÷\ e\düyê<ä+, kÕÁe÷»´yê<ä+ e\¢ @s¡Œ&˚ Á|üe÷<Ûë
sê»˙‹ XÊÁkÕÔìï sê»´ n<Ûä´j·Tq XÊÁdüÔ+>± |ü]>∑ ñ<ëVü≤s¡D:- J$+#˚ Vü≤≈£îÿ, kÕ«‘·+Á‘·´|ü⁄ Vü≤≈£îÿ, ÄdæÔ ne>±Vü≤qqT n+~düTÔ+~. \qT ‘Ó*|æ, Á|ü|ü+#· XÊ+‹ neX¯´ø£‘·qT $e]düTÔ+~.
DÏ+#ês¡T. sêC≤´ìøÏ bÂs¡T\øÏ eT<Ûä´>∑\ düìïVæ≤‘· Vü≤≈£îÿ. uÛ≤<Ûä´‘·\˝À |üqTï\ #˚*¢+|ü⁄, XÊdüq ii). Á|üuÛÑT‘·«+-|ü]bÕ\q |ü]C≤„D+: sê»´+˝À Á|üuÛÑT‘·« viii). sê»ø°j·T ne>±Vü≤qqT ô|+b˛+~+#·&É+: sê»
dü+ã+<Ûëìï á XÊÁdüÔ+ ‘Ó\T|ü⁄‘·T+~. n˝≤π> sê»´ $<Û˚j·T‘· ˝≤+{Ï$ ñHêïsTT. |ü]bÕ\q |ü{Ïwüºe+‘·+>± ìs¡«Væ≤+#˚+<äT≈£î bÕ\ ˙‹ XÊÁdüÔ n<Ûä´j·Tq+ <ë«sê ùd«#·Ã¤, düe÷q‘·«+,
ne‘·s¡D dæ<ë›+‘ê\qT $e]düTÔ+~. sê»´+ vi). C≤rj·T, n+‘·sê®rj·T n+XÊ\ n<Ûä´j·Tq+: sê»˙‹ ≈£î\≈£î sê»ø°j·T Hêj·T≈£î\≈£î sê»˙‹ XÊÁdüÔ Hê´j·T+ e+{° sê»ø°j·T n+XÊ\≈£î dü+ã+~+∫q
>∑‘·+˝À m˝≤ ñ+&˚~, es¡Ôe÷q+˝À m˝≤ ñ+~, XÊÁdüÔ+ 20e X¯‘êãΔ+ qT+&ç Á|ü|ü+#·+˝Àì $$<ä |ü]C≤„q+ m+‘√ nedüs¡+. |ü]bÕ\q j·T+Á‘ê+>∑+, |ü]C≤„Hêìï Á|ü‹ ˇø£ÿs¡T bı+<äe#·TÃ. n˝≤π>
uÛÑ$wü´‘Y ˝À m˝≤ ñ+&ÉuÀ‘·T+~ nH˚ $wüj·÷\qT <˚XÊ\ eT<Ûä´ dü+ã+<Ûë\qT #·]ÃdüTÔ+~. Á|ü#·Ãqï Á|üC≤dü+ã+<ë\ ìs¡«Vü≤D, kÕúìø£ dü+düú\ ìs¡«Vü≤D bòÕdæ»+, kÕeT´yê<ä+, ø£eT÷´ì»+ ˝≤+{Ï uÛ≤eC≤
sê»˙‹ XÊÁdüÔ+ $X‚¢wædüTÔ+~. j·TT<ä›+, ÁbÕã\´düeT‘Í\´+, ìsêj·TT~Ûø£s¡D, ‘·~‘·s¡ n+XÊ\ô|’ á XÊÁdüÔ+ #·ø£ÿ{Ï ne>±Vü≤q ˝≤\ düeTÁ>∑yÓTÆq ne>±Vü≤qqT bı+<äe#·TÃ. |òü*‘·+
iii) Á|üuÛÑT‘·« n<Ûä´j·Tq+: sê»˙‹ XÊÁdüÔ+ Á|üuÛÑT‘·« n<Ûä´j·T <Í‘·´˙‹ ‘·~‘·s¡ n+XÊ\qT $e]düTÔ+~. Ä<ÛäTìø£ ø£*ŒdüTÔ+~. >± Á|ü»\˝À sê»ø°j·T #Ó’‘·q´+ ô|s¡T>∑T‘·T+~.
HêìøÏ dü+ã+~+∫q XÊÁdüÔeTì d”ºô|òHé ©ø±ø˘ sêC≤´\T ˇ+≥] sêC≤´\T ø±e⁄. edüTÔ ùde\ iii). Á|üC≤kÕ«eT´ $\Te\ düe÷#ês¡+: sê»˙‹ XÊÁdüÔ
eT]j·TT d”©\T ù|s√ÿHêïs¡T. Á|üuÛÑT‘·«eTH˚~ sê»´|ü⁄ m>∑TeT‹, ~>∑TeT‹ $wüj·T+˝À Ç‘·s¡ sêC≤´\ô|’ n<Ûä´j·Tq+ sê»´+, Á|üuÛÑT‘·«+, C≤‹, C≤rj·T‘·,
to be continued

Sarla Devi Scholarship kÕÿ\sYwæ|t‡ ry, carpentry, gardening and tailoring.

The applicant's family income shouldn't
students looking forward to study in any
of the listed 200+ partner universities of
Category: Means Based be more than INR 4.5L p.a. and in 1st MPOWER in US or Canada to support
Scholarship: Sarla Devi Scholarship year of UG program. their career aspirations.
for Students in Crisis 2019 http://www.b4s.in/nava/SDS6
Category: Merit cum Means Based Prizes & Rewards: Candidates pursuing Eligibility: The applicant must be 18
Description: Students undergoing any science and engineering will be facili- years or older, who has been accepted or
kind of personal crisis causing hindra Scholarship: Sarla Devi Scholarship
2019 tated with INR 75K, arts stream stu- enrolled for a full-time program at US
nce in their higher education are invited dents will be granted INR 20K, com- or Canada based partnered college or
by Dharampal Satyapal Charitable Trust Description: Dharampal Satyapal Char
itable Trust announces this scholarship merce and CA, CS students who have university. Short essay response of 1000
to apply for this scholarship. This call cleared the first level examination will words is expected from the applying
aims to support families dwindling with open to meritorious but financially
under-privileged students to allow them be given INR 25K and INR 15K for candidates on “What impact do you
finances and help them with their wards' vocational studies. The applicant must want to make in the world with your
higher studies post class 9. a chance for a better future. Students of
all the streams are welcomed under this be in first year of under-graduation pro- U.S. or Canadian education?” and “If
Eligibility: Applicants who are orphan, gram in any recognized college or uni- you could improve integration between
physically challenged, cancer patients, scholarship invite.
Eligibility: For science related courses, versity in Delhi NCR. international students and domestic stu-
or have single parent (only mother) or Last Date to apply: March 31, 2019 dents (from the U.S./Canada) on your
have any chronic illness (candidate or the applicant must be pursuing MBBS,
Engineering or Nursing from any reput- Application: Only online applications campus, how would you do so?”
immediate family member) can send are accepted. Prizes & Rewards: Scholarship worth
their entries. The applicant's highest ed college with minimum 75% in class
12. Arts students who are studying LLB, Short Source URL: USD 5000 p.a. is offered to the selected
family income should be INR 4.5L p.a. http://www.b4s.in/nava/SDS5 scholars.
and can be pursuing any stream of study. Psychology or Mass Communication
and minimum 75% in class 12 can Last Date to apply: April 15, 2019
Prizes & Rewards: 100 desirous and MPOWER Global Citizen Application: Online applications only.
apply. Commerce related students must
needy students will be awarded with Scholarship Short Source URL:
INR 10K. be aspiring for CA, CS and minimum
Category: International Level http://www.b4s.in/nava/MGC1
Last Date to apply: March 31, 2019 80% in class 12. For vocational courses,
Scholarship: MPOWER Global Citizen
Application: Applications can be sent the candidate who is a drop-out looking
Scholarship Program 2019
online only. forward to pursue employability with Courtesy: www.buddy4study.com /
Description: MPOWER Financing has
Short Source URL: mechanics, electrician, plumber, mason-
proposed this scholarship opportunity to Call: 08448709545, 08527484563
 eT+>∑fi¯yês¡+ 22
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f¯]lÐ]lÇ l 22 l 2019 11

Find the distance between the centres A, B of..

Ķæ*Æý‡ÏVýSyýlz Ð]l¯]l…Æ>k
çÜ*PÌŒæ AíÜòÜt…sŒæ,
{糿¶æ$™èlÓ E¯]l²™èl ´ëuý‡Ô>Ë,
¼.M>Å…‹³, MýSÆý‡*²Ë$.
Tangents and Secants to a circle
1. If the circle and a line
have no point in
O
common, then we say
that the line is a non-
intersecting line with Ð]l*Å£ýlÐðl$sìæMŠæÞ (C….Ò$.)
respect to the circle.
2. If the circle and a line BN2 + NQ2 = BQ2 Let AB be a chord of the larger circle
O
external point A. If PAQ = 80°
have two common (∵ Pythagoras theorem) which touches the small circle at P. Then
points, then we say that then Find OPQ, OQP AM2 + 152 = 172, AP = PB, OP ⊥ AB
line is a secant of the Sol: Given that PAQ = 80° ⇒ AM = 17 2 − 152 = 8cm
circle. AP = AQ BN2+122= 132
3. If the circle and a line (∵ The lengths of the tangents drawn from BN = 132 − 122 = 5cm
have only one point in O
an external point to a circle are equal) But AM = AC, BN = BC cm O
common, then we say ∴ AC = 8cm; BC = 5cm 13
12cm
P
that the line is a (∵ Radii of the same circle are equal) A B
tangent to the circle. AB = AC+BC = 8+5 = 13cm
P
80°
O
A
The common point of the tangent, and the Applying Pythagoras theorem in ΔOPA,
Model Problem:
circle is called the point of contact. Q
we have
» The tangent at any point of � The length of a tangent from a point A at OA2 = OP2 + AP2
a circle is perpendicular to ΔAPQ is an isosceles triangle distance 5cm from the centre of the circle ⇒132 = 122 + AP2
the radius through the is 4cm. Find the radius of the circle. ⇒ AP = 13 2 − 12 2 = 5cm
i.e. APQ = AQP .......... (1)
O
point of contact. Ans: 3cm ∴AB=2AP = 2×5=10cm.
» No tangent can be drawn to a circle from a also APQ + AQP + PAQ = 180° 4. In the figure, a circle touches all the four
Model Problem:
point lying inside the circle. APQ + APQ + 80 ° = 180 ° (∵ from 1)
sides of a quadrilateral ABCD with
» There is one and only one tangent to a AB= 6cm, BC=7cm CD= 4cm. Find AD. � In two concentric circles, prove that all
circle passing through a point lying on the ⇒ APQ = 50° .......... (2) chords of the outer circle which touch the
R
circle. i.e. OP⊥AP D C inner are of equal length.
» Exactly two tangents can be drawn to a 6. A circle is inscribed in a ΔABC having
OPA = 90°
circle from a point lying outside the circle. sides 8cm, 10cm, 12cm. as shown in
» The length of tangents drawn from an ⇒ OPQ + APQ = 90 � S Q figure. Find AD, BE, CF.
O
external point to a circle are equal. OPQ + 50° = 90° (∵ from 2) Sol: We know the tangents drawn from an
external point to a circle are equal.
Important Problems OPQ = 40° C
1. In figure, If AP, AQ are the two tangents to A P B z
similarly we can prove that z
a circle with centre 'O'. Sol: Given that AB = 6cm, BC = 7cm CD= F
OQP = 40°
E
so that POQ = 130° then 4cm.
x
Since tangents drawn from an exterior y
PAQ is equal to ––––– Model Problems:
A
point to a circle are equal in length. A
P
� Find AQB, AMB from the figure. ∴AP = AS, BP = BQ, x D y B
Ans: 60°, 120° CR= CQ, DR= DS Adding AD = AF = x say;
130° A (AP+BP) + (CR+DR) = AS+BQ+CQ+DS BD = BE = y say;
O
Q AB + CD = (AS+DS)+(BQ+CQ) CE = CF = z say;
P 60° M O Q AB + CD = AD+ BC Given that AB = 12cm, BC = 8cm,
6+4 = AD+7 (∵ Given) CA = 10cm
AD = 3cm. i.e., x+y=12, y+z=8, z+x=10
Sol: Given POQ = 130° B
Adding
Model Problems:
we know that OPA = 90° � If PTR = 50° then find (x+y) + (y+z) + (z+x) = 12+8+10
OQA = 90 ° A circle touches all the four sides of a 2(x+y+z) = 30, x+y+z=15.
PSQ , OPQ
�
Ans: 65°, 25° quadrilateral ABCD. Prove that AB+CD= x = (x+y+z) – (y+z) = 15 – 8 =7
(∵ Angle between tangent, radius through Q BC+DA. similarly we get y= 5cm, z=3cm
the point of contact is 90°) � If all the side of a parallelogram touch a i.e. x = AD= 7cm, y=BE= 5cm
In quadrilateral OPAQ circle. Show that the parallelogram is a z= CF= 3cm
S O T
POQ + OQA + OPA + PAQ = 360 ° rhombus.
130° + 90°+ 90°+ PAQ = 360 ° � Prove that opposite sides of
P a quadrilateral circu-
⇒ PAQ = 50° 3. From the figure find the distance between mscribing a circle subtend
the centres A, B of circles. supplementary angles at the
N centre of the circle.
Model Problem:
5. Two concentric circles are
12
cm

17 cm
� If Tangents PA, PB from a point P to a P C B 13 cm Q of radii 13cm, 12cm. Find
A
circle with centre O are inclined to each 15
cm
the length of the chord of
other at angle of 60°, then POB = M the larger circle which
Ans: 60° touches the smaller circle.
–––––––.
2. Two tangents AP, AQ are drawn to a circle Sol: AMP, BNQ = 90 ° Sol: Let 'O' be the centre of the
with centre O from an (∵Angle between radius, tangent) concentric circles of radii
AM2 + PM2 = AP2, 13cm, 12cm respectively.