The total number of rectangles in the given figure is,

Rectangles: ABCD, AMND, ALKD, AIJD, MLKN, LIJK, MIJN, IBCJ, LBCK, MBCN, EFGH, EFLI, EFJK, KJGH, LIHG, NKOP, MLPO

Hence, “17” is the correct answer.

The paper when folded appears like,

Hence, figure 3 is the correct answer.

According to the given information,

Let the printed price of the product be ‘x’,

After giving 20% discount on the printed price,

The product price will be 0.8x.

The shopkeeper gave an additional 10 % discount on 0.8x.

Hence, the product price will become 0.72x.

A customer bought the product at Rs. 468

∴ 0.72x = 468

∴ x = 650

Hence, printed price of the given product is Rs. 650.

Bread is made from wheat.

Similarly,

Jaggery is made from unrefined sugar and is obtained by boiling raw, concentrated sugarcane.

Hence, “jaggery” is the correct answer.

The pattern followed here is,

1537 + 2 = 1539;

1539 + 4 = 1543;

1543 + 6 = 1549;

1549 + 8 = 1557;

1557 + 10 = 1567

Hence, “1549” is the correct answer.

According to the alphabetical positions of the letters,

The pattern followed here is,

D + 2 = F; F + 2 = H; H + 2 = J; J + 2 = L

A + 2 = C; C + 2 = E; E + 2 = G; G + 2 = I

G + 2 = I; I + 2 = K; K + 2 = M; M + 2 = O

Hence, “HEK” is the correct answer.

The logic follows here is :-

According to the given figures,

‘*’ is opposite to ‘+’

‘#’ is opposite to ‘%’

And,

‘=’ will be on the face opposite to ‘$’

Hence, “$” is the correct answer.

The pattern followed here is,

72 : 14

7 × 2 = 14

And,

96 : 54

9 × 6 = 54

Similarly,

87 : ?

8 × 7 = 56

Hence, “56” is the correct answer.

The pattern followed here is,

(First number + 7) = Second number, (Second number + 21) = Third number

(17, 24, 45) → 17 + 7 = 24, 24 + 21 = 45

Similarly,

Option 1 → (12, 19, 40) → 12 + 7 = 19, 19 + 21 = 40

Option 2 → (15, 20, 43) → 15 + 5 = 20, 20 + 23 = 43

Option 3 → (18, 23, 46) → 18 + 5 = 23, 23 + 23 = 46

Option 4 → (19, 26, 34) → 19 + 7 = 26, 26 + 8 = 34

‘(12, 19, 40)’ shares the same relationship as that shared by given number set.

Hence, “(12, 19, 40)” is the correct answer.

According to the given information,

The shaded part represents people who are married, but not engineers = 16 + 13 = 29.

Hence, “29” is the correct answer.

According to the given information,

A + B means ‘A is the wife of B’;

A - B means ‘B is the daughter of A’;

A × B means ‘B is the brother of A’;

A ÷ B means ‘A is the father of B’;

"P + R × T - Q + S ÷ U" imply that P is the wife of R, T is the brother of R, Q is the daughter of T, Q is the wife of S and S is the father of U.

Drawing the diagram for P + R × T – Q + S ÷ U

Q is the mother of U and P is the aunt of Q.

Thus, P is the aunt to the mother of U.

Hence, “aunt” is the correct answer.

Mistake Points

We have to find out the relation between P and mother of U, not P and U.

The pattern followed is,

Similarly,

Hence, “NFOUTOE” is the correct answer.

Arranging the states according to the population,

Uttar Pradesh has the highest population.

Uttar Pradesh is followed by Madhya Pradesh.

Madhya Pradesh is followed by Andhra Pradesh.

Andhra Pradesh is followed by Himachal Pradesh.

Himachal Pradesh is followed by Arunachal Pradesh.

The correct logical order is,

Uttar Pradesh (4) → Madhya Pradesh (2) → Andhra Pradesh (1) → Himachal Pradesh (5) → Arunachal Pradesh (3)

Hence, option 1 is the correct answer.

Interchanging the signs and using BODMAS rule,

Option 1 → × and –

121 + 11 × 42 - 6 ÷ 7 = 83

121 + 11 × 42 – 0.85 = 83

121 + 462 – 0.85 = 83

583 – 0.85 is not equal to 83

Option 2 → + and ÷

121 ÷ 11 – 42 × 6 + 7 = 83

11 – 42 × 6 + 7 = 83

11 – 252 + 7 = 83

18 – 252 is not equal to 83

Option 3 → ÷ and ×

121 + 11 – 42 ÷ 6 × 7 = 83

121 + 11 – 7 × 7 = 83

121 + 11 – 49 = 83

132 – 49 = 83

Option 4 → – and ÷

121 + 11 ÷ 42 × 6 - 7 = 83

121 + 0.26 × 6 - 7 = 83

121 + 1.56 – 7 = 83

122.56 – 7 is not equal to 83

Hence, option 3 is the correct answer.

The pattern followed here is,

Number: (Number + 1)² + 2

Option 1 → 14 : 227 → 14 : 15² + 2

Option 2 → 11 : 146 → 11 : 12² + 2

Option 3 → 17 : 290 → 17 : 17² + 1

Option 4 → 10 : 123 → 10 : 11² + 2

All follow the same pattern, except ’17 : 290’

Hence, “17 : 290” is the different pair.

Chennai is the capital of Tamil Nadu.

Gangtok is the capital of Sikkim.

Hyderabad is the capital of Telangana.

Whereas,

Aurangabad is a city in Maharashtra.

Hence, “Aurangabad” is the odd one out.

The pattern followed here is the given symbols in the images are rotating in an anti-clockwise direction and circle is rotating 90º clockwise.

The arrow and pentagon are getting flipped, The triangle is shifted as it is.

The image which will come next in the pattern is,

Hence, figure 4 is the correct answer.

According to the alphabetical positions of the letters,

The pattern followed here is,

Similarly,

Hence, “20921815” is the correct answer.

The least possible Vann diagram is as follows,

Conclusions:

1: No plumber is a singer → True, as all singers are dancers, no dancer is a plumber.

2: Some singers are plumbers → False, as all singers are dancers, no dancer is a plumber.

3: Some dancers are singers → True, as all singers are dancers

Hence, only conclusion 1 and 3 follow.

The pattern followed here is,

1. ABF → A + 1 = B; B + 4 = F

2. FGL → F + 1 = G; G + 5 = L

3. MNR → M + 1 = N; N + 4 = R

4. IJN → I + 1 = J; J + 4 = N

All follow the same pattern, except ‘FGL’.

Hence, “FGL” is the odd one out.

The correct mirror image of the given figure when a mirror is placed on the right side of the figure is,

Hence, figure 4 is the correct answer.

Dentist is a doctor who is qualified to treat diseases and other conditions that affect the teeth and gums.

Similarly,

Option 1 → Chemistry: Science → Chemistry is the branch of science concerned with the substances of which matter is composed.

Option 2 → Line: Circle → Line is a long narrow mark and circle is a geometric figure.

Option 3 → Biology: Astrology → Biology is the study of living organisms and Astrology is the study of celestial bodies.

Option 4 → Algebra: Geometry → Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations and Geometry is a branch in mathematics.

‘Chemistry: Science’ share the same relationship as that shared by the given pair of words.

Hence, “chemistry: science” is the correct answer.

The pattern followed here is,

4 + 6 = 10

10 + 10 = 20

20 + 14 = 34

34 + 18 = 52

52 + 22 = 74

74 + 26 = 100

100 + 30 = 130

Hence, “74” is the correct answer.

The pattern followed here is,

A + 4 = E;

E + 5 = J;

J + 6 = P;

P + 7 = W

Hence, “P” is the correct answer.

The given figure is embedded in,

Hence, figure 2 is the correct answer.

• The International Monetary Fund was formed in 1944 and started on 27 December 1945 with the aim to ensure the stability of the international monetary system.
• It is an organization of 189 countries (members) and accountable to the members.
• The IMF ensures stability of following ways: Tracking Global Economy, Lending to countries with a balance of payment difficulties, providing practical help to members.
• It also oversees the international monetary system and monitors its member countries' economic and financial policies.
• It also has Special Drawing Rights (SDR) which are an international reserve asset. It is helpful to supplement its member countries’ official reserves.
• IMF HQ - Washington, D.C.
• IMF Managing Director and Chairman of the Executive Board - Kristalina Georgieva

• Iranian Major General Qasem Soleimani was recently (2020) assassinated by the US military in Iraq.
• He was killed in a U.S. airstrike in Baghdad.
• Qasem Soleimani had been a key figure of Iranian and the Middle East politics. He led a special force unit of Iran’s elite Revolutionary Guards.
• It was under his leadership when Iran expanded its military presence in Iraq and Syria.
• His killing caused a major escalation in tensions between the USA and Iran.
• As per the statement of the government of the USA, "Soleimani had been "developing plans to attack American diplomats and service members in Iraq and throughout the region".

• Alpenglow is the name of the phenomena (driven by the scattering of light) in which mountain tops acquire a rosy or orange hue around sunrise and sunset.
• In simpler terms, it means "redirected light".
• Alpenglow can be caused by direct or indirect sunlight. It mainly occurs when red light waves in sunlight reflect off of mountains back to an observer.
• The best time to observe this phenomenon in the summer months. During this time, the air is relatively still and clouds are minimal.
• Brillouin scattering is the interaction of light with the material waves in a medium.
• A circle of confusion is the measurement of a point of light that falls on the focal plane (camera sensor).
• Barrel distortion occurs when straight lines bend outward from the center of the image.

• Light energy to chemical energy: It is the energy conversion that happens in the process called photosynthesis.
• Photosynthesis is a phenomenon used by plants, algae and certain bacteria.
• There are two kinds of photosynthetic processes: Oxygenic photosynthesis and Anoxygenic photosynthesis.
• Light energy transfers electrons from water to carbon dioxide in order to produce carbohydrates during oxygenic photosynthesis. It functions as a counterbalance to respiration because it takes in the carbon dioxide which is produced by all breathing organisms and it reintroduces oxygen to the atmosphere.
• A slight difference in anoxygenic photosynthesis is that it uses electron donors other than water. It more commonly occurs in purple bacteria and green sulfur bacteria. It does not produce oxygen.

• Karun Nair has scored a triple century in test cricket other than Virender Sehwag.
• He struck the triple century on day four of the fifth Test in Chennai against England. He was only playing his third test and he scored 303 runs, not out in December 2016.
• Karun, although an International Player, plays domestic cricket from Karnataka side. He has also played in different teams in IPL.
• Virendra Sehwag became the first Indian to score a triple century on March 29. 2004 against Pakistan. The second time he hit a triple century on March 29, 2008, against South Africa.

• Frederick Sanger- the only scientist in the world to have won the Nobel prize in Chemistry twice.
• He was a medical practitioner by profession, born in Gloucestershire in South West England.
• His primarily work, however, focused on working with problems related to the determination of the structure of proteins. He later determined the structure of insulin.
• First Nobel Prize (in chemistry) - It was awarded for his work on the structure of proteins, especially that of insulin in 1958.
• Second Nobel Prize(in chemistry) - It was awarded into two halves. The first half went to Paul Berg "for his fundamental studies of the biochemistry of nucleic acids, with particular regard to recombinant - DNA". The other half was divided between Walter Gilbert and Frederick Sanger "for their contributions concerning the determination of base sequences in nucleic acids." It was in the year 1980.

• Bhavai and Kalbelia as traditional dance forms owe their genesis to-Rajasthan.
• Bhavai - It is a swaying and twirling movements performed by women but can also be performed by men. It involves tricky balancing acts such as balancing seven to nine brass pots on the head while standing on narrow or sharp objects.
• Kalbelia - It is more often associated with the tribe - Kalbelia of Rajasthan. The women sway, twirl and gyrate to the music and use acrobatic dance steps to showcase the flexibility and the different instruments are played by men.
• Assam Traditional Folk Dance - Bihu.
• Punjab Traditional Folk Dance - Bhangra and Giddha
• Haryana traditional Folk Dance - Dhamal, Phag, Jhumar, etc.

• The 3rd Khelo India Youth Games 2020 is hosted in Guwahati, Assam.
• It was held from 10 January 2020 to 22 January 2020. The event witnessed the participation of over 10,000 students.
• The event was organized by the Ministry for Youth Affairs and Sports in partnership with the School Games Federation of India, Indian Olympic Association and Assam State government.
• Mascots - Vijay the Tiger and Jaya the Blackbuck.
• Maharastra stood first with a total of 256 medals (78 gold, 77 silver, and 101 bronze).
• Haryana stood second with the total of 200 medals (68 gold, 60 silver, and 72 bronze).
• Delhi stood third with a total of 122 medals (39 gold, 36 silver, and 47 bronze).
• Torchbearer of opening ceremony - Sprinter Hima Das

• Zeeman Effect:- It is the name of the phenomena in physics and astronomy which involves the splitting of a spectral line into two or more components of slightly different frequencies when the light source is placed in a magnetic field.
• It is named after the Dutch physicist Pieter Zeeman.
• The effect is caused due to the distortion of the electron orbitals because of the magnetic field.
• This effect is very useful n investigating the charge/mass ratio and magnetic moment of an electron.
• Raman Effect:- It is nothing but the change in the wavelength of light that occurs whenever a light beam is deflected by molecules.
• Alpenglow Effect:- It is an atmospheric optical phenomenon. It makes mountains appear to glow in a light pink or red hue after the Sun sets or just before it rises.

• Governor of Kerala - Arif Mohammed Khan
• He had previously been a cabinet minister in the Union government handling the portfolios of Energy and Civil Aviation among others.
• Chief Minister of Kerala- Pinarayi Vijayan. Capital of Kerala-Thiruvananthapuram
• B.D. Mishra - He is the Governor of Arunachal Pradesh. (July 2021)
• Shri P.S. Sreedharan Pillai  - He is the Governor of Goa.
• Mangubhai C. Patel - He is the Governor of Madhya Pradesh.

• Four multi-storied illegal constructions - Jain Coral Cove, H20 Holy Faith, Alfa Serene, and Golden Kayaloram - were razed to the ground in January 2020 following Supreme Court instructions. It happened in Kerala.
• Around 850 kg of explosives were required for the demolition of the buildings. It was a controlled implosion.
• There were almost 343 waterfront flats which were built in the Maradu municipality. It violated the Coastal Regulation Zone norms.
• These buildings were constructed in a Coastal Regulation Zone. This zone was a part of the tidally-influenced water body in Kerala.
• The Supreme Court had directed the concerned authorities in September 2019 to demolish apartment complexes within 138 days.

• Ayushmann Khurrana and Vicky Kaushal - shared the National Best Actor Award (Male) in the 66th National Film Awards 2019.
• It was the 66th edition of the National Film Awards.
• Ayushmann Khurana won for his performances in 'Andhadhun' whereas Vicky Kaushal won for his performance in 'Uri: The Surgical strike".
• South Indian star Keerthi Suresh won the the National award for Best Actress for her performance in ‘Mahanati’. This movie is a biopic of the legendary Telugu actress Savitri.
• Best Feature Film Award-Gujarati film ‘Hellaro’
• Best Popular Film-‘Badhaai Ho’
• Best Film on Social Issues-‘Padman’ starring Akshay Kumar
• National award for Best Director-Aditya Dhar. He directed "Uri: The Surgical strike".

• The city of Vijayawada lies on the banks of the Krishna river.
• Vijayawada is in Andhra Pradesh.
• Krishna River originates from Mahabaleshwar near Jor village in the Satara District, Maharashtra.
• It drains into the Bay of Bengal.
• It has a total length of about 1,290 km. It also has a very highly fertile delta.
• Left Tributaries of Krishna river-Bhima, Dindi, Peddavagu, Halia, Musi, Paleru.
• Right Tributaries of Krishna river-Venna, Koyna, Panchganga, Dudhganga, Ghataprabha, Malaprabha, Tungabhadra.
• Dams on Krishna river -Almatti Dam, Nagarjuna Sagar Dam,Srisailam Dam, etc.

• Saikat Majumdar is the author of the book - 'The Scent of God'.
• It is a fictional story of students studying in a boarding school run by a Hindu monastic order in late-twentieth-century in India.
• Saikat Majumdar has written two other books - Silverfish and The Firebird. He has also written a self-help book - College: Pathways of Possibility
• Books by Githa Hariharan - The Thousand Faces of Night, Battling for India: A Citizen's Reader, I Have Become the Tide, Fugitive Histories, etc.
• Books by Himanjali Sankar - The Lies we tell, Talking of Muskaan
• Books by Nayantara Sehgal - Plans for Departure, Mistaken Identity, Lesser Breeds, etc.

• Ebix Inc. has acquired an online travel portal Yatra Online Inc., for an enterprise value of $337.8 million, in an all-stock transaction.
• Ebix Inc.is a supplier of on-demand software and e-commerce services. It is more focused on Business to Business Segment whereas Yatra Online is more focused on Business to Customer segment.
• However, it was decided that Yatra would continue to be independently run and would retain its brand and status in India.
• EbixCash is an  Indian subsidiary of the company. It had bought Mumbai-based Mercury Travels and Delhi-based Leisure Corp with an aim to create a travel division which would be greatly focused on luxury-, events- and sports-related travellers.
• Ebix chairman, president and CEO- Robin Raina

• The recently revoked 'Article 370' is associated with Jammu and  Kashmir.
• Under this article, the state of J&K had certain autonomy- separate constitution, separate flag and freedom to make laws. However, Foreign affairs, defense and communications of the state were still under the authority of the Central Government.
• After the scrapping of Article 370, it would have to abide by the Indian constitution much like any other state.
• All Indian laws are automatically applicable to people residing there and people from outside the state would be able to buy property there as it was not allowed as per the state's constitution.
• This move is welcomed by many as the Central Government would now be able to run academic institutions, build proper infrastructure and construct factories for production, etc.
• Now the former state has been divided into Union Territories - Jammu & Kashmir and Ladakh.

• Lord William Bentinck was the then Governor-General of British India, when ‘Sati Pratha’ became illegal and punishable.
• Sati was a practice predominant in some regions of South Asia in which a widow was burnt alive (either forcefully or voluntarily)with her deceased husband's funeral pyre.
• On December 4, 1829, Governor-General Lord William Bentinck passed the Bengal Sati Regulation which made this horrifying practice illegal in all jurisdictions of India and subject to prosecution.

Key Points

• He is regarded as the ‘Father of Modern Western Education in India’.
• Abolition of Sati in 1829 AD.
• Suppression of Thugi (1830 AD), curbed by colonel Sleeman.
• Deposition of Raja of Mysore and annexation of his territories (1831 AD).
• He was the first Governor-General of Bengal.
• First Medical College was opened in Calcutta in 1835 AD.
• Treaty of Friendship with Ranjit Singh (1831 AD).

Additional Information

 Lord Dalhousie (1848 AD-1856 AD):

• Abolished Titles and pensions, Widow Remarriage Act (1856 AD).
• Wars: Introduced Doctrine of Lapse; Captured Satara (1848 AD), Jaitpur and Sambalpur (1849 AD), Baghat (1850 AD), Udaipur (1852 AD), Jhansi (1853 AD), and Nagpur (1854 AD); Fought 2nd Anglo-Sikh War (1848 AD-1849 AD) and annexed the whole of Punjab; Annexation of Berar in 1853 Annexation of Awadh in 1856 AD on charges of mal-administration.
• Santhal uprisings (1855 AD-1856 AD).
• Nationalist Educational Institutions were founded.
• Administrative Reforms: Raised Gurkha regiments.
• Educational Reforms:
• Wood’s Educational Despatch of 1854 AD.
• An Engineering College was established at Roorkee.
• Public Works: Started the first railway line in 1853 AD; Started electric telegraph service.
• Laid the basis of the modern Postal System (1854 AD); A separate public works department was set up for the first time.

Lord Cornwallis (1786 AD-1793 AD )

• First-person to codify laws (1793 AD).
• He introduced Izaredari System in 1773 AD.
• He started the permanent settlement of Bengal.
• He created the post of District Judge. He is called the Father of Civil Services in India.
• Third Anglo-Mysore War and the Treaty of Srirangapatnam.
• He undertook police reforms.

Lord Wellesley (1798 AD-1803 AD ):

• Introduced the system of the Subsidiary alliance. Madras Presidency was formed during his tenure.
• Fourth Anglo-Mysore War in 1799 AD, Tipu Sultan died.
• First subsidiary treaty with Nizam of Hyderabad.
• In 1800 AD, he set up Fort William College in Calcutta. He was famous as Bengal Tiger.
• He brought the censorship of the Press Act, 1799 AD.
• Treaty of Bassein in 1802 AD.

• Ramachandra Guha is the author of the delightful and anecdotal history of Indian cricket titled - 'A Corner of a Foreign Field: The Indian History of a British Sport '.
• In this book, he included how cricket, a foreign sport, has inculcated itself in every street and corner of our country.
• He shares biography and history of the sport with several funny anecdotes.
• There are also snippets of many cricketers' life and their interviews.
• Guha has a special research interest in cricket, polity, environment, etc.
• Other books by him - Gandhi before India, India after Gandhi, Savaging the civilized, etc.
• Books by Romila Thapar - Somanatha, Ancient Indian social history, The Past before us, etc.
• Books by Bipin Chandra - India's struggle for Independence, India since Independence, India after Independence, etc.

• The correct sequence of important Mughal rulers: Babur ⇒ Humayun ⇒ Akbar ⇒ Jahangir ⇒ Shah Jahan ⇒ Aurangzeb.
• Mughal Dynasty was predominant in Northern India and ruled from the early 16th to the mid-18th century.
• Babur was the founder of the dynasty. He overthrew Lodhi dynasty in the First Battle of Panipat.
• It was during the Mughal dynasty when famous monuments were constructed in India- Taj Mahal, Lal Quilla, Humayun's Tomb, Bibi ka Maqbara, Safdarjung Tomb.
• However, Mughals had to fight several battles in order to maintain their ruling in Northern India.

• On the Day of equinox, the entire earth experience equal days and nights.
• An equinox occurs when the plane of Earth's equator passes through the center of the Sun. It takes place twice in a year-March 21 and September 23. However, the dates can sometimes vary by a day or two because of difference in leap and non-leap years.
• Winter Solstice-It is the shortest period of daylight and longest night of the year. In this, the Earth's poles has its maximum tilt away from the Sun.
• It occurs twice a year. It occurs once in Northern Hemisphere and once in Southern Hemisphere.
• Northern Hemisphere Winter Solstice: 21st December, Southern Hemisphere Winter Solstice: 21st June. Again, the dates can sometimes vary by a day or two because of difference in leap and non-leap years.
• Summer Solistice-It is the longest period of daylight and shortest night of the year. In this, the Earth's poles has its maximum tilt towards the Sun.
• It occurs twice a year. It occurs once in Northern Hemisphere and once in Southern Hemisphere.
• Northern Hemisphere Summer Solstice: 21st June, Southern Hemisphere Summer Solstice: 22nd December. Again, the dates can sometimes vary by a day or two because of difference in leap and non-leap years.

• Maitri is an Indian Research Station in the Antarctica Region.
• This station has been in operation in Antarctica Region since 1989.
• Bharati station has also been established in 2013.
• Several research and investigations are undertaken to understand the Polar processes and phenomenon at both the stations.
• Himadri is an Indian Arctic station.
• Orcadas is an Argentina based scientific station in Antarctica.
• Mawson station is Australia’s first continental station and the longest continuously operating station south of the Antarctic Circle.
• Hope Bay is the name of the bay in Antarctica.

• 2019 Padma Vibhushan winner, Teejan Bai hails from Chhattisgarh.
• She is a folk singer. She enacts tales from the Mahabharata performing a traditional art form Pandavani.
• Besides Padma Vibhushan, she had also been awarded the Padma Shri in 1987 and Padma Bhushan in 2003.
• She is a very respectable and known figure in Sangeet Natak Akademi.

• Former union minister and former Karnataka CM, D.V. Sadananda Gowda is a cabinet minister in the Chemicals and Fertilizers in current central cabinet (February-2020).
• Union Minster of Law and Justice-Ravi Shankar Prasad. He is also the Union Minister of Communications and Electronics and Information Technology.
• Union Minister of Consumer Affairs, Food and Public Distribution - Ramvilas Paswan (26 May 2014 – 8 October 2020)
• Union Minister of Tribal Affairs - Arjun Munda

• Nawab Abdul Latif Khan- is known as the 'father of Muslim renaissance' in Bengal.
• He has been compared to Raja Ram Mohan Roy as Nawab Abdul was considered as the greatest Muslim Reformer of his times.
• He persuaded Muslim Community who were opposed to English education due to a combination of social, economic and religious reasons to understand the importance of English education and take part in politics.
• He also founded Mohammadan Literary Society in April 1863.
• He also proposed a new syllabus for the madrassah that included arithmetic, algebra, geometry, philosophy with moral philosophy.

• Edwin Hubble's name is associated with "The Big Bang Theory".
• This theory gives an explanation of how our universe might have started in the first place.
• As per this theory, the universe that we know today started with a small singularity and then inflated over the next 13.8 billion years (present times).
• Edwin Hubble was an American Astronomer. His work involved extragalactic astronomy.
• The Hubble telescope, the first major optical telescope to be placed in space, is named after him.

7 cos²θ + 3sin²θ = 6,

⇒ 4 cos² θ + 3 cos²θ + 3sin²θ = 6

⇒ 4 cos² θ + 3 = 6

⇒ 4 cos² θ = 6 – 3

⇒ 4 cos² θ = 3

⇒ cos² θ = 3/4

⇒ cos θ = √(3/4) = √3/2

⇒ cos θ = cos 30

⇒ θ = 30

Now,

$\Rightarrow \frac{{{\rm{co}}{{\rm{t}}^2}2{\rm{\theta }} + {\rm{se}}{{\rm{c}}^2}2{\rm{\theta }}}}{{{\rm{ta}}{{\rm{n}}^2}2{\rm{\theta }} - {\rm{si}}{{\rm{n}}^2}2{\rm{\theta }}}}$

$\Rightarrow \frac{{{\rm{co}}{{\rm{t}}^2}60 + {\rm{se}}{{\rm{c}}^2}60}}{{{\rm{ta}}{{\rm{n}}^2}60 - {\rm{si}}{{\rm{n}}^2}60}}$

$\Rightarrow \frac{{{{\left( {\frac{1}{{\sqrt 3 }}} \right)}^2} + {2^2}}}{{{{\left( {\sqrt 3 } \right)}^2} - {{\left( {\frac{{\sqrt 3 }}{2}} \right)}^2}}}$

$\Rightarrow \frac{{\frac{1}{3} + 4}}{{3 - \frac{3}{4}}}$

$\Rightarrow \frac{{\frac{{13}}{3}}}{{\frac{9}{4}}}$

⇒ (13/3) × (4/9)

⇒ 52/27

Let efficiency of 1 man and 1 woman be x and y respectively.

Total work = (4x + 6y) × 5     ------ (1)

Total work = (3x + 4y) × 7     ------ (2)

From equation (1) and equation (2)

(4x + 6y) × 5 = (3x + 4y) × 7

⇒ 20x + 30y = 21x + 28y

⇒ 21x – 20x = 30y – 28y

⇒ x = 2y

⇒ x : y = 2 : 1

Efficiency of 1 man = 2

Efficiency of 1 woman = 1

Total work = (4x + 6y) × 5 = (4 × 2 + 6 × 1) × 5 = 70

2

 times of total work = 70 × 5/2 = 175

Let the number of men be n, then

According to the question

(2n + 1 × 25) 5 = 175

⇒ 2n + 25 = 175/5 = 35

⇒ 2n = 35 – 25 = 10

⇒ n = 10/2 = 5

So, the number of men are 5.

Let A buy an article be 100 unit,

A sell the article to B = 100 × (80/100) = 80 unit

B sell the article to C = 80 × (9/8) = 90 unit

C buy the article = 90 unit

C sell the article to D = 90 × (92/100) = 82.8 unit

D buy the article = 82.8 unit

Loss of A = 100 – 80 = 20 unit

Loss of C = 90 – 82.8 = 7.2 unit

Difference of loss A and C = 20 – 7.2 = 12.8 unit

⇒ 82.8 unit = Rs. 248.40

⇒ 1 unit = 248.40/82.8 = Rs. 3

⇒ 12.8 unit = Rs. 3 × 12.8 = 38.40

Concept Used:

cos(90° - θ) = sin θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

sin 2θ = 2sin θ cos θ

Calculation:

$\Rightarrow \frac{{\left( {cos9^\circ + sin81^\circ } \right)\left( {sec9^\circ + cosec81^\circ } \right)}}{{\left( {2si{n^2}63^\circ + 1 + 2si{n^2}{{27}^0}} \right)}}$

$\Rightarrow \frac{{\left[ {cos9^\circ + \cos \left( {90 - 81} \right)]} \right[sec9^\circ + \sec \left( {90 - 81} \right)]}}{{2si{n^2}63^\circ + 1 + 2[si{n^2}(90 - {{63}^0})]}}$

$\Rightarrow \frac{{\left[ {cos9^\circ + \cos 9]} \right[sec9^\circ + \sec 9]}}{{2;si{n^2}63^\circ + 1 + 2;co{s^2}{{63}^0}]}}$

$\Rightarrow \frac{{2cos9; \times 2sec9}}{{2\left( {si{n^2}63 + {\rm{co}}{{\rm{s}}^2}63} \right) + 1}}$

As we know,

⇒ cosθ secθ = 1

⇒ sin2θ + cos2 θ = 1

$\Rightarrow \frac{{2cos9; \times 2sec9}}{{2\left( {si{n^2}63 + {\rm{co}}{{\rm{s}}^2}63} \right) + 1}}$

⇒ 4/3

12x² – 21x+1 = 0

Divide by 4x

⇒ 3x – 21/4 + 1/4x = 0

⇒ 3x + 1/4x = 21/4

Squaring on both sides

⇒ 9x² + 1/16x² + 2 × 3x × (1/4x) = 441/16

⇒ 9x² + 1/16x² = 441/16 – 3/2

⇒ 9x² + 1/16x² = (441 – 24)/16

⇒ 9x² + 1/16x² = 417/16

From the following figure,

AP = 7 cm, PB = 5 cm, and AB = 10 cm

Let AE = x cm, so EB = (10 – x) cm

The length of the common chord PQ

In right angled ΔAPE,

AP² = PE² + AE²

PE² = 7² – x²      –––(1)

In right angled ΔPBE,

PB² = PE² + BE²

PE² = 5² – (10 – x)²

PE² = 25 – 100 – x² + 20x      ------ (2)

From equation (1) and equation (2)

7² – x² = 25 – 100 – x² + 20x

⇒ 20x = 100 + 49 – 25

⇒ x = 124/20

⇒ x = 31/5

From equation (1)

PE² = 49 – (31/5)² = 49 – 961/25

⇒ PE² = (1225 – 961)/25 = 264/25

⇒ PE = √264/25

⇒ PE = 2√66/5

As we know,

PQ = 2PE

⇒ PQ = 2 × 2√66/5

⇒ PQ = 4√66/5

Total number from 200 and 800 = 800 – 200 + 1 = 601      ----(including 200)

Total number from 1  to 200 (200 excluding) which are divisible by 5 = 199/5 ≈ 39

Total number from 1 to 800 (800 including) which are divisible by 5 = 800/5 = 160

Total number from 200 to 800 which are divisible by 5 = 160 – 39 = 121

Total number from 1  to 200 (200 excluding) which are divisible by 7 = 199/7 ≈ 28

Total number from 1 to 800 (800 including) which are divisible by 7 = 800/7 ≈ 114

Total number from 200 to 800 which are divisible by 7 = 114 – 28 = 86

Similarly, Total number from 200 to 800 which are divisible by 35 = 800/35 – 200/35 = 22 – 5 = 17

Number neither divisible by 5 nor 7 = 601 – 121 – 86 + 17 = 411

∴ Number neither divisible by 5 nor 7 is 411.

Given:

P = 8000

A = 9261

r = 10% (Interest is compounded half-yearly)

Formula Used:

A = $P(1+\frac{r}{n})^{nt}$

Where t = Time (in years) and n = Number of times interest is credited annually

Calculation:

Interest is compounded half-yearly

⇒ n = 2

According to the question,

9261 = 8000 × $(1+\frac{10}{100 × 2})^{2t}$

⇒ (9261/8000) = $(1+\frac{10}{200})^{2t}$

⇒ (9261/8000) = (210/200)^2t

⇒ (21/20)³ = (21/20)2t

Comparing the powers

t = 3/2 = 1.5 years

Now, CI on sum of Rs. 8000 for 1.5 years at the rate of 20% p.a

For simplicity, solving CI by tree method

⇒ CI = 1600 + 800 + 160 = Rs. 2560

Shortcut Trick

As we know,

P : A for 1 year = ∛8000 : ∛9261 = 20 : 21

Percentage = 1/20 × 100 = 5% (satisfied)

Time = 1 year 6 months

Now, rate = 20%

Rate for 1 year 6 month = 20 + 10 + (20 × 10)/100 = 30 + 2 = 32%

⇒ 32% of 8000

⇒ 2560

\( \Rightarrow \frac{{8 \div \left[ {\left( {8 - 3} \right) \div \left{ {\left( {4 \div 4of8} \right) + 4 - 4 \times 4 \div 8} \right} - 2} \right]}}{{8 \times 8 \div 4 - 8 \div 8;of2 - 7}}\)

\( \Rightarrow \frac{{8 \div \left[ {5 \div \left{ {\left( {4 \div 32} \right) + 4 - 4 \times \frac{1}{2}} \right} - 2} \right]}}{{8 \times 2 - 8 \div 16 - 7}}\)

\( \Rightarrow \frac{{8 \div \left[ {5 \div \left{ {\frac{1}{8} + 4 - 2} \right} - 2} \right]}}{{16 - \frac{1}{2} - 7}}\)

$\Rightarrow \frac{{8 \div \left[ {5 \div \frac{{17}}{8} - 2} \right]}}{{9 - \frac{1}{2}}}$

$\Rightarrow \frac{{8 \div \left[ {5 \times \frac{8}{{17}} - 2} \right]}}{{2 - \frac{1}{2} - 7}}$

$\Rightarrow \frac{{8 \div \left[ {\frac{{40}}{{17}} - 2} \right]}}{{\frac{{17}}{2}}}$

$\Rightarrow \frac{{8 \div \frac{6}{{17}}}}{{\frac{{17}}{2}}}$

$\Rightarrow \frac{{8 \times \frac{{17}}{6}}}{{\frac{{17}}{2}}}$

$\Rightarrow 8 \times \frac{{17}}{6} \times \frac{2}{{17}}$

⇒ 8/3

As we know

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

3² = 101 + 2 (xy + yz + zx)

⇒ 2 (xy + yz + zx) = 9 – 101

⇒ 2 (xy + yz + zx) = –92

⇒ (xy + yz + zx) = –92/2

⇒ (xy + yz + zx) = –46

Again, As we know,

x³ + y³ + z³ – 3xyz = (x + y + z) [(x + y + z)² – 3(xy + yz + zx)]

⇒ x³ + y³ + z³ – 3xyz = 3 [3² – 3 × (–46)] = 3 [9 + 138] = 3 × 147 = 441

⇒ √[x³ + y³ + z³ – 3xyz] = √441 = 21

Distance between A and B = 144

Let the speed of first and second car be x km/hr and y km/hr

If both cars going same direction, then relative speed = (x - y) km/hr

If both cars going opposite direction, then relative speed = (x + y) km/hr

As we know,

Speed = Distance/Time

According to the question

(x + y) = 144/(9/8)

⇒ (x + y) = 128      –––(1)

⇒ (x – y) = 144/12

⇒ (x – y) = 12      –––(2)

From equation (1) and equation (2)

⇒ x = 70

So, speed of first car = 70 km/hr

Ratio of sum of A, B, C and D = 1/3 : 1/5 : 1/6 : 1/9 = 90/3 : 90/5 : 90/6 : 90/9 = 30 : 18 : 15 : 10

According to the question

⇒ 18 – 10 = 8 unit

⇒ 8 unit = 832

⇒ 1 unit = 104

⇒ Total sum = 30 + 18 + 15 + 10 = 73 unit

⇒ 73 unit = 104 × 73 = 7592

So, total sum is 7592.

Short Trick:

Ratio of CP to MP = (100% – Discount) : (100 + Profit) = (100 – 10) : (100 + 8) = 90 : 108 = 5 : 6

Detailed Method:

Let the cost price be Rs. 100

SP of the article = 100 × (108/100) = Rs. 108

MP of the article = 108 × (100/90) = 120

∴ Ratio of CP to MP = 100 : 120 = 5 : 6

sec θ + tan θ = P       ----(1)

As we know,

(sec²θ – tan² θ) = 1

⇒ (sec θ + tan θ) (sec θ – tan θ) = 1

⇒ sec θ – tan θ = 1/p      ----(2)

From equation (1) and equation (2)

⇒ 2 sec θ = p + 1/p

⇒ 2 sec θ = (p² + 1)/P

⇒ sec θ = (P² + 1)/2p

Perpendicular = P, Hypotenuse = H and Base = B.

⇒ sec θ = H/B = (p² + 1)/2p

⇒ H = (p² + 1) and B = 2p

As we know,

H² = P² + B²

⇒ (p² + 1)² = (2p)² + P²

⇒ p² = p⁴ + 1 + 2p² – 4p²

⇒ p² = p⁴ + 1 – 2p²

⇒ p² = (p² – 1)²

⇒ p = (p² – 1)

As we know

⇒ sin θ = (p² – 1)/(p² + 1)

Let the number of persons in the group be x, then

Wrong average of age of persons in the group = 35

Wrong sum of age of person in the group = 35x

Correct average of age of persons in the group = 35 – 2.5 = 32.5

Correct sum of age of persons in the group = 32.5x

According to the question

35x – 32.5x = 38.5 + 40 – 29 – 22

⇒ 2.5x = 27.5

⇒ x = 27.5/2.5

⇒ x = 11

Short Trick:

Total number of persons in the group = (38.5 + 40 – 29 – 22)/2.5 = 11

As we know,

(a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca

⇒ 2x² + y² + 8z²– 2√2xy + 4√2yz – 8zx = (Ax + y + Bz)²

⇒ (√2 x)² + y² + (2√2 z)² – 2 × (√2 x) × y + 2 × y × 2√2z – 2 × 2√2 z × √2 x = (Ax + y + Bz)²

⇒ (√2x –y – 2√2 z)² = (Ax + y + Bz)²

Comparing on both sides

A = √2 and B = – 2√2

Now,

(A² + B² – AB)

⇒ [(√2)² + (–2√2)² – √2 × (–2√2)]

⇒ [2 + 8 + 4]

⇒ 14

Let salaries of A and B be Rs. x and salary of C be Rs. y, then

8x/100 – 7x/100 = 259

⇒ x/100 = 259

⇒ x = 25900

Salaries of A and B is Rs. 25900.

Donation of A and B = 25900 × (8 + 7)/100 = 25900 × 15/100 = 3885

Donation of C = 3885 – 1185 = 2700

y × (9/100) = 2700

⇒ y = 30,000

Sum of salaries of A, B and C = 25900 + 25900 + 30000 = 81800

Donation of A = 259 × (8/100) = 2072

Total donation of A and C = 2072 + 2700 = 4772

Percentage of total donation of total salaries of A, B and C = (4772/81800) × 100 = 5.8%

 Shortcut Trick

Let salaries of A, B be 100%

⇒ 8% – 7% = 259

⇒ 1% = 259

⇒ 8% = 2072

⇒ 15% = 3885

⇒ 100% = 25900

Donation of C = 3885 – 1185 = 2700

⇒ 9% = 2700

⇒ 100% = 30,000

Total donation of A and C = 2072 + 2700 = 4772

Total salaries of A, B and C = 25900 + 25900 + 30000 = 81800

∴ Required percentage = 4772/81800 × 100 = 5.8%

As we know,

Perimeter of the regular hexagon = 6a

Area of regular hexagon = 6 × (√3/4) × a²

6 × (√3/4) × a² = 2400√3

a²= 2400√3 × 4/√3 × 1/6

⇒ a² = 1600

⇒ a = √1600

⇒ a = 40

Perimeter of regular hexagon = 6 × 40 = 240 m

Cost of fencing 1 m = Rs. 16.80

∴ Cost of fencing 2400 m = 240 × 16.80 = Rs. 4032

From the following figure,

If ∠ACP = ∠B and ∠A common, then

ΔACP∼ΔABC

As we know,

AC/AB = PC/BC

⇒ 8.4/AB = 11.2/14

⇒ AB = (8.4 × 14)/11.2 = 10.5

Again,

AP/AC = PC/BC

⇒ AP/8.4 = 11.2/14

⇒ AP = (11.2 × 8.4)/14 = 6.72

As we know,

∴ BP = AB – AP = 10.5 – 6.72 = 3.78

Let DE = x and CE = 3x

If chord AB and CD intersect at point E, then

As we know,

AE × BE = CE × DE

⇒ 12 × 20.25 = 3x × x

⇒ x² = 4 × 20.25

⇒ x = √(2 × 2 × 4.5 × 4.5)

⇒ x = 2 × 4.5

⇒ x = 9

⇒ CE = 3 × 9 = 27

Short trick:

∠B = 72° and ∠C = 44°

∠A = 180° – 72° – 44° = 64°

∠BEC = ∠A/2 = 64°/2 = 32°

Detailed Method:

⇒ ∠ACD = 180° – ∠C = 180° – 44° = 136°

⇒ ∠ACE = ∠ACD/2 = 136°/2 = 68°

⇒ ∠BCE = ∠BCA + ∠ACE = 44° + 68° = 112°

⇒ ∠EBC = ∠B/2 = 72°/2 = 36°

In ΔBCE,

∠EBC + ∠BCE + ∠BEC = 180°

⇒ ∠BEC = 180° – 36° – 112° = 32°

Total number of motorcycles of type B produced in 2016 and 2018 = 88 + 92 = 180

Total number of motorcycles of types D produced in 2013, 2015 and 2016 = 103 + 102 + 95 = 300

Required ratio = 180 : 300 = 3 : 5

Total production of type A motorcycles over six years = 95 + 84 + 85 + 89 + 80 + 98 = 531

Total production of all types of motorcycles in 2013 and 2016 = 95 + 98 + 104 + 103 + 89 + 88 + 92 + 95 = 764

∴ Required percentage = [(764 – 531)/764] × 100 = 30.5%

Total production of motorcycles in 2017 = 80 + 96 + 100 + 104 = 380

⇒ 25% of 380

⇒ 380 × 1/4

⇒ 95

In 2014, D = 100 is only more than 95

Now it clear in 2014 production of type D was more than 25% more than total production in 2017.

Total production of motorcycles in 2014 = 84 + 87 + 89 + 100 = 360

Total production of motorcycles in 2018 = 98 + 92 + 110 + 120 = 420

∴ Required percentage = [(420 – 360)/360] × 100 = $16\frac{2}{3}$%

The correct answer is Option 1.

It is an interrogative statement in Active Voice. It is in the Simple Past Tense. Its structure is: Did + Subject phrase (you + past perfect tense + infinitive form) + base form of the verb  + Object ?

It must be changed into Passive Voice. Its structure is: Were + Subject (object of the Active Voice) + past tense of verb + by + Object (subject phrase) of the active voice?

Note: The tense of the verb does not undergo any change when it is changed from Active to Passive or vice-versa.

Thus the correct sentence will be: Were you discouraged by the problems you had to face?

While converting from active to Passive, the sentence undergoes the following changes:

Rules of Conversion from Active to Passive Voice:

• Identify the subject, the verb, and the object: S+V+O
• Change the object into subject
• The active verb does not have an object. Therefore, the passive form should begin with you.
• Change the verb into past participle of the verb
• Change the subject into object

Given below are the rules of tense conversion from Active to Passive Voice and vice verse:

Other options do not fit as per the rules of conversion.

The correct answer is Option 4.

 Key Points

Itinerant is wrongly spelled as Itinerent. It means travelling from one place to another.(एक स्थान से दूसरे स्थान की यात्रा करना)

Other words are spelled correctly. Their meanings are:

• Influence: the capacity to have an effect on the character, development, or behaviour of someone or something(किसी या किसी चीज के चरित्र, विकास या व्यवहार पर प्रभाव डालने की क्षमता)
• Impeach: call into question the integrity or validity of (a practice); charge (someone) with treason or another crime against the state((एक अभ्यास) की अखंडता या वैधता पर सवाल उठाएं; राज्य के खिलाफ राजद्रोह या किसी अन्य अपराध का आरोप (कोई)
• Ignorance: lack of knowledge or information or understanding(ज्ञान या जानकारी या समझ की कमी)

The correct answer is Option 1.

• Sparse: thinly dispersed or scattered; scanty; in short supply
• Abundant: existing or available in large quantities; plentiful
• Tranquil: free from disturbance; calm
• Costly: costing a lot; expensive
• Scanty: small or insufficient in quantity or amount

Thus Abundant is antonym of sparse.

The correct answer is Option 4.

• Egoist: a person who is excessively conceited or absorbed in themselves.
• Optimist: a person who tends to be hopeful and confident about the future or the success of something
• Pessimist: a person who tends to see the worst aspect of things or believe that the worst will happen.
• Atheist: a person who disbelieves or lacks belief in the existence of God.

Egotist matches with One who is preoccupied with his own interests

The correct answer is Option 2.

When we put the sentences in order, it is important to establish a connection between them.

• The subject 'Vaikom Muhammad Basheer' is introduced in line B. The line gives information that he was a Malayalam fiction writer from Vaikom in Kerala.
• A follows B. It connects with B, that when the writer returned back from travel, he was forced to live in a house where his large family lived.
• D is the following line after A. It describes the large family where myriads of domestic animals also lived in the house and treated the house as their own.
• C follows D. Because there were family members and animals, the household was always noisy and full of chaos.

Thus the correct order is BADC.

Vaikom Muhammad Basheer was a Malayalam fiction writer from Vaikom in Kerala. Upon returning from his travels, he was forced to live in a house where his large family lived. Apart from family members, myriads of domestic animals also treated the house as their own. The household was always noisy and full of chaos, no place for a writer surely!

Other options are ruled out as they do not fit in structurally.

The correct answer is Option 3.

• Conducive: making a certain situation or outcome likely or possible
• Expensive: costing a lot of money
• Luxurious: extremely comfortable or elegant, especially when involving great expense; affluent, expensive
• Destitute: extremely poor and lacking the means to provide for oneself

Thus the correct fit is luxurious. Being a millionaire, he is leading a luxurious life.

The correct answer is Option 4.

The preposition 'in' is used when we refer to a year, month, etc.

Other options' usage makes the sentence incorrect.

In is the most suitable option for blank 1.

The correct answer is Option 1.

• Needed: (past tense) require (something) because it is essential or very important rather than just desirable
• Considered: (past tense) having been thought about carefully.
• Kneaded: (past tense) work (moistened flour or clay) into dough or paste with the hands; make (bread or pottery) by kneading flour or clay.
• Distributed: (past tense) give a share or a unit of (something) to each of a number of recipients.

The correct blank fit is needed.

The correct answer is Option 2.

The pronoun 'that' is the correct fit for blank 3.

'That' is used in a restrictive clause. The part of the sentence before 'that' does not have any meaning on its own.

Thus it is correct to use 'that' Other words are incorrect to use.

The correct answer is Option 3.

Adjectives qualify nouns or pronouns. It is formulated in degrees - positive, comparative and superlative.

The superlative degree largest is the best fit for the blank**.** It is used when one is selected among many.

The other forms are:

large - positive degree

larger - comparative degree

largest - superlative degree

Note: 'Most largest' form is not so commonly used in today's time. 'Most' here, acts as a quantifier.

Thus the correct sentence is .....the largest chocolate factory in the world.

The correct answer is Option 1.

• Favourites: a thing that is preferred to all others of the same kind or is especially well-liked.
• Collections: the action or process of collecting someone or something.
• Choices: an act of choosing between two or more possibilities.
• Selections: the action or fact of carefully choosing someone or something as being the best or most suitable.

Other options are incorrect for the blank.

The correct answer is DACB

Key Points

When we combine the sentences to form a paragraph, it is important to establish a connection between them.

• A sentence cannot start with words like - thus, generally, hence. They are concluding words, which denotes end of the idea. Thus B is ruled out.
• A sentence cannot start with a conjunction in the beginning. It must have a reference to the subject or the idea in the previous sentence. Thus sentence C cannot form the beginning of the sentence.
  

Eg: *But in the West, people have jobs.

• The subject of the paragraph is clearly mentioned in sentence D. The curies had to work in extreme poverty.
• A is followed after D. It shows a connection with the word 'even'. It tells that  Even the most ill-equipped laboratory would have been better than their shed.
• A contradiction is showed by the conjunction 'but' which is in line C. It states that the mind of the curies was set upon the discovery of radium.
• Finally, the conclusion is drawn by line B. It gives information that they did not allow any difficulties in their way.

Thus the correct order is DACB.

The Curies had to work in extreme poverty. Even the most ill-equipped laboratory would have been better than their shed. But their mind was set upon the discovery of radium. Thus, they did not allow any difficulties to come in their way.

When other options are used to frame the sentences, it does not fit structurally. There is no proper connection between them.

The correct answer is Option 3.

Hold water means to be correct, valid, sound or reasonable.

Eg: This argument just does not hold water.

Other phrases do not fit in the context.

The correct answer is Option 4.

• Exempted: (past tense) free from an obligation or liability imposed on others
• Acquitted: (past tense) free (someone) from a criminal charge by a verdict of not guilty
• Escaped: (past tense) having broken free from confinement or control
• Allowed: (past tense) let (someone) have or do something

Thus the correct answer is exempted. Other options do not fit in the context.

The correct answer is Option 3.

Key Points

• Octagon: a plane figure with eight straight sides
• Hexagon: a plane figure with six straight sides
• Pentagon: a plane figure with five straight sides
• Heptagon: a plane figure with seven straight sides

The other options are incorrect in their usage in the context.

The correct answer is "Option 4". 

Key Points

• The sentence uses the verb "to believe" in its present continuous form ("I am believing"), which is not typically used in English for state verbs.
• State verbs describe a state rather than an action and usually do not have continuous forms.
• Verbs of perception, emotion, and thinking (such as "believe") are generally not used in the continuous form because they do not describe physical actions.
• The correct form of the verb "to believe" in this context is the simple present tense ("I believe") because it expresses a general opinion or conviction.
• Changing "I am believing" to "I believe" corrects the grammatical error, making the sentence "I believe you have submitted all the necessary documents."
• Therefore, the correct answer is "Option 4".