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F F F F F F F F F F F F F F F F

17ñªëÅ]î¦ô¢Ù šúšídÙñô¢ª 8, 2021 n e-mail:pratibhadesk@eenadu.net

Y-Break

On 1 September 2021, the Union

AYUSH Minister Sarbananda Sonowal

launched ‘Y Break’ mobile application

in New Delhi.

The Y-Break (Yoga Break) mobile

application is a unique 5 minute Yoga

protocol app, designed especially for

working professionals to de-stress,

refresh and re-focus at their workplace

to increase their productivity, and con-

sists of Asanas, Pranayam and

Dhyana.

Parliamentary Outreach

Programme

On 27 August 2021 Lok Sabha

Speaker Om Birla inaugurated the

Parliamentary Outreach Programme

for the empowerment of Panchayat Raj

Institutions of the Union Territory of

Ladakh at the Sindhu Sanskritik

Kendra, Leh.

DefExpo 2022

On 30 July 2021, the Union Ministry of

Defence declared that India’s flagship

military exhibition, DefExpo, will be held

at Gandhinagar in Gujarat in March

2022.

Recently Government of India noti-

fied a list of 108 defence items that can-

not be imported by the armed forces.

The list, called the ‘positive indigenisa-

tion list’, will be implemented progres-

sively from December 2021 to

December 2025.

The aim is to build upon the vision to

achieve ‘Aatmanirbharta’ in defence

and reach the $5bn defence exports

target by 2024.

‘e-Source’

On 30 August 2021, in a press release

the Indian Institute of Technology (IIT)

Madras said that it is working on devel-

oping an innovative digital model to

address the problem of e-waste (elec-

tronic wastes).

The digital platform, named ‘e-Source,’

will act as an exchange platform, to serve

as an online marketplace for Waste

Electrical and Electronic Equipment

(WEEE). According to studies, currently,

the world annually generates 53.6 million

tonnes of e-waste, which is expected to

double in the next 16 years.

SVEEP

The Election Commission of India

organized the Systematic Voters’

Education and Electoral Participation

(SVEEP) Consultation Workshop

between August 25-26, 2021.

The workshop reviews state SVEEP

plans, conduct extensive deliberations

on the important aspects of it and for-

mulate a comprehensive strategy for

future elections. The SVEEP program

is the flagship program of the Election

Commission of India for voter educa-

tion and awareness.

Writer

Amara Veeraswamy

Subject Expert

India’s first smog tower was installed in...

GeneralAwareness

GeneralAwareness

GeneralAwarenessGeneral

AwarenessGeneral

AwarenessGeneral

AwarenessGeneral

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AwarenessGeneral

AwarenessGeneral

AwarenessGeneral

AwarenessGeneral

Awareness

National

Eastern Economic Forum

On 3 September 2021, Prime Minister

Narendra Modi addressed the 6th

Eastern Economic Forum (EEF) virtu-

ally. The plenary session of the forum

was organized in Vladivostok, Russia.

At the meeting, PM reiterated

India’s commitment as part of its "Act

East Policy’ of being a reliable partner

of Russia in this regard. He also men-

tioned India’s partnership with Russia

in space exploration through the

Gaganyaan Programme.

BRICS meeting on

National Security

On 24 August 2021, the 11th Meeting

of the BRICS High Representatives

Responsible for National Security was

held via video conferencing.

The National Security Advisor of

India Ajit Doval hosted the meeting

since India is the chair for the 2021

BRICS summit.

‘WHO Hub for Pandemic and

Epidemic Intelligence’

On 1 September 2021, ‘WHO Hub for

Pandemic and Epidemic Intelligence’

was set up by WHO under its Health

Emergencies Programme. It was inau-

gurated in Berlin by Germany by

Chancellor Angela Merkel and WHO

chief Tedros Adhanom Ghebreyesus.

This hub will perform the work to

analyze information on emerging pan-

demic threats, filling the gaps exposed

by COVID-19.

“KAZIND-21”

On 1 September 2021, the India-

Kazakhstan joint military exercise

“KAZIND-21” at Training Node Aisha

Bibi, Kazakhstan. Both the contin-

gents will share their expertise and

skills in the field of counter-terror oper-

ations.

India-UK Economic and

Financial Dialogue

On 2 September 2022, the 11th India-

UK Economic and Financial Dialogue

was held virtually. Finance Minister

Nirmala Sitharaman and the UK

Treasury Chancellor Rishi Sunak co-

chaired the meeting.

The Dialogue covered discussions

on economic cooperation on multilater-

al issues, including G-20 and COP-26.

The dialogue made special empha-

sis on Fin-Tech and GIFT City, annual

India-UK Financial Market Dialogue

and measures underway to reform

financial markets.

International

Stockholm World Water Week

Between 23-27 August 2021,

Stockholm World Water Week was held

in online by the Stockholm International

Water Institute (SIWI).

‘Building Resilience Faster’ is the

theme of the week.

Surat Municipal Corporation the civic

body from India participated in the

panel discussion on the topic “Zero

Liquid Discharge Cities”.

Tokyo Paralympics 2020

Tokyo Paralympics held between 24

August and 5 September 2021.

Someity is the mascot of Tokyo

Paralympics 2020. India is represented

by its biggest contingent with 54 ath-

letes competing in 9 sports - archery,

athletics, badminton, canoeing, shoot-

ing, swimming, powerlifting, table ten-

nis and taekwondo.

Indian shot-putter Tek Chand was

the flag-bearer for the Indian contin-

gents at the opening ceremony.

China got a total of 207 medals in

which 96 Gold, 60 Silver and 51 Bronze

are there.

India got 5 Gold medals, 8 Silver

medals and 6 Bronze medals with a

total of 19 medals.

Avani Lekhara got a Gold in Shooting

Women's 10m Air Rifle Standing SH1.

She also won bronze in Women's 50m

Rifle 3 Positions SH1.

Sumit Antil got a Gold medal in Men's

Javelin Throw- F64. Pramod Bhagat

and Krishna Nagar got gold medals in

Badminton in Men’s Singles SL3 and

SL6 respectively.

Manish Narwal also got Gold medal in

shooting in P4 - Mixed 50m Pistol SH1.

The Paralympics featured around 4500

athletes from 163 countries competing in

540 events in 22 sports. On 5 September

2021, the closing ceremony was held

where Avani Lekhara was India's flag

bearer during the athlete’s parade.

International Day for People of

African Descent

The inaugural International Day for

People of African Descent is being cel-

ebrated on August 31, 2021, by the

United Nations.

This day was created to promote the

extraordinary contributions of Africans

and members of the African diaspora

around the world and is an opportunity

to focus on eliminating all forms of dis-

crimination against people of African

descent.

Others

1. Hakainde Hichilema became the

new president of........

1) South Africa 2) Peru

3) Zambia 4) Sweden

5) Haiti

2. New Chairperson of “Stop TB

Partnership Board” is......

1) Narender Modi

2) Ramnath Kovind

3) Kiren Rijiju

4) Mansukh Mandaviya

5) Dharmendra Pradhan

3. International Army Games 2021

hosted by......

1) India 2) USA 3) France

4) Russia 5) UK

4. India’s first smog tower was

installed in........

1) Mumbai 2) Pune 3) Chennai

4) Delhi 5) Kolkata

5. The book “Balakot Air Strike: How

India Avenged Pulwama” authored

by.......

1) Ved Prakash Malik

2) Sudha Murthy

3) Mukul Deva

4) Shivam Shankar Singh

5) Manan Bhatt

6. Army Sports Institute’s stadium

named after Neeraj Chopra located

in........

1) Pune 2) Nagpur

3) Kolkata 4) Chennai

5) Hyderabad

7. National Sports day observed on......

1) 18 August 2) 31 August

3) 1 September 4) 29 August

5) 15 August

8. Which animal named as the state

animal by the Union Territory of

Ladakh?

1) Black buck 2) Jaguar

3) Snow leopard 4) Red panda

5) Stag

9. Pankaj Kumar Singh is the new

Director General of........

1) BSF 2) CISF 3) CRPF

4) RPF 5) Assam Rifles

10. Who was appointed as the new

Governor of Manipur?

1) L.Murugan 2) TK Rangarajan

3) La.Ganesan 4) Vaidhyanathan

5) N.S.R.Pillai

11. Barcelona Open chess tourna-

ment won by........

1) Aravindh Chithambaram

2) S.P.Sethuraman

3) Krishnan Sasikiran

4) Srinath Narayanan

5) A.Balasundaram

12. World Mosquito Day observed

on........

1) 12 August 2) 23 August

3) 20 August 4) 24 August

5) 25 August

13. World Water Week observed

globally in 2021 from ........ (The

annual celebration is organised by

international organisation SIWI)

1) August 3 to 7

2) August 13 to 17

3) August 23 to 27

4) August 5 to 10

5) August 7 to 11

Model Questions

1-3 2-4 3-4 4-4 5-5 6-1 7-4

8-3 9-1 10-3 11-2 12-3 13-3

Key

For more Information see

today’s Telangana Edition

1. tanθ =

1 11) 2)

√

√

cosec2θ + 1 cosec2θ − 1

1 13) 4)

√

√

1 − cosec2θ − 1 − cosec2θ

þ§-ëÅ]ì: cosec2θ − cot2θ = 1

⇒ cot2θ = cosec2θ − 1

⇒ cotθ = √

cosec2θ − 1

1⇒ tanθ =

√cosec2θ − 1 ú£-÷«-ëůìÙ: 2

2. ∆ABC ÖÚÛ õÙñ-ÚÁé vAòÅ¡ªáÙ, ∠B = 90°, a = 24

óŸ´Eåªx, b = 25 óŸ´Eåªx, ∠BAC = θ Íô³ê¶cosθ = ........

25 24 7 71) 2) 3) 4)

24 25 24 25

þ§-ëÅ]ì: ∆ABC ÖÚÛ õÙñ-ÚÁé vAòÅ¡ªáÙ, ∠B = 90°,

a = 24 óŸ´Eåªx ⇒ BC = 24 óŸ´Eåªx b = 25 óŸ´Eåªx ⇒ AC = 25 óŸ´Eåªx í£åÙ ìªÙ#,AC2 = AB2 + BC2

⇒ 252 = AB2 + 242

⇒ AB2 = 625 − 576 = 49

⇒ AB = √49 = 7 óŸ´Eåªx

AB 7cosθ = =

AC 25 ú£-÷«-ëůìÙ:- 43. sinθ.cosθ (tanθ + cotθ) = ........

1) 0 2) 3 3) 2 4) 1

þ§-ëÅ]ì: sinθ.cosθ (tanθ + cotθ)

sinθ cosθ= sinθcosθ( + )

cosθ sinθ

sin2θ + cos2θ= sinθcosθ()

sinθcosθ

= sin2θ + cos2θ = 1

ú£-÷«-ëůìÙ:- 41 − tan2θ

4. = ........ cot2θ − 1

1) tanθ 2) tan2θ 3) cot2θ 4) cotθ

1 − tan2θ 1 − tan2θþ§-ëÅ]ì: =

cot2θ − 1 1 − 1 tan2θ

1 − tan2θ= = tan2θ

1 − tan2θ

tan2θ ú£-÷«-ëůìÙ:- 2

√

√

1 + sinθ 1 − sinθ5. + = ......

1 − sinθ 1 + sinθ

1) 2secθ 2) secθ + tanθ

3) cosθ − cotθ 4) 2tanθ

√

√

1 + sinθ 1 + sinθ 1 + sinθþ§-ëÅ]ì: = ×

1 − sinθ 1 − sinθ 1 + sinθ

√

√

(1 + sinθ)2 (1 + sinθ)2= =

1 − sin2θ cos2θ

1 + sinθ 1 sinθ= = +

cosθ cosθ cosθ

= secθ + tanθ ú£-÷«-ëůìÙ:- 21

6. cos4θ − sin4θ = Íô³ê¶ θ Nõª÷? 2

1) 30° 2) 60° 3) 45° 4) 90°

þ§-ëÅ]ì: cos4θ − sin4θ = (cos2θ)2 − (sin2θ)2

= (cos2θ + sin2θ) (cos2θ − sin2θ)

= 1 × cos2θ = cos2θ

1Ï-í£±pè[ª cos4θ − sin4θ =

2

⇒ cos2θ = cos60° ⇒ 2θ = 60°

60°⇒ θ = = 30°

2 ú£-÷«-ëůìÙ:- 1sin30°. sec60° + cos30°. cosec60°

7. = ......sec45°. cot45°. cose45°

1) 2 2) 1 3) √3 4) √

6

sin30°. sec60° + cos30°. cosec 60°þ§-ëÅ]ì:

sec45°. cot45°. cose45°

1 √3 2

× 2 + × 2 2 √

3 1 + 1 2

= = = = 1√

2 × 1 × √

2 2 2

ú£-÷«-ëůìÙ:- 28. tan24°. tan42°. tan48°. tan66° N-õª÷ ÓÙêŸ?-

11) √

3 2) 3) 1 4) 0

√3

þ§-ëÅ]ì: tan24°. tan42°. tan48°. tan66°

= tan24°. tan42°. tan(90° − 42°). tan(90° − 24°)

= tan24°. tan42°. cot42°. cot24°

1 1= tan24°. tan42°. . = 1

tan42° tan24°

ú£-÷«-ëůìÙ:- 39. sin25°. cos65° + cos25°. sin65° Nõª÷ .-.-.-.-

1) sin40° 2) sin60° 3) 1 4) 0

þ§-ëÅ]ì: sin25°. cos65° + cos25°. sin65°

sinAcosB + cosAsinB = sin(A + B)

= sin (25° + 65°) = sin 90° = 1

ú£-÷«-ëůìÙ :- 310. sin6A + cos6A + 3sin2A.cos2A = ......

1) 1 2) −1 3) 8 4) 2

þ§-ëÅ]ì: sin6A + cos6A + 3sin2A.cos2A

= (sin2A)3 + (cos2A)3 + 3sin2A cos2A

(sin2A + cos2A)

a3 + b3 + 3ab(a + b) = (a + b)3

= (sin2A + cos2A)3

= 13 = 1 ú£-÷«-ëůìÙ:- 111. tan30°. tan45°. tan60°õª Ú¨ÙC î¦æ¨ö˺ Ô

vøË™-èÅ…ö˺ ÑÙæ°ô³? 1) ÍÙÚÛ-vøË™èÅ… 2) Þœªé-vøË™èÅ… 3) ô¦-êŸtÚÛvøË™èÅ… 4) ÔD-Ú¥ë]ª

þ§ëÅ]ì: a = tan30°, b = tan45°, c = tan60°

1a = tan30° = ,

√3

b = tan45° = 1

c = tan60° = √3

1b2 = (1)2 = 1 ac = () √

3 = 1 √

3

b2 = ac Í-ô³ê¶ a, b, cõª Þœªé-vøË™èÅ… (G.P)ö˺ÑÙæ°ô³.

ú£-÷«-ëůìÙ:- 212. sin (60° − x) = cos (y + 60°) Í-ô³ê¶

sin (x − y) N-õª÷....

1 1 √3

1) 2) 1 3) 4) √

2 2 2

þ§-ëÅ]ì: sinA = cosB ⇒ A + B = 90°

sin (60° − x) = cos (y + 60°)

⇒ (60° − x) + (y + 60°) = 90°

⇒ 120° − x + y = 90°

⇒ (x − y) = 120° − 90° = 30°

∴ x − y = 30°

sin (x − y) = sin 30°

1=

2 ú£-÷«-ëůìÙ:- 3

√secθ − 1

13. = ......secθ + 1

1) cotθ + cosecθ 2) secθ − tanθ

3) cosecθ − cotθ 4) tanθ − secθ

√

√

secθ − 1 (secθ − 1) (secθ − 1)þ§-ëÅ]ì: = ×

secθ + 1 (secθ + 1) (secθ − 1)

√ (secθ − 1)2 secθ − 1 secθ − 1

= = = sec2 − 1 √

tan2θ tanθ

secθ 1 = −

tanθ tanθ

1 cosθ 1 1 1

= − = − sinθ tanθ sinθ tanθcosθ

= cosecθ − cotθ ú£-÷«-ëůìÙ:- 3

vAÚÁ-é-NªA÷³Ý°uÙ-ø‹õª

vAÚÁ-é-NªA Eù£p-꟪hõª (Trigonometric Ratios)

∆ ABC ÖÚÛ õÙñÚÁé vAòÅ¡ªáÙ, ∠B = 90° Í-ô³ê¶

Ó-ë]ªæ¨ òÅ¡ªáÙ BCsinA = - =

ÚÛô¢gÙ AC

Î-ú£-ìo-òÅ¡ªáÙ ABcosA = =

ÚÛô¢gÙ AC

Ó-ë]ªæ¨ òÅ¡ªáÙ BCtanA = =

Îú£ìo òÅ¡ªáÙ AB

ÚÛô¢gÙ ACcosecA = =

Óë]ªæ¨ òÅ¡ªáÙ BC

ÚÛô¢gÙ ACsecA = =

Îú£ìo òÅ¡ªáÙ AB

Îú£ìo òÅ¡ªáÙ ABcotA = =

Óë]ªæ¨ òÅ¡ªáÙ BC

vA-ÚÁ-é-NªA Eù£p-꟪hõ ÷ªëÅ]u ú£ÙñÙëÅ]Ù (Relation between trigonometric ratios):

1 1sinA = öË¶ë¯ cosecA =

cosecA sinA

1 1cosA = öË¶ë¯ secA =

secA cosA

1 1tanA = öË¶ë¯ cotA =

cotA tanA

sinA cosAtanA = öË¶ë¯ cotA =

cosA sinA

÷«uëÇ]æ̈ÚÂq

ô¢àŸô³êŸúˆšï°àÂ. ô¦ëůÚÛ”ù£gNù£óŸª Eí£±éªõª

sin6A + cos6A + 3sin2A.cos2A = ...

÷«-CJ ú£÷ª-ú£uõª

ð¼MúÃ, ÏêŸô¢ ð¼æ© í£K¤Ûõ ví£ê¶uÚÛÙ

C

B A

C

25

θ

24

B A

vA-ÚÁ-é-Nª-BóŸª ú£ô¢y-ú£-Oª-ÚÛ-ô¢-é°õª1) sin2 A + cos2A = 1

2) sec2A − tan2A = 1

3) cosec2A − cot2A = 1

√

F sin2A + cos2A = 1 ⇒ sinA = 1 − cos2A

ö˶ë¯

√

cosA = 1 − sin2A

√

F sec2A − tan2A = 1 ⇒ secA = 1 + tan2A

ö˶ë¯

√

tanA = sec2A − 1

F cosec2A − cot2A = 1 ⇒ cosecA

√

= 1 + cot2A

ö˶ë¯

√

cotA = cosec2A − 1

* sin(A + B) = sinA cosB + cosA sinB

* sin(A − B) = sinA cosB − cosA sinB

* cos(A + B) = cosA cosB − sinA sinB

* cos(A − B) = cosA cosB + sinA sinB

tanA + tanB* tan(A + B) =

1 − tanA tanB

tanA − tanB* tan(A − B) =

1 + tanA tanB

* sin(A + B) + sin(A − B) = 2 sinA cosB

* sin(A + B) − sin (A − B) = 2 cosA sinB

* cos(A + B) + cos(A − B) = 2 cosA cosB

* cos(A − B) − cos(A + B) = 2 sinA sinB

C + D C − D* sinC + sinD = 2 sin () cos ()

2 2

C + D C − D* sinC − sinD = 2 cos () sin ()

2 2

C + D C − D* cosC + cosD = 2 cos () cos ()

2 2

C + D D − C* cosC − cosD = 2 sin () sin ()

2 2

2 tanA* sin 2A = 2 sinA cosA =

1 + tan2A

* cos2A = cos2A − sin2A = 2 cos2A − 1

1 − tan2A= 1 − 2 sin2A =

1 + tan2A

2tanA* tan2A =

1 − tan2A

* sin3A = 3sinA − 4sin3A

* cos3A = 4cos3A − 3cosA

* sin (90° − A) = cosA, * cos (90° − A) = sinA

* tan (90° − A) = cotA

* sin (180° − A) = sinA

* cos (180° − A) = −cosA

* tan (180° − A) = −cotA

÷³-Üu-iì ú£«vê¦õª

Pankaj Kumar Singh

GH

F F F F F F F F F F F F F F F F

17ñªëÅ]î¦ô¢Ù šúšídÙñô¢ª 8, 2021 n e-mail:pratibhadesk@eenadu.net

NADS

On 31 August 2021, the Indian Navy

signed a contract with Bharat

Electronics Limited (BEL) for the supply

of the first indigenous comprehensive

Naval Anti Drone System (NADS), with

both 'hard kill' and 'soft kill' capabilities.

The NADS, developed by DRDO and

manufactured by BEL, is the first

indigenously developed anti-drone sys-

tem to be inducted into the Indian

Armed Forces.

The NADS can instantly detect and

jam micro drones and use a laser-based

kill mechanism to terminate targets.

Renovated Jallianwala

Bagh memorial complex

On 28 August 2021 PM Modi inaugu-

rated renovated Jallianwala Bagh

memorial complex in Amritsar.

The complex pays tribute to the great

martyrs so that history may always

remember their sacrifice and our pre-

sent and future generations can draw

inspiration from their patriotism.

A crowd of thousands had gathered

at Jallianwala Bagh in Amritsar on April

13, 1919, to peacefully protest against

the arrest of two nationalist leaders.

Hundreds of Indians were killed that

day when British soldiers blocked all

exits and started firing on them.

General Dyer ordered the soldiers not

to stop firing until all their ammunition

was exhausted.

INS Vigraha

On 28 August 2021, Defence minister

Rajnath Singh commissioned the

indigenously built Coast Guard Ship

‘Vigraha’ in Chennai, Tamil Nadu.

The ship was designed and built

indigenously by the Larsen & Toubro

Ship Building Limited will have a per-

manent base in Visakhapatnam,

Andhra Pradesh.

The ICGS Vigraha is the seventh and

final ship in the series of Offshore

Patrol Vessels (OPVs).

‘Gandiv’

National Security Guard(NSG) con-

ducted a counter-terrorism exercise

called ‘Gandiv’ from August 22 to

August 28, 2021, The exercise was

held at multiple cities in Uttar Pradesh,

Madhya Pradesh, Gujarat and the

national capital region.

The exercise aimed at enhancing the

counter-terror skills of the state adminis-

tration, police, anti-terrorist squads and

also create awareness among the public.

The counter-terrorism exercise

aimed to "validate the planning para-

meters" of the commando force in the

event of a terrorist attack, hijack of an

aircraft or hostage situation.

SAMRIDH programme

On 25 August 2021, Union Minister of

Electronics & Information Technology

Ashwini Vaishnaw launched the

“Startup Accelerator of MeitY for

Product Innovation, Development and

Growth (SAMRIDH)” programme.

The objective of this programme is to

create a conducive platform for Indian

Software Product start-ups to enhance

their products and securing invest-

ments for scaling their business.

This programme will focus on accel-

erating the 300 start-ups by providing

customer connect, investor connect,

and international immersion in the next

three years. Investment of up to Rs.40

lakh will be provided to the start-up

through selected accelerators.

Writer

Amara Veeraswamy

Subject Expert

National

2021 Ramon Magsaysay Awards

2021 Ramon Magsaysay Awards

were declared on 31 August 2021.

Established in 1957, the award is con-

sidered Asia’s highest honour.

Dr. Firdausi Qadri, the senior scien-

tist of Bangladesh was selected for her

commitment to vaccine development

that saved millions of lives. she is

credited with developing an affordable

oral cholera vaccine and the typhoid

conjugate vaccine for adults, children,

and infants.

Dr. Amjad Saqib a prominent micro-

finance pioneer from Pakistan has

been chosen for the outstanding

efforts in restoring human dignity, alle-

viating poverty and helping over 3 mil-

lion families to come out of poverty

through the zero-interest lending

model.

Dr. Steven Muncy was chosen for the

Award helped the displaced refugees of

South-East Asia and has rebuilt their

lives. He belongs to the USA.

Philippines fisherman and commu-

nity environmentalist Roberto "Ka

Dodoy" Ballon was named as the

awardee for conservation of natural

resources and help communities

improve their income through liveli-

hood programs.

Watchdoc Indonesia was conferred

with the award for emergent leader-

ship and for using journalism and film-

making to give voice to the people.

World Athletics U20

Championships

2021 World Athletics U20

Championships also known as the

World Junior Championships, was

held from 18 to 22 August 2021 at the

Moi International Sports Centre in

Nairobi, Kenya.

Host nation Kenya won the highest

number of 16 medals (8 Gold, 1 Silver

and 7 Bronze).

India won a total of 3 Medals ( 2 sil-

ver and 1 bronze), stood in 21st place

in the medal tally table.

17-year-old Shaili Singh won the sil-

ver medal in the women's long jump,

missed the gold medal by 1 cm, Maja

Askag of Sweden won the gold medal.

Amit Khatri won the silver medal in

the men's 10,000 m race walk at the

World Athletics U20 Championships in

Nairobi. He finished with a time of 42

minutes 17.94 seconds.

India’s 4 x 400m mixed relay team

comprises Bharat Sridhar, Priya

Mohan, Summy and Kapil won the

bronze medal .

Cali city in Colombia will host the

2022 World Athletics U20

Championships.

International Climate Summit

On 3 September 2021 India hosted the

“International Climate Summit”. The key

conference was held in a bid to strength-

en the dialogue for the country’s transi-

tion to clean energy with a focus on

‘Powering India's Hydrogen Ecosystem'.

International Experts, Policymakers,

regulators, experts and industry lead-

ers participated in the summit.

India elected to UPU bodies

On 27 August 2021, India was elected

to the Council of Administration(CA) at

the 27th Universal Postal Union (UPU)

Congress in Abidjan, Ivory Coast.

India with 134 votes got the highest

votes in CA elections from South Asia

and the Oceania region. India was also

elected with 106 votes out of 156 coun-

tries to the Postal Operations Council

(POC). Universal Postal Union head-

quarters is located in Switzerland.

Ants, Avocados, robotic arm to ISS

On August 29, 2021, SpaceX launched

shipments of ants, avocados, human-

sized robotic arm toward the

International Space Station (ISS). This

is SpaceX’s 23rd mission for NASA in a

decade.

The recycled Falcon rocket was

used, which carrying the Dragon cap-

sule is carrying more than 2,170 kilo-

grams of supplies and experiments as

well fresh food like avocados, lemons

and even ice cream for seven astro-

nauts of the space station.

"Operation Devi Shakti"

India's mission to evacuate its citizens

and Afghan partners from Kabul has

been named "Operation Devi Shakti".

India evacuated the Indian embassy

staff from Kabul on 17 August 2021.

G20 Conference on Women’s

Empowerment

On 26 August 2021, Union Minister of

Women & Child Development Smriti Irani

addressed the First ever G20 Ministerial

Conference on Women’s Empowerment

held in Italy in a hybrid format.

Times World University

Rankings 2022

On 2 September 2021, in the Times

World University Rankings 2022, three

Indian educational institutes were

placed in the top 400.

While IISc Bangalore has been

placed at 301-350, the Indian Institute

of Technology (IIT) Ropar has been

placed in the 351-400 bracket. IIT

Indore is in the 401-500 bracket.

UK's University of Oxford has topped

the list for the sixth consecutive year.

It include more than 1,600 universi-

ties across 99 countries and territories.

The University of Delhi has been

ranked on the 801-1,000 bracket.

New Head of Estonia

On 31 August 2021, Estonia’s parlia-

ment elected Alar Karis as the coun-

try's new President after the second

round of voting.

He will succeed Kersti Kaljulaid,

Estonia’s first female president.

Others

International

Model Questions

Key

1-3 2-3 3-2 4-3 5-3 6-1 7-3

1. World’s second-largest National

Gene Bank was inaugurated in......

1) Mumbai 2) Pune 3) Delhi

4) Kochi 5) Bhubaneswar

2. Who was appointed as the Deputy

Leader of the House in Rajya

Sabha?

1) Dharmendra Pradhan

2) Bhupender Yadav

3) Mukhtar Abbas Naqvi

4) JP Nadda 5) S.Jai Shankar

3. 2021 Spilimbergo Open chess

tournament won by.....

1) Viswanath Anand

2) Raunak Sadhwani

3) Anurag Mhamal

4) Nihal Sarin 5) Gukesh D

4. Muhyiddin Yassin resigned as the

Prime Minister of......

1) Afghanistan 2) Turkey

3) Malaysia 4) Kuwait

5) Maldives

5. Which state launched the

‘Prarthana Scheme’ providing

Rs.1 lakh to the next of kin of

deceased Covid19 patients?

1) Meghalaya

2) Arunachal Pradesh

3) Assam 4) Manipur 5) Tripura

6. First ever Indian selected as the

Honorary Member of the

International Astronomical Union

(IAU) is.....

1) Dorje Angchuk

2) Arjun Tatake

3) Chrisphin Karthick

4) K.L.Pradip Kumar

5) Rangaraya K

7. Which city’s World Heritage status

in the UK was stripped of after a

UN committee found develop-

ments threatened the value of the

city's waterfront?

1) London 2) Birmingham

3) Liverpool 4) Cambridge

5) Nottingham

For more Information see

today’s AP Edition

Which nation hosted

International Climate Summit?

Dr. Firdausi

Qadri

1. tanθ =

1 11) 2)

√

√

cosec2θ + 1 cosec2θ − 1

1 13) 4)

√

√

1 − cosec2θ − 1 − cosec2θ

þ§-ëÅ]ì: cosec2θ − cot2θ = 1

⇒ cot2θ = cosec2θ − 1

⇒ cotθ = √

cosec2θ − 1

1⇒ tanθ =

√cosec2θ − 1 ú£-÷«-ëůìÙ: 2

2. ∆ABC ÖÚÛ õÙñ-ÚÁé vAòÅ¡ªáÙ, ∠B = 90°, a = 24

óŸ´Eåªx, b = 25 óŸ´Eåªx, ∠BAC = θ Íô³ê¶cosθ = ........

25 24 7 71) 2) 3) 4)

24 25 24 25

þ§-ëÅ]ì: ∆ABC ÖÚÛ õÙñ-ÚÁé vAòÅ¡ªáÙ, ∠B = 90°,

a = 24 óŸ´Eåªx ⇒ BC = 24 óŸ´Eåªx b = 25 óŸ´Eåªx ⇒ AC = 25 óŸ´Eåªx í£åÙ ìªÙ#,AC2 = AB2 + BC2

⇒ 252 = AB2 + 242

⇒ AB2 = 625 − 576 = 49

⇒ AB = √49 = 7 óŸ´Eåªx

AB 7cosθ = =

AC 25 ú£-÷«-ëůìÙ:- 43. sinθ.cosθ (tanθ + cotθ) = ........

1) 0 2) 3 3) 2 4) 1

þ§-ëÅ]ì: sinθ.cosθ (tanθ + cotθ)

sinθ cosθ= sinθcosθ( + )

cosθ sinθ

sin2θ + cos2θ= sinθcosθ()

sinθcosθ

= sin2θ + cos2θ = 1

ú£-÷«-ëůìÙ:- 41 − tan2θ

4. = ........ cot2θ − 1

1) tanθ 2) tan2θ 3) cot2θ 4) cotθ

1 − tan2θ 1 − tan2θþ§-ëÅ]ì: =

cot2θ − 1 1 − 1 tan2θ

1 − tan2θ= = tan2θ

1 − tan2θ

tan2θ ú£-÷«-ëůìÙ:- 2

√

√

1 + sinθ 1 − sinθ5. + = ......

1 − sinθ 1 + sinθ

1) 2secθ 2) secθ + tanθ

3) cosθ − cotθ 4) 2tanθ

√

√

1 + sinθ 1 + sinθ 1 + sinθþ§-ëÅ]ì: = ×

1 − sinθ 1 − sinθ 1 + sinθ

√

√

(1 + sinθ)2 (1 + sinθ)2= =

1 − sin2θ cos2θ

1 + sinθ 1 sinθ= = +

cosθ cosθ cosθ

= secθ + tanθ ú£-÷«-ëůìÙ:- 21

6. cos4θ − sin4θ = Íô³ê¶ θ Nõª÷? 2

1) 30° 2) 60° 3) 45° 4) 90°

þ§-ëÅ]ì: cos4θ − sin4θ = (cos2θ)2 − (sin2θ)2

= (cos2θ + sin2θ) (cos2θ − sin2θ)

= 1 × cos2θ = cos2θ

1Ï-í£±pè[ª cos4θ − sin4θ =

2

⇒ cos2θ = cos60° ⇒ 2θ = 60°

60°⇒ θ = = 30°

2 ú£-÷«-ëůìÙ:- 1sin30°. sec60° + cos30°. cosec60°

7. = ......sec45°. cot45°. cose45°

1) 2 2) 1 3) √3 4) √

6

sin30°. sec60° + cos30°. cosec 60°þ§-ëÅ]ì:

sec45°. cot45°. cose45°

1 √3 2

× 2 + × 2 2 √

3 1 + 1 2

= = = = 1√

2 × 1 × √

2 2 2

ú£-÷«-ëůìÙ:- 28. tan24°. tan42°. tan48°. tan66° N-õª÷ ÓÙêŸ?-

11) √

3 2) 3) 1 4) 0

√3

þ§-ëÅ]ì: tan24°. tan42°. tan48°. tan66°

= tan24°. tan42°. tan(90° − 42°). tan(90° − 24°)

= tan24°. tan42°. cot42°. cot24°

1 1= tan24°. tan42°. . = 1

tan42° tan24°

ú£-÷«-ëůìÙ:- 39. sin25°. cos65° + cos25°. sin65° Nõª÷ .-.-.-.-

1) sin40° 2) sin60° 3) 1 4) 0

þ§-ëÅ]ì: sin25°. cos65° + cos25°. sin65°

sinAcosB + cosAsinB = sin(A + B)

= sin (25° + 65°) = sin 90° = 1

ú£-÷«-ëůìÙ :- 310. sin6A + cos6A + 3sin2A.cos2A = ......

1) 1 2) −1 3) 8 4) 2

þ§-ëÅ]ì: sin6A + cos6A + 3sin2A.cos2A

= (sin2A)3 + (cos2A)3 + 3sin2A cos2A

(sin2A + cos2A)

a3 + b3 + 3ab(a + b) = (a + b)3

= (sin2A + cos2A)3

= 13 = 1 ú£-÷«-ëůìÙ:- 111. tan30°. tan45°. tan60°õª Ú¨ÙC î¦æ¨ö˺ Ô

vøË™-èÅ…ö˺ ÑÙæ°ô³? 1) ÍÙÚÛ-vøË™èÅ… 2) Þœªé-vøË™èÅ… 3) ô¦-êŸtÚÛvøË™èÅ… 4) ÔD-Ú¥ë]ª

þ§ëÅ]ì: a = tan30°, b = tan45°, c = tan60°

1a = tan30° = ,

√3

b = tan45° = 1

c = tan60° = √3

1b2 = (1)2 = 1 ac = () √

3 = 1 √

3

b2 = ac Í-ô³ê¶ a, b, cõª Þœªé-vøË™èÅ… (G.P)ö˺ÑÙæ°ô³.

ú£-÷«-ëůìÙ:- 212. sin (60° − x) = cos (y + 60°) Í-ô³ê¶

sin (x − y) N-õª÷....

1 1 √3

1) 2) 1 3) 4) √

2 2 2

þ§-ëÅ]ì: sinA = cosB ⇒ A + B = 90°

sin (60° − x) = cos (y + 60°)

⇒ (60° − x) + (y + 60°) = 90°

⇒ 120° − x + y = 90°

⇒ (x − y) = 120° − 90° = 30°

∴ x − y = 30°

sin (x − y) = sin 30°

1=

2 ú£-÷«-ëůìÙ:- 3

√secθ − 1

13. = ......secθ + 1

1) cotθ + cosecθ 2) secθ − tanθ

3) cosecθ − cotθ 4) tanθ − secθ

√

√

secθ − 1 (secθ − 1) (secθ − 1)þ§-ëÅ]ì: = ×

secθ + 1 (secθ + 1) (secθ − 1)

√ (secθ − 1)2 secθ − 1 secθ − 1

= = = sec2 − 1 √

tan2θ tanθ

secθ 1 = −

tanθ tanθ

1 cosθ 1 1 1

= − = − sinθ tanθ sinθ tanθcosθ

= cosecθ − cotθ ú£-÷«-ëůìÙ:- 3

vAÚÁ-é-NªA÷³Ý°uÙ-ø‹õª

vAÚÁ-é-NªA Eù£p-꟪hõª (Trigonometric Ratios)

∆ ABC ÖÚÛ õÙñÚÁé vAòÅ¡ªáÙ, ∠B = 90° Í-ô³ê¶

Ó-ë]ªæ¨ òÅ¡ªáÙ BCsinA = - =

ÚÛô¢gÙ AC

Î-ú£-ìo-òÅ¡ªáÙ ABcosA = =

ÚÛô¢gÙ AC

Ó-ë]ªæ¨ òÅ¡ªáÙ BCtanA = =

Îú£ìo òÅ¡ªáÙ AB

ÚÛô¢gÙ ACcosecA = =

Óë]ªæ¨ òÅ¡ªáÙ BC

ÚÛô¢gÙ ACsecA = =

Îú£ìo òÅ¡ªáÙ AB

Îú£ìo òÅ¡ªáÙ ABcotA = =

Óë]ªæ¨ òÅ¡ªáÙ BC

vA-ÚÁ-é-NªA Eù£p-꟪hõ ÷ªëÅ]u ú£ÙñÙëÅ]Ù (Relation between trigonometric ratios):

1 1sinA = öË¶ë¯ cosecA =

cosecA sinA

1 1cosA = öË¶ë¯ secA =

secA cosA

1 1tanA = öË¶ë¯ cotA =

cotA tanA

sinA cosAtanA = öË¶ë¯ cotA =

cosA sinA

÷«uëÇ]æ̈ÚÂq

ô¢àŸô³êŸúˆšï°àÂ. ô¦ëůÚÛ”ù£gNù£óŸª Eí£±éªõª

sin6A + cos6A + 3sin2A.cos2A = ...

÷«-CJ ú£÷ª-ú£uõª

ð¼MúÃ, ÏêŸô¢ ð¼æ© í£K¤Ûõ ví£ê¶uÚÛÙ

C

B A

C

25

θ

24

B A

vA-ÚÁ-é-Nª-BóŸª ú£ô¢y-ú£-Oª-ÚÛ-ô¢-é°õª1) sin2 A + cos2A = 1

2) sec2A − tan2A = 1

3) cosec2A − cot2A = 1

√

F sin2A + cos2A = 1 ⇒ sinA = 1 − cos2A

ö˶ë¯

√

cosA = 1 − sin2A

√

F sec2A − tan2A = 1 ⇒ secA = 1 + tan2A

ö˶ë¯

√

tanA = sec2A − 1

F cosec2A − cot2A = 1 ⇒ cosecA

√

= 1 + cot2A

ö˶ë¯

√

cotA = cosec2A − 1

* sin(A + B) = sinA cosB + cosA sinB

* sin(A − B) = sinA cosB − cosA sinB

* cos(A + B) = cosA cosB − sinA sinB

* cos(A − B) = cosA cosB + sinA sinB

tanA + tanB* tan(A + B) =

1 − tanA tanB

tanA − tanB* tan(A − B) =

1 + tanA tanB

* sin(A + B) + sin(A − B) = 2 sinA cosB

* sin(A + B) − sin (A − B) = 2 cosA sinB

* cos(A + B) + cos(A − B) = 2 cosA cosB

* cos(A − B) − cos(A + B) = 2 sinA sinB

C + D C − D* sinC + sinD = 2 sin () cos ()

2 2

C + D C − D* sinC − sinD = 2 cos () sin ()

2 2

C + D C − D* cosC + cosD = 2 cos () cos ()

2 2

C + D D − C* cosC − cosD = 2 sin () sin ()

2 2

2 tanA* sin 2A = 2 sinA cosA =

1 + tan2A

* cos2A = cos2A − sin2A = 2 cos2A − 1

1 − tan2A= 1 − 2 sin2A =

1 + tan2A

2tanA* tan2A =

1 − tan2A

* sin3A = 3sinA − 4sin3A

* cos3A = 4cos3A − 3cosA

* sin (90° − A) = cosA, * cos (90° − A) = sinA

* tan (90° − A) = cotA

* sin (180° − A) = sinA

* cos (180° − A) = −cosA

* tan (180° − A) = −cotA

÷³-Üu-iì ú£«vê¦õª

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