SSC CGL 2024 Tier-I Official Paper (Held On: 10 Sept, 2024 Shift 1) (Previous Year Paper)

The logic followed here is :-

From the given two different cubes the adjacent sides are as shown below :

As T is common on both the dices, it is clear that S, M, P, and R are adjacent to it, and the remaining i.e. C will be opposite to it.

Opposite pairs:

M → R

S → P

T → C

So, the opposite side of "M" will be side "R".

Hence, the correct answer is "Option 1".

The logic followed here is:

1. All the elements shift clockwise including the center element.

Thus, the final series is:

Hence, "Option 1" is the correct answer.

The logic followed here is:

Logic: Opposites.

Here,

Blossom : Wither → Blossom is the opposite of Wither.

Similarly,

Stagnate : Flow → Stagnate is the opposite of Flow.

Hence, the correct answer is "Option 4".

Given: 18 - 23 × 455 ÷ 7 + 28 = 377

According to the question, after interchanging the signs:

Option 1) + and ×

18 - 23 × 455 ÷ 7 + 28 = 377 ⇒ 18 - 23 + 455 ÷ 7 × 28 = 377

⇒ 18 - 23 + 455 ÷ 7 × 28 = 377

⇒ 18 - 23 + 65 × 28 = 377

⇒ 18 + 1820 - 23 = 377

⇒ 1838 - 23 = 377

⇒ 1815 ≠ 377 (LHS ≠ RHS)

Option  2) − and ×

18 - 23 × 455 ÷ 7 + 28 = 377 ⇒ 18 × 23 - 455 ÷ 7 + 28 = 377

⇒ 18 × 23 - 455 ÷ 7 + 28 = 377

⇒ 18 × 23 - 65 + 28 = 377

⇒ 414 + 28 - 65 = 377

⇒ 442 - 65 = 377

⇒ 377 = 377 (LHS = RHS)

Option  3) − and +

18 - 23 × 455 ÷ 7 + 28 = 377 ⇒ 18 + 23 × 455 ÷ 7 - 28 = 377

⇒ 18 + 23 × 455 ÷ 7 - 28 = 377

⇒ 18 + 23 × 65 - 28 = 377

⇒ 18 + 1495 - 28 = 377

⇒ 1513 - 28 = 377

⇒ 1485 ≠ 377 (LHS ≠ RHS)

Option  4) ÷ and −

18 - 23 × 455 ÷ 7 + 28 = 377 ⇒ 18 ÷ 23 × 455 - 7 + 28 = 377

⇒ 18 ÷ 23 × 455 - 7 + 28 = 377

⇒ 0.78 × 455 - 7 + 28 = 377

⇒ 354.9 + 28 - 7 = 377

⇒ 382.9 - 7 = 377

⇒ 375.9 ≠ 377 (LHS ≠ RHS)

Hence, "Option 2" is the correct answer.

Given: 4 − 12 + 3 ÷ 12 × 6 = ?

According to the question, after ‘+’ and ‘÷’ are interchanged and ‘×’ and ‘–’ are interchanged:

4 − 12 + 3 ÷ 12 × 6 = ? ⇒ 4 × 12 ÷ 3 + 12 - 6 = ?

⇒ 4 × 12 ÷ 3 + 12 - 6 = ?

⇒ 4 × 4 + 12 - 6 = ?

⇒ 16 + 12 – 6 = ?

⇒ 28 – 6 = ?

⇒ 22 = ?

Hence, "Option 4" is the correct answer.

Given word: 'PRANKED'

After arranged in alphabetical order:

 

Now,

there are None of the alphabets will remain unchanged if each of the letters in the word 'PRANKED' is arranged in English alphabetical order.

Hence, the correct answer is "Option 3".

The logic followed here is:

10 is related to 15

10 + 10 ÷ 2 = 15

10 + 5 = 15

15 = 15 (LHS = RHS)

And,

42 is related to 63

42 + 42 ÷ 2 = 63

42 + 21 = 63

63 = 63 (LHS = RHS)

Similarly,

22 is related to ?

22 + 22 ÷ 2 = ?

22 + 11 = ?

33 = ?

Hence, "Option 3" is the correct answer.

The number of squares in the given figure is shown below:

Thus, there are total 14 squares.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Logic: Opposites

Here,

Temerity : Shyness  → Temerity is the opposite of Shyness.

Similarly,

Pique : Delight  → Pique is the opposite of Delight.

Hence, the correct answer is "Option 3".

Given series: L M B A _ H O L M B A C _ O L M _ A C H O L _ B A C H O L _

Option 1) CHMBM→ L M B A C H O L M B A C H O L M M A C H O L B B A C H O L M → (It doesn't follow any pattern)

Option 2) CMHBM → L M B A C H O L M B A C M O L M H A C H O L B B A C H O L M → (It doesn't follow any pattern)

Option 3) CHBMM → L M B A C H O L M B A C H O L M B A C H O L M B A C H O L M → (It follows the pattern of: LMBACHO - LMBACHO - LMBACHO - LMBACHO - LM)

Option 4) CHMMB → L M B A C H O L M B A C H O L M M A C H O L M B A C H O L B → (It doesn't follow any pattern)

Hence, "Option 3" is the correct answer.

The logic followed here is:

(7, 9, 48)

(7 + 9) × 3 = 48 

16 × 3 = 48

48 = 48 (LHS = RHS)

And,

(6, 8, 42)

(6 + 8) × 3 = 42 

14 × 3 = 42

42 = 42 (LHS = RHS)

So, checking each option one by one:

Option 1) (8, 10, 47)

(8 + 10) × 3 = 47 

18 × 3 = 47

54 ≠ 47 (LHS ≠ RHS)

Option 2) (9, 11, 56)

(9 + 11) × 3 = 56 

20 × 3 = 56

60 ≠ 56 (LHS ≠ RHS)

Option 3) (11, 13, 72)

(11 + 13) × 3 = 72 

24 × 3 = 72

72 = 72 (LHS = RHS)

Option 4) (10, 12, 54)

(10 + 12) × 3 = 54 

22 × 3 = 54

66 ≠ 54 (LHS ≠ RHS)

Thus, out of all the options, '(11, 13, 72)' follows the same logic as given.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Option 1) 2 – 4 – 8

2 × 4 = 8

8 = 8 (LHS = RHS)

Option 2)  2 – 2 – 4

2 × 2 = 4

4 = 4 (LHS = RHS)

Option 3) 3 – 2 – 5

3 × 2 = 5

6 ≠ 5 (LHS ≠ RHS)

Option 4) 1 – 3 – 3

1 × 3 = 3

3 = 3 (LHS = RHS)

Thus, out of all the options, '3 – 2 – 5' is the odd one out.

Hence, "Option 3" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. No box is a utensil → Does not follow (As some boxes are cartons, all cartons are papers and some cartons are utensils. As no direct relation is given between boxes and utensils, therefore it is false.)

II. Some papers are utensils → Follow (As all cartons are papers and some cartons are utensils. As whole cartons comes in papers and has some part common with utensils, therefore it is true.)

∴ Here, Only conclusion (II) follows.

Hence, "Option 3" is the correct answer.

The logic followed here is:

(79, 58, 62)

(58 + 62) ÷ 2 + 19 = 79 

120 ÷ 2 + 19 = 79

60 + 19 = 79

79 = 79 (LHS = RHS)

And,

(54, 33, 37)

(33 + 37) ÷ 2 + 19 = 54 

70 ÷ 2 + 19 = 54

35 + 19 = 54

54 = 54 (LHS = RHS)

So, checking each option one by one:

Option 1) (59, 33, 43)

(33 + 43) ÷ 2 + 19 = 59 

76 ÷ 2 + 19 = 59

38 + 19 = 59

57 ≠ 59 (LHS ≠ RHS)

Option 2) (47, 36, 41)

(36 + 41) ÷ 2 + 19 = 47 

77 ÷ 2 + 19 = 47

38.5 + 19 = 47

57.5 ≠ 47 (LHS ≠ RHS)

Option 3) (66, 45, 49)

(45 + 49) ÷ 2 + 19 = 66 

94 ÷ 2 + 19 = 66

47 + 19 = 66

66 = 66 (LHS = RHS)

Option 4) (43, 25, 30)

(25 + 30) ÷ 2 + 19 = 43 

55 ÷ 2 + 19 = 43

47.5 + 19 = 43

66.5 ≠ 43 (LHS ≠ RHS)

Thus, out of all the options, '(66, 45, 49)' follows the same logic as given.

Hence, "Option 3" is the correct answer.

According to the given information,

Thus, according to the given codes the code 'mi' is for the word 'chair'.

Hence, the correct answer is "Option 3".

The logic followed here is:

1. First box series →

1st column elements interchange positions.

2nd column elements shift to the 3rd column.

3rd column elements shift to the 3rd column diagonally.

2) Second box series →

Left bottom corner elements is replaced by new element.

2nd column elements shift to the 3rd column.

First element of the 3rd column shift to the 2nd column.

1st row element shift to the bottom center position.

Second element of 3rd column shift to the opposite side diagonally.

Thus, the final series is:

Hence, "Option 2" is the correct answer.

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

The logic followed here is:

'BELL' is written as 'MKFA'

And,

'BOND' is written as 'EMPA'

Similarly,

'BLEW’ will be written as

Hence, "Option 4" is the correct answer.

The figure which will look like the paper when unfolded is shown below:

Thus, option Figure 3 will look like the paper when unfolded.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Option 1) LQV

Option 2) DIN

Option 3) PTX

Option 4) AFK

Thus, out of all the options, 'PTX' is the odd one out.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Hence, "Option 4" is the correct answer.

The logic followed here is:

Hence, "Option 2" is the correct answer.

Decoding symbols:

Given: P & Q % L # R & K

P & Q → P is the wife of Q.

Q % L → Q is the father of L.

L # R → L is the sister of R.

R & K → R is the wife of K.

Thus, Q is R's father.

Hence, the correct answer is "Option 3"

The logic followed here is:

Hence, "Option 1" is the correct answer.

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 4".

The correct answer is Imphal.

Key Points

• Jawaharlal Nehru Manipur Dance Academy:
• It is a constituent unit of the Sangeet Natak Akademi, New Delhi.
• The institute is a premier institution for teaching Manipuri Dance and Music and allied subjects like Thang-Ta.
• It was established in 1954.

 Additional Information

• Sangeet Natak Akademi:
• ​The Sangeet Natak Akademi was founded on 31 May 1952.
• It is also known as the National Academy for Music, Dance, and Drama.
• In India, it functions as the headquarters of performing arts.
• It presents the Sangeet Natak Akademi Awards which is the highest Indian recognition given to performing/practising artists.

The correct answer is Champaran.

Key Points

• Champaran Satyagraha of 1917 
• It was the first civil disobedience movement in India.
• Champaran district is a historic region in Bihar, India, that is now divided into the East Champaran district and the West Champaran district.
• Farmers in this region revolted against the British in 1914 and 1916 over the conditions imposed on indigo cultivation.
• Earlier, similar conditions existed in Bengal, but as a result of a major uprising in 1859-61, the peasants there won their freedom from the indigo planters.
• Champaran is a district in the state of Bihar where tens of thousands of landless serfs, indentured labourers, and poor farmers were forced to grow indigo and other cash crops instead of food crops.
• The European planters had forced the peasants to grow indigo on 3/20 of the total land area (called the tinkathia system).
• When German synthetic dyes replaced indigo at the end of the nineteenth century, European planters demanded high rents and illegal dues from the peasants in order to maximize their profits before the peasants could switch to other crops.
• The first Satyagraha movement, inspired by Mahatma Gandhi, took place in the Bihar district of Champaran in 1917.
• The Champaran Satyagraha was the first to be launched, but the term Satyagraha was first used in the Anti-Rowlatt Act agitation.

 Additional Information

• Kheda Satyagrah-
• 1918 was a year of failed crops in the Kheda district of Gujarat due to droughts.
• As per law, the farmers were entitled to remission if the produce was less than a quarter of the normal output.
• But the government refused any remission from paying land revenue.
• Sardar Vallabhbhai Patel, under Gandhi’s guidance, led the farmers in protest against the collection of taxes in the wake of the famine.
• The protest was peaceful and people showed remarkable courage even in the face of adversities like confiscation of personal property and arrest.
• Finally, the authorities gave in and gave some concessions to the farmers.
• Kheda peasant struggle :
• The Kheda Satyagraha of 22 march 1918, in the Kheda district of Gujarat.
• India during the period of the British Raj, was a Satyagraha movement organized by Mohandas Gandhi.
• It was a major revolt in the Indian independence movement.
• It was the third Satyagraha movement after Champaran Satyagraha and Ahmedabad mill strike.
• Bardoli Satyagraha:
• The Bardoli satyagraha took place in Gujarat and was led by Sardar Vallabhbhai Patel.
• It was a tax revolt that took place in 1928 in the taluka (sub-district) of Bardoli, located in the district of Surat in south Gujarat.
• Patel had gathered all the Muslims and Hindus in Bardoli and performed non-violent satyagraha in a revolt against the British.
• The protest achieved its principal objective, which included forcing the colonial state to withdraw a policy aimed at raising the rate of land revenue paid by peasants.

The correct answer is Madhya Pradesh Government.

Key Points

Tansen Music Festival:

• It began with a traditional style in Gwalior (Madhya Pradesh).
• It is a five-day festival that is celebrated every year.
• It is celebrated in the memory of ‘Mian Tansen’.
• It is the oldest and esteemed classical music festival in India is organized by Ustad Alauddin Khan Kala Evam Sangeet Academy.

The correct answer is Fifth Five-Year Plan.

Key Points

• Fifth Five Year Plan: Its duration was 1974 to 1978. This plan focussed on Garibi Hatao, employment, justice, agricultural production, and defense.
• The Electricity Supply Act was amended in 1975, a Twenty-point program was launched in 1975, the Minimum Needs Programme (MNP), and the Indian National Highway System was introduced.

Additional Information

• First Five Year Plan: It was launched for the duration of 1951 to 1956, under the leadership of Jawaharlal Nehru. It was based on the Harrod-Domar model with a few modifications. Its main focus was on the agricultural development of the country.
• Second Five Year Plan: It was made for the duration of 1956 to 1961, under the leadership of Jawaharlal Nehru. It was based on the P.C. Mahalanobis Model made in the year 1953. Its main focus was on the industrial development of the country.
• Third Five-Year Plan: It was made for the duration of 1961 to 1966, under the leadership of Jawaharlal Nehru. This plan is also called ‘Gadgil Yojna’, after the Deputy Chairman of Planning Commission D.R. Gadgil. The main target of this plan was to make the economy independent. The stress was laid on agriculture and the improvement in the production of wheat.
• Plan Holidays: Due to the failure of the previous plan, the government announced three annual plans called Plan Holidays from 1966 to 1969. The main reason behind the plan holidays was the Indo-Pakistani war and the Sino-India war, leading to the failure of the third Five Year Plan.
• Fourth Five-Year Plan: Its duration was from 1969 to 1974, under the leadership of Indira Gandhi. There were two main objectives of this plan i.e. growth with stability and progressive achievement of self-reliance.
• Rolling Plan: After the termination of the fifth Five Year Plan,  the Rolling Plan came into effect from 1978 to 1980. In 1980, Congress rejected the Rolling Plan, and a new sixth Five Year Plan was introduced.
• Sixth Five Year Plan: Its duration was from 1980 to 1985, under the leadership of Indira Gandhi. The basic objective of this plan was economic liberalization by eradicating poverty and achieving technological self-reliance.
• Seventh Five Year Plan: Its duration was from 1985 to 1990, under the leadership of Rajiv Gandhi. The objectives of this plan include the establishment of a self-sufficient economy, opportunities for productive employment, and up-gradation of technology.
• Eighth Five Year Plan: Its duration was from 1992 to 1997, under the leadership of P.V. Narasimha Rao. In this plan, the top priority was given to the development of human resources i.e. employment, education, and public health.
• Ninth Five Year Plan: Its duration was from 1997 to 2002, under the leadership of Atal Bihari Vajpayee. The main focus of this plan was “Growth with Social Justice and Equality”.
• Tenth Five Year Plan: Its duration was from 2002 to 2007, under the leadership of Atal Bihari Vajpayee and Manmohan Singh. II. This plan aimed to double the Per Capita Income of India in the next 10 years.
• Eleventh Five Year Plan: Its duration was from 2007 to 2012, under the leadership of Manmohan Singh. It was prepared by C. Rangarajan.Its main theme was “rapid and more inclusive growth”. Twelfth Five Year Plan: Its duration is from 2012 to 2017, under the leadership of Manmohan Singh. Its main theme is “Faster, More Inclusive and Sustainable Growth”.

The correct answer is Grass.

Key Points

• A species of grass named Hubbardia heptaneuron is on the verge of extinction due to its insensitivity towards the environment.
• At least 10 per cent of India’s recorded wild flora and 20 per cent of its mammals are on the threatened list.
• Many of the flora and fauna on the threatened list would now be categorised as ‘critical’, like:
• the cheetah,
• pink-headed duck,
• mountain quail,
• forest spotted owlet, and
• plants like madhuca insignis (a wild variety of mahua) and
• hubbardia heptaneuron,(a species of grass).

The correct answer is Do.

Key Points

• John Newlands, a British chemist, proposed the Law of Octaves in 1864.
• This law stated that when elements are arranged in order of increasing atomic mass, every eighth element has properties similar to the first.
• He compared this pattern to the octaves of music, where every eighth note is similar to the first.
• However, Newlands' Law of Octaves had limitations.
• It didn't work for all elements, especially those with higher atomic masses.
• To accommodate all the known elements, he had to place two elements in the same slot in some cases.
• One such instance was with cobalt (Co) and nickel (Ni), which he placed in the same slot as the halogens.
• This placement highlights the limitations of Newland's classification system.
• While it was a significant step towards understanding the periodic relationships between elements, it was not entirely accurate and was eventually superseded by more comprehensive periodic tables, such as Mendeleev's Periodic Table.

The correct answer is Punjab.

Key Points

• Humayun (1530–1540, 1555–1556):
• Humayun was the son of Babur.
• Humayun means "fortune" but he remained the most unfortunate ruler of the Mughal Empire.
• Sher shah suri became powerful in the east.
• Humayun marched against him and in the Battle of Chausa in 1539, Sher Khan destroyed the Mughal army and Humayun fled.
• In 1540, at the Battle of Bilgram or Ganga, also known as the Battle of Kannauj, Humayun was forced to fight alone with Sher Khan and after losing his kingdom, Humayun was exiled for the next fifteen years.

 Additional Information

• Rulers During The Mughal Empire:
• Babur (1526-1530)
• Humayun (1530-1540 & 1555-56)
• Akbar (1556-1605)
• Jahangir (1605- 1627)
• Shah Jahan (1628- 1658)
• Aurangzeb (1658- 1707)

Akbar’s Coronation at Kalanaur – Key Details

• Context:
• Humayun died on 24 January 1556 after falling from the stairs of his library (Purana Qila, Delhi).
• At that time, Akbar was in Punjab, campaigning with his guardian and regent Bairam Khan against the Afghans.
• Place of Coronation:
• Akbar was crowned at Kalanaur, a small town in present-day Gurdaspur district, Punjab.
• A special platform (chabutra) was constructed there, on which the coronation took place.
• Age & Date:
• Akbar was about 13 years old at the time.
• He was formally proclaimed Emperor in 1556 at Kalanaur, though real power initially lay with Bairam Khan as Wakil / Regent.
• Political Situation:
• The Mughals were still not fully secure in India.
• The Sur Empire (Afghans) remained a threat; within the same year, Akbar would face Hemu in the Second Battle of Panipat (1556).
• Akbar’s early reign (till about 1560) is often called the “regency period under Bairam Khan.”
• Why Kalanaur? (Significance):
• Kalanaur was chosen because:
• The Mughal army was already stationed in Punjab, a frontier and conflict zone.
• It was not safe to immediately go to Delhi or Agra after Humayun’s death.
• So, Akbar’s empire formally began from Kalanaur, not from Delhi/Agra.

The correct answer is Type of soil.

Key Points

• The first scientific classification of soil was done by Vasily Dokuchaev.
• In India, the Indian Council of Agricultural Research (ICAR) has classified soils into 8 categories.
• They are:
• Alluvial Soil
• Black Cotton Soil
• Red & Yellow Soil
• Laterite Soil
• Mountainous or Forest Soil
• Arid or Desert Soil
• Saline and Alkaline Soil
• Peaty and Marshy Soil

 Additional Information

• Alluvial Soil:
• Alluvial soils are widespread in the northern plains and river valleys.
• It covers about 40% of the total land area of the country.
• These soils are mainly derived from the debris brought down from the Himalayas.
• In the Peninsular region, they are found in deltas of the east coast and in the river valleys.
• The colour of the alluvial soil varies from light grey to ash grey.
• Red & Yellow Soil:
• Also known as the “omnibus group”.
• It covers about 18.5 % of the total land area of the country.
• It is found in regions of low rainfall (eastern and southern parts of the Deccan Plateau).
• Along the piedmont zone of the Western Ghats, a long stretch of area is occupied by red loamy soil.
• This soil is also present in parts of Odisha and Chattisgarh and in the southern parts of the Middle Ganga Plain.
• Black or Regur Soil:
• Black soil is also known as “Regur Soil” or the “Black Cotton Soil”.
• It covers about 15% of the total land area of the country.
• It covers most of the Deccan Plateau – parts of Maharashtra, Madhya Pradesh, Gujarat, Andhra Pradesh and some parts of Tamil Nadu. In the upper reaches of the Godavari and Krishna, and the north-western part of Deccan Plateau, the black soil is very deep.
• The colour of these soils varies from deep black to grey.
• Desert Soil:
• Also known as arid soil, it accounts for over 4.42 % of the total land area of the country.
• The colour ranges from red to brown.
• Desert soils are sandy to gravelly in texture, have low moisture content and low water-retaining capacity.
• These soils are saline in nature and in certain regions the salt content is so high that common salt is obtained by evaporating water.
• Laterite Soil:
• The name has been derived from the Latin word “later” which means brick.
• It accounts for about 3.7% of the total area of the country.
• These are typical soils of the monsoon climate which is characterised by seasonal rainfall.
• With rain, lime and silica are leached away, and soil rich in iron oxide and aluminium are left leading to the formation of laterite soil.
• Laterite soil is deficient in organic matter, nitrogen, phosphate and calcium, however iron oxide and potash are in abundance.
• Mountain Soil:
• This type of soil is found in the forest regions where rainfall is sufficient.
• The texture of the soil depends on the mountain environment where they are found.
• These soils are coarse-grained in the upper slopes and loamy and silty on valley sides.
• Peaty and Marshy Soils:
• These soils are found in regions of heavy rainfall and high humidity, and it supports the good growth of vegetation.
• Peaty soils are rich in humus and organic matter.
• These soils are generally heavy and black in colour.
• In many places, these soils are alkaline.
• These are found in southern Uttarakhand, the northern part of Bihar, and the coastal areas of West Bengal, Odisha and Tamil Nadu.
• Saline and Alkaline Soils:
• These soils have high percentages of sodium, magnesium and potassium, and hence are infertile.
• The high salt content is mainly because of the dry climate and poor drainage.
• The texture ranges from sandy to loamy.
• These soils are found in arid and semi-arid areas, and in waterlogged and swampy regions.
• These soils are deficient in calcium and nitrogen.

The correct answer is Dadabhai Naoroji divided the Indian economy into two parts: primary sector and secondary sector.

Key Points

• National income is the total income of all the residents of the country.
• Thus, it is the total income of the country.
• It is considered to be one of the most important attributes for comparing the development of countries.
• The National Income formula equals GDP + Foreign Production by National Residents – Domestic Production by Non-National Residents.

 Additional Information

• Dadabhai Naoroji
• Dadabhai Naoroji, known as the Grand Old Man of India, was a leading figure in the early Indian independence movement and a founder of the Indian National Congress.
• His activities were mainly in the late 19th and early 20th centuries.
• The Indian Independence League, however, was primarily active during the 1940s, significantly after Naoroji's most impactful years and following his death in 1917.
• This organization sought to obtain independence for India from British rule during World War II, with notable connections to Subhas Chandra Bose and the Indian National Army (INA).
• Naoroji founded or was associated with several organizations, like the Bombay Association (the oldest political organization in Bombay), the East India Association (established in London to discuss Indian affairs), and Gyan Prasarak Mandali (for the education of adult men in Bombay).
• These affiliations reflected his focus on political advocacy, economic theories (notably the Drain Theory), and social reform both in India and the UK.
• The Indian Independence League's context and methods, focused on leveraging the geopolitical situations during World War II for India's independence through more direct action, were distinct and not within Naoroji's lifetime or direct activities.
• ​PC Mahalanobis
• The Second five-year plan was based on the Mahalanobis model, an economic development model developed by the Indian statistician Prasanta Chandra Mahalanobis in 1953.
• PC Mahalanobis established the Indian Statistical Institute (ISI) in Calcutta in 1931 and for the same reason, he is known as the father of Indian Statistics. He was a member of the first Planning Commission of free India.
• The second five-year plan was for the year 1956-1961.
• The second five-year plan focused on the development of the public sector and rapid Industrialisation.  The plan attempted to determine a balanced allocation of investment between productive sectors in order to maximize long-run economic growth.
• As a result, hydroelectric power projects across the country and five steel plants at Bhilai, Durgapur, and Rourkela were established with the help of Russia, Britain (the U.K), and West Germany respectively. Coal production was increased. More railway lines were added all over the country.
• The Tata Institute of Fundamental Research and Atomic Energy Commission of India was established to promote research in India.

The correct answer is Only Karaikal

Key Points

• Karaikal is a district of the Union Territory of Puducherry that is closest to Sri Lanka.
• It is located on the southeastern coast of India, just across the Palk Strait from Sri Lanka.
• Yanam, Mahe, and Puducherry are the other districts of the Union Territory of Puducherry,
• but they are located on the eastern and western coasts of India, respectively, and are not in close proximity to Sri Lanka.

                                                                    

Additional Information

• Karaikal and Yanam
• While Karaikal is indeed the closest district of the Union Territory of Puducherry to Sri Lanka, Yanam is located on the eastern coast of India, near the Bay of Bengal.
• It is not geographically close to Sri Lanka.
• Mahe
• It is another district of the Union Territory of Puducherry located on the western coast of India, near the Arabian Sea.
• It is also not geographically close to Sri Lanka.

The correct answer is They promote welfare of individuals. Hence, they are personal and individualistic.

Key Points

• The Directive Principles of State Policy of India are the guidelines or principles given to the institutes for governing India and are provided in Part IV of the Constitution of India.
• Directive Principles of State Policy aim to create social and economic conditions under which the citizens can lead a good life.
• They also aim to establish social and economic democracy through a welfare state. Directive principles are classified under the following categories; economic, socialistic, political, executive, justice and legal, administrative, environmental, protection of monuments, peace, and security.

Hence, it can be said that the Directive Principles of State Policy are fundamental for the Governance of the state.

Additional Information

• Features of DPSPs
• The phrase ‘Directive Principles of State Policy’ denotes the ideals that the State should keep in mind while formulating policies and enacting laws.
• These are the constitutional instructions or recommendations to the State in legislative, executive, and administrative matters. According to Article 36, the term ‘State’ in Part IV has the same meaning as in Part III dealing with Fundamental Rights.
• Therefore, it includes the legislative and executive organs of the central and state governments, all local authorities, and all other public authorities in the country.
• The Directive Principles resemble the ‘Instrument of Instructions’ enumerated in the Government of India Act of 1935.
• In the words of Dr. B.R. Ambedkar, ‘the Directive Principles are like the instrument of instructions, which were issued to the Governor-General and to the Governors of the colonies of India by the British Government under the Government of India Act of 1935. What is called Directive Principles is merely another name for the instrument of instruction. The only difference is that they are instructions to the legislature and the executive’.
• The Directive Principles constitute a very comprehensive economic, social and political program for a modern democratic State.
• They aim at realizing the high ideals of justice, liberty, equality, and fraternity as outlined in the Preamble to the Constitution.
• They embody the concept of a ‘welfare state’ and not that of a ‘police state’, which existed during the colonial era3. In brief, they seek to establish economic and social democracy in the country.
• The Directive Principles are non-justiciable in nature, that is, they are not legally enforceable by the courts for their violation.
• Therefore, the government (Central, state, and local) cannot be compelled to implement them.
• Nevertheless, the Constitution (Article 37) itself says that these principles are fundamental in the governance of the country and it shall be the duty of the State to apply these principles in making laws.
• The Directive Principles, though nonjusticiable in nature, help the courts in examining and determining the constitutional validity of a law.
• The Supreme Court has ruled many times that in determining the constitutionality of any law if a court finds that the law in question seeks to give effect to a Directive Principle, it may consider such law to be ‘reasonable’ in relation to Article 14 (equality before law) or Article 19 (six freedoms) and thus save such law from unconstitutionality.

The correct answer is Gurpurab.

Key Points

• Guru Nanak Dev
• Guru Nanak Dev was born on 15 April 1469 at Talwaṇḍi village in the Lahore province of the Delhi Sultanate (present-day Nankana Sahib, Punjab, Pakistan).
• He was the founder of Sikhism and the first of the ten Sikh Gurus.
• Around the age of 55, Nanak Devji settled in Kartarpur and lived there until his death in September 1539.
• Recently Kartarpur Corridor was opened by the Indian and Pakistan government for the Sikh devotees.
• The teaching of Guru Nanak Devji can be found in the Sikh scripture Guru Granth Sahib, as a collection of verses recorded in Gurmukhi which was written by Guru Arjan Dev.

Additional Information

• Guru Nanak Dev Ji- He started the practice of "Langer".
• Guru Angad Dev Ji developed the Gurumukhi script.
• Guru Amar Das- Composed the Anand Sahib, introduced the simple Anand Karaj marriage and abolished Sati among the Sikhs.
• Guru Ram Das- He laid the foundation of the holy city of Amritsar and initiated the construction of the Golden Temple.
• Guru Arjan Dev- Compiled the Adi Granth and constructed the Golden Temple.
• Guru Hargobind- Created the Sikh martial art called Gatka. Built the Akal Takhat.
• Guru Har Rai- He was known as the "tender-Hearted Guru".
• Guru Har Kishan- Youngest Sikh Guru, installed at the age of 5.
• Guru Teg Bahadur- Refused to convert to Islam and was beheaded by Aurangzeb.
• Guru Gobind Singh- He was the last Sikh Guru. He established the military force of "Khalsa".

The correct answer is 44

Key Points

• The National Sports and Adventure Awards-2022 ceremony held on 30 November 2022 at Rashtrapati Bhavan recognized the achievements of outstanding individuals in the fields of sports and adventure.
• A total of 44 awards were presented across various categories.
• The awards were presented by President Droupadi Murmu at the Rashtrapati Bhavan.
• The event celebrated a wide range of sports and adventure activities, highlighting India's growing stature in both areas on the global stage.
• Athletes from different disciplines, including cricket, boxing, wrestling, and athletics, were honored.

Additional Information

• Types of Awards in Event
• Major Dhyan Chand Khel Ratna Award:
• India’s highest sporting honor, awarded for the most outstanding performance in sports over the past four years.
• 1 recipient in 2022.
• Arjuna Award:
• Given to athletes for consistently excellent performances over the past four years.
• 25 recipients in 2022.
• Dronacharya Award (Lifetime and Regular categories):
• Presented to coaches for producing exceptional sportspersons.
• 7 recipients in total (both lifetime and regular categories).
• Dhyan Chand Award for Lifetime Achievement:
• Honors sportspersons for their lifetime contribution to sports.
• 4 recipients in 2022.
• Rashtriya Khel Protsahan Puruskar:
• Recognizes corporate entities, sports promotion boards, or individuals who have contributed to the promotion and development of sports.
• 2 recipients in 2022.
• Maulana Abul Kalam Azad (MAKA) Trophy:
• Awarded to the best performing university in sports.
• 1 recipient in 2022.
• Tenzing Norgay National Adventure Award:
• Given for excellence in adventure activities on land, water, and air, as well as lifetime achievements in adventure sports.
• 4 recipients in 2022.

The correct answer is West Bengal.

Key Points

• Pandit Anindo Chatterjee:-
• Anindo Chatterjee is known for his mastery of tabla, excelling in various styles including khayal and thumri.
• He has performed with many renowned classical musicians and has participated in international music festivals.
• His awards:
• Sangeet Natak Akademi Award (2012).
• Acknowledged as a leading tabla player in various music festivals and competitions.

Additional Information

• Sangeet Natak Akademi Award
• The Sangeet Natak Akademi is India's national academy for music, dance, and drama. It was established in 1952 by the Indian government.
• The Akademi Awards are the highest national recognition given to practicing artists.
• Bharatanatyam has its roots in the Natya Shastra, an ancient treatise on performing arts written by Bharata Muni.
• This dance form is traditionally performed by women, but in contemporary times, it is also performed by men.
• The themes of Bharatanatyam performances are usually derived from Hindu mythology and epics.

The correct answer is Uttar Pradesh.

Key Points

• According to the Indian Railways (2019-2020), Uttar Pradesh has the largest railway track in India. 
• As of 31.03.2020, the route length of Indian Railways covered 67,956 km with a running track length of 99,235 km.
• The total trackage including yards, sidings etc. stood at 1,26,366 km.
• In India, the largest route kilometers are in Uttar Pradesh with 8726 kilometres. This comes out to be 13.6% of India’s total Route Kilometers. The Five states with the largest route kilometres are as follows:
• Uttar Pradesh: 8,808.
• Rajasthan: 5,894
• Maharashtra: 5,784
• Karnataka:  3,542
• Jharkhand: With 2,596

 Additional Information

• The first railway line in India was opened for public traffic on 16 April 1853 between Mumbai and Thane over a distance of 34 km.
• Meanwhile, the construction of the railway lines was going on in the eastern part of the country, and the first section of the East Indian Railway, from Howrah to Hugli, a distance of 37 km was inaugurated on 15 August 1854.

The correct answer is SS Dubey.

Key Points

• Controller General of Accounts (CGA):
• Bharati Das, a 1988-batch Indian Civil Accounts Service Officer, took charge as the new Controller General of Accounts in New Delhi on 18 October 2022.
• She is the 27th Controller General of Accounts, Ministry of Finance.
• CGA is responsible for establishing and maintaining a technically sound Management Accounting System.
• The CGA is not a constitutional body.
• The CGA is the Principal Accounting Adviser to the Government of India.
• It is also in charge of the exchequer control and internal audits.​

The correct answer is Lord William Bentinck.

Key Points

• William Bentinck was the first Governor-General of India.
• Macaulay’s Minute is a historical document in Indian Education.
• It was prepared under the guidance of Thomas Babington Macaulay in 1835.
• Macaulay promotes English literature in India through his Minute on Indian Education.

Additional Information

• Lord Ripon
• Lord Ripon's Resolution on Local Self-Government, passed in 1882, was a landmark reform that marked the beginning of democratic decentralization in India.
• This resolution aimed to promote local self-government by strengthening municipal institutions and providing a framework for involving Indians in the local administration.
• Key Aspects of the Resolution:
• Empowerment of Local Bodies: The resolution emphasized the need for establishing and nurturing local self-governing bodies such as municipalities and district boards, with significant Indian participation.
• Decentralization of Authority: It advocated for the transfer of administrative functions from the central and provincial governments to local bodies, thus laying the foundation for a decentralized administrative structure.
• Encouragement of Local Leadership: By involving locals in governance, it aimed to foster a sense of responsibility and leadership among Indian citizens, thereby paving the way for a more representative and participatory form of governance.​
• Lord Curzon:-
• Lord Curzon served as the Viceroy of India from 1899 to 1905 under Queen Victoria.
• He was born on January 11, 1859, in Kedleston, Derbyshire, England.
• Curzon was a British Conservative statesman who later served as the Foreign Secretary from 1919 to 1924.
• He declared the partition of Bengal in 1905.
• The partition, was implemented for administrative reasons according to the British.
• It was a controversial action that led to widespread protests as it was seen by Indians as an attempt to "divide and rule," given that it separated the largely Muslim eastern areas from the Hindu western areas.
• Lord Cornwallis:
• Lord Cornwallis was the Governor-General of Fort William (Bengal) and the Commander-in-Chief of British India.
• The Third Anglo-Mysore War took place in his tenure.
• Cornwallis introduced the Permanent Settlement of land revenue in Bengal and other parts of India.
• He established thanas to maintain law and order.
• He founded a Sanskrit College at Benaras for Hindus and this is today the Government Sanskrit College in Benares.
• He also established a mint at Calcutta.

The correct answer is The Lighthouse Act,1927

Key Points

• The Marine Aids to Navigation Bill, 2021 repealed and replaced the Lighthouse Act of 1927.
• Lighthouse Act of 1927
• This act regulated lighthouses and navigational techniques in India for almost nine decades.
• Marine Aids to Navigation Bill, 2021
• This bill was introduced in Lok Sabha on March 15, 2021.
• It aims to: 
• Provide a framework for the development, maintenance, and management of aids to navigation in India
• Improve the legal framework for matters related to aids to navigation and vessel traffic services
• Manage vessel traffic services to enhance the safety and efficiency of shipping
• Develop lighthouses for education, culture, and tourism
• The Ministry of Ports, Shipping and Waterways adopted this initiative to repeal colonial laws and replace them with legislation that caters to the modern needs of the maritime industry.

Additional Information

• The Dangerous Machines (Regulation) Act, 1983
• This act was enacted to regulate the manufacture, import, and use of dangerous machines and to ensure safety measures to protect workers from injuries while operating such machinery.
• It mandates the registration of dangerous machines, along with guidelines for their operation, maintenance, and safety protocols to prevent accidents and injuries.
• The Personal Injuries (Emergency Provisions) Act, 1962
• This act was introduced to provide immediate relief and compensation to individuals who suffer personal injuries during emergencies, such as natural disasters or accidents.
• It establishes a framework for assessing and compensating victims, ensuring timely financial support for medical expenses and rehabilitation.
• The Dourine Act, 1910
• This act was enacted to control and prevent the spread of dourine, a venereal disease affecting horses, which could have significant implications for equine health and the livestock industry.
• It includes provisions for the examination, isolation, and treatment of infected animals, as well as guidelines for the disposal of affected livestock to prevent the disease's transmission.

The correct answer is Pulses.

Key Points

• Perishable foods are foods that spoil quickly within one or two days e.g., milk, curds, fish and meat.
• Semi perishable foods can last for 1–2 weeks. Examples are fruits and vegetables. Root crops like onions and potatoes last for 2–4 weeks.
• Non-perishable foods are those foods that generally last for one year e.g., grains like rice, wheat, pulses and dals, oilseeds.

The correct answer is 1980.

Key Points

• Kabaddi:
• Kabaddi is a popular contact sport in Southern Asia that first originated in Ancient India.
• It is played across the country and is the official game in the states of Punjab, Tamil Nadu, Bihar, Telangana, and Maharashtra.
• In the kabaddi game, each side has 7 players when the game starts.
• There will be 5 substitutes for each side as well.
• The games in Kabaddi are 40 minutes long.

The correct answer is Olly.

Key Points

• Men’s Hockey World Cup 2023 (15th edition) is hosted by India.
• It was held in Bhubaneswar and Rourkela, Odisha, and Olly (Turtle) was its mascot.
• Belgium won the 14th edition of FIH by defeating the Nwetherlands and India was at 6th place.
• The Men’s Hockey World Cup 2023 (15th edition) was won by Germany.
• Pakistan is the most successful team with the four-time winner of this tournament.
• India had won only once in 1975.

Additional Information

• Hockey:
• The Hockey World cup is organised by the International Hockey Federation (FIH) once in four years.
• The headquarters of FIH is located in Lausanne, Switzerland.
• ‘Blackheath Rugby and Hockey Club’ was the first hockey club in the world that was set up in the year 1861 in England.
• The first Hockey World Cup was organised in Barcelona (Spain) in 1971 and the winner was Pakistan.
• India took part in Olympics for the first time in the year 1928 (in Amsterdam Olympics).
• In Olympics, India has won the hockey title a maximum of eight times.

The correct answer is Pallava.

Key Points

• Pallava
• Simhavishnu was the founder of the Pallava dynasty.
• The Pallava dynasty was a Tamil dynasty that ruled parts of South India from the 3rd century to the 9th century.
• Simhavishnu's reign marked the rise of the Pallavas to prominence in the region.
• The Pallavas are known for their patronage of architecture, particularly the rock-cut temples of Mahabalipuram.
• The dynasty played a crucial role in the cultural and political history of South India.

Additional Information

• Chola
• Vijayalaya Chola (reigned 848–871 CE) is widely regarded as the founder of the Chola dynasty that ruled over a significant portion of the Indian subcontinent from the 9th to the 13th centuries.
• Before Vijayalaya, the Cholas were a minor dynasty ruling from Uraiyur (modern-day Tiruchirappalli).
• Vijayalaya established the Chola capital at Thanjavur (modern-day Tanjore) after defeating the Muttaraiyar chieftain and establishing Chola supremacy in the region.
• He is credited with reviving the Chola power and laying the foundation for the future expansion of the Chola empire under his successors.
• Chera
• The Chera kingdom of Mahodayapuram was established in the ninth century in the south-western part of the peninsula, part of the present-day Kerala.
• The Chera kingdom played a pivotal role in the history of South India.
• The region's significant historical and cultural contributions include the introduction of Malayalam, which became prominent in inscriptions during this period.
• The association of the Chera kingdom with the South-Western part of the Indian peninsula highlights the geocultural significance of Kerala in Indian history, especially in relation to the development of regional languages and script.
• Moreover, the geographical location of the Chera kingdom contributed to its maritime commerce and cultural exchanges with distant lands, enhancing the spread of its cultural and linguistic influence.
• Chalukya dynasty
• They ruled in southern India from the 6th century.
• There were three types of Chalukyas,
• Badami Chalukyas
• Eastern Chalukyas
• Western Chalukyas
• Their capital was Badami.
• They followed Hinduism, Buddhism, and Jainism as well.
• They had a monarchial structure with King as its head.
• The chief queen was known as Tattamahish.
• King had all the judicial as well as administrative powers

The correct answer is zero.

Key Points

• Ammonia (NH3):

• Ammonia is produced in large quantities worldwide for use as a fertilizer and as a primary source of nitrogen in the production of many chemicals. Liquid ammonia can be sprayed directly into the soil as a fertilizer, or it can be converted to ammonium salts (nitrate, sulfate or phosphate) for solid fertilizers.

• It is a colourless gas with a pungent smell and can be toxic when exposed to higher concentration levels. It is highly soluble in water and basic in nature

• Nitrogen (Z=7), its ground state electronic configuration is $1s^2 2s^2 2p^3$ and the observed valency is 3. At the time of hybridization, one 2s and three 2p orbitals mix to form four sp3 hybridized orbitals. Out of the four hybrid orbitals, one contains a lone pair of electrons and the other three are half filled.

• Three half-filled sp3 hybrid orbitals of N atom overlap with half-filled 1s orbital of three hydrogen atoms to form three sigma covalent bonds. Thus, in NH3 molecule, there are three bonding pairs of electrons and one lone pair of electrons.

​

• This single lone pair distorts the bond angle of H-N-H to107.5°. and therefore due to the lone pair effect, the structure of the NH3 molecule is pyramidal with nitrogen at the tip.

The correct answer is independent judicial system.

Key Points

• The Government of India Act, 1935 was a British law that established a federal system of government in India.
• It was the most important source of inspiration for the framers of the Indian Constitution.
• The Act provided for a bicameral legislature, a federal court, and a system of provincial autonomy.
• It also guaranteed fundamental rights to the people of India.
• The other options are incorrect.
• The Constitution of Britain is a unitary system of government, while the Constitution of Ireland is a republic.
• The Indian Council Act, 1919 was a less comprehensive document than the Government of India Act, 1935.
• The Government of India Act, 1935 was not without its flaws.
• However, it was a significant step forward in the development of self-government in India.
• It provided the basis for the Indian Constitution, which is one of the most progressive constitutions in the world.

Additional Information

Here are some of the specific provisions of the Government of India Act, 1935 that were incorporated into the Indian Constitution:

• The federal system of government
• The bicameral legislature
• The federal court
• The system of provincial autonomy
• The guarantee of fundamental rights

The Indian Constitution also incorporated some provisions from other sources, such as the American Constitution, the Irish Constitution, and the British Magna Carta. However, the Government of India Act, 1935 was the most important source of inspiration for the framers of the Indian Constitution.

The correct answer is Birds.

Key Points

• Aves is a class of vertebrates that comprises the birds' species.
• Birds are egg-laying animals, have fur, two legs, and are endothermic.
• They are found worldwide, and their size ranges from 5 cm (bee hummingbird) to 2.75 m (ostrich).
• Ostrich is the organism that belongs to the Aves as they have those characteristics that are found in the members of the class Aves.
• The ostrich is typical of a group of flightless birds called ratites.
• There are two living species of ostrich: the common ostrich (Struthio camelus) and the Somali ostrich (Struthio molybdophanes).​

Additional Information

• Aves:
• The class Aves includes birds or flying vertebrates.
• A characteristic feature of Aves is the presence of feathers.
• The birds are unique in having a coat of feathers and in resting on the hindlimbs alone.
• Respiration takes place only by the lungs.
• Lungs are respiratory organs, which are spongy, inelastic, and are connected to large air sacs.
• Endoskeleton is bony, but delicate and light.
• Bones are pneumatic i.e., contain air cavities to reduce weight.
• These are warm-blooded animals.
• Example - Covus (Crow), Neophron (Vulture)

Shortcut Trick

Let Ram's age be x. So, Mayadevi's age = 3x.

Assuming Ravi's age is x − 7 to obtain an integer value.

Soham's age = (x − 7) ÷ 4.

Given difference: 3x − (x − 7) ÷ 4 = 65 ⇒ 11x + 7 = 260 ⇒ 11x = 253 ⇒ x = 23.

Mayadevi's age = 3 × 23 = 69 years.

∴ The correct answer is 69 years.

Alternate Method

Given:

⇒ Difference between the ages of Ram and Ravi = 7 years

⇒ Mayadevi's age = 3 × Ram's age

⇒ Ravi's age = 4 × Soham's age

⇒ Difference between the ages of Mayadevi and Soham = 65 years

Calculations:

⇒ Let the age of Ram be x years.

⇒ Age of his mother, Mayadevi = 3x years.

⇒ Since Ram and Ravi differ by 7 years, let's assume Ravi's age is x − 7. (Assuming x + 7 will result in a fractional age).

⇒ Age of Ravi's brother, Soham = (x − 7) ÷ 4.

⇒ The difference between the ages of Mayadevi and Soham is 65 years:

⇒ 3x − [(x − 7) ÷ 4] = 65

⇒ (12x − x + 7) ÷ 4 = 65

⇒ 11x + 7 = 260

⇒ 11x = 253

⇒ x = 23

⇒ Therefore, Ram's age = 23 years.

⇒ Mayadevi's age = 3 × 23 = 69 years.

⇒ (Note: If we take Ravi's age as x + 7, we get 11x − 7 = 260 ⇒ x = 267 ÷ 11, which does not yield an integer value for age.)

∴ The correct answer is 69 years.

Additional Information

Linear Equations in Age Problems

Most age problems can be modeled using single-variable linear equations by expressing all ages in terms of one common person's age. Choosing the right central character minimizes complex fractional calculations.

Extraneous or Fractional Values

In quantitative aptitude, if solving an age equation yields a non-integer value where a whole number is expected, try alternating the assumed condition (e.g., swapping which person is older in an absolute difference relationship).

Constant Age Difference

The age difference between any two individuals remains constant regardless of the time elapsed, whereas the ratio of their ages changes continuously over the years.

Calculation:

The male research scholars in Mathematics = 18 - 4 = 14

The male research scholars in Statistics = 12 - 3 = 9

According to the question,

The ratio of male research scholars in Mathematics to Statistics is 14 : 9.

Shortcut Trick

Given the proportion: A : C = B : D

Substitute the values from the table: 25 : 30 = 40 : D

Simplify the ratio of A to C: 25 / 30 = 5 / 6

Equating the parts: 5 units = 40 → 1 unit = 8

Therefore, D = 6 units = 6 × 8 = 48

∴ The correct answer is 48.

Alternate Method

Given:

Value of A = 25

Value of B = 40

Value of C = 30

Formula Used:

Proportion rule: If four values are in proportion such that A : C = B : D, then A × D = B × C

Calculation:

⇒ A / C = B / D

⇒ 25 / 30 = 40 / D

⇒ 5 / 6 = 40 / D

Cross-multiplying the terms:

⇒ 5 × D = 40 × 6

⇒ 5 × D = 240

⇒ D = 240 ÷ 5

⇒ D = 48

∴ The correct answer is 48.

Additional Information

Proportion Rule

When two ratios are set equal to each other, they form a proportion. In any valid proportion a : b = c : d, the product of the extreme terms equals the product of the mean terms (a × d = b × c).

Fourth Proportional

If four variables a, b, c, and d are in proportion, then d is known as the fourth proportional to a, b, and c. It can be easily calculated using the direct relation d = (b × c) ÷ a.

Mean Proportional

The mean proportional between two distinct quantities a and b is given by √(a × b). It acts as the middle value x in a continuous proportion where a : x = x : b.

Shortcut Trick

For two triangles to be exactly identical (congruent), at least one corresponding side length must be equal.

Standard criteria like SSS, SAS, ASA, and AAS all evaluate at least one side length.

The AAA (Angle-Angle-Angle) criterion only compares angles and completely lacks a side measurement.

Thus, AAA only proves that triangles have the same shape (similarity), but not necessarily the same size.

∴ The correct answer is angle-angle-angle.

Alternate Method

Given:

Criteria for proving triangle congruency.

Concept Used:

Congruent triangles must have exactly the same size and shape. All corresponding angles and sides must be equal.

A B C

D E F

60° 60° 60°

60° 60° 60°

AAA guarantees Similarity, NOT Congruence

Calculations:

⇒ ASA (Angle-Side-Angle): Valid criterion. If two angles and the included side of one triangle equal the corresponding parts of another, they are congruent.

⇒ SSS (Side-Side-Side): Valid criterion. If all three sides of one triangle are equal to the corresponding sides of another, they are congruent.

⇒ SAS (Side-Angle-Side): Valid criterion. If two sides and the included angle of one triangle are equal to the corresponding parts of another, they are congruent.

⇒ AAA (Angle-Angle-Angle): Invalid criterion for congruence. Having all three corresponding angles equal only ensures the triangles are proportional (similar). For instance, an equilateral triangle with side lengths of 2 cm has angles 60°, 60°, 60°, and an equilateral triangle with side lengths of 4 cm also has angles 60°, 60°, 60°. They have identical angles but different sizes.

⇒ Therefore, AAA cannot be used to prove that two triangles are congruent.

∴ The correct answer is angle-angle-angle.

Additional Information

Congruence vs Similarity

Congruent figures have exactly the same shape and size (ratio of corresponding sides is 1). Similar figures have the same shape but may differ in size (ratio of corresponding sides is constant).

RHS Congruence Criterion

RHS (Right Angle-Hypotenuse-Side) applies only to right-angled triangles. If the hypotenuse and one side of a right triangle are equal to the hypotenuse and corresponding side of another right triangle, they are congruent.

AAS Congruence Criterion

If any two angles and a non-included side of one triangle are exactly equal to the corresponding angles and side of another triangle, the two triangles are geometrically congruent.

Shortcut Trick

For two circles touching externally, the length of the Direct Common Tangent (DCT) is given directly by the property:

DCT = 2√(r₁ × r₂)

⇒ DCT = 2 × √(18 × 12)

⇒ DCT = 2 × √216 = 2 × 6√6

⇒ DCT = 12√6 cm

∴ The correct answer is 12√6 cm.

Alternate Method

Given:

Radius of the larger circle (R) = 18 cm

Radius of the smaller circle (r) = 12 cm

Since the circles touch each other externally, the distance between their centres (D) = R + r = 18 + 12 = 30 cm.

18 cm 12 cm C₁ C₂ A B DCT = ? P 6 cm 30 cm

Formula Used:

Length of Direct Common Tangent = √[D² − (R − r)²]

Calculations:

⇒ D = 30 cm

⇒ R − r = 18 − 12 = 6 cm

⇒ Length of DCT = √[30² − 6²]

⇒ Length of DCT = √[900 − 36]

⇒ Length of DCT = √864

⇒ Length of DCT = √(144 × 6) = 12√6 cm

∴ The correct answer is 12√6 cm.

Additional Information

Direct Common Tangent (DCT)

The general formula for the length of a direct common tangent is DCT = √[D² − (R − r)²], where D is the distance between the centres.

Transverse Common Tangent (TCT)

When two circles are separated, the length of the Transverse Common Tangent is given by TCT = √[D² − (R + r)²].

Number of Common Tangents

Circles touching externally have exactly 3 common tangents. Circles touching internally have exactly 1 common tangent. Separated disjoint circles have 4 common tangents.

Shortcut Trick

Put 2A = 45° for quick evaluation.

Numerator = 2 + tan² 45° + cot² 45° = 2 + 1² + 1² = 4

Denominator = sec 45° × cosec 45° = √2 × √2 = 2

Expression value = 4 ÷ 2 = 2

Check Option 4: sec 45° × cosec 45° = √2 × √2 = 2 (Matches perfectly)

∴ The correct answer is sec 2A × cosec 2A.

Alternate Method

Given:

Expression = (2 + tan² 2A + cot² 2A) ÷ (sec 2A × cosec 2A)

Formula Used:

1 + tan² θ = sec² θ

1 + cot² θ = cosec² θ

sec² θ + cosec² θ = sec² θ × cosec² θ

Calculation:

⇒ Split 2 into 1 + 1 to use standard trigonometric identities.

⇒ Numerator = (1 + tan² 2A) + (1 + cot² 2A)

⇒ Numerator = sec² 2A + cosec² 2A

⇒ Using the standard identity: sec² 2A + cosec² 2A = sec² 2A × cosec² 2A

⇒ Substitute this back into the given expression.

⇒ Expression = (sec² 2A × cosec² 2A) ÷ (sec 2A × cosec 2A)

⇒ Cancel out the common terms from numerator and denominator.

⇒ Expression = sec 2A × cosec 2A

∴ The correct answer is sec 2A × cosec 2A.

Additional Information

Pythagorean Identities

The three fundamental identities are: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Sum to Product Conversion Identity

A very useful direct result in trigonometry is sec²θ + cosec²θ = sec²θ × cosec²θ.

Double Angle Formulas

Important double angle relations: sin 2θ = 2sinθcosθ and cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ.

Shortcut Trick

Overall rate of interest = (352 ÷ 4000) × 100 = 8.8%

By applying the rule of alligation on the interest rates:

Ratio of principal (8% part : 10% part) = (10 − 8.8) : (8.8 − 8) = 1.2 : 0.8 = 3 : 2

Sum lent at 10% corresponds to 2 parts out of (3 + 2) = 5 total parts.

Sum lent at 10% = (2 ÷ 5) × 4000 = ₹1,600

∴ The correct answer is ₹1,600.

Alternate Method

Given:

Total Principal (P) = ₹4000

Total Simple Interest (SI) = ₹352

Time (T) = 1 year

Formula Used:

Simple Interest = (P × R × T) ÷ 100

Part 1 Part 2 Overall Rate

8% 10% 8.8% 1.2 0.8

Ratio of Amounts = 1.2 : 0.8 = 3 : 2

Calculations:

Let the amount lent at 10% be x.

Then, the amount lent at 8% will be (4000 − x).

⇒ SI from 10% part + SI from 8% part = 352

⇒ [(x × 10 × 1) ÷ 100] + [((4000 − x) × 8 × 1) ÷ 100] = 352

⇒ 10x + 32000 − 8x = 35200

⇒ 2x + 32000 = 35200

⇒ 2x = 35200 − 32000

⇒ 2x = 3200

⇒ x = 1600

∴ The correct answer is ₹1,600.

Additional Information

Rule of Alligation

It is a quick method to find the ratio in which two components of different rates or concentrations are mixed to obtain a mixture of a desired average rate.

Simple Interest (SI)

SI = (P × R × T) ÷ 100, where P is the Principal, R is Rate per annum, and T is Time in years.

Average / Effective Rate of Interest

When a single total interest is given for parts lent at different rates, the effective rate is calculated as: R = (Total SI × 100) ÷ (Total Principal × Time).

Shortcut Trick Check the sum of the two smaller distances: XY + YZ = 11 + 13 = 24 cm. The largest distance is XZ = 24 cm. Since XY + YZ = XZ, the three points X, Y, and Z lie on a straight line (they are collinear). A circle can only pass through three non-collinear points. ∴ The correct answer is 0.

Alternate Method Given: Distance XY = 11 cm Distance YZ = 13 cm Distance XZ = 24 cm

X Y Z

11 cm

13 cm

24 cm

Concept Used: For any three points to form a triangle (non-collinear), the sum of the distances between any two pairs of points must be strictly greater than the third distance. A circle can uniquely pass through three non-collinear points, but zero circles can pass through three collinear points.

Calculations: ⇒ Sum of two smaller distances = XY + YZ ⇒ Sum = 11 cm + 13 cm = 24 cm ⇒ The given largest distance XZ = 24 cm ⇒ Since XY + YZ = XZ, the point Y perfectly lies on the straight line segment connecting X and Z. ⇒ Therefore, points X, Y, and Z are collinear. ⇒ Because no circle can pass through three points lying on a single straight line, the number of such circles is zero. ∴ The correct answer is 0.

Additional Information Triangle Inequality Theorem For any three points to form a valid triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side (a + b > c). Circumcircle A circumcircle is a unique circle that passes through all three non-collinear vertices of a triangle. The intersection of the perpendicular bisectors of the sides acts as its center. Collinear Points When three or more points lie entirely on the same straight line, they are known as collinear points. They cannot form a closed geometrical 2D figure like a circle or a triangle.

Given:

Expression: 15 ÷ 3 of 2 × 2 + 9 ÷ 18 of 2 × 4 − 4 ÷ 8 × 2

Formula Used:

Order of operations (PEMDAS/BODMAS): Parentheses, Exponents (i.e. powers and roots, etc.), Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)

Calculation:

First, we solve the expression step-by-step according to the order of operations:

15 ÷ 3 of 2 × 2 + 9 ÷ 18 of 2 × 4 − 4 ÷ 8 × 2

⇒ 15 ÷ (3 × 2) × 2 + 9 ÷ (18 × 2) × 4 − 4 ÷ 8 × 2

⇒ 15 ÷ 6 × 2 + 9 ÷ 36 × 4 − 4 ÷ 8 × 2

⇒ (15 ÷ 6) × 2 + (9 ÷ 36) × 4 − (4 ÷ 8) × 2

⇒ 2.5 × 2 + 0.25 × 4 − 0.5 × 2

⇒ 5 + 1 − 1

⇒ 5

∴ The correct answer is option (2).

Shortcut Trick

Effective successive discount of 30% and 40% = 30 + 40 − (30 × 40) ÷ 100 = 58%

Difference in discount percentages = 65% − 58% = 7%

Given difference in Selling Prices = ₹105

⇒ 7% of Marked Price = 105

⇒ Marked Price = 105 × (100 ÷ 7) = 1500

∴ The correct answer is ₹1,500.

Alternate Method

Given:

First discount (D₁) = 65%

Second discount series = Two successive discounts of 30% and 40%

Difference in Selling Prices = ₹105

Formula Used:

Effective discount for two successive discounts A% and B% = A + B − (A × B) ÷ 100

Calculations:

⇒ Effective second discount (D₂) = 30 + 40 − (30 × 40) ÷ 100

⇒ D₂ = 70 − 12 = 58%

⇒ Let the Marked Price (MP) be 100%.

⇒ Selling Price 1 (SP₁) = 100% − 65% = 35% of MP

⇒ Selling Price 2 (SP₂) = 100% − 58% = 42% of MP

⇒ Difference in Selling Prices = SP₂ − SP₁

⇒ Difference in SP = 42% − 35% = 7% of MP

⇒ We are given that the difference is ₹105.

⇒ 7% of MP = ₹105

⇒ 1% of MP = 105 ÷ 7 = ₹15

⇒ 100% of MP (Marked Price) = 15 × 100 = ₹1,500

∴ The correct answer is ₹1,500.

Additional Information

Successive Discounts

When two discounts D₁% and D₂% are given successively, the effective single discount is calculated as [D₁ + D₂ − (D₁ × D₂) ÷ 100]%.

Marked Price & Discount Relation

Discount is always computed on the Marked Price (MP). The final Selling Price (SP) is calculated as MP − Discount Amount. Therefore, Discount % = (Discount ÷ MP) × 100.

Effective Equivalent Pricing

A single larger discount percentage generally offers more savings to the buyer compared to two successive discounts that add up to the same numerical sum (e.g., 65% is a bigger discount than 30% and 35% successively).

Shortcut Trick

Using basic trigonometry, the distance from the wall (Base) can be calculated directly as:

Base = Hypotenuse × cos(θ)

Base = 10 × cos(30°)

Base = 10 × (√3 / 2) = 5√3 m

∴ The correct answer is 5√3 m.

Alternate Method

Given:

Length of the pole (Hypotenuse AC) = 10 m

Angle made by the pole with the ground (∠ACB) = 30°

Formula Used:

In a right-angled triangle, cos(θ) = Base / Hypotenuse

Calculations:

Let the vertical wall be AB and the pole be AC.

The distance of the foot of the pole from the wall is represented by BC.

⇒ In ΔABC, cos(30°) = BC / AC

⇒ √3 / 2 = BC / 10

⇒ BC = 10 × (√3 / 2)

⇒ BC = 5√3 m

∴ The correct answer is 5√3 m.

Additional Information

Trigonometric Ratios

In a right-angled triangle, sin(θ) = Perpendicular / Hypotenuse, cos(θ) = Base / Hypotenuse, and tan(θ) = Perpendicular / Base.

Common Values

For standard angles: sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: Hypotenuse² = Base² + Perpendicular².

Calculation:

The average sales of the schools A, B and C in 2014

= (70 + 35 + 80)/3

⇒ 61.66

The average sales of schools A, C, and F in 2013

⇒ (50 + 60 + 40)/3

⇒ 50

According to the question,

⇒ (50/61.66)× 100

⇒ 81.08%

∴ The correct answer is 81.08%.

Shortcut Trick

Value ratio of ₹1, ₹5, and ₹10 coins = (1 × 3) : (5 × 5) : (10 × 7) = 3 : 25 : 70

Total value units = 3 + 25 + 70 = 98 units

Since 98 units = ₹980, 1 unit = ₹10

Total value of ₹10 coins = 70 units = ₹700

Number of ₹10 coins = 700 ÷ 10 = 70

∴ The correct answer is 70 coins.

Alternate Method

Given:

Ratio of number of ₹1, ₹5, and ₹10 coins = 3 : 5 : 7

Total amount in the bag = ₹980

Formula Used:

Total Value = Denomination × Number of coins

Calculations:

⇒ Let the number of ₹1, ₹5, and ₹10 coins be 3x, 5x, and 7x respectively.

⇒ Value of ₹1 coins = 1 × 3x = 3x

⇒ Value of ₹5 coins = 5 × 5x = 25x

⇒ Value of ₹10 coins = 10 × 7x = 70x

⇒ Total value = 3x + 25x + 70x = 98x

⇒ 98x = 980

⇒ x = 980 ÷ 98 = 10

⇒ Number of ₹10 coins = 7x = 7 × 10 = 70

∴ The correct answer is 70 coins.

Additional Information

Coins based on Value Ratio

When the total number of coins is given but their value ratio is provided, divide the value ratio by the respective denomination to find the ratio of the number of coins.

Coins based on Number Ratio

When the total value is given and the ratio of the number of coins is provided, multiply the number ratio by the respective denomination to find the value ratio.

Proportional Division

If a total quantity T is to be divided in the ratio a : b : c, the individual parts are given by T × a / (a+b+c), T × b / (a+b+c), and T × c / (a+b+c).

Shortcut Trick The largest 5-digit number is 99999. Divide 99999 by 47 to find the remainder. 99999 ÷ 47 gives a quotient of 2127 and a remainder of 30. Subtract the remainder from 99999 to find the exact largest multiple. Required number = 99999 − 30 = 99969. ∴ The correct answer is 99969.

Alternate Method Given: We need to find the largest 5-digit number divisible by 47. Concept Used: According to the division algorithm: Dividend = (Divisor × Quotient) + Remainder. To make a number perfectly divisible by a divisor without exceeding it, we must subtract the remainder from the dividend. Calculations: ⇒ The largest 5-digit number is 99999. ⇒ Let us divide 99999 by 47. ⇒ First, 99 ÷ 47 = 2 with a remainder of 5. ⇒ Bringing down 9, we get 59. 59 ÷ 47 = 1 with a remainder of 12. ⇒ Bringing down 9, we get 129. 129 ÷ 47 = 2 with a remainder of 35. ⇒ Bringing down the final 9, we get 359. 359 ÷ 47 = 7 with a remainder of 30. ⇒ The final remainder is 30. ⇒ To make it exactly divisible by 47, we subtract the remainder from 99999. ⇒ Required Number = 99999 − 30 ⇒ Required Number = 99969 ∴ The correct answer is 99969.

Additional Information Division Algorithm Dividend = (Divisor × Quotient) + Remainder. Subtracting the remainder from the dividend makes the result perfectly divisible by the divisor. Smallest n-digit number divisible by 'd' Divide the smallest n-digit number (e.g., 10000 for 5 digits) by the divisor 'd' to find the remainder 'r'. The required smallest divisible number is (Smallest n-digit number) + (d − r). Nearest number to 'N' divisible by 'd' Find the remainder 'r' when N is divided by d. If r < d/2, subtract r from N. If r > d/2, add (d − r) to N.

Shortcut Trick

Volume = (1/3)πr²h = 1232 ⇒ (1/3) × (22/7) × r² × 24 = 1232

⇒ r² = (1232 × 7) ÷ 176 = 49 ⇒ r = 7 cm.

Using the standard Pythagorean triplet (7, 24, 25), the slant height l = 25 cm.

TSA = πr(r + l) = (22/7) × 7 × (7 + 25) = 22 × 32 = 704 cm².

∴ The correct answer is 704 cm².

Alternate Method

Given:

Height of the cone (h) = 24 cm

Volume of the cone (V) = 1232 cm³

Formula Used:

Volume of a cone = (1/3)πr²h

Slant height (l) = √(r² + h²)

Total Surface Area (TSA) = πr(r + l)

Calculations:

⇒ 1232 = (1/3) × (22/7) × r² × 24

⇒ 1232 = (176/7) × r²

⇒ r² = (1232 × 7) ÷ 176

⇒ r² = 49

⇒ Radius (r) = 7 cm

⇒ Slant height (l) = √(7² + 24²)

⇒ l = √(49 + 576)

⇒ l = √625 = 25 cm

⇒ TSA = (22/7) × 7 × (7 + 25)

⇒ TSA = 22 × 32 = 704 cm²

∴ The correct answer is 704 cm².

Additional Information

Curved Surface Area of a Cone

The curved surface area (CSA) covers only the lateral curved side of the cone. Formula: CSA = πrl.

Volume of a Cone

The internal capacity of a cone is exactly one-third of the volume of a cylinder with the same base and height. Formula: V = (1/3)πr²h.

Pythagorean Triplets

Standard sets of integers that satisfy a² + b² = c² speed up geometry computations considerably. Common examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Shortcut Trick For any angle t < 45º, we can easily verify the relation by substituting a standard value like t = 30º. From trigonometric tables, we know that sin(30º) = 0.5 and cos(30º) = √3/2 ≈ 0.866. Comparing both results, it is clearly evident that 0.5 < 0.866. This confirms that sin(30º) < cos(30º), establishing that sin(t) < cos(t) for all t < 45º. ∴ The correct answer is sin(t) < cos(t) when t < 45º.

Alternate Method Given: ⇒ The angle t lies in the first quadrant where 0º < t < 90º

Calculations: ⇒ In the first quadrant (0º to 90º), the sine function is strictly increasing while the cosine function is strictly decreasing. ⇒ At the exact midpoint t = 45º, both functions intersect and are equal: sin(45º) = cos(45º) = 1/√2. ⇒ For the interval 0º < t < 45º, the cosine curve lies above the sine curve. Therefore, sin(t) < cos(t). ⇒ For the interval 45º < t < 90º, the sine curve lies above the cosine curve. Therefore, sin(t) > cos(t). ⇒ Checking all given statements against these intervals, only the condition for t < 45º matches correctly. ∴ The correct answer is sin(t) < cos(t) when t < 45º.

Additional Information Behavior of Tangent Function For 0º < t < 90º, tan(t) is strictly increasing from 0 to infinity. At the midpoint t = 45º, tan(45º) = 1. Complementary Angle Properties Trigonometric functions of complementary angles are directly related by: sin(90º − t) = cos(t) and cos(90º − t) = sin(t). Trigonometric Identities The foundational Pythagorean identity holds true for all possible values of t: sin2(t) + cos2(t) = 1.

Shortcut Trick

Bill 1 (10 Nov to 19 Nov = 9 days gap) is eligible for a 2% prompt payment discount.

Bill 2 (purchased and paid on 19 Nov) is eligible for a 5% instant cash discount.

Total Amount = (9800 × 0.98) + (5480 × 0.95)

Total Amount = 9604 + 5206 = 14810

∴ The correct answer is ₹14,810.

Alternate Method

Given:

Bill 1 value = ₹9,800 (Purchased on 10 November 2019)

Bill 2 value = ₹5,480 (Purchased on 19 November 2019)

Payment date for both the bills = 19 November 2019

Discount scheme 1 = 2% for payments cleared within 10 days.

Discount scheme 2 = 5% for payments made in cash at the time of purchase.

Calculations:

⇒ Time gap for clearing Bill 1 = 19 November − 10 November = 9 days.

⇒ Since Bill 1 is cleared within 10 days, a 2% discount is applicable on it.

⇒ Payable amount for Bill 1 = 9800 × (100 − 2) / 100 = 9800 × 98 / 100 = ₹9,604

⇒ Bill 2 is purchased and paid on the same day in cash, making it eligible for the 5% discount.

⇒ Payable amount for Bill 2 = 5480 × (100 − 5) / 100 = 5480 × 95 / 100 = ₹5,206

⇒ Total amount paid by Ramesh = Payable for Bill 1 + Payable for Bill 2

⇒ Total amount paid = 9604 + 5206 = ₹14,810

∴ The correct answer is ₹14,810.

Additional Information

Cash Discount vs Trade Discount

A trade discount is offered directly on the catalog price to encourage bulk buying, whereas a cash discount is offered on the invoice price to encourage early or prompt payments by the customer.

Effective Successive Discount

When two successive discounts of a% and b% are applied sequentially on a product, the effective overall discount is calculated using the formula: Effective Discount = a + b − (a × b) / 100 %.

Selling Price after Discount

The final selling price of an item after offering a discount of d% on the marked price (MP) is determined by the relation: SP = MP × (100 − d) / 100.

Shortcut Trick

Since Gain is 1/6th of the Selling Price (SP), let us assume SP = 6 units.

This means Gain = 1 unit.

Cost Price (CP) = SP − Gain = 6 − 1 = 5 units.

Gain % = (Gain ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Selling Price (SP) = ₹42564

Gain = 1/6 of Selling Price

Formula Used:

Cost Price (CP) = SP − Gain

Gain % = (Gain ÷ CP) × 100

Calculations:

⇒ Gain = (1 ÷ 6) × 42564 = ₹7094

⇒ Cost Price (CP) = 42564 − 7094 = ₹35470

⇒ Gain % = (7094 ÷ 35470) × 100

⇒ Gain % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Profit & Loss Base Rule

Profit and Loss percentages are always calculated on the Cost Price (CP) unless specifically stated otherwise in the question.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). The Selling Price is obtained as: SP = MP − (Discount % × MP ÷ 100).

Dishonest Dealer Concept

If a trader sells goods at Cost Price but uses false weights, the profit percentage is given by: [Error ÷ (True Value − Error)] × 100.

Successive Discounts

A single equivalent discount for two successive discounts of a% and b% is given by the formula: (a + b − a × b ÷ 100)%.

Shortcut Trick

If a + 1/a = k, then a² + 1/a² = k² − 2.

Here, the given value of k = 12.

⇒ a² + 1/a² = 12² − 2

⇒ a² + 1/a² = 144 − 2 = 142

∴ The correct answer is 142.

Alternate Method

Given:

a + 1/a = 12

Formula Used:

(x + y)² = x² + y² + 2xy

Calculations:

Squaring both sides of the given equation:

⇒ (a + 1/a)² = 12²

⇒ a² + 1/a² + 2 × a × (1/a) = 144

⇒ a² + 1/a² + 2 = 144

⇒ a² + 1/a² = 144 − 2

⇒ a² + 1/a² = 142

∴ The correct answer is 142.

Additional Information

Sum of Cubes Formula

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Negative Counterpart Formula

If x − 1/x = k, then x² + 1/x² = k² + 2.

Difference of Squares

For finding the difference, use the identity: x² − 1/x² = (x + 1/x)(x − 1/x).

Shortcut Trick

The sum of the three 60° sectors at the vertices of an equilateral triangle forms exactly one semicircle.

Required Area = Area of Equilateral Triangle − Area of Semicircle

Required Area = [(√3 ÷ 4) × side²] − [(1 ÷ 2) × π × radius²]

Required Area = [(√3 ÷ 4) × 28²] − [(1 ÷ 2) × (22 ÷ 7) × 14²]

Required Area = (196 × 1.73) − 308 = 339.08 − 308 = 31.08 cm²

∴ The correct answer is 31.08 cm².

Alternate Method

Given:

Side of the equilateral triangle, a = 28 cm

Radius of the circle, r = a ÷ 2 = 14 cm

Formula Used:

Area of an equilateral triangle = (√3 ÷ 4) × a²

Area of a sector of angle θ = (θ ÷ 360°) × πr²

Calculation:

⇒ Area of the equilateral triangle = (√3 ÷ 4) × 28²

⇒ Area of triangle = (√3 ÷ 4) × 784 = 196√3 cm²

⇒ Taking √3 ≈ 1.73, Area = 196 × 1.73 = 339.08 cm²

⇒ The triangle has three identical sectors, each with a central angle of 60°.

⇒ Total area of 3 sectors = 3 × (60° ÷ 360°) × π × 14²

⇒ Total sector area = 3 × (1 ÷ 6) × (22 ÷ 7) × 196

⇒ Total sector area = (1 ÷ 2) × 22 × 28 = 11 × 28 = 308 cm²

⇒ Area of the remaining part = Area of triangle − Area of 3 sectors

⇒ Area of the remaining part = 339.08 − 308 = 31.08 cm²

∴ The correct answer is 31.08 cm².

Additional Information

Properties of Equilateral Triangle

All three sides are equal, and all internal angles are 60°. The height is given by (√3 ÷ 2) × a.

Area of a Circular Sector

The area of a sector with central angle θ and radius r is given by the formula: A = (θ ÷ 360°) × πr².

Incircle and Circumcircle

For an equilateral triangle of side a, the inradius r = a ÷ (2√3) and circumradius R = a ÷ √3. Their area ratio is always 1 : 4.

Calculation:

The average production of B-type cars

⇒ (680 + 750 + 620 + 540)/4 

⇒ 647.5

The combined average production of A-type cars and C-types cars

⇒ (840 + 900 + 760 + 800)/4 + (890 + 960 + 1000 + 1200)/4

⇒ (825 + 1012.5)/2 

⇒ 918.75

According to the question, 

⇒ 918.75 - 647.5 

⇒ 271.25

∴ The correct answer is 271.25.

Shortcut Trick

When running in the same direction, relative speed is the difference between the two speeds.

Relative Speed = 13.5 − 9 = 4.5 km/h

Convert to m/s: 4.5 × (5/18) = 1.25 m/s

Time to meet = Track Length ÷ Relative Speed = 750 ÷ 1.25 = 600 seconds.

∴ The correct answer is 600 seconds.

Alternate Method

Given:

Distance of the circular track = 750 m

Speed of runner X = 9 km/h

Speed of runner Y = 13.5 km/h

Formula Used:

Time to meet for the first time = Distance ÷ Relative Speed

Speed in m/s = Speed in km/h × (5/18)

Calculations:

⇒ Speed of X in m/s = 9 × (5/18) = 2.5 m/s

⇒ Speed of Y in m/s = 13.5 × (5/18) = 3.75 m/s

⇒ Relative speed (same direction) = Speed of Y − Speed of X

⇒ Relative speed = 3.75 − 2.5 = 1.25 m/s

⇒ Time to meet = 750 ÷ 1.25

⇒ Time to meet = 75000 ÷ 125 = 600 seconds

∴ The correct answer is 600 seconds.

Additional Information

Meeting Anywhere on the Track (Same Direction)

Time to meet for the first time anywhere on the track = Track Length ÷ (S₁ − S₂).

Meeting Anywhere on the Track (Opposite Direction)

Time to meet for the first time anywhere on the track = Track Length ÷ (S₁ + S₂).

Meeting at the Starting Point

To find when runners will meet exactly at the starting point, calculate the time taken by each to complete one round, then take their LCM: LCM(T₁, T₂).

Number of Distinct Meeting Points

If the ratio of speeds is a:b (in lowest terms), the number of distinct meeting points is |a − b| for same direction, and a + b for opposite direction.

Shortcut Trick

Let each 1/3 segment of the journey be the LCM of the given speeds (40, 50, and 60) = 600 km.

Total distance of the journey = 3 × 600 = 1800 km.

Total time taken = (600 ÷ 40) + (600 ÷ 50) + (600 ÷ 60) = 15 + 12 + 10 = 37 hours.

Average Speed = Total Distance ÷ Total Time = 1800 ÷ 37 ≈ 48.65 km/h.

∴ The correct answer is 48.65 km/h.

Alternate Method

Given:

The speeds for three equal segments of the journey are 40 km/h, 50 km/h, and 60 km/h.

Formula Used:

When a journey is divided into three equal parts with speeds x, y, and z, the formula for average speed is:

Average Speed = 3xyz ÷ (xy + yz + zx)

Calculations:

⇒ Let x = 40, y = 50, and z = 60

⇒ Average Speed = (3 × 40 × 50 × 60) ÷ (40 × 50 + 50 × 60 + 60 × 40)

⇒ Average Speed = 360000 ÷ (2000 + 3000 + 2400)

⇒ Average Speed = 360000 ÷ 7400

⇒ Average Speed = 1800 ÷ 37

⇒ Average Speed ≈ 48.648 km/h

⇒ Rounding off to two decimal places, we get 48.65 km/h.

∴ The correct answer is 48.65 km/h.

Additional Information

Average Speed for 2 Equal Distances

When a body travels two equal distances at speeds x and y, the average speed over the entire journey is given by 2xy ÷ (x + y).

General Average Speed Concept

When distances are unequal, the average speed is strictly calculated by dividing the Total Distance Travelled by the Total Time Taken.

Relative Speed

If two objects move in the same direction at speeds x and y, their relative speed is |x − y|. If they move in opposite directions, their relative speed is (x + y).

Shortcut Trick

Total capacity of the cistern = LCM(40, 80, 120) = 240 units.

Efficiency of P = 6, Q = 3, and R = 2 units/min.

Since P works for the entire 30 minutes, work done by P = 30 × 6 = 180 units.

Remaining work to be done by Q and R together = 240 − 180 = 60 units.

Time pipes Q and R were open = 60 ÷ (3 + 2) = 12 minutes.

∴ The correct answer is 12 minutes.

Alternate Method

Given:

Time taken by P = 40 minutes

Time taken by Q = 80 minutes

Time taken by R = 120 minutes

Total time to fill the cistern = 30 minutes (half an hour)

Calculations:

⇒ The cistern is filled in exactly 30 minutes, which means pipe P was kept open for the entire duration of 30 minutes.

⇒ Work done by pipe P alone in 30 minutes = Efficiency × Time = 6 × 30 = 180 units.

⇒ Total work remaining = Total capacity − Work done by P

⇒ Remaining work = 240 − 180 = 60 units.

⇒ This remaining work must have been completed by pipes Q and R before they were turned off.

⇒ Combined efficiency of pipes Q and R = 3 + 2 = 5 units/min.

⇒ Time for which Q and R were kept open = Remaining work ÷ Combined efficiency

⇒ Time = 60 ÷ 5 = 12 minutes.

∴ The correct answer is 12 minutes.

Additional Information

Emptying Pipes (Negative Work)

If a pipe or a leak empties a tank, its efficiency is considered negative. The net filling efficiency becomes (Sum of inlet efficiencies) − (Sum of outlet efficiencies).

Time and Efficiency Relation

Time taken is always inversely proportional to efficiency. The ratio of time taken by two pipes A and B to fill a tank is the exact inverse of their efficiency ratio.

Pipes Working on Alternate Minutes

When pipes operate alternately, calculate the net work done in a complete cycle (e.g., 2 minutes or 3 minutes). Find out how many such cycles fit into the total work and manually add the remaining efforts for the last fractions.

Shortcut Trick

Using the direct formula: a³ + b³ + c³ − 3abc = [(a + b + c) ÷ 2] × [3(a² + b² + c²) − (a + b + c)²]

Substitute the given values: Sum = 18, Sum of squares = 36

⇒ Required value = (18 ÷ 2) × [3 × 36 − 18²]

⇒ Required value = 9 × [108 − 324] = 9 × (−216) = −1944

∴ The correct answer is −1944.

Alternate Method

Given:

Sum of three numbers, a + b + c = 18

Sum of their squares, a² + b² + c² = 36

Formula Used:

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

a³ + b³ + c³ − 3abc = (a + b + c) × [a² + b² + c² − (ab + bc + ca)]

Calculation:

⇒ (18)² = 36 + 2(ab + bc + ca)

⇒ 324 − 36 = 2(ab + bc + ca)

⇒ 288 = 2(ab + bc + ca)

⇒ ab + bc + ca = 144

Now, substitute the values into the second formula:

⇒ a³ + b³ + c³ − 3abc = 18 × [36 − 144]

⇒ a³ + b³ + c³ − 3abc = 18 × (−108)

⇒ a³ + b³ + c³ − 3abc = −1944

∴ The correct answer is −1944.

Additional Information

Alternative Algebraic Identity

a³ + b³ + c³ − 3abc = [(a + b + c) ÷ 2] × [(a − b)² + (b − c)² + (c − a)²].

Special Conditional Case

If the sum of the variables a + b + c = 0, then the identity reduces to a³ + b³ + c³ = 3abc. This property is frequently tested in competitive exams.

Square of a Trinomial

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca). It helps in finding the sum of the product of variables taken two at a time.

The correct answer is: Option 3.

Key Points

• The idiom "on the ball" means being aware of any changes or developments and quick to react to them. (जागरूक और तत्पर)
• Example: She is always on the ball and never misses a deadline.
• Explanation: The idiom indicates someone who is quick to understand and react in a situation, showing alertness and efficiency. (जागरूकता और कुशलता दिखाना)
• Example: Our new manager is really on the ball when it comes to handling customer complaints.

Therefore, the correct answer is: Option 3.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: Unaware of any changes or developments and lazy to react to them (परिवर्तनों या विकासों से अनजान और प्रतिक्रिया करने में आलसी)
• Example: He missed the opportunity because he was unaware of any changes or developments and lazy to react to them.
• Option 2: Aware of any changes or developments but slow to react to them (परिवर्तनों या विकासों से अवगत लेकिन प्रतिक्रिया करने में धीमा)
• Example: Though he knew about the changes, he was aware of any changes or developments but slow to react to them.
• Option 4: Aware of any changes or developments but doing nothing (परिवर्तनों या विकासों से अवगत लेकिन कुछ नहीं करना)
• Example: She was aware of any changes or developments but doing nothing, leading to missed opportunities.

The correct answer is: Option 3.

Key Points

• The word "Timid" means showing a lack of courage or confidence; easily frightened. (डरपोक)
• Example: She was too timid to ask for what she wanted.
• "Daring" refers to being adventurous or bold; showing courage. (साहसी)
• Example: His daring rescue of the child from the burning building was commendable.
• Hence, we can infer that the opposite of 'timid' is 'daring'.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Meek (नम्र): Quiet, gentle, and easily imposed on; submissive.
• Example: He was always meek and never stood up for himself.
• Anxious (चिंतित): Experiencing worry, unease, or nervousness.
• Example: She felt anxious before the big presentation.
• Spooky (डरावना): Sinister or ghostly in a way that causes fear and unease.
• Example: The abandoned house looked spooky at night.

The correct answer is Option 4.

 

Key Points

• The primary issue in the sentence is the incorrect use of the present tense "am" after "since." The correct past tense form should be used.
• The correct phrase should be "since I was born," ensuring grammatical correctness and clarity.
• The use of "since" indicates a point in time from the past until now, requiring the past tense "was" to be grammatically correct.

Therefore, the correct answer is Option 4.

 

Correct Sentence: "I have been living in Hyderabad since I was born."

 

Additional Information

• Option 1: "since I have born," – Incorrect because "have born" is not a correct tense form.
• Option 2: "since I will be born," – Incorrect because "will be born" indicates future tense, which is not applicable here.
• Option 3: "since I born," – Incorrect because "born" without "was" is grammatically incorrect.

The correct answer is: Option 2.

Key Points

• The word "descendent" is the incorrectly spelled word that should be corrected to "descendant".
• English meaning: A person, plant, or animal that is descended from a particular ancestor. (वंशज)
• Example: Many people believe they are descendants of famous historical figures.
• "Descendant" is the correct spelling.
• Hindi meaning: वंशज
• Example: She is a descendant of a long line of artists.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Dicsendent (डिसेंडेंट): This is an incorrect spelling and does not have a meaning.
• Discendent (डिसेंडेंट): This is also an incorrect spelling and does not have a meaning.
• Decendant (डिसेंडेंट): This is an incorrect spelling and does not have a meaning.

The correct answer is: Option 2.

Key Points

• The word "clique" means a small group of people who spend their time together and do not welcome other people into that group. (गुट)
• Example: The school was divided into several cliques, each with its own interests and activities.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Squad (दस्ता): A small group of people having a particular task.
• Example: The rescue squad arrived promptly at the scene of the accident.
• Employees (कर्मचारी): People who work for someone.
• Example: The company has over 200 employees working in various departments.
• Lobby (लॉबी): A group of people seeking to influence politicians or public officials on a particular issue.
• Example: The environmental lobby is pushing for new legislation to protect wildlife.

The correct answer is: Option 2.

Key Points

• The word "deplorable" means deserving strong condemnation or shockingly bad in quality. (निंदनीय)
• Example: The living conditions in the slum were deplorable.
• "Despicable" refers to something that is deserving hatred and contempt. (घृणित)
• Example: Stealing from the poor is a despicable act.

Hence, we can infer that the synonym of 'deplorable' is 'despicable'.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Jubilation (उल्लास): A feeling of great happiness and triumph.
• Example: The team was in jubilation after winning the championship.
• Complaining (शिकायत करना): Expressing dissatisfaction or annoyance about something.
• Example: She kept complaining about the cold weather.
• Rejoicing (आनंदित होना): Feeling or showing great joy or delight.
• Example: There was much rejoicing at the news of the victory.

The correct answer is: Shivering.

Key Points

• The word "trembling" means to shake involuntarily, typically as a result of anxiety, excitement, or frailty. (कांपना)
• Example: She was trembling with cold as she stood outside in the snow.
• "Shivering" refers to the act of shaking or trembling uncontrollably as a result of cold, fear, or excitement. (कांपना)
• Example: He was shivering with fear as he heard the strange noises from the dark alley.

Hence, we can infer that the synonym of 'trembling' is 'shivering'.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Weeping (रोना): Shedding tears; crying.
• Example: She was weeping after hearing the sad news.
• Running (दौड़ना): Moving at a speed faster than a walk.
• Example: He was running to catch the bus.
• Intrepid (निर्भीक): Fearless; adventurous.
• Example: The intrepid explorer ventured into unknown territories.

The correct answer is Option 2.

 

Key Points

• The primary issue in the sentence is the incorrect verb form "go" instead of "went".
• The sentence is in the past tense, so the verb should also be in the past tense.
• The correct phrase should be "they went to a restaurant".
• Thus, the error is located in Option 2 of the sentence.

Therefore, the correct answer is Option 2.

 

Correct Sentence: "After the concert was over, they went to a restaurant, had dinner together, and talked until midnight."

 

Additional Information

• Option 1: "had dinner together" – This part of the sentence is grammatically correct and does not require any changes.
• Option 3: "After the concert was over" – This part of the sentence is grammatically correct and does not require any changes.
• Option 4: "and talked until midnight" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is: Option 4.

Key Points

• The word "Florist" refers to a person who sells and arranges cut flowers. (फूलवाला)
• Example: The florist arranged a beautiful bouquet for the wedding.

Therefore, the correct answer is: Option 4 i.e. Florist.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Horticulturist (उद्यान विशेषज्ञ): Someone involved in the science or art of cultivating gardens.
• Example: The horticulturist explained the best methods for growing roses.
• Botanist (वनस्पतिशास्त्री): A scientist who studies plants.
• Example: The botanist discovered a new species of plant in the rainforest.
• Gardener (माली): A person who tends and cultivates a garden as a pastime or for a living.
• Example: The gardener spends hours every day taking care of the plants.

The correct answer is Option 3.

 

Key Points

• The primary issue in the sentence is the incorrect verb form "replaced" used in place of "replace".
• The correct future tense form should be "will replace" instead of "will replaced".
• This ensures grammatical correctness and clarity.
• Thus, the error is located in Option 3 of the sentence.

Therefore, the correct answer is Option 3.

 

Correct Sentence: "Renewable energy sources like solar and wind power will replace traditional fossil fuels and help reduce carbon emissions."

 

Additional Information

• Option 1: "and help reduce carbon emissions," – This part of the sentence is grammatically correct and does not require any changes.
• Option 2: "Renewable energy" – This part of the sentence is grammatically correct and does not require any changes.
• Option 4: "sources like solar and wind power" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is Option 3.

 

Key Points

• The paragraph begins by introducing the central concept of literacy and illiteracy in India, setting a contrasting tone regarding the existence of both more literate and illiterate people today: Sentence D.
• Sentence A follows, providing a specific example of the paradoxical situation in today's context, more children attending school yet also more children out of school: "More children go to school than at any time in the past. But more children today are out of school than any time in the past."
• Sentence B continues by stating that it is not sufficient to blame only the high birth rate for this situation: "But it is not enough to blame the high birth rate for this state of affairs."
• Sentence C concludes the argument by presenting the idea that illiteracy may actually be the cause of the high birth rate rather than the effect: "Indeed, it can be reasonably argued that continued mass illiteracy is not the result but the cause of the high birth rate."

Therefore, the correct answer is Option 3.

 

Rearranged Paragraph: "There are more literate people in India today than ever before. But there are also more illiterates than ever before. More children go to school than at any time in the past. But more children today are out of school than any time in the past. But it is not enough to blame the high birth rate for this state of affairs. Indeed, it can be reasonably argued that continued mass illiteracy is not the result but the cause of the high birth rate."

The correct answer is: Option 4.

Key Points

• The word "alleviate" means to make (suffering, deficiency, or a problem) less severe. (कम करना)
• Example: He took painkillers to alleviate his headache.
• "Relieve" means to cause (pain, distress, or difficulty) to become less severe or serious. (राहत देना)
• Example: She applied a cold compress to relieve the swelling.
• Hence, we can infer that the synonym of 'alleviate' is 'relieve'.

Therefore, the correct answer is: Option 4.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Swell (सूजना): To increase in size, typically as a result of an accumulation of fluid.
• Example: Her ankle began to swell after she twisted it.
• Add (जोड़ना): To put something together with something else to increase the number or amount.
• Example: You can add sugar to your coffee if you like it sweet.
• Intensify (तीव्र करना): To become or make something stronger or more extreme.
• Example: The conflict began to intensify as both sides refused to compromise.