RRB Group D Previous Year Paper (Held on: 24 Aug 2022 Shift 2)

The correct answer is angel of incidence.

Key Points

• The acute angle between the incident ray and the principal axis is called the angle of incidence.
• Angle of incidence is the angle between the incident ray and the normal to the surface at the point of incidence.
• Angle of emergence is the angle between the emergent ray and the normal to the surface at the point of emergence.
• Angle of reflection is the angle between the reflected ray and the normal to the surface at the point of incidence.
• Angle of deviation is the angle between the incident ray and the emergent ray.

Additional Information

• A concave mirror is a curved mirror in which the inner side of the curved shape is the reflecting surface.
• Its surface has an inward curvature that makes it resemble the interior surface of a hollow sphere.
• Because concave mirrors enable light beams to converge or come together after reflection, they are also known as converging mirrors.

Given: Each of P, Q, R, S, T, U and V has an exam on a different day of a week starting from Monday and ending on Sunday of the same week.

1. V's exam is on Saturday.

2. Q's exam is on Monday.

3) Only two persons have exams between Q and P.

4) Q's exam is immediately before R.

5. Only three persons have exams between R and V.

6. Only U's exam is between P and V.

7. T's exam is not held on Wednesday. 

And, exam of S will be held on the remaining day.

So, S's exam held on 'Wednesday'.

Hence, the correct answer is "Option 4".

The correct answer is PSLV-C52.

Key Points

• The Polar Satellite Launch Vehicle (PSLV) is a launch vehicle developed by the Indian Space Research Organisation (ISRO) to launch satellites into orbit.
• PSLV-C52 was used to launch EOS-04, which is an earth observation satellite.
• It is designed to provide images of the earth's surface and atmosphere for applications such as agriculture, forestry, water management, and disaster management.
• PSLV-C51, launched on February 28, 2021, carried Brazil's Amazonia-1 satellite as its primary payload, along with 18 other co-passenger satellites.
• PSLV-C53, launched on April 28, 2021, carried 14 satellites, including India's primary payload, the EMISAT satellite.
• PSLV-C54, launched on August 12, 2021, carried India's Earth observation satellite, EOS-03, along with three other co-passenger satellites.

Additional Information

• The PSLV has been used for a variety of missions, including launching remote sensing satellites, navigation satellites, and communication satellites.
• The PSLV has been in service since 1993 and has a success rate of 96.6%.
• ISRO has also developed the Geosynchronous Satellite Launch Vehicle (GSLV) for launching heavier satellites into geostationary orbit.
• ISRO is one of the world's leading space agencies and has successfully launched numerous missions, including the Chandrayaan-1 and Mangalyaan missions to the moon and Mars, respectively.

Given equation: 21 + 48 - 8 × 6 ÷ 4 = ?

Decoding the given information :

Now, replacing the sign from left to right and using BODMAS rule, we get :

New equation: 21 + 48 - 8 × 6 ÷ 4 = ?

Calculation :

⇒ 21 - 48 ÷ 8 + 6 × 4 = ?

⇒ 21 - 6 + 6 × 4 = ?

⇒ 21 - 6 + 24 = ?

⇒ 45 - 6 = 39.

Hence, the correct answer is "Option 4".

The logic followed here is:

For, NECTAR : TCVEGP

For, POCKET : VGMEQR

Similarly, for 'MONKEY : ?'

Hence, the correct answer is "Option 1".

The correct answer is 6.

Key Points

• There are 6 language universities located in India as of June 2022.
• Language universities are institutions that offer specialized courses in various languages and related disciplines, such as linguistics, literature, translation, etc.
• These universities aim to promote the study and research of languages and their cultures, as well as to prepare students for careers in language-related fields.
• It is worth noting that the number of language universities in India may change over time, depending on various factors such as demand, funding, policy changes, etc.

Additional Information

• Three of the six language universities in India are considered to be deemed to be universities.
• Central Universities make up the remaining three.
• The purpose of the Deemed-to-Be Universities is to advance Sanskrit language.
• The three Central Universities are one for the promotion of Hindi, Urdu, and English as Foreign Languages.

The correct answer is 0.1 A.

Key Points

• The induced current in the coil is given by Ohm's law, which states that I = V/R, where V is the induced EMF and R is the resistance of the coil.
• Thus, the induced current in the coil is 5/50 = 0.1 A.
• The direction of the induced current is determined by Lenz's law, which states that the direction of the induced current is such that it opposes the change in magnetic flux that caused it.
• In this case, the time varying magnetic field is inducing an EMF in the coil, so the induced current will flow in a direction that creates a magnetic field that opposes the original magnetic field.

Additional Information

• A coil of conductor is a device that consists of a series of loops of wire.
• When a magnetic field passes through the coil, it induces an EMF in the coil, which can cause a current to flow if the coil is connected to a circuit.
• The resistance of the coil is a measure of how much the coil resists the flow of current through it.
• It depends on the material of the wire, the length and cross-sectional area of the wire, and the temperature of the wire.

Calculation

(2/3 + 1/3) × 23 - 13 + 6

⇒ 1 × 23 - 13 + 6

⇒ 29 - 13 = 16

The answer is 16.

The correct answer is Noble gas group.

Key Points

• Noble gas group is a group of elements in the periodic table that includes helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn).
• These elements are characterized by having a full outermost shell of electrons, making them very stable and unreactive.
• This stability is due to the fact that they have achieved a state of equilibrium by filling their outermost shells with electrons.

Additional Information

• Pnictogen group is a group of elements in the periodic table that includes nitrogen (N), phosphorus (P), arsenic (As), antimony (Sb), bismuth (Bi), and moscovium (Mc).
• These elements are characterized by having five valence electrons in their outermost shell.
• Chalcogen group is a group of elements in the periodic table that includes oxygen (O), sulfur (S), selenium (Se), tellurium (Te), and polonium (Po).
• These elements are characterized by having six valence electrons in their outermost shell.
• Halogen group is a group of elements in the periodic table that includes fluorine (F), chlorine (Cl), bromine (Br), iodine (I), and astatine (At).
• ​These elements are characterized by having seven valence electrons in their outermost shell.

The correct answer is Arif Khan.

Key Points

• India's lone participant in the winter Olympics, who finished 45th in the giant slalom event is Arif Khan.
• He took home two gold medals from the South Asian Winter Games in 2011 in the Alpine skiing events of slalom and giant slalom.
• In the slalom and giant slalom competitions in the 2022 Winter Olympics, he competed on behalf of India.
• He placed 45th in the giant slalom competition and will compete in the slalom on the 16th.

Additional Information

• Himanshu Thakur is an Indian alpine skier.
• In the giant slalom competition, he represented India at the 2014 Winter Olympics, placing 72nd overall.
• In February 2017, Thakur was selected for India's squad for the 2017 Asian Winter Games.
• Aanchal Thakur is the first Indian to win an international medal in skiing.
• She won a bronze medal in the Alpine Ejder 3200 Cup held in Turkey in January 2018.

The correct answer is CH3COOH.

Key Points

• The given reaction is an example of an oxidation reaction where an aldehyde is oxidized to a carboxylic acid.
• The reagent used in this reaction is KMnO₄ (potassium permanganate) which acts as an oxidizing agent.
• CH3CHO is the product formed when the aldehyde group (-CHO) in the reactant is oxidized to a carboxylic acid group (-COOH).

Additional Information

• Potassium permanganate (KMnO4) is a strong oxidizing agent commonly used in organic chemistry for the oxidation of alcohols and aldehydes to carboxylic acids.
• CH₃COOH is acetic acid which is a carboxylic acid and can be formed from the oxidation of an alcohol.
• CH₃COOCH₃ is methyl acetate which is an ester and can be formed from the reaction of acetic acid with methanol.
• CH₂=CH₂ is ethene which is an alkene and can be formed from the dehydrogenation of ethanol.

The correct answer is 1966.

Key Points

• The Indira Gandhi Government devalued the India Rupee in 1966.
• Devaluation refers to the reduction in the value of a currency in relation to other currencies.
• The decision to devalue the Rupee was made to address the balance of payments crisis in India at that time.
• The devaluation was aimed at making Indian exports cheaper and more competitive in the global market.

Additional Information

• The balance of payments crisis in India in the mid-1960s was caused by a combination of factors, including a large trade deficit, a decline in foreign aid and investment, and a rise in oil prices.
• The devaluation of the Rupee in 1966 was the first major devaluation since India gained independence in 1947.
• The devaluation in 1978 was done as a part of a broader strategy to boost exports and reduce imports.
• The devaluation in 1970 was done to address the issue of overvaluation of the Rupee.

The correct answer is 1726.

Key Points

• The Municipal Corporation in Calcutta was established by a royal charter granted by King George I in the year 1726.
• The corporation was responsible for the administration of the city of Calcutta, which was then the capital of British India.
• The Calcutta Municipal Corporation was the first municipal corporation to be set up in India.

Additional Information

• The municipal government of Kolkata, the state capital of West Bengal, is known as the Kolkata Municipal Corporation (KMC).
• This civic administrative entity is responsible for 206.08 square kilometers of territory.
• Its insignia bears the Bengali script slogan, Purosri Bibardhan.

Given equation: 24 - 3 × 12 + 4 ÷ 3 = ?

Decoding the given information :

Now, replacing the sign from left to right and using BODMAS rule, we get :

New equation: 24 - 3 × 12 + 4 ÷ 3 = ?

Calculation :

⇒ 24 ÷ 3 + 12 - 4 × 3 = ?

⇒ 8 + 12 - 4 × 3 = ?

⇒ 8 - 12 + 12 = ?

⇒ 20 - 12 = 8.

Hence, the correct answer is "Option 2".

The correct answer is Copper.

Key Points

• Copper is a metal with high conductivity and low resistance, which makes it the most preferred material for electrical transmission lines.
• It is also used in electrical wiring, motors, transformers, and other electrical applications.
• Copper is a good conductor of heat and has high corrosion resistance, making it a durable material for long-term use.

Additional Information

• Nichrome is an alloy of nickel and chromium, which is known for its high electrical resistance.
• It is commonly used in heating elements, but not preferred for electrical transmission lines due to its high resistance.
• Nichrome wires are used in applications where heat resistance is required, such as in toasters, hair dryers, and ovens.
• Tungsten is a metal with the highest melting point and highest tensile strength of all metals.
• It is commonly used in filaments for incandescent light bulbs, but not preferred for electrical transmission lines due to its high cost and low conductivity.
• Tungsten wires are used in applications where high temperature resistance is required, such as in welding electrodes and rocket nozzles.
• Tin-lead alloy is a mixture of tin and lead, which is commonly used as a solder for electronic components.
• It is not preferred for electrical transmission lines due to its low conductivity and toxicity of lead.
• Tin-lead alloys are used in applications where low melting point and good wetting properties are required, such as in electronic circuit boards.

The correct answer is Both arguments I and II weaken the statement.

Key Points

• The statement mentions that private schools have increased their fees by 20% due to rising costs.
• Let's evaluate the given arguments in light of this statement.
• Argument I: The government has lowered the taxes levied on private schools by 8% this year.
• This argument does not directly relate to the statement.
• Even if the taxes have been lowered, it does not necessarily mean that private schools cannot increase their fees.
• Hence, argument I weakens the statement.
• Argument II: The Teacher's Union is protesting against the decision of private schools to cut salaries by 15%.
• This argument also does not directly relate to the statement.
• It talks about private schools cutting salaries, which is not mentioned in the statement.
• Hence, argument II weakens the statement.
• Therefore, both arguments I and II weaken the statement.

Additional Information

• Private schools are educational institutions that are not run by the government but are owned and managed by private individuals or organizations.
• Instead of depending on taxes for public (government) funding, private schools maintain the autonomy to choose their students and are entirely or partially supported by the tuition fees paid by their students.
• Certain religious denominations or faiths, such Roman Catholicism, different branches of Protestantism, or Judaism, are linked to certain private schools.

Concept used

Average

Calculation

Total sale = 2.6 + 1.2 + 5.5 + 4.2 + 6.3 + 5.4 = 25.2

Average = 25.2/6 = 4.2

The sale is below the average only in 2014 and 2015.

Shortcut Trick

Area = 100 m², Side 1 = 20 m

Side 2 = 100 ÷ 20 = 5 m

Perimeter = 2 × (20 + 5) = 50 m

Total Cost = 50 × 30 = ₹1,500

∴ The correct answer is ₹1,500.

Alternate Method

Given:

Area of rectangular land = 100 m²

Length (l) = 20 m

Rate of fencing = ₹30 per meter

Formula Used:

Area = Length × Breadth

Perimeter = 2 × (Length + Breadth)

Total Cost = Perimeter × Rate

Calculations:

⇒ 100 = 20 × Breadth (b)

⇒ Breadth (b) = 100 ÷ 20 = 5 m

⇒ Perimeter = 2 × (20 + 5)

⇒ Perimeter = 2 × 25 = 50 m

⇒ Total Cost = 50 × 30

⇒ Total Cost = ₹1,500

∴ The correct answer is ₹1,500.

Additional Information

Diagonal of a Rectangle

The distance between opposite corners is given by d = √(l² + b²).

Cost of Leveling

Leveling is done on the Area, so Total Cost = Area × Rate per sq. unit.

Fencing for Square

For a square land with side 'a', the perimeter is 4a and the area is a².

The logic followed here is:

Logic: (1st number)² + 3 = 2nd number.

Now,

For, 8 : 67

→ (8)² + 3

→ 64 + 3 = 67.

For, 10 : 103

→ (10)2 + 3

→ 100 + 3 = 103.

Similarly, for 17 : ?

→ (17)2 + 3

→ 289 + 3 = 292.

Hence, the correct answer is "Option 2".

The correct answer is Rajasthan.

Key Points

• The Bishnoi community is known for their beliefs in protecting the environment and wildlife in Rajasthan.
• They follow the principle of "Jeev Daya" which means compassion towards all living beings.
• The community has a long history of conservation efforts, dating back to the 15th century, when their founder Guru Jambheshwar preached about the importance of protecting nature.

Additional Information

• In Rajasthan, the Bishnoi community has played a significant role in protecting the Khejri trees, which are considered sacred by them and are also essential for the ecosystem.
• They also protect the blackbuck antelope, which is a symbol of their community, and have even sacrificed their lives to protect them from hunters.
• Apart from Rajasthan, the Bishnoi community is also present in parts of Haryana, Punjab, Gujarat, and Madhya Pradesh, where they continue to contribute to conservation efforts.

Shortcut Trick

Sum of roots (S) = (1 + √3) + (1 − √3) = 2.

Product of roots (P) = (1 + √3) × (1 − √3) = 1² − (√3)² = 1 − 3 = −2.

The quadratic equation is given by x² − (Sum)x + (Product) = 0.

Substituting the values: x² − 2x + (−2) = 0 ⇒ x² − 2x − 2 = 0.

∴ The correct answer is x² − 2x − 2 = 0.

Alternate Method

Given roots of the quadratic equation: α = 1 + √3 and β = 1 − √3.

Formula: x² − (Sum of roots)x + (Product of roots) = 0

⇒ Sum of roots (α + β) = (1 + √3) + (1 − √3)

⇒ Sum of roots = 1 + 1 + √3 − √3 = 2

⇒ Product of roots (α × β) = (1 + √3)(1 − √3)

⇒ Product of roots = (1)² − (√3)² [Using identity (a + b)(a − b) = a² − b²]

⇒ Product of roots = 1 − 3 = −2

⇒ Required Equation: x² − (2)x + (−2) = 0

⇒ x² − 2x − 2 = 0

∴ The correct answer is x² − 2x − 2 = 0.

Additional Information

Nature of Roots

For a quadratic equation ax² + bx + c = 0, the nature of roots depends on the discriminant D = b² − 4ac. If D > 0 and not a perfect square, the roots are irrational and conjugate pairs.

Irrational Conjugate Roots

If one root of a quadratic equation with rational coefficients is p + √q, then the other root must be p − √q.

Relation between Coefficients and Roots

If α and β are the roots, then the sum of roots is −b/a and the product of roots is c/a.

Shortcut Trick

When two triangles are similar, the ratio of their corresponding sides is equal.

Given ΔABC ~ ΔDEF and 2AB = DE ⇒ DE/AB = 2.

The scale factor from ΔABC to ΔDEF is 2.

∴ EF = 2 × BC = 2 × 8 = 16 cm.

∴ The correct answer is 16 cm.

Alternate Method

Given:

ΔABC ~ ΔDEF

2AB = DE ⇒ AB/DE = 1/2

BC = 8 cm

Formula Used:

If two triangles are similar, the ratio of their corresponding sides is equal:

AB/DE = BC/EF = AC/DF

A B C D E F AB DE = 2AB BC = 8 cm EF = ? ΔABC ~ ΔDEF   →   Scale Factor = 2

Calculations:

⇒ AB/DE = BC/EF

⇒ 1/2 = 8 / EF

⇒ EF = 8 × 2

⇒ EF = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. ΔABC ~ ΔDEF ⇒ AB/DE = BC/EF = AC/DF.

Area of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area(ΔABC) / Area(ΔDEF) = (AB/DE)².

Perimeter of Similar Triangles

The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides: Perimeter(ΔABC) / Perimeter(ΔDEF) = AB/DE.

Shortcut Trick Convert dozen to individual units: 1 dozen = 12 units. Cost Price (CP) of 12 pens = ₹150. Selling Price (SP) of 12 pens = 15 × 12 = ₹180. Profit = 180 − 150 = ₹30. Profit% = (30 ÷ 150) × 100 = 20%. ∴ The correct answer is 20% profit.

Alternate Method Given: CP of 1 dozen = ₹150, SP of 1 pen = ₹15. Formulas: CP per unit = Total CP ÷ Quantity | Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ CP of 1 pen = 150 ÷ 12 = ₹12.50. ⇒ SP of 1 pen = ₹15. ⇒ Profit per pen = 15 − 12.50 = ₹2.50. ⇒ Profit% = (2.50 ÷ 12.50) × 100 = 20%. ∴ The correct answer is 20% profit.

Additional Information Unitary Method Always ensure both Cost Price and Selling Price are calculated for the same number of articles before finding profit or loss percentage. Profit and Loss Basics If SP > CP, it is a Profit. If CP > SP, it is a Loss. Percentages are always calculated on the Cost Price (CP) unless stated otherwise.

Given equation: [{(36 × 16) - (5 ÷ 3)} + (2 - 5)]

Decoding the given information :

Now, replacing the sign from left to right and using BODMAS rule, we get :

New equation: [{(36 × 16) - (5 ÷ 3)} + (2 - 5)]

Calculation :

⇒ [{(36 - 16) + (5 × 3)} ÷ (2 + 5)]

⇒ [{(36 - 16) + 15)} ÷ (2 + 5)]

⇒ [{20 + 15)} ÷ (2 + 5)]

⇒ [35 ÷ (2 + 5)]

⇒ 35 ÷ 7 = 5.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: (1st number + 2)² = 2nd number.

Now,

For, 3 : 25

→ (3 + 2)² 

→ 5² = 25.

For, 7 : 81

→ (7 + 2)2 

→ 92 = 81.

Similarly, for 11 : ?

→ (11 + 2)2 

→ 132 = 169.

Hence, the correct answer is "Option 3".

Shortcut Trick

Total distance to cover = Sum of lengths of both trains = 320 + 380 = 700 m.

Relative speed (same direction) = Difference of speeds = 84 − 42 = 42 km/h.

Relative speed in m/s = 42 × (5/18) = 35/3 m/s.

Time taken = Distance ÷ Relative Speed = 700 ÷ (35/3) = 20 × 3 = 60 sec.

∴ The correct answer is 60 sec.

Alternate Method

Given:

Length of first train (L₁) = 320 m

Length of second train (L₂) = 380 m

Speed of first train (S₁) = 84 km/h

Speed of second train (S₂) = 42 km/h

Formula Used:

Relative Speed (Same direction) = S₁ − S₂

Time = (L₁ + L₂) ÷ Relative Speed

Conversion: 1 km/h = 5/18 m/s

Calculations:

⇒ Total distance = 320 + 380 = 700 m

⇒ Relative speed = 84 − 42 = 42 km/h

⇒ Relative speed in m/s = 42 × 5/18 = 7 × 5/3 = 35/3 m/s

⇒ Time taken = 700 ÷ (35/3)

⇒ Time taken = (700 × 3) ÷ 35

⇒ Time taken = 20 × 3 = 60 seconds

∴ The correct answer is 60 sec.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Train Crossing a Pole or Point Object

The distance covered to cross a pole or stationary person is equal to the length of the train itself.

Unit Conversion Shortcut

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

The correct answer is 2002. 

Key Points

• Fundamental Duties were added to the Constitution of India through the 42nd Amendment Act, 1976.
• The 11th Fundamental Duty was added to the Indian Constitution in 2002.
• The 11th Fundamental Duty of Indian citizens is to provide educational opportunities to children between the ages of 6 and 14, and to ensure that parents provide these opportunities to their children.
• The inclusion of Fundamental Duties was inspired by the Constitution of the former USSR and aims to promote a sense of patriotism and civic responsibility among citizens.

Additional Information

• Fundamental Duties were recommended by the Swaran Singh Committee in 1976, and their importance became apparent during the internal emergency of 1975–1977.
• Ten Fundamental Duties were added to the Indian Constitution by the 42nd Amendment Act of 1976.
• Later, the 11th Fundamental Duty was added to the list by the 86th Amendment Act of 2002.
• In Article 51A of Part IV A of the Indian Constitution, the Fundamental Duties are addressed.
• Indian Fundamental Duties include maintaining public property, honoring our flag, listening to the national song with reverence, and abiding by the Indian Constitution.

Calculation

The total spending is spent on sector S = 95°

360° → 100%

95° → 100/360 × 95 = 26.38% ≈ 26.40%

The answer is 26.40%

The logic followed here is:

Given Sequence: G _ Q _ A _ T _ N _ G T Q N _ G _ Q N D

Now, putting the letters in the blank spaces given in the option one by one:

Option 1) T N G Q B Q C →  G T Q N A / G T Q N B / G T Q N Q / G C Q N D

Option 2) T G G Q B C Q → G T Q G A / G T Q N B / G T Q N C / G Q Q N D

Option 3) T N G Q B C T → G T Q N A / G T Q N B / G T Q N C / G T Q N D

Here, the last letter is increasing by +1 and it's followed a pattern.

Option 4) T A G N B C T → G T Q A A / G T N N B / G T Q N C / G T Q N D

Hence, the correct answer is "Option 3".

The correct answer is 3.

Key Points

• The chemical equation N2 + xH2  → 2NH3 represents the reaction between nitrogen and hydrogen to form ammonia.
• The balanced equation states that two molecules of ammonia are formed for every one molecule of nitrogen and three molecules of hydrogen.
• Therefore, the value of x represents the number of hydrogen molecules required to react with one molecule of nitrogen.
• By comparing the number of hydrogen atoms on each side of the equation, it can be determined that x = 3.
• This is because there are six hydrogen atoms on the product side (2NH₃) and only two hydrogen atoms on the reactant side (xH₂).

Additional Information

• Nitrogen fixation is the process of converting atmospheric nitrogen into a usable form by living organisms.
• This reaction is an example of nitrogen fixation, as ammonia can be used by plants as a source of nitrogen for growth.
• The Haber-Bosch process is an industrial method used to produce ammonia on a large scale.
• It involves the reaction of nitrogen and hydrogen under high pressure and temperature, using a catalyst.

Shortcut Trick

Given x = 3, substitute directly into the expressions.

The numbers are: (3² + 7×3) = 30, (3 + 5) = 8, and (3 + 6) = 9.

Fourth proportional d = (b × c) ÷ a.

d = (8 × 9) ÷ 30 = 72 ÷ 30 = 2.4.

Wait, checking calculations: x²+7x = 9+21=30; x+5=8; x+6=9. Fourth proportional = (8×9)/30 = 2.4. Let us re-read the first term: x + 7x as given in the prompt.

If first term is x + 7x: 3 + 21 = 24. Then d = (8 × 9) ÷ 24 = 72 ÷ 24 = 3.

∴ The correct answer is 3.

Alternate Method

Given: x = 3. Expressions: x + 7x, x + 5, x + 6.

Formula Used: If a, b, c, d are in proportion, then a/b = c/d or d = (b × c) / a.

⇒ First term (a) = x + 7x = 3 + 7(3) = 3 + 21 = 24

⇒ Second term (b) = x + 5 = 3 + 5 = 8

⇒ Third term (c) = x + 6 = 3 + 6 = 9

⇒ Let the fourth proportional be d.

⇒ 24 ÷ 8 = 9 ÷ d

⇒ 3 = 9 ÷ d

⇒ d = 9 ÷ 3 = 3

∴ The correct answer is 3.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is given by √(a × b).

Third Proportional

The third proportional of two numbers a and b is such that a, b, c are in continued proportion, where c = b² ÷ a.

Compounded Ratio

The compounded ratio of (a : b) and (c : d) is (ac : bd).

Given Statements:

I. Some monkeys are pandas.

II. Some pandas are giraffes.

III. All giraffes are lions.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some lions are pandas - True (Because, Those part of lions which is giraffes will pandas also).

II. Some lions are monkeys - False (Because, No definite relation given between lions and monkeys, so it can be possible only).

III. Some giraffes are monkeys - False (Because, No definite relation given between giraffes and monkeys, so it can be possible only).

So, Only conclusion I follows.

Hence, the correct answer is "Option 1".

The correct answer is common water hyacinth.

Key Points

• Aerenchyma cells are specialized cells that have large air spaces in them, which allows for gas exchange and buoyancy in aquatic plants.
• These cells are present in the common water hyacinth, which is a floating aquatic plant that can be found in many freshwater ecosystems.
• Aerenchyma is especially in the stem of the hyacinth.
• The stem part becomes buoyant and due to this, the water hyacinth floats on the surface of the water.

Additional Information

• Neem is a tree species that is native to the Indian subcontinent and is widely used in traditional medicine and agriculture.
• Neem does not typically have aerenchyma cells, as it is not an aquatic plant.
• Pea is a common garden plant that is a member of the legume family and is often used as a source of protein in human and animal diets.
• Pea plants do not have aerenchyma cells either, as they are not adapted to living in water.
• Cactus is a group of plants that are adapted to living in arid environments and have specialized water storage structures and spines to protect against herbivores.
• Cactus plants also do not have aerenchyma cells, as they are adapted to living in dry environments and do not need to exchange gases in the same way that aquatic plants do.

Shortcut Trick

Split the given expression: (a + 1)/a = 1 + 1/a = k ⇒ 1/a = k − 1.

Split the target expression: (a² − 1)/a² = 1 − (1/a)².

Substitute 1/a: 1 − (k − 1)² = 1 − (k² − 2k + 1).

On simplifying: 1 − k² + 2k − 1 = 2k − k².

∴ The correct answer is 2k - k².

Alternate Method

Given: (a + 1)/a = k

Formula Used: (x − y)² = x² − 2xy + y²

Calculations:

⇒ (a + 1)/a = k

⇒ a/a + 1/a = k

⇒ 1 + 1/a = k

⇒ 1/a = k − 1

Now, evaluate the required expression:

⇒ (a² − 1)/a² = a²/a² − 1/a²

⇒ 1 − (1/a)²

Substituting the value of 1/a = (k − 1):

⇒ 1 − (k − 1)²

⇒ 1 − [k² − 2(k)(1) + 1²]

⇒ 1 − [k² − 2k + 1]

⇒ 1 − k² + 2k − 1

⇒ 2k − k²

∴ The correct answer is 2k - k².

Additional Information

Algebraic Identities

The expansion (x − y)² = x² − 2xy + y² is fundamental for simplifying expressions involving squares of differences.

Reciprocal Relations

Equations in the form (x + 1)/x = k can be simplified to 1 + 1/x = k, allowing for easy substitution of the reciprocal term 1/x.

Difference of Squares

The identity x² − y² = (x − y)(x + y) is another way to factorize the numerator before division.

The correct answer is Sunlight.

Key Points

• The given reaction is a photochemical reaction, which means it requires light energy to proceed.
• Sunlight provides the necessary energy to break the bond between Ag and Cl atoms in AgCl.
• This reaction is an example of a decomposition reaction, where a single compound breaks down into two or more simpler substances.
• The Cl₂ gas produced in this reaction is highly reactive and can be used for various industrial purposes.

Additional Information

• Sunlight refers to the energy provided by the sun in the form of light.
• Sunlight is essential for the survival of all living organisms on Earth and is also used as a source of energy in various applications.
• The bonds that hold the atoms of the initial substance together break down during a decomposition reaction.
• After that, the atoms rearrange to create new bonds, creating new substances with characteristics distinct from the original component.
• Metal extraction from their ores is a significant use of decomposition processes.

Shortcut Trick

In the same time, the ratio of distances is equal to the ratio of speeds.

Ratio of speeds = 60 : 40 = 3 : 2

Difference in distance ratio = 3 − 2 = 1 unit

Given 1 unit = 60 km

Total distance ratio = 3 + 2 = 5 units ⇒ 5 × 60 = 300 km

∴ The correct answer is 300 km.

Alternate Method

Given:

Speed of Train 1 (v₁) = 60 km/h

Speed of Train 2 (v₂) = 40 km/h

Difference in distance covered = 60 km

Formula:

Time (t) = Difference in Distance ÷ Difference in Speed

Calculations:

⇒ Difference in speed = 60 − 40 = 20 km/h

⇒ Time taken to meet (t) = 60 ÷ 20 = 3 hours

⇒ Distance traveled by Train 1 = 60 × 3 = 180 km

⇒ Distance traveled by Train 2 = 40 × 3 = 120 km

⇒ Total distance between A and B = 180 + 120 = 300 km

∴ The correct answer is 300 km.

Additional Information

Relative Speed Concept

When two objects move towards each other (opposite direction), relative speed = S₁ + S₂. When they move in the same direction, relative speed = |S₁ − S₂|.

Meeting Time Formula

Time taken to meet = Initial distance between objects ÷ Relative speed. In this problem, Total Distance = (S₁ + S₂) × t.

Constant Time Property

If time is constant, the distance covered is directly proportional to the speed (D₁/D₂ = S₁/S₂).

• According to Fleming's right-hand rule, the forefinger is pointed in the direction of Magnetic Field.
• The thumb is pointed in the direction of the motion of the conductor related to the Magnetic Field while the second finger represents the direction of the induced or generated current.
• Fleming's Right-hand Rule gives the direction of induced current when a conductor moves inside a Magnetic Field.

The logic followed here is:

Position of Krish from the left = 19th

Position of Maya from the right = 30th

After interchanging, Position of Krish from the right = 36th

So,

Total number of people = Position of Maya from the right + Position of Krish from the right after interchange - 1

= 30 + 36 - 1

= 66 - 1 = 65.

Hence, the correct answer is "Option 3".

The correct answer is 12.5 kWh.

Key Points

• The unit of energy is measured in kWh (kilowatt-hour), which is the amount of energy consumed by a 1000 W device in one hour.
• The electric oven is rated at 2500 W, which means it consumes 2.5 kWh of energy in one hour.
• To calculate the energy used by the oven in 5 hours, we need to multiply the power rating by the time duration: 2500 W x 5 hours = 12,500 Wh.
• Since 1 kWh = 1000 Wh, we need to convert the energy from watt-hours to kilowatt-hours: 12,500 Wh x 1000 = 12.5 kWh.
• Therefore, the energy used by the electric oven in 5 hours is 12.5 kWh.

Additional Information

• Kilowatt hours (kWh) are the unit of measurement for energy consumption, which is the entire amount of energy used for a specific process.
• This covers the usage of fuels such as oil, gas, diesel, electricity, and biomass.
• E = P × (t/1000) is the formula for energy consumption, where P is the power unit in watts, t is the amount of time that the power or energy was consumed, and E is the energy measured in Joules or kilowatt-hours (kWh).

Shortcut Trick

To find the next simultaneous occurrence, calculate the LCM of the given intervals.

LCM of 30, 45, and 60 = 180 seconds.

Convert 180 seconds to minutes: 180 ÷ 60 = 3 minutes.

Next simultaneous red light = 8:30 a.m. + 3 minutes = 8:33 a.m.

∴ The correct answer is 8:33 a.m.

Alternate Method

Given: Intervals are 30 sec, 45 sec, and 60 sec. Initial time is 8:30 a.m.

Formula: Time for next simultaneous event = Initial Time + LCM(intervals).

Step-by-step calculations:

⇒ 30 = 2 × 3 × 5

⇒ 45 = 3² × 5

⇒ 60 = 2² × 3 × 5

⇒ LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

⇒ Time interval in minutes = 180 ÷ 60 = 3 minutes

⇒ New time = 8:30 a.m. + 0:03 = 8:33 a.m.

∴ The correct answer is 8:33 a.m.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers. Used for periodic events like bells ringing or lights flashing.

Time Conversion Units

1 minute = 60 seconds and 1 hour = 60 minutes. Always ensure units are consistent before adding to the clock time.

Prime Factorization Method

LCM can be found by taking the highest power of all prime factors present in the numbers: LCM = Product of highest powers of primes.

According to the given Venn diagram:

R = Students, P = Boys, W = Girls, and Z = Psychology.

The shaded area shows "the girls who are psychology students?".

So, there are 16 girls who are psychology students.

Hence, the correct answer is "Option 1".

The correct answer is more than 1.

Key Points

• Refractive index is a measure of how much a ray of light bends when it passes through a medium.
• Absolute refractive index is the ratio of the speed of light in a vacuum to the speed of light in the medium.
• The value of absolute refractive index of a medium is always more than 1.

Additional Information

• The characteristics of a medium affect the speed of light within it.
• The optical density of the medium affects the speed of electromagnetic waves.
• The atoms' propensity to replenish the electromagnetic energy they have received is known as optical density.
• The speed of light decreases with increasing optical density.
• The refractive index is one such measure of a medium's optical density.
• The refractive index of a medium depends on the density and composition of the medium.
• When the refractive index of a medium is more than 1, it means that the speed of light is slower in that medium than in a vacuum.

The correct answer is Reserve Bank of India.

Key Points

• Reserve Bank of India (RBI) is the central bank of India and was established on April 1, 1935.
• It is responsible for regulating the monetary policy of India, managing the country's foreign exchange reserves, and issuing currency notes and coins.
• The RBI also acts as a banker to the government and other commercial banks in India.

Additional Information

• Small Industries Development Bank of India (SIDBI) is a financial institution established in 1990.
• It was made with the aim of promoting and developing small-scale industries in India.
• Insurance Regulatory and Development Authority of India (IRDAI) is a regulatory body that oversees the insurance sector in India.
• It operated under Ministry of Finance, Government of India.
• Securities and Exchange Board of India (SEBI) is a regulatory body that oversees the securities market in India. 
• The chairperson of SEBI is Madhabi Puri Buch.
• She is SEBI's first female chairperson.

The correct answer is 100°C.

Key Points

• Plaster of Paris is a white powder obtained by heating gypsum at a temperature of 100-150^oC.
• When gypsum is heated to this temperature, it loses about three-quarters of its water of crystallization and forms a fine powder known as Plaster of Paris.
• This powder can be mixed with water to form a paste, which can be used to cast molds, sculptures, and other decorative objects.
• The setting time of Plaster of Paris is relatively short, making it an ideal material for casting small objects.

Additional Information

• The quantity of water molecules that combines chemically in a specific molecular proportion with the relevant salt in a crystalline state is known as the water of crystallization.
• The crystals' geometric form and color are the result of this water.
• It is a crystal structure made of water bound together chemically.

Given: 

Anuj started to walk from a point A and goes 10 m towards North,

And then turned left and walked 5 m.

He then turned right and walked a distance of 5 m,

And then again turned right and walked a distance of 5 m before stopping at point P.

So, Point A is '15 m towards South' from Point P.

Hence, the correct answer is "Option 4".

The logic followed here is:

'DOVE' is written as 'PEFW'

'CROW' is written as 'SDXP'

Similarly, for MYNA:

'MYNA' will be written as 'ZNBO'

Hence, the correct answer is "Option 2".

The correct answer is Temperature.

Key Points

• In some animals, the sex of a developing organism is determined by the temperature at which the eggs are incubated.
• This phenomenon is known as temperature-dependent sex determination (TSD).
• In TSD, the sex of the offspring is determined by the temperature experienced during a critical period of embryonic development.
• This critical period is different for different species.

Additional Information

• In some species, such as turtles and crocodiles, warmer temperatures during incubation result in the development of more females, while cooler temperatures result in more males.
• In other species, such as some lizards, the opposite is true.
• TSD is an adaptive mechanism that allows species to adjust the sex ratio of their offspring to match the prevailing environmental conditions.
• However, it also makes them vulnerable to climate change, as even small changes in temperature can have significant effects on the sex ratio of their offspring.

Given: Today is Wednesday.

Now, after 75 days from today.

As, we know 7 days = 1 week.

So, 75 days = 75/7 = 10 week and 5 odd days.

Then,

After 75 days from today will be Wednesday + 5 = Monday.

Hence, the correct answer is "Option 3".

Given Statements:

1. All parrots are doves.
2. No dove is canary.
3. All flamingoes are canaries.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some flamingoes are parrots - False (Because, no definite relation given between flamingoes and parrots, so it can be possible only).

II. No canary is parrot - True (Because, All parrots are Doves, and no Doves is canary, it gives no canary is parrot).

III. No parrot is flamingo - True (Because, All parrots are Doves, and no Doves is canary, and all the flamingoes are canary, so no parrot will be the part of flamingo).

IV. Some canaries are parrots - False (Because, All parrots are Doves, and no Doves is canary, it gives no canary is parrot).

So, Only conclusions II and III follow.

Hence, the correct answer is "Option 4".

The correct answer is In red blood cells.

Key Points

• The respiratory pigment in the human body is hemoglobin, which is a protein found in the red blood cells.
• Hemoglobin is responsible for carrying oxygen from the lungs to the different parts of the body and also helps in transporting carbon dioxide from the tissues to the lungs for exhalation.
• The red color of the blood is due to the presence of hemoglobin in the red blood cells.
• The concentration of hemoglobin in the blood can affect the oxygen-carrying capacity of the blood, and any abnormality in the hemoglobin levels can lead to various health conditions like anemia, sickle cell disease, etc.

Additional Information

• White blood cells, also known as leukocytes, are responsible for the immune response in the body.
• They travel throughout the bloodstream, initiating cellular and inflammatory reactions to infections or wounds.
• Plasma is the liquid component of the blood that carries various nutrients, hormones, and waste products.
• ​Water and salts that are absorbed through the digestive tract combine to make plasma.

Calculation:

Let the fraction be x/y

First case,

(x + 2)/(y + 5) = 1/2

⇒ 2x + 4 = y + 5→ equation 1

Second case,

(x – 2)/(y - 2) = 1/3

⇒ 3x – 6 = y - 2→ equation 2

On solving equations 1 and 2, we get

x = 3 and y = 5

∴ The fraction is 3/5

The correct answer is 2017.

Key Points

• Union of India Budget is the annual financial statement of the Government of India that outlines the estimated revenues and expenditures for the upcoming fiscal year.
• In 2017, the Union of India Budget was presented on 1 February for the first time, marking a departure from the earlier practice of presenting it on the last working day of February.
• In 2015, the Union of India Budget was presented on the last working day of February, which was the practice until 2016.
• In 2021, the Union of India Budget was presented on 1 February.

Additional Information

• On November 26, 1947, Sir R.K. Shanmugham Chetty, the country's first finance minister, unveiled the nation's first union budget.
• Article 112 of the Indian Constitution refers to the Union Budget as the Annual Financial Statement.
• Revenue Budget and Capital Budget are two categories of the Union Budget.
• Since 2019, Nirmala Sitharaman has held the positions of Minister of Finance and Minister of Corporate Affairs in the Indian government.

Shortcut Trick

Area of a regular hexagon = 6 × Area of an equilateral triangle

Area = 6 × (√3/4) × side²

Area = 6 × (√3/4) × 6² = 6 × (√3/4) × 36

Area = 6 × 9√3 = 54√3 cm²

∴ The correct answer is 54√3 cm².

Alternate Method

Given:

Side length of regular hexagon (a) = 6 cm

Formula Used:

Area of Regular Hexagon = [3√3 / 2] × a²

Calculations:

⇒ Area = [3√3 / 2] × (6)²

⇒ Area = [3√3 / 2] × 36

⇒ Area = 3√3 × 18

⇒ Area = 54√3 cm²

∴ The correct answer is 54√3 cm².

Additional Information

Perimeter of Regular Hexagon

The total boundary length is calculated as P = 6 × a, where 'a' is the side length.

Interior Angles

Each interior angle of a regular hexagon is 120°. The sum of all interior angles is 720°.

Circumradius and Inradius

Circumradius (R) = a and Inradius (r) = (√3/2) × a.

Shortcut Trick

Given ratio a : b = 4 : 3. Let a = 4k and b = 3k.

Substitute in 3a + 2b = 24 → 3(4k) + 2(3k) = 24 → 12k + 6k = 18k = 24.

Find k = 24 ÷ 18 = 4/3.

Value of a = 4k = 4 × (4/3) = 16/3.

∴ The correct answer is 16/3.

Alternate Method

Given: a/b = 4/3 and 3a + 2b = 24

Formula Used: Substitution Method for Linear Equations

⇒ From a/b = 4/3, we get b = 3a/4.

⇒ Substitute b in 3a + 2b = 24:

⇒ 3a + 2(3a/4) = 24

⇒ 3a + 3a/2 = 24

⇒ (6a + 3a)/2 = 24

⇒ 9a/2 = 24

⇒ 9a = 48

⇒ a = 48/9 = 16/3

∴ The correct answer is 16/3.

Additional Information

Ratio and Proportion

If a/b = c/d, then a = bk where k is a constant. This helps in reducing variables in linear equations.

Componendo and Dividendo

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). Useful for solving complex fractional equations.

Substitution Method

Express one variable in terms of another from the simplest equation and substitute it into the other to find the unknown values.

The correct answer is Rowlatt Act.

Key Points

• Mahatma Gandhi decided to launch nationwide Satyagraha against the Rowlatt Act, which was passed by the British government in 1919.
• The Rowlatt Act allowed the British government to imprison any person without trial, and it was seen as a direct attack on the civil liberties of the Indian people.
• The Satyagraha movement led by Mahatma Gandhi against the Rowlatt Act was one of the first mass movements against British rule in India, and it played a significant role in the Indian independence movement.

Additional Information

• The Regulating Act of 1773 was the first major British law on the governance of India.
• It was an Act of the British Parliament designed to restructure the East India Company's administration of its dominion over India.
• The Inland Emigration Act of 1859 regulated the movement of Indian laborers.
• It made it illegal for Assamese plantation laborers to leave the tea estate without authorization.
• The Pitt's India Act of 1784 reformed the East India Company's governance of India.
• In order to address the shortcomings of the Regulating Act of 1773, it was passed by the British Parliament.

Shortcut Trick

For fourth proportion (d) of a, b, c, the logic is d = (b × c) ÷ a.

Given values are 12, 24, 45.

Fourth proportion = (24 × 45) ÷ 12

Fourth proportion = 2 × 45 = 90

∴ The correct answer is 90.

Alternate Method

Given: Three numbers 12, 24, and 45.

Formula Used: If a, b, c, d are in proportion, then a : b = c : d, which implies a × d = b × c.

Let the fourth proportional be x.

⇒ 12 : 24 = 45 : x

⇒ 12 ÷ 24 = 45 ÷ x

⇒ 1 ÷ 2 = 45 ÷ x

⇒ x = 45 × 2

⇒ x = 90

∴ The correct answer is 90.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is calculated as a : b = b : c, or c = b² ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Continued Proportion

Numbers a, b, c are in continued proportion if a : b = b : c.

The correct answer is Oceans.

Key Points

• Panthalassa refers to the vast global ocean that surrounded the supercontinent Pangaea during the Paleozoic and Mesozoic eras.
• The name comes from the Greek words "pan" meaning all and "thalassa" meaning sea.
• Panthalassa covered about 70% of the Earth's surface and was home to a diverse range of marine life.
• The breakup of Pangaea and the formation of modern continents caused the oceanic basins to expand and Panthalassa gradually disappeared.

Additional Information

• The atmosphere refers to the layer of gases surrounding the Earth.
• About 78% of the gases in Earth's atmosphere are nitrogen, 21% are oxygen, and 1% are other gases.
• Mountains are large landforms that rise above the surrounding terrain.
• Typically, they have ridges that are sharp or rounded, steep, sloping sides, and a high point known as a peak or summit.
• Soil is the mixture of organic and inorganic materials that support plant growth.
• They obtain their nutrients and water from the soil.

The correct answer is Wetland's action for people and nature.

Key Points

• Wetlands are vital for the survival of various species of plants and animals.
• They also provide numerous ecosystem services, including water purification, flood control, and carbon sequestration.
• The theme of this year's World Wetland Day, "Wetland's action for people and nature", emphasizes the importance of wetlands in providing benefits to people and nature.

Additional Information

• "Wetlands and Biodiversity," also highlights the significance of wetlands in supporting biodiversity.
• Wetlands are home to various plant and animal species, many of which are endangered.
• "Wetlands and climate change," emphasizes the role of wetlands in mitigating the impacts of climate change.
• Wetlands store carbon and help reduce greenhouse gas emissions.
• "Wetlands and water," highlights the importance of wetlands in providing clean water.
• ​Wetlands act as natural filters, removing pollutants and impurities from water.

Given:

Now, the group of numbers and symbols is: % + < 2 5 =

Here, Third conditions followed in the given group,

So, Codes of the First and last elements are to be interchanged.

Then, the code for '% + < 2 5 =' is → 'L P P M B J'.

Hence, the correct answer is "Option 3".

Shortcut Trick

Total Interest in percentage = (3/5) × 100 = 60%

In Simple Interest, interest is distributed equally over the Time Period.

Annual Rate (R) = Total Interest % ÷ Time

R = 60% ÷ 12 = 5%

∴ The correct answer is 5 %.

Alternate Method

Given:

Time (T) = 12 years

Simple Interest (SI) = 3/5 × Principal (P)

Formula Used:

Rate (R) = (SI × 100) ÷ (P × T)

Calculations:

⇒ Let Principal (P) = 5 units

⇒ Simple Interest (SI) = 3/5 × 5 = 3 units

⇒ R = (3 × 100) ÷ (5 × 12)

⇒ R = 300 ÷ 60

⇒ R = 5%

∴ The correct answer is 5 %.

Additional Information

Amount Formula

The total sum after interest is added is called Amount (A), where A = P + SI = P(1 + RT/100).

Simple Interest Property

Unlike Compound Interest, Simple Interest is calculated only on the original Principal for every year.

Relationship between Rate and Time

For a fixed amount of interest, the Rate of Interest is inversely proportional to the Time Period.

The correct answer is 11 A.

Key Points

• The current drawn by the appliance is directly proportional to the potential difference applied across its terminals, according to Ohm's law.
• Therefore, the new current can be calculated as I2 = (V2/V1) * I1, where I1 and V1 are the initial current and potential difference, and I2 and V2 are the new current and potential difference, respectively.
• Substituting the given values, we get I2 = (110/50) * 5 = 11 A.
• Hence, the correct answer is 11 A.

Additional Information

• Only when the given temperature and the other physical parameters stay constant does Ohm's law apply.
• In some components, the temperature rises when the current is increased.
• A light bulb's filament serves as an illustration of this, wherein increasing current causes the filament's temperature to rise.
• Ohm's law cannot be applied in this case.
• The filament of the lightbulb defies Ohm's Law.

Shortcut Trick

The ratio difference between 7 : 6 and 8 : 7 is exactly 1 unit (8−7 = 1 and 7−6 = 1).

Since the time gap is 6 years, 1 unit = 6 years.

Present ages: Man = 7 × 6 = 42 years, Wife = 6 × 6 = 36 years.

Marriage ratio was 4 : 3; let marriage be x years ago: (42−x)/(36−x) = 4/3.

By inspection: (42−18)/(36−18) = 24/12 = 2/1, but if we check 18 years, 42−18=24 and 36−18=18; 24:18 = 4:3.

∴ The correct answer is 18 years.

Alternate Method

Given: Present ratio = 7 : 6, Future ratio (after 6 yrs) = 8 : 7, Marriage ratio = 4 : 3.

Formula: Ratio = Age₁ / Age₂

⇒ Let present ages be 7k and 6k.

⇒ (7k + 6) / (6k + 6) = 8 / 7

⇒ 49k + 42 = 48k + 48

⇒ k = 6

⇒ Present ages: Man = 42, Wife = 36

⇒ Let marriage occur n years ago: (42 − n) / (36 − n) = 4 / 3

⇒ 126 − 3n = 144 − 4n

⇒ n = 144 − 126 = 18

∴ The correct answer is 18 years.

Additional Information

Ratio Difference Method

When the absolute difference between two ages remains constant, ensure the ratio difference is also equalized by multiplying the ratios by the difference of the other ratio.

Age Calculations

Age after n years = Present Age + n. Age n years ago = Present Age − n.

Constant Age Gap

The difference between the ages of two people never changes regardless of the number of years passed.

The correct answer is Odisha.

Key Points

• Mahendragiri Hills is located in the Paralakhemundi region of Gajapati district in Odisha.
• Mahendragiri Hills is a part of the Eastern Ghats mountain range, and it is also known as the Mahendra Parvat.
• It is considered to be a sacred place by the locals, and it is believed that Lord Parashurama, an incarnation of Lord Vishnu, meditated here.

Additional Information

• At 3,636 meters, Sandakphu is the highest hill in West Bengal.
• At 3,600 meters, Phalut, sometimes known as Falut, is the second-highest peak in West Bengal, India.
• Jharkhand is home to several hill ranges, including the Chota Nagpur Plateau and the Parasnath Hills.
• Chhattisgarh has several hills and mountain ranges, including the Maikal Hills and the Satpura Range.

The correct answer is iii, iv.

Key Points

• Statement (ii) is correct as  Starting from lithium (Li), counting every element up to sodium (Na), sodium indeed is the eighth element (Li, Be, B, C, N, O, F, Na).
• Statement (iv) is also incorrect as Newlands' Law of Octaves was found to be inaccurate for elements beyond calcium, not magnesium. The significant deviation happens after calcium.
• Statement (iii) is incorrect as Calcium (Ca) is actually the 17th element after Beryllium (Be), not the eighth. Therefore, statement (iii) is incorrect.
• Statement (i) is also correct as Newland did arrange the elements in the order of increasing atomic masses and gave the law of Octaves.

Additional Information

• John Newlands was an English chemist who arranged the elements in the order of increasing atomic masses and gave the law of Octaves in 1864.
• The law of Octaves states that every eighth element has properties similar to that of the first element when arranged in the order of increasing atomic masses.
• However, this law was applicable only up to Calcium and not beyond that as after Calcium, every eighth element did not possess properties similar to that of the first element.
• The modern periodic table is based on the atomic number of the elements and not their atomic masses.
• The elements are arranged in increasing order of atomic number in the modern periodic table.

Shortcut Trick

Rationalize the numerator: √((1 + sin θ) / (1 − sin θ)) × √((1 + sin θ) / (1 + sin θ))

⇒ √((1 + sin θ)² / (1 − sin² θ)) = (1 + sin θ) / cos θ

⇒ (1 / cos θ) + (sin θ / cos θ) = sec θ + tan θ

∴ The correct answer is sec θ + tan θ.

Alternate Method

Given: expression √((1 + sin θ) / (1 − sin θ))

Formula Used: sin² θ + cos² θ = 1; sec θ = 1/cos θ; tan θ = sin θ/cos θ

⇒ Multiply numerator and denominator by (1 + sin θ) under the square root.

⇒ √[((1 + sin θ)(1 + sin θ)) / ((1 − sin θ)(1 + sin θ))]

⇒ √[(1 + sin θ)² / (1 − sin² θ)]

⇒ √[(1 + sin θ)² / cos² θ]

⇒ (1 + sin θ) / cos θ

⇒ 1/cos θ + sin θ/cos θ

⇒ sec θ + tan θ

∴ The correct answer is sec θ + tan θ.

Additional Information

Trigonometric Identities

Standard identities include 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ.

Rationalization Rule

When simplifying √((1 ± sin θ)/(1 ∓ sin θ)), multiplying by the conjugate of the denominator converts the expression into a perfect square.

Shortcut Trick

Total Sum = Mean × n = 21 × 7 = 147.

Sum of terms = (11 + 17 + 1 + 19 − 4 + 5) + (x + 3x + 2x + x) = 49 + 7x.

Equating: 49 + 7x = 147 ⇒ 7x = 98 ⇒ x = 14.

Data set becomes: 11, 17, 15, 42, 19, 24, 19.

The value 19 repeats most frequently (twice).

∴ The correct answer is 19.

Alternate Method

Given:

Data: 11, 17, x + 1, 3x, 19, 2x − 4, x + 5

Mean = 21, Total terms (n) = 7

Formula Used:

Mean = (Sum of all observations) ÷ (Total number of observations)

Mode = The observation with the highest frequency.

Calculations:

⇒ Mean = (11 + 17 + x + 1 + 3x + 19 + 2x − 4 + x + 5) ÷ 7

⇒ 21 = (49 + 7x) ÷ 7

⇒ 21 × 7 = 49 + 7x

⇒ 147 − 49 = 7x

⇒ 98 = 7x ⇒ x = 14

Now, finding all observations:

⇒ x + 1 = 14 + 1 = 15

⇒ 3x = 3 × 14 = 42

⇒ 2x − 4 = 2(14) − 4 = 24

⇒ x + 5 = 14 + 5 = 19

Observations list: 11, 17, 15, 42, 19, 24, 19

⇒ Frequency of 19 = 2; Frequency of others = 1

∴ The correct answer is 19.

Additional Information

Arithmetic Mean

The average of a set of numbers, calculated by dividing the sum of the values by the number of values.

Mode

The value that appears most frequently in a data set. A set can be unimodal, bimodal, or multimodal.

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Median

The middle value of a data set when it is arranged in ascending or descending order.

The logic followed here is:

So, '66' will replace the question mark in the given series.

Hence, the correct answer is "Option 2".

Given Statement: A < Q, Q < X, E = A < T

After combining the statements: T > E = A < Q < X

Now,

Conclusions:

1. T < E → False (E = A < T It gives T > E)
2. A = X → False (A < Q < X It gives A < X)

Here, Neither conclusion 1 nor 2 is true.

Hence, the correct answer is "Option 4".

The correct answer is X-chromosome is bigger, whereas Y is smaller.

Key Points

• The X chromosome is larger than the Y chromosome in humans.
• The X chromosome contains more genes than the Y chromosome.
• The X chromosome plays a crucial role in determining the sex of an individual, as females have two copies of the X chromosome (XX) while males have one X and one Y chromosome (XY).
• The Y chromosome is responsible for the development of male-specific characteristics.

Additional Information

• There are 23 pairs of chromosomes in humans.
• Only one pair, the "Sex Chromosome," which actively participates in the process of determining a person's sex, is among these 23 pairings; the other 22 pairs are autosomes.
• Both sex chromosomal sets are carried by males and females.
• A few plant species, animals, insects, and humans all have the XY sex-determination system.
• Regardless of gender, every kid will receive an X chromosome from their mother.
• Therefore, the type of chromosome that the offspring get from their father will decide their sex.
• A child with the X chromosome will be a girl, and a child with the Y chromosome will be a male.

Shortcut Trick

For any quadratic equation of the form ax² + bx + c = 0, the sum of the roots is given by the formula −b/a.

In the given equation 5x² − 11x + 6 = 0, we have a = 5 and b = −11.

Sum of roots = −(−11) / 5 = 11/5.

∴ The correct answer is 11/5.

Alternate Method

Given:

The quadratic equation is 5x² − 11x + 6 = 0.

Formula Used:

If α and β are the roots of the equation ax² + bx + c = 0, then:

Sum of roots (α + β) = −(Coefficient of x) / (Coefficient of x²) = −b/a

Calculations:

⇒ Comparing 5x² − 11x + 6 = 0 with ax² + bx + c = 0:

⇒ a = 5

⇒ b = −11

⇒ c = 6

⇒ Sum of roots = −b/a

⇒ Sum of roots = −(−11) / 5

⇒ Sum of roots = 11/5

∴ The correct answer is 11/5.

Additional Information

Product of Roots

The product of the roots (αβ) of a quadratic equation ax² + bx + c = 0 is given by c/a. For the given equation, the product is 6/5.

Discriminant and Nature of Roots

The discriminant (D) is calculated as b² − 4ac. If D > 0, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, the roots are imaginary.

Quadratic Formula

The roots of the equation can be found using the formula x = [−b ± √(b² − 4ac)] / 2a.

The correct answer is Bodh Gaya.

Key Points

• Gautam Buddha, also known as Siddhartha Gautama, was a spiritual leader and founder of Buddhism.
• He attained enlightenment under a Bodhi tree in Bodh Gaya, Bihar, India.
• Bodh Gaya is one of the most important pilgrimage sites for Buddhists around the world.
• The Mahabodhi Temple, located in Bodh Gaya, is a UNESCO World Heritage Site and attracts millions of visitors every year.

Additional Information

• Lumbini, located in Nepal, is the birthplace of Gautam Buddha.
• It is also a UNESCO World Heritage Site and a popular destination for Buddhist pilgrims.
• Sanchi, located in Madhya Pradesh, India, is famous for its Buddhist stupas and monuments.
• These were built by the Mauryan emperor Ashoka in the 3rd century BCE.
• Kaushambi, located in Uttar Pradesh, India, is an ancient city that was an important center for Buddhism in the 6th century BCE.
• ​It is believed that Gautam Buddha visited Kaushambi several times during his lifetime.

Shortcut Trick

Present Age Ratio of Rekha and Sister = 4 : 7

Total Ratio parts = 4 + 7 = 11 units.

Since the total age must be a multiple of 11, check the options.

Only 44 and 66 are multiples of 11. Using 12 years ago ratio 2 : 5:

If Total = 66, then ages are 24 and 42. 12 years ago: (24−12) : (42−12) = 12 : 30 = 2 : 5. (Matches)

∴ The correct answer is 66 years.

Alternate Method

Given:

Present age ratio = 4 : 7

Age 12 years ago = 2/5 of sister's age

Let Rekha's present age be 4x and Sister's present age be 7x.

⇒ 12 years ago, Rekha's age = 4x − 12

⇒ 12 years ago, Sister's age = 7x − 12

⇒ (4x − 12) = (2/5) × (7x − 12)

⇒ 5 × (4x − 12) = 2 × (7x − 12)

⇒ 20x − 60 = 14x − 24

⇒ 20x − 14x = 60 − 24

⇒ 6x = 36

⇒ x = 6

⇒ Total present age = 4x + 7x = 11x

⇒ Total present age = 11 × 6 = 66 years

∴ The correct answer is 66 years.

Additional Information

Ratio Difference Method

If the age difference between two ratios is constant, make the difference between ratio parts equal to solve for 1 unit value.

Age Difference Property

The difference between the ages of two persons remains constant throughout their lives, regardless of the years passed.

Total Age Calculation

Sum of ages after 'n' years = Present Sum + (Number of persons × n).

Shortcut Trick

Let Marked Price (MP) = 100.

Cost Price (CP) = 64% of 100 = 64.

Selling Price (SP) = MP × (100 − Discount%) ÷ 100 = 100 × 96 ÷ 100 = 96.

Profit = SP − CP = 96 − 64 = 32.

Gain % = (Profit ÷ CP) × 100 = (32 ÷ 64) × 100 = 50%.

∴ The correct answer is 50 %.

Alternate Method

Given: CP = 64% of MP, Discount = 4%.

Formulas: SP = MP × (1 − Discount%); Profit% = [(SP − CP) ÷ CP] × 100.

⇒ Let Marked Price be M. Then Cost Price = 0.64M.

⇒ Selling Price after 4% discount = M × (100 − 4) ÷ 100 = 0.96M.

⇒ Gain = SP − CP = 0.96M − 0.64M = 0.32M.

⇒ Gain % = (0.32M ÷ 0.64M) × 100 = (1 ÷ 2) × 100 = 50%.

∴ The correct answer is 50 %.

Additional Information

Relation between CP and MP

The ratio of Cost Price to Marked Price can be expressed as CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Marked Price & Discount

Marked Price is the price printed on the label, and Discount is always calculated on this price to arrive at the Selling Price.

The correct answer is Haryana.

Key Points

• Haryana topped the 4th Khelo India Youth Games 2021 medals tally by winning a total of 102 medals, including 38 gold, 26 silver, and 38 bronze medals.
• The Khelo India Youth Games is an initiative by the Ministry of Youth Affairs and Sports, Government of India, to identify and promote young talent across various sports disciplines in the country.
• The 4th edition of the Khelo India Youth Games was held in Guwahati, Assam, from 10th to 22nd January 2020, and saw participation from over 6,000 athletes from 37 states and union territories.

Additional Information

• Rajasthan secured the 2nd position in the medals tally with a total of 73 medals, including 22 gold, 27 silver, and 24 bronze medals.
• Maharashtra stood at the 3rd position with 56 medals, including 18 gold, 19 silver, and 19 bronze medals.
• Maharashtra became victorious overall in the Khelo India Youth Games - 2022, having won 161 medals in total, comprising 56 gold, 55 silver, and 50 bronze.
• Conversely, Haryana secured the second spot with a total of 128 medals, including of 41 gold, 32 silver, and 55 bronze.
• With 96 medals, including 39 gold, the host state of Madhya Pradesh came in third.

The correct answer is a - Neutrophil, b - Eosinophil, c - Basophil, d - Lymphocyte, e - Monocyte.

Key Points

• Neutrophil, Eosinophil, Basophil, Lymphocyte, Monocyte correctly labels all the white blood cells in the diagram.
• Neutrophil is labelled as 'a', Eosinophil is labelled as 'b', Basophil is labelled as 'c', Lymphocyte is labelled as 'e' and Monocyte is labelled as 'd'.
• Another name for white blood cells is leukocytes.
• As a component of the immune system, they take part in the humoral as well as innate immunological responses.
• They travel throughout the bloodstream, initiating cellular and inflammatory reactions to infections or wounds.

Additional Information

• Neutrophils are the most common type of white blood cells and play a crucial role in fighting off bacterial infections.
• Eosinophils are involved in the body's response to parasitic infections and allergic reactions.
• Basophils release histamine during an allergic reaction and play a role in the body's immune response.
• Lymphocytes are responsible for recognizing and attacking specific foreign invaders, such as viruses and bacteria.
• Monocytes are involved in the body's immune response by engulfing and digesting foreign substances.

The logic followed here is:

So, '122' will replace the question mark in the given series.

Hence, the correct answer is "Option 3".

The correct answer is Only (iii).

Key Points

• The salt which will give an aqueous solution having pH of almost 7 is K2SO4.
• K2SO4 is a neutral salt as it is formed by the combination of a strong base (KOH) and a strong acid (H₂SO₄).
• K2SO4 will not affect the pH of the aqueous solution as it does not produce any H+ or OH- ions.
• K2SO4 is used as a fertilizer in agriculture as it provides essential nutrients to the plants.

Additional Information

• The keyword mentioned in the question is pH, which is a measure of the acidity or basicity of a solution.
• NH4Cl is commonly used as a buffer in laboratory experiments as it can maintain a constant pH in the solution.
• NH4Cl is an acidic salt as NH4+ ion is acidic in nature and it will produce H+ ions in aqueous solution, resulting in a lower pH.
• Na2CO3 is a basic salt as CO32- ion is basic in nature and it will produce OH- ions in aqueous solution, resulting in a higher pH.
• Na₂CO₃ is used in the manufacturing of glass, detergents, and paper as it can act as a cleaning agent due to its basic nature.

Given word: 'PARTICLE'

Now, checking the pairs in both forward and backward directions:

The pairs in forward direction are: PR and CE

And, no pairs formed in backward direction.

So, Total 'Two' pairs are formed from the given word.

Hence, the correct answer is "Option 2".

The correct answer is six.

Key Points

• Students from six minority communities under the Maulana Azad Fellowship Scheme (MANF) get five-year fellowships in the form of financial assistance notified by the Central Government, to pursue M. Phil and Ph.D.
• The minority communities are Buddhist, Christian, Jain, Muslim, Parsi and Sikh.
• The MANF scheme was launched by the Ministry of Minority Affairs, Government of India, in 2003 to provide financial assistance to students from minority communities to pursue higher education.
• The scheme aims to provide opportunities to minority students to undertake advanced studies and research in science, humanities, social sciences, and engineering and technology.

Additional Information 

• The Maulana Azad Fellowship Scheme (MANF) is named after Maulana Abul Kalam Azad, who was an Indian scholar and the first education minister of independent India.
• The MANF scheme is implemented through the University Grants Commission (UGC), which is the apex body for higher education in India.
• The MANF scheme provides financial assistance to students from minority communities to pursue M. Phil and Ph.D. in universities, deemed universities, and research institutions recognized by the UGC.
• The MANF scheme provides financial assistance in the form of a monthly stipend, contingency grant, and other benefits.
• The scheme also provides a separate ad-hoc grant to the universities and institutions to provide infrastructure and other facilities to the MANF scholars.

Given: Six teachers, C, D, E, F, G and H, are sitting around a circular table.

They were facing the centre of the table.

1. Exactly two people sit between H and D.

2. E is sitting to the immediate right of H.

3) F is sitting to the immediate left of C. 

4. D is an immediate neighbour of both G and F.

Here, 'G' is the third person to the left of C.

Hence, the correct answer is "Option 1".

Shortcut Trick

A parallelogram with all sides equal is either a rhombus or a square.

If the diagonals are equal, it is a square.