RRB Group D Previous Year Paper (Held on: 6 Oct 2022 Shift 3)

The correct answer is closed curves.

Key Points

Magnetic field lines:

• A magnetic field line is an imaginary line such that tangent to it at any point gives the direction of the magnetic field at that point in space.
• Magnetic field lines are drawn to represent the magnetic fields.
• Magnetic field lines can be drawn with the help of a magnetic compass.
• Magnetic field lines are also called magnetic lines of force.

Additional Information

Properties of magnetic field lines:

1. The magnetic field lines of a magnet (or a solenoid) form continuous closed loops.
2. The magnetic field lines start from the north pole and end on a south pole outside the magnet. Inside the magnet, the magnetic field lines start from the south pole and end on the north pole.
3. The tangent to the field line at a given point represents the direction of the net magnetic field B at that point.
4. The larger the number of field lines crossing per unit area, the stronger is the magnitude of the magnetic field B.
5. The magnetic field lines do not intersect with each other if they did, there will be two directions of the magnetic field at that point which is not possible.

The correct answer is The Second Five-Year Plan was based on the PC Mahalanobis model..

Key Points

• The duration of the second five-year plan was 1956–61.
• The five-year plans were one of the central plans.
• The plans were formulated and were financed by the central government.
• These were launched in 1951, with fist five-year plans covering the years 1951-56.
• There were three breaks in five-year plans during 1966-69, 1978-80, and 1991-92.
• During 1966-69, consequences of war, drought, and a misbalanced economy lead to the abandonment of a five-year plan and three annual plans were launched.
• The breaks of 1978-80 and 19991-92 were due to a change of government and economic crisis respectively.

Main objectives of the second five-year plan:

• It was also called Mahalanobis Plan after the name of its architect.
• Rapid industrialization was the main objective of this plan.
• Imports were encouraged that later led to the emptying of forex reserves.
• The Agricultural sector was not given much importance.
• As a result of all this Shortage of food, the increased price of goods and capital shortage was observed.

Additional Information

• The Planning Commission
• The Planning Commission, although no longer active (since it was replaced by the NITI Aayog), was an important feature of policymaking and governance in India.
• The Planning Commission was reporting directly to the Prime Minister of India.
• The Planning Commission was established in March 1950 by an executive resolution of the Government of India, on the recommendation of the Advisory Planning Board constituted in 1946, under the chairman of KC Neogi. Thus the Planning Commission was neither constitutional nor a statutory body. It was an advisory body.
• It is a non-constitutional (not created by the Constitution) and a non-statutory body (not created by an act of Parliament).
• In India, it was the supreme organ of planning for social and economic development.
• It was not responsible for making and implementing decisions.
• This responsibility rests with the Central and State governments.
• The Planning Commission was only a staff agency- an advisory body and has no executive responsibility.
• The critics have described it as a 'Super Cabinet', an 'Economic Cabinet' a 'Parallel Cabinet', the 'Fifth Wheel of the Coach' etc.

Shortcut Trick

By Basic Proportionality Theorem (Thales's Theorem), since DE is parallel to BC, the ratio of segments on AB is equal to the ratio of segments on AC.

AE ÷ AC = AD ÷ AB = AD ÷ (AD + DB)

AE ÷ 8.4 = 2 ÷ (2 + 5) = 2 ÷ 7

AE = (2 ÷ 7) × 8.4 = 2.4 cm

∴ The correct answer is 2.4 cm.

Alternate Method

Given:

In triangle ABC, DE is parallel to BC.

D is a point on AB and E is a point on AC.

The ratio of AD to DB is: AD : DB = 2 : 5

The length of AC is: AC = 8.4 cm

Formula Used:

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

Therefore, if DE || BC, then:

AD ÷ DB = AE ÷ EC

Calculations:

Let AD = 2x and DB = 5x.

⇒ AB = AD + DB = 2x + 5x = 7x

By Thales's Theorem, we have:

⇒ AE ÷ AC = AD ÷ AB

Substitute the given values into the ratio:

⇒ AE ÷ 8.4 = 2x ÷ 7x

⇒ AE ÷ 8.4 = 2 ÷ 7

Solve for AE:

⇒ AE = (2 ÷ 7) × 8.4

⇒ AE = 2 × 1.2

⇒ AE = 2.4 cm

Option Analysis:

⇒ Option (1) 2.8 cm: This would correspond to a ratio of 1 ÷ 3, which is incorrect.

⇒ Option (2) 2.4 cm: This is the correct value obtained using the ratio 2 ÷ 7.

⇒ Option (3) 1.4 cm: This would correspond to a ratio of 1 ÷ 6, which is incorrect.

⇒ Option (4) 1.2 cm: This is the value of 1 part out of 7 (8.4 ÷ 7 = 1.2 cm), not 2 parts.

∴ The correct answer is 2.4 cm.

Additional Information

Basic Proportionality Theorem (Thales's Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio: AD ÷ DB = AE ÷ EC.

Similarity of Triangles

Since DE || BC, ΔADE is similar to ΔABC by AA similarity. Thus, the ratio of their corresponding sides is equal: AD ÷ AB = AE ÷ AC = DE ÷ BC.

Ratio of Areas 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(ΔADE) ÷ Area(ΔABC) = (AD ÷ AB)².

The correct answer is NaCl.

Key Points

• washing soda
• Sodium Carbonate is the chemical name for ‘washing soda’ (Na2CO3.10H2O.).
• It is the water-soluble sodium salt of carbonic acid.
• It is well known domestically for its everyday use as a water softener.
• Washing soda is produced using Solvay's process.
• The process includes the reaction of Sodium Chloride with ammonia, and carbon dioxide, in water.
• Carbon Dioxide is produced through calcium carbonate from which calcium oxide that gets left is used in ammonia recovery from ammonium chloride.

Additional Information

• NaCl (Sodium Chloride)
• Common Name: Table salt
• Composition: An ionic compound consisting of sodium (Na+) and chloride (Cl-) ions in a 1:1 ratio.
• Properties: White crystalline solid, readily dissolves in water, essential for human and animal life.
• Uses: Flavor enhancer, food preservative, de-icing roads, industrial processes.
• MgCO3 (Magnesium Carbonate)
• Composition: An inorganic compound containing magnesium, carbon, and oxygen.
• Properties: White solid, slightly soluble in water, decomposes upon heating.
• Uses: Antacid, dietary supplement, fire retardant, filler in plastics and rubber.
• NaHCO3 (Sodium Bicarbonate)
• Common Name: Baking soda
• Composition: A chemical compound containing sodium, hydrogen, carbon, and oxygen.
• Properties: White crystalline solid, slightly alkaline, decomposes upon heating to release carbon dioxide.
• Uses: Baking leavening agent, antacid, cleaning agent, deodorizer.
• NaOH (Sodium Hydroxide)
• Common Name: Caustic soda
• Composition: A strong alkali consisting of sodium and hydroxide ions.
• Properties: White solid, highly corrosive, readily absorbs moisture from the air.
• Uses: Soap and detergent production, paper making, water treatment, drain cleaner.

Shortcut Trick

Current age ratio Raja : Vikram = 13 : 1 (Difference = 12 units)

Age ratio after 10 years = 3 : 1 (Difference = 2 units)

Make differences equal: Multiply second ratio by 6 → 18 : 6

Increase in units = 18 − 13 = 5 units or 6 − 1 = 5 units

5 units = 10 years ⇒ 1 unit = 2 years

∴ The correct answer is 2 years.

Alternate Method

Given: Present age of Raja is 13 times Vikram. After 10 years, Raja will be 3 times as old as Vikram.

Let Vikram's present age = x years.

Then Raja's present age = 13x years.

⇒ According to the question: (13x + 10) = 3 × (x + 10)

⇒ 13x + 10 = 3x + 30

⇒ 13x − 3x = 30 − 10

⇒ 10x = 20

⇒ x = 20 ÷ 10 = 2

∴ The correct answer is 2 years.

Additional Information

Ratio Difference Method

In age problems, the difference between the ages of two people remains constant regardless of time. Always equate the difference in ratios to simplify calculations.

Cross Multiplication Shortcut

For equations like a/b = c/d, use cross multiplication: (Initial Ratio Product − Final Ratio Product) × x = Constant Term × Difference in Ratios.

Future and Past Ages

If present age is x, age after n years is (x + n) and age n years ago was (x − n).

Given:

Number of patients admitted in June = 25

Number of patients admitted in July = 15

Number of patients admitted in August = 20

Number of patients admitted in September = 30

Number of patients admitted in October = 40

Formula used:

Total number of patients = Sum of patients admitted in all months

Calculation:

Total number of patients = 25 + 15 + 20 + 30 + 40

⇒ Total = 130

∴ The total number of patients admitted in the given five months is 130.

Shortcut Trick

Let the height be h cm. Since the base is double the height, base = 2h.

Area = (1/2) × base × height = (1/2) × 2h × h = h²

Given Area = 81 cm²

⇒ h² = 81 ⇒ h = 9 cm

∴ The correct answer is 9 cm.

Alternate Method

Given:

Area of the triangle = 81 cm²

Base = 2 × Height

Base = 2h Height = h Area = 81 cm²

Formula Used:

Area of a triangle = (1/2) × Base × Height

Calculations:

⇒ Let the height of the triangle be h cm.

⇒ Then, the base of the triangle = 2h cm.

⇒ Area = (1/2) × 2h × h

⇒ 81 = h²

⇒ h = √81

⇒ h = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Area of Equilateral Triangle

For a triangle with all three sides equal to a, Area = (√3 ÷ 4) × a².

Heron's Formula

For a triangle with sides a, b, and c, Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2 is the semi-perimeter.

Right-Angled Triangle Area

If the triangle contains a 90° angle, Area = (1/2) × Base × Perpendicular.

Shortcut Trick

Difference in Selling Price per litre = 60 − 50 = Rs. 10.

Total difference in Profit/Loss amount = Gain − (−Loss) = 1,500 − (−2,000) = Rs. 3,500.

Total Quantity of milk = Total difference in amount ÷ Difference in rate per litre.

Quantity = 3,500 ÷ 10 = 350 litres.

∴ The correct answer is 350 litres.

Alternate Method

Given → Case 1: SP₁ = Rs. 50, Loss = Rs. 2,000 | Case 2: SP₂ = Rs. 60, Gain = Rs. 1,500. Formulas: CP = SP × Quantity + Loss | CP = SP × Quantity − Gain.

⇒ Let the total quantity of milk be Q litres. ⇒ In Case 1: Total CP = 50 × Q + 2,000. ⇒ In Case 2: Total CP = 60 × Q − 1,500. ⇒ Equating both: 50Q + 2,000 = 60Q − 1,500. ⇒ 60Q − 50Q = 2,000 + 1,500. ⇒ 10Q = 3,500 ⇒ Q = 350.

∴ The correct answer is 350 litres.

Additional Information

Total Cost Price calculation

Once quantity is known, CP can be found using CP = (50 × 350) + 2,000 = Rs. 19,500.

Relationship between Gain and Loss

The shift from a loss of Rs. 2,000 to a gain of Rs. 1,500 represents a total recovery of Rs. 3,500.

The correct answer is 0.5 amp.

Key Points

• Given:
• Power (P) = 10 Watts
• Voltage (V) = 20 Volts
• Rearranging the formula to find current (I):
• Current (I) = Power (P) / Voltage (V)
• Substituting the given values:
• Current (I) = 10 Watts / 20 Volts
• Current (I) = 0.5 Amperes
• Therefore, the current flowing through the fan is 0.5 Amperes.

Shortcut Trick

Ratio of Kamal and his son at present = 4 : 1 (Difference = 3 parts)

Ratio after 18 years = 2 : 1 (Difference = 1 part)

To equate the age difference, multiply the future ratio by 3: New ratio = 6 : 3

Increase in parts = 6 − 4 = 2 units or 3 − 1 = 2 units

2 units = 18 years ⇒ 1 unit = 9 years

∴ The correct answer is 9 years.

Alternate Method

Given:

Present age of Kamal = 4 × Present age of son

After 18 years, Son's age = 1/2 × Kamal's age

Let the present age of the son be x years.

∴ Present age of Kamal = 4x years.

After 18 years:

Age of son = x + 18

Age of Kamal = 4x + 18

According to the question:

⇒ x + 18 = 1/2 × (4x + 18)

⇒ 2 × (x + 18) = 4x + 18

⇒ 2x + 36 = 4x + 18

⇒ 4x − 2x = 36 − 18

⇒ 2x = 18

⇒ x = 9

∴ The correct answer is 9 years.

Additional Information

Constant Age Difference

The mathematical difference between the ages of two individuals always remains constant regardless of how many years pass.

Ratio Adjustment Rule

To solve age problems using ratios, ensure the difference between ratio parts is equal for both time periods to represent the constant age gap.

Age After 'n' Years

If the present age is x, then the age after n years is x + n, and the age n years ago was x − n.

The correct answer is Tehsil Parishad.

Key Points

• Balwant Rai Mehta Committee suggested the establishment of a democratic decentralized local government which came to be known as the Panchayati Raj.
• Three-tier Panchayati Raj system: Gram Panchayat, Panchayat Samiti, and Zila Parishad.

 Additional Information

• Ashok Mehta Committee was appointed in 1977 to suggest measures to revive and strengthen the declining Panchayati Raj system in India.
• The key recommendations are:
• The three-tier system should be replaced with a two-tier system: Zila Parishad (district level) and the Mandal Panchayat (a group of villages).
• The district level is the first level of supervision after the state level.
• Zila Parishad should be the executive body and responsible for planning at the district level.
• The institutions (Zila Parishad and the Mandal Panchayat) have compulsory taxation powers to mobilize their own financial resources.

The correct answer is CH2 unit.

Key Points

• Homologous series is defined as a series of compounds in which the same functional group substitutes for hydrogen in a carbon chain.
• This series will be differing with a successive member of CH2.
• Some examples are methane (CH4), ethane (C2H6), propane (C3H8) as they differ by CH2 and they have the same functional group called alkanes (General formula CnH2n+2).
• As series differ by CH2 , Atomic mass units of CH2 = Atomic mass of carbon + Atomic mass of Hydrogen

          12 + 2 = 14 (as there are 2 atoms of hydrogen, therefore the atomic mass of H2 is taken as 2)

Hence, subsequent members of the homologous series differ by 14 atomic mass units.

The correct answer is It marks the resurrection of Jesus Christ from the dead..

Key Points

• Easter is a religious festival celebrated by Christians.
• It commemorates the resurrection of Jesus Christ from the dead, described in the New Testament as having occurred on the third day after his burial following his crucifixion by the Romans at Calvary c. 30 AD.
• Easter is the culmination of the Passion of Jesus, preceded by Lent (or Great Lent), a 40-day period of fasting, prayer, and penance.
• The week before Easter is known as Holy Week, and it contains the days of the Easter Triduum, including Maundy Thursday (commemorating the Maundy and Last Supper), and Good Friday (commemorating the crucifixion and death of Jesus).

Additional Information

• Easter is linked to the Jewish Passover by much of its symbolism, as well as by its position in the calendar.
• In many languages, the words for "Easter" and "Passover" are identical or very similar.
• Easter customs vary across the Christian world, and include sunrise services, exclaiming the Paschal greeting, clipping the church, and decorating Easter eggs.
• The Easter lily, a symbol of the resurrection in Western Christianity, traditionally decorates the chancel area of churches on Easter Day and throughout the Eastertide season.
• Additional customs include egg hunting, the Easter Bunny, and Easter parades.

Given: A, B, C, D, E and F live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it is numbered 2 and so on till the topmost floor is numbered 6.

1) A lives on floor number 5. 

2) Only two people live between A and C.

3) E lives immediately below C.

3) Neither D nor B lives on the topmost floor. 

4) B does not live on an even-numbered floor.

After positioning D and B only one position is left which will be occupied by the only person left i.e. F.

Thus. according to the final arrangement, B lives on floor number 3.

Hence, "Option 1" is the correct answer.

Given:

2.$\overline{6}$

Formula used:

Let x = 2.$\overline{6}$

To get rid of the repeating decimal, multiply by 10:

10x = 26.$\overline{6}$

Subtract the original x from this equation:

10x - x = 26.$\overline{6}$ - 2.$\overline{6}$

Calculation:

10x - x = 26.$\overline{6}$ - 2.$\overline{6}$

⇒ 9x = 24

⇒ x = 24/9

⇒ x = 8/3

∴ The correct answer is option (3).

The correct answer is Luni.

Key Points

• Luni River:
• The Luni River is an important river in Rajasthan, India. It originates in the Aravalli Range near Ajmer and flows southwestwards through the Thar Desert.
• Unlike many rivers, the Luni River does not flow into a sea or ocean. Instead, it ends in the marshy lands of the Rann of Kutch in Gujarat, making it an inland draining river.

Additional Information

• Ghaggar
• The Ghaggar River rises in Shivalik hills in the Solan district of Himachal Pradesh.
• It enters Haryana from the Panchkula district.
• Then after the river flows making the border between Punjab and Haryana for more than 100 Km.
• And at the end, the river leaves Sirsa District of Haryana State and reaches Sri Ganga Nagar district, Rajasthan.
• ​Chambal
• The Chambal river originates in Manpura near Indore in Madhya Pradesh.
• The length of the river is 900 km and it is a tributary of Yamuna river.
• The Chambal River remains one of North India’s unpolluted rivers, home to a rich diversity of flora and fauna.
• The National Chambal Gharial Wildlife Sanctuary is famous for the rare Ganges river dolphin.
• Apart from the Ganges river dolphin, the other inhabitants of the sanctuary include magar (crocodile) and gharial (alligator). Migratory birds from Siberia form part of its rich avian fauna.
• Banas
• Banas river originates from Khamnor Hills of Aravalli Range, near Kumbhalgarh in Rajsamand, Rajasthan.  '
• Banas river is a tributary of the Chambal River and ii is approximately 512 kilometers in length.
• This river is also called ‘Van Ki Asha‘.
• There is another river in Rajasthan with the name of Banas.
• The Banas river entirely flows within the Rajasthan in:-
• Ajmer, Bhilwara, Bundi, Chittorgarh, Dausa, Jaipur, Pali, Rajsamand, Sawai Madhopur, Sirohi, Tonk, and Udaipur districts of Rajasthan.
• It flows northeast through the Mewar region of Rajasthan and meets the Chambal near the village of Rameshwar in Sawai Madhopur district.

The correct answer is Beyond centre of curvature .

Key Points 

• A concave mirror is a type of curved mirror where the reflective surface bulges inward.
• When light rays that are initially parallel to the principal axis reflect off the surface, they converge at a point known as the focal point.

Explanation: The behavior of a concave mirror depends on the position of the object:

• Object at infinity: Produces a real image at the focal point.
• Object beyond the focal point: Produces a real, inverted image between the focal point and the center of curvature.
• Object at the focal point: No image is formed as rays reflect parallel and never meet.
• Object between the focal point and the mirror: Produces a virtual, upright, and magnified image behind the mirror.

Additional Information

Different positions of the object and their image by a concave mirror: 

The logic followed here is: 

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

Now,

For (75, 16, 67):

⇒ (16 ÷ 2) + 67

= 8 + 67 = 75.

For (33, 14, 26)

⇒ (14 ÷ 2) + 26

= 7 + 26 = 33.

Similarly, for Option 4: (29, 6, 26)

⇒ (6 ÷ 2) + 26

= 3 + 26 = 29.

Hence, the correct answer is Option 4.

Shortcut Trick

Let the Selling Price (SP) of 1 kg rice be x.

Total SP of 38 kg = 38x and Profit = SP of 8 kg = 8x.

Cost Price (CP) of 38 kg = SP − Profit = 38x − 8x = 30x.

Given CP of 38 kg = 1710 ⇒ 30x = 1710 ⇒ x = 57.

SP of 5 kg rice = 5 × 57 = 285.

∴ The correct answer is ₹ 285.

Alternate Method

Given → CP of 38 kg rice = 1710 | Profit = SP of 8 kg. Formulas: Profit = SP − CP | SP = CP + Profit.

⇒ Let SP per kg be S. ⇒ 38 × S − 1710 = 8 × S ⇒ 38S − 8S = 1710 ⇒ 30S = 1710 ⇒ S = 1710 ÷ 30 = 57. ⇒ SP of 5 kg = 5 × 57 = 285.

∴ The correct answer is ₹ 285.

Additional Information

Profit and Loss in terms of Quantity

When profit is expressed as the selling price of certain items, the effective CP is the difference in quantity (Total units − Profit units).

Basic Relation

Profit % = [Profit units ÷ (Total units − Profit units)] × 100.

Given:

$[65 + {31 - (45 \text{ of } 3 \div 5)} \times 4]$

Formula used:

Order of operations (BODMAS): Brackets, Orders (i.e. powers and square roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)

Calculation:

$[65 + {31 - (45 \times 3 \div 5)} \times 4]$

$[65 + {31 - (135 \div 5)} \times 4]$

$[65 + {31 - 27} \times 4]$

⇒ $[65 + 4 \times 4]$

⇒ $[65 + 16]$

⇒ 81

∴ The correct answer is option (4).

Given: A, B, C, D, E and F live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it is numbered 2, and so on till the topmost floor is numbered 6.

1) C lives on an even numbered floor but not on the topmost floor. 

2) B lives immediately below C. 

3) Only three people live between B and D.

As there is no possible position for D in case 2, therefore is eliminated.

3) A lives immediately above F.

After positioning A and F only one position is left which will be occupied by the only person left i.e. E.

Thus. according to the final arrangement, A lives on floor number 4.

Hence, "Option 4" is the correct answer.

The correct answer is Sodium carbonate.

Key Points

• Sodium carbonate
• It is commonly known as Na2CO3 or Washing Soda.
• It is used in the softening of hard water i.e it is used to remove the temporary and permanent hardness of the water.
• It is commonly used as a cleaning agent for domestic purposes.
• It is also used in the manufacturing of paper, soap, textiles, paints, etc.

Additional Information

• Sodium hydroxide​
• Sodium hydroxide (NaOH) dissociates completely in water to give Na+ and OH−
• Sodium hydroxide is considered a strong base as all the available [OH] in NaOH is present in the solution as [OH−] and available to accept protons.
• Sodium hydroxide is the strongest base because it totally dissociates to form sodium ions and hydroxide ions.
• These hydroxide ions are treated with hydrogen ions further from the acid and totally ionize the hydrogen ions.
• Producers may utilize sodium hydroxide to manufacture soaps, rayon, paper, products that explode, dyes, and petroleum products.
• Other works that may utilize sodium hydroxide comprise preparing cotton fabric, metal cleaning and preparing, oxide coating, electroplating, and electrolytic extraction.
• ​Sodium chloride
• Sodium Chloride is also known as Salt.
• It is an ionic compound with the chemical formula NaCl.
• It has a ratio of 1:1 of sodium and chloride ions, With molar masses of 22.99 and 35.45 g/mol respectively.
• 100 g of NaCl contains 39.34 g Na and 60.66 g Cl.
• Sodium chloride is used in the Solvay process to produce sodium carbonate and calcium chloride.
• In the Mannheim process and in the Hargreaves process, sodium chloride is used for the production of sodium sulfate and hydrochloric acid.
• Sodium hydrogencarbonate
• Sodium hydrogencarbonate is also known as baking soda.
• Baking soda is a white crystalline powder that is commonly used as a leavening agent in baking.
• It is also used for cleaning and as a natural deodorizer.

Shortcut Trick

Average Speed for same distance = (2 × S₁ × S₂) ÷ (S₁ + S₂)

Average Speed = (2 × 40 × 5) ÷ (40 + 5) = 400 ÷ 45 = 80/9 km/h

Total Distance = Average Speed × Total Time

⇒ Total Distance = (80/9) × 4.5 = (80/9) × (9/2) = 40 km

∴ The correct answer is 40 km.

Alternate Method

Given:

Speed of bus (S₁) = 40 km/h

Speed of walking (S₂) = 5 km/h

Total time (T) = 4 hours 30 minutes = 4.5 hours

Calculations:

Let the one-way distance be d km.

⇒ Time taken by bus + Time taken walking = Total Time

⇒ (d/40) + (d/5) = 4.5

⇒ (d + 8d) / 40 = 4.5

⇒ 9d / 40 = 4.5

⇒ 9d = 4.5 × 40

⇒ 9d = 180

⇒ d = 20 km (One-way distance)

⇒ Total distance covered = 2 × d = 2 × 20 = 40 km

∴ The correct answer is 40 km.

Additional Information

Average Speed Formula

When a person travels equal distances at speeds x and y, the Average Speed is given by (2xy) / (x + y).

Speed-Time Relationship

For a fixed distance, Speed is inversely proportional to Time (S₁/S₂ = T₂/T₁).

Total Distance vs Displacement

In a round trip, Total Distance is the sum of both paths, while Displacement is zero as the starting and ending points are the same.

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

Conclusions:

(I) Some ducks are birds → Does not follow (As some animals are plants, some plants are birds and no duck is an animal. So it is possible but not definite.)

(II) No bird is an animal → Does not follow (As some animals are plants and some plants are birds. As no direct relation is given between birds and animals, therefore it is false.)

(III) Some animals are birds → Does not follow (As some animals are plants and some plants are birds. So it is possible but not definite.)

∴ Here, None of the conclusions follow.​

Hence, the correct answer is "Option 2".

Given number: '56137428'

After arranged in ascending order:

 

Now,

there is One digit "8" that will remain unchanged if each of the digits in the number '56137428' is arranged in ascending order.

Hence, the correct answer is "Option 2".

The correct answer is (a), (b), (d).

Key Points

Magnetic field lines:

• A magnetic field line is an imaginary line such that tangent to it at any point gives the direction of the magnetic field at that point in space.
• Magnetic field lines are drawn to represent the magnetic fields.
• Magnetic field lines can be drawn with the help of a magnetic compass.
• Magnetic field lines are also called magnetic lines of force.

Additional Information 

Properties of magnetic field lines:

1. The magnetic field lines of a magnet (or a solenoid) form continuous closed loops.
2. The magnetic field lines start from the north pole and end on a south pole outside the magnet. Inside the magnet, the magnetic field lines start from the south pole and end on the north pole.
3. The tangent to the field line at a given point represents the direction of the net magnetic field B at that point.
4. The larger the number of field lines crossing per unit area, the stronger is the magnitude of the magnetic field B.
5. The magnetic field lines do not intersect with each other if they did, there will be two directions of the magnetic field at that point which is not possible.

The correct answer is Tungsten .

Key Points

• Tungsten
• Atomic Number: 74
  

Atomic Mass: 183.84 u

• Highest Melting Point of Any Metal: This makes it incredibly resistant to heat, crucial for applications like light bulb filaments.
• High Density: One of the densest metals, making it useful in applications where weight is a factor, such as in counterweights and projectiles.
• Excellent Strength: Exhibits high tensile strength, even at elevated temperatures.
• Applications:
• Light Bulb Filaments: Its high melting point allows it to withstand the intense heat generated during operation.
• Electrical Contacts: Used in applications requiring high current and resistance to wear and tear.
• Hard Metals: Tungsten carbide is extremely hard and used in cutting tools and industrial machinery.
• Medical Applications: Found in radiation shielding and some medical implants.

Additional Information 

• Copper
• Atomic Number: 29
  

Atomic Mass: 63.546 u

• Excellent Electrical Conductivity: Copper is an excellent conductor of electricity, making it ideal for electrical wiring and electronics.
• High Thermal Conductivity: It efficiently conducts heat, making it useful in heat exchangers and cooling systems.
• Malleable and Ductile: Can be easily shaped into various forms like wires, sheets, and tubes.
• Corrosion Resistance: Relatively resistant to corrosion, though it can react with certain chemicals.
• Applications:
• Electrical Wiring: Widely used in electrical wiring for homes, buildings, and power grids.
• Electronics: Essential component in many electronic devices, including printed circuit boards and integrated circuits.
• Plumbing: Used in pipes and fittings for water and other fluids.
• Industrial Applications: Used in various industrial processes, such as heat exchangers and manufacturing.
• Gold
• Atomic Number: 79
  

Atomic Mass: 196.967 u

• High Malleability and Ductility: Can be easily shaped into thin sheets (gold leaf) or drawn into fine wires.
• High Luster and Corrosion Resistance: Retains its shine and resists corrosion, making it valuable for jewelry and decorative purposes.
• High Electrical Conductivity: A good conductor of electricity, though not as conductive as copper.
• Chemical Inertness: Relatively unreactive with most chemicals.
• Applications:
• Jewelry: The most common use, prized for its beauty and durability.
• Investments: Gold is considered a valuable investment asset due to its rarity and stability.
• Industrial Applications: Used in electronics, dentistry, and medical applications.
• Monetary Purposes: Historically used as a medium of exchange and currency.
• Nichrome
• It's primarily composed of nickel (Ni) and chromium (Cr).​​
• High Electrical Resistance: A nickel-chromium alloy known for its high electrical resistance.
• High Melting Point: Can withstand high temperatures without melting.
• Good Oxidation Resistance: Resists oxidation at high temperatures.
• Applications:
• Heating Elements: Widely used in heating elements for appliances like toasters, heaters, and hair dryers.
• Resistance Wire: Used in resistors and other electrical components where high resistance is required.
• Thermocouples: Used in thermocouples to measure temperature.

Shortcut Trick

Total students passed in 5 classes = (50 × 0.20) + (60 × 0.25) + (25 × 0.20) + (40 × 0.10) + (70 × 0.30)

Total passed = 10 + 15 + 5 + 4 + 21 = 55 students

Average passed students per class = Total passed ÷ Number of classes

Average = 55 ÷ 5 = 11 students

∴ The correct answer is 11.

Alternate Method

Given: Students in 5 classes = 50, 60, 25, 40, 70. Pass % = 20%, 25%, 20%, 10%, 30%.

Formula Used: Average = ∑(Class Students × Pass Percentage) ÷ Total Classes

⇒ Passed in Class 1 = 20% of 50 = (20/100) × 50 = 10

⇒ Passed in Class 2 = 25% of 60 = (25/100) × 60 = 15

⇒ Passed in Class 3 = 20% of 25 = (20/100) × 25 = 5

⇒ Passed in Class 4 = 10% of 40 = (10/100) × 40 = 4

⇒ Passed in Class 5 = 30% of 70 = (30/100) × 70 = 21

⇒ Total Passed Students = 10 + 15 + 5 + 4 + 21 = 55

⇒ Average Students Passed = 55 ÷ 5 = 11

∴ The correct answer is 11.

Additional Information

Weighted Average

It calculates the average by considering the weight (number of students) of each group: Average = ∑(w_i × x_i) ÷ ∑w_i.

Percentage to Fraction Conversion

Common conversions include 20% = 1/5, 25% = 1/4, and 10% = 1/10 to speed up calculations.

Overall Pass Percentage

Overall Pass % = (Total students passed ÷ Total students in all classes) × 100.

Shortcut Trick

The third side of a triangle must always be less than the sum and greater than the difference of the other two sides.

Range: (26 − 24) < Third Side < (26 + 24)

Range: 2 < Third Side < 50

Among the given options, only 48 cm lies within this range.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Side 1 (a) = 24 cm

Side 2 (b) = 26 cm

Formula Used:

Triangle Inequality Theorem: For any triangle with sides a, b, and c, the following condition must be satisfied: |a − b| < c < a + b.

Calculations:

⇒ Let the third side be c.

⇒ Sum of the two given sides = 24 + 26 = 50 cm.

⇒ Difference of the two given sides = 26 − 24 = 2 cm.

⇒ According to the Triangle Inequality Theorem, the third side must be greater than 2 cm and less than 50 cm.

⇒ 2 cm < c < 50 cm.

⇒ Checking the options: (1) 54 cm, (2) 55 cm, (3) 48 cm, (4) 52 cm.

⇒ Only 48 cm satisfies the condition 2 < 48 < 50.

∴ The correct answer is 48 cm.

Additional Information

Triangle Inequality Theorem

Given: if the time is 1 ∶ 20 p.m

Current time is 1 ∶ 20 p.m

Adding 58 minutes: 20 Minutes + 58 Minutes = 78 Minutes.

So, it can be written as 1 hours : 18 Minutes.

Now, adjusting the time.

Add 1 hour to 1:20 p.m. → 2:20 p.m.

Add 18 minutes to 2:20 p.m. → 2:38 p.m.

Thus, time after 58 minutes,  if the time is 1 ∶ 20 p.m. is "2 ∶18 p.m".

Hence, the correct answer is "Option 1".

The correct answer is CNG and LPG do not contain carbon particles.

Key Points

• Composition: CNG is primarily methane (CH₄) and LPG is mainly propane (C₃H₈) and butane (C₄H₁₀). Both are hydrocarbons — i.e., they contain carbon and hydrogen. (Statement 3: Correct)
• Clean fuel label: Because CNG and LPG burn with lower particulate and NOx emissions compared with petrol/diesel, they are commonly promoted as 'clean fuels' for transport and cooking. This is a regulatory and policy designation in many countries. (Statement 1: Correct)
• Piped supply to homes: CNG is often supplied as PNG (Piped Natural Gas) for domestic cooking in urban areas. LPG is usually distributed in cylinders, but piped LPG systems or bulk pipeline arrangements do exist in select locations and projects. Thus both can be delivered via pipelines in specific contexts. (Statement 2: Correct (context-dependent))
• Carbon particles vs carbon atoms: The fuels themselves contain carbon atoms (they are hydrocarbons). Saying they “do not contain carbon particles” is misleading—while combustion of CNG/LPG produces far fewer soot/particulate emissions than diesel, the fuels inherently contain carbon. Therefore the absolute statement is incorrect. (Statement 4: False)

Additional Information

• Applications: CNG is widely used in public transport and NG-based PNG for households; LPG is predominantly used for cooking, some rural heating and as an auto-fuel (autogas).
• Safety & storage: Both require proper pressure vessels and leak detection. CNG is stored at high pressure; LPG is stored as a liquid under moderate pressure.
• Environmental note: Switching to CNG/LPG reduces particulate matter and SOx emissions; however, being carbon-based, they still emit CO₂ on combustion.

Important Points

• The official answer key had incorrectly marked Option 2 as the correct answer. This has now been corrected, and the answer has been updated to reflect the accurate option.

The correct answer is Average Variable Cost + Average Fixed Cost.

Key Points

• In cost theory, the point where the marginal cost curve intersects the average cost curve at its minimum indicates optimal production efficiency.
• The intersection of LRMC and LRAC at the minimum point of LRAC signals the firm's most efficient scale in the long run.
• This concept is essential in understanding cost minimization and firm behavior over time.

Additional Information

• Long Run Marginal Cost (LRMC) and Long Run Average Cost (LRAC):
• The long run marginal cost (LRMC) curve represents the additional cost incurred by producing one more unit of output in the long run.
• The long run average cost (LRAC) curve reflects the per-unit cost of production when all inputs are variable in the long run.
• The LRMC curve always intersects the LRAC curve at the minimum point of the LRAC curve, indicating the point of optimal efficiency.
• At this point, the cost of producing an additional unit equals the average cost, which means the firm is operating at the most efficient scale of production.
• The short run marginal cost (SRMC) curve does pass through the minimum point of the short run average cost (SRAC) curve, indicating the optimal output level in the short run.
• The short run marginal cost curve does not necessarily relate to the upward-rising portion of the long run average cost curve.
• The short run average cost curve may not necessarily touch the minimum point of the long run average cost curve.

The correct answer is +5 D.

Key Points

• F = R/2
• Where, F = focal length, R = radius of curvature
• So, F = 40/2 = 20 cm = 0.2 m
• Calculate Power:
• Power (P) = 1/f (where f is in meters)
• P = 1 / 0.2 m = 5 Diopters

The correct answer is NABARD.

Key Points

• NABARD
• NABARD is a Bank.
• NABARD refers to the National Bank for Agriculture and Rural Development.
• NABARD was formed on the recommendation of the committee formed on 30 March 1979, under the Chairmanship of Shri B. Sivaraman.
• It came into existence on 12 July 1982 by transferring the agricultural credit functions of RBI and refinance functions of the then Agricultural Refinance and Development Corporation (ARDC).
• NABARD today is fully owned by the Government of India.
• NABARD Facts:-
• | Founded | 12 July 1982 | | --- | --- | | Headquarter | Mumbai, India | | Chairman | Dr. G.R. Chintala |

Additional Information

• PNB
• Punjab National Bank:
• It commenced its operations on April 12, 1895, from Lahore.
• Tag Line: The Name You Can Bank Upon.
• During the long history of the Bank, 9 banks have been merged with Punjab National Bank
• The financial year 2019- 20 was significant in view of the announcement of amalgamation of Public sector Banks wherein Oriental Bank of Commerce and United Bank of India amalgamated with Punjab National Bank with effect from April 1, 2020.
• Founder: Lala Lajpat Rai.
• Products: Consumer Banking, Corporate Banking, Finance and Insurance, Investment Banking, Mortgage Loans, etc.
• Headquarters: New Delhi.

​

• RBI
• RBI was nationalized in 1949.
• The Reserve Bank of India was nationalised with effect from 1st January, 1949 on the basis of the Reserve Bank of India (Transfer to Public Ownership) Act, 1948.
• All shares in the capital of the Bank were deemed transferred to the Central Government on payment of a suitable compensation.
• SIDBI
• SIDBI stands for Small Industries Development Bank of India.
• Small Industries Development Bank of India (SIDBI) was set up on 2nd April 1990 under an Act of the Indian Parliament.
• It was formed in the year 1990.
• It is the apex regulatory body for the licensing and regulation of micro, small and medium enterprise finance companies in India.
• It comes under the jurisdiction of the Ministry of Finance.
• Its headquarters is located at Lucknow, Uttar Pradesh.
• SIDBI is regulated and supervised by the Reserve Bank of India.

The correct answer is oxidation and reduction occur simultaneously.

Key Points

Redox reaction-

• A redox (reduction-oxidation) reaction is a type of chemical reaction that involves the transfer of electrons between species.
• It consists of two simultaneous processes: reduction and oxidation.
• Reduction: In a redox reaction, reduction refers to the gain of electrons by a species. During reduction, the oxidation state of the species decreases, indicating a decrease in its positive charge or an increase in its negative charge.
• Oxidation: Oxidation, on the other hand, involves the loss of electrons by a species. It leads to an increase in the oxidation state of the species, indicating an increase in its positive charge or a decrease in its negative charge.

Additional Information 

Example of redox reaction-

• MnO2 + 4HCl → MnCl2 + 2H2O + Cl2
• ​MnO2 is manganese dioxide, and HCl is hydrochloric acid.
• In this reaction, manganese dioxide (MnO2) reacts with hydrochloric acid (HCl) to produce manganese(II) chloride (MnCl2), water (H2O), and chlorine gas (Cl2).
• MnO2 acts as the oxidizing agent, while HCl acts as the reducing agent.
• The manganese dioxide undergoes reduction, meaning it gains electrons. It is reduced from an oxidation state of +4 in MnO2 to +2 in MnCl2.
• The hydrochloric acid undergoes oxidation, meaning it loses electrons. The chlorine atoms in HCl are oxidized from an oxidation state of -1 to 0 in Cl2.
• As a result of the reaction, two hydrogen atoms from the hydrochloric acid combine with one oxygen atom from the manganese dioxide to form water (H2O).
• Simultaneously, the remaining chlorine atoms from the hydrochloric acid combine to form chlorine gas (Cl2).
• Overall, the reaction involves the transfer of electrons from manganese dioxide to hydrochloric acid, resulting in the formation of new compounds and the liberation of chlorine gas.

Given:

29.99 + 15.01 + 44.99 × 15.01 = ?

Formula used:

Basic arithmetic operations

Calculation:

29.99 + 15.01 + 44.99 × 15.01 = ?

⇒ 29.99 + 15.01 + 675.299

⇒ 45 + 675.299

⇒ 720 (approximately)

∴ The correct answer is option (4).

Given: Seven boxes M, N, O, P, Q, R and S are kept one over the other.

1) Only three boxes are kept above P.

2) Only two boxes are kept between P and M. 

3) Only N is kept between O and M. 

4) Only R is kept between S and Q. 

5) Neither Q nor M is at the top most position.

As M cannot be position at the top, therefore case 1 is eliminated.

Thus. according to the final arrangement, P box is kept between Q and O.

Hence, "Option 1" is the correct answer.

The logic followed here is:

LJ - HF - DB

And,

WU - SQ - OM

So, checking each option one by one:

Option 1) ZW - OR - FC 

Option 2) TR - QS – GI

Option 3) US - QO - MK

Option 4) SV - XV - TR

Thus, out of all the options, 'US - QO - MK' follows the same logic as given.

Hence, "Option 3" is the correct answer.

The correct answer is 1948 .

Key Points

• Industrial Finance Corporation of India Ltd. (IFCI)
• IFCI is one of the oldest development banks in India, primarily focused on providing financial assistance for industrial development.
• The organization plays a significant role in promoting entrepreneurship and supporting large-scale projects across various sectors.
• Rahul Bhave has been recommended for the position of MD & CEO at Industrial Finance Corporation of India Ltd. (IFCI).
• Palash Srivastava has been recommended for the Deputy Managing Director position at India Infrastructure Finance Company Limited (IIFCL).
• FSIB is responsible for recruiting chairpersons of public sector banks, insurers, and financial institutions.
• These recommendations are part of FSIB's mandate to ensure effective leadership in financial institutions.

 Additional Information

• Financial Services Institutions Bureau (FSIB)
• FSIB is a government body responsible for the recruitment and recommendation of leadership positions in financial institutions.
• The bureau aims to enhance governance and operational efficiency in the financial sector through effective leadership appointments.

The correct answer is Double displacement and precipitation reaction.

Key Points

• Double displacement 
• Double displacement reaction occurs between ionic compounds.
• In a double displacement reaction, two ionic compounds exchange their ions to form two new compounds.
• The reaction takes place in aqueous solutions or in the molten state.
• The products of the reaction are usually a precipitate, water, or a gas.
• Ionic compounds are composed of positively charged metal ions and negatively charged non-metal ions, and they dissociate into their ions in water or in the molten state.

Additional Information

• Precipitation reactions
• Precipitation reactions occur when two aqueous solutions containing soluble salts are mixed, and the resulting mixture leads to the formation of an insoluble solid product. This insoluble solid is called a precipitate.
• In this specific reaction:
• Reactants:
• Sodium carbonate (Na2CO3) is dissolved in water (aq)
• Calcium chloride (CaCl2) is also dissolved in water (aq)
• Products:
• Calcium carbonate (CaCO3) is formed as a solid precipitate (s).
• Sodium chloride (NaCl) remains dissolved in water (aq).
• The driving force behind this reaction is the formation of the insoluble calcium carbonate. When the solutions of sodium carbonate and calcium chloride are mixed, the calcium ions (Ca2+) from calcium chloride combine with the carbonate ions (CO3^2-) from sodium carbonate to form calcium carbonate, which is insoluble in water and thus precipitates out of the solution.

Given: Q, R, S, T, U and V are six family members sitting around a circular table, facing the centre in a restaurant.

1) U is seated to the immediate left of S.

2) R is seated second to the right of S.

 

3) V is seated third to the left of Q and is the immediate neighbour of U.

Thus, according to the final arrangement, T is seated second to the left of U.

Hence, "Option 3" is the correct answer.

The correct answer is Kaimur Hills - Bihar.

Key Points

Kaimur hills:

• Kaimur Hills, also called Kaimur Range, the eastern portion of the Vindhya Range
• Starting near Katangi in the Jabalpur district of Madhya Pradesh it running generally east for a distance of about 300 miles (480 km) to Sasaram in Bihar.
• Its maximum width is about 50 miles (80 km).

 Additional Information

• Cardamom Hills
• The famous Cardamom hill is located in the Western Ghats.
• Cardamom Hills is in southeastern Kerala.
• The Cardamom Hills region is famous for the production of tea, coffee, teak, and bamboo as well as cardamom.
• The name cardamom hills come from the cardamom spice grown in much of the hills.
• ​Mahadeo Hills
• The Mahadeo Hills are a range of hills in Madhya Pradesh state of central India.
• These are located in the northern part of the Satpura Range.
• Highest peak is Dhoopgarh.
• Satpura Mountain Range is southward of Vindhya Hills and parallel to it.
• Mahadeo Hills are located in Gujarat.
• Satpura range is a range of hills in Central India.
• The range rises in eastern Gujarat, through the border of Maharashtra and Madya Pradesh to the east of Chhattisgarh.
• ​Garhjat Hills
• The Garhjat Hills is a mountain range formed by a series low lying hills, plateaux, ridges, and meadows that stretch into Odisha from the Utkal Plains in the Chotanagpur region of Jharkhand and the Chhattisgarh Plains.
• The Garhjat hill is also known as Odisha Highland.
• Middle Mountain and Highland Region are well marked by a number of interfluves or watersheds.
• The Eastern Ghats is interrupted by a number of broad and narrow river valleys and flood plains.
• The average height of this region is about 900 meters above the mean sea level.
• The highest peak of Odisha is Deomali.

The correct answer is President.

Key Points

• The President of India submits his resignation to the Vice President.
• The President is elected by members of an electoral college consisting of elected members of both Houses of Parliament and Legislative Assemblies of the states in accordance with the system of proportional representation, by means of a single transferable vote.
• The President must be a citizen of India, not less than 35 years of age, and qualified for election as a member of the Lok Sabha.
• His term of office is five years, and he is eligible for re-election.
• The President's removal from office is to be in accordance with the procedure prescribed in Article 61 of the Constitution.​

Additional Information

• The Vice-President is elected by members of an electoral college consisting of members of both Houses of Parliament in accordance with the system of proportional representation by means of a single transferable vote.
• He must be a citizen of India, not less than 35 years of age, and eligible for election as a member of the Rajya Sabha.
• The Vice-President's term of office is five years, and he is eligible for re-election.
• His removal from office is to be in accordance with the procedure prescribed in Article 67(b).

The correct answer is third trophic.

Key Points

• Producers are the base of the food chain and support all other trophic levels.
• They are the organisms that can make their own food through the process of photosynthesis or chemosynthesis.
• Therefore, they have the highest number of individuals in the pyramid of numbers.
• Producers are also known as autotrophs, which means "self-feeders."

 Additional Information

• Primary consumers are the herbivores that eat producers.
• They have fewer individuals than producers because they need to consume a larger quantity of producers to meet their energy requirements.
• Secondary consumers are the carnivores that eat herbivores.
• They have even fewer individuals than primary consumers because they need to consume a larger quantity of primary consumers to meet their energy requirements.
• Tertiary consumers are the top predators that eat other carnivores.
• They have the fewest individuals in the pyramid of numbers because they need to consume a larger quantity of secondary consumers to meet their energy requirements.

The correct answer is F and Cl.

Key Points

• Valence is the number of electrons an atom must lose or gain to attain the nearest noble gas or inert gas electronic configuration.
• “Electrons in the outer shells that are not filled are called valence electrons”.
• The valence electrons are part of most of the chemical reactions because they contain more energy compared to the electrons present in inner orbits.
• Meanwhile, the number of valence electrons present also helps us determine a specific element’s chemical properties, such as its valence or valency, the formation of bonds with other elements.
• It also gives us an idea of how readily the atoms can form bonds, the number of unpaired electrons, and how many atoms can take part.

 Additional Information

• Some key characteristics of a valence electron are:
• For the main group elements, the valence electron exists only in the outermost electron shell.
• A valence electron can exist in the inner shell of a transition metal.
• An atom consisting of a closed shell of valence electrons will usually be chemically inert.
• A valence electron can either absorb or release energy in the form of a photon.
• Valence electrons also determine the electrical conductivity of an element.

Given:

Find the value of 3 × 6 + 5 - 6 ÷ 2 - 9 + 72 ÷ (3 × 4) + 42

Formula used:

Order of operations (BODMAS/BIDMAS): Brackets, Orders (i.e. powers and square roots, etc.), Division and Multiplication, Addition and Subtraction

Calculation:

3 × 6 + 5 - 6 ÷ 2 - 9 + 72 ÷ (3 × 4) + 42

⇒ 18 + 5 - 3 - 9 + 72 ÷ 12 + 42

⇒ 18 + 5 - 3 - 9 + 6 + 42

⇒ 23 - 3 - 9 + 6 + 42

⇒ 11 + 6 + 42

⇒ 59

∴ The correct answer is option (4).

The logic followed here is:

MONKS’ is written as ‘LMMIR'

And,

'PRICE’ is written as ‘OPHAD'

Similarly,

‘FROCK' is written as

Hence, "Option 4" is the correct answer.

Shortcut Trick

We know that 1/x + 1/y = (x + y) / (x × y)

Product of two numbers (x × y) = LCM × HCF

Given Sum (x + y) = 48 and LCM × HCF = 286 × 2

Direct Calculation: 48 / (286 × 2) = 48 / 572 = 12 / 143

∴ The correct answer is 12/143.

Alternate Method

Given:

LCM of x and y = 286

HCF of x and y = 2

Sum (x + y) = 48

Calculations:

⇒ x × y = LCM × HCF

⇒ x × y = 286 × 2 = 572

⇒ Required value = 1/x + 1/y

⇒ 1/x + 1/y = (y + x) / (x × y)

⇒ Substitute (x + y) = 48 and (x × y) = 572

⇒ 1/x + 1/y = 48 / 572

⇒ Dividing both numerator and denominator by 4: 12 / 143

∴ The correct answer is 12/143.

Additional Information

Product Rule

For any two numbers a and b, Product (a × b) = HCF(a, b) × LCM(a, b).

HCF and Ratios

If two numbers are in the ratio a:b and their HCF is h, then the numbers are ah and bh, and their LCM is abh.

Co-prime Numbers

The HCF of two co-prime numbers is always 1, and their LCM is the product of the numbers themselves.

Given:

1) In an examination, Raman scored more marks than Prince. 

Raman > Prince

2) Abhishek scored more marks than Deepak but less marks than Prince.

Raman > Prince > Abhishek > Deepak

3) Deepak scored more marks than Keshav.

Raman > Prince > Abhishek > Deepak > Keshav

4) Robin scored less marks than Keshav and Deepak.

Raman > Prince > Abhishek > Deepak > Keshav > Robin

Thus, according to the final arrangement, Abhishek scored the third highest marks in the exam.

Hence, "Option 3" is the correct answer.

Given: All turns are 90 degree turns only

Archit starts from Point A and drives 6 km towards the east.

He then takes a left turn, drives 24 km, turns right and drives 8 km.

He then takes a right turn and drives 12 km.

He takes a final right turn, drives 14 km and stops at Point B.

Thus, Archit should drive 12 km in the south direction to reach Point A from Point B.

Hence, "Option 1" is the correct answer.

The correct answer is the glomerulus.

Key Points

• Bowman’s capsules are found in the Kidney of Mammals.
• Bowman’s capsules is a cup-like sack at the nephron in the mammalian's kidney that performs the filtration of blood to form urine.
• The Bowman's capsule is a part of the excretion.

 Additional Information

• Digestion
• Biomacromolecules in food cannot be utilized by our body in their original form.
• They have to be broken down and converted into simple substances in the digestive system.
• This process of conversion of complex food substances to simple absorbable forms is called digestion and is carried out by our digestive system by mechanical and biochemical methods.
• Respiration
• This process of exchange of O2 from the atmosphere with CO2 produced by the cells is called breathing, commonly known as respiration.
• The biological process involved in the removal of harmful metabolic wastes from the body is called excretion.
• Animals accumulate ammonia, urea, uric acid, carbon dioxide, water, and ions like Na+, K+, Cl–, phosphate, sulfate, etc., either by metabolic activities or by other means like excess ingestion.
• These substances have to be removed totally or partially. The mechanisms of elimination of these substances with special emphasis on common nitrogenous wastes.
• Ammonia, urea, and uric acid are the major forms of nitrogenous wastes excreted by animals.

Given:

Find the value of (6 × 16 - 9 ÷ 3) + 17 - 12.

Formula used:

BODMAS/BIDMAS rules (Brackets, Orders, Division/Multiplication, Addition/Subtraction)

Calculation:

(6 × 16 - 9 ÷ 3) + 17 - 12

⇒ (96 - 3) + 17 - 12

⇒ 93 + 5

⇒ 98

∴ The correct answer is option (2).

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

Conclusions:

I. All plants are greens → Does not follow (As all trees are plants, all leaves are trees and all greens are leaves. So it is possible but not definite.)

II. All trees are leaves → Does not follow (As all leaves are trees. So it is possible but not definite.)

III. Some plants are leaves → Follow (As all trees are plants and all leaves are trees. As whole plants comes in trees which whole comes in plants, therefore it is true.)

∴ Here, Only conclusion III follows.​

Hence, the correct answer is "Option 3".

The correct answer is 1.24 ×108 m/s .

Key Points

• Refractive Index: The refractive index of a material is the ratio of the speed of light in a vacuum to the speed of light in that material.
• Calculation:
• Given:
• Refractive index of diamond (n) = 2.42
• Speed of light in vacuum (c) = 3 x 10^8 m/s
• Formula:
• Refractive index (n) = Speed of light in vacuum (c) / Speed of light in the medium (v)
• Rearranging the formula to find the speed of light in diamond (v):
• v = c / n
• Substituting the values:
• v = (3 x 10^8 m/s) / 2.42
• v ≈ 1.24 x 10^8 m/s
• Therefore, the speed of light in diamond is approximately 1.24 x 10^8 meters per second

Given: All turns are 90 degree turns only

Riya walks 25 m towards north.

Then she turns right and walks 30 m, then turns right and walks 40 m.

She then turns left and walks 15 m.

Finally, she turns left and walks 15 m.

Thus, Riya is (30 m + 15 m) = 45 m far from the starting point in the east direction.

Hence, "Option 3" is the correct answer.

Shortcut Trick

A quadratic equation can be directly formed using its roots m and n with the standard formula:

x² − (Sum of Roots)x + (Product of Roots) = 0

Sum of roots = m + n

Product of roots = m × n = mn

Substituting these values gives: x² − (m + n)x + mn = 0.

∴ The correct answer is x² − (m + n)x + mn = 0.

Alternate Method

Given:

The roots of the quadratic equation are m and n.

Formula Used:

If α and β are the roots of a quadratic equation, then the equation is given by:

(x − α)(x − β) = 0

Calculations:

⇒ (x − m)(x − n) = 0

⇒ x(x − n) − m(x − n) = 0

⇒ x² − nx − mx + mn = 0

⇒ x² − (n + m)x + mn = 0

⇒ x² − (m + n)x + mn = 0

∴ The correct answer is x² − (m + n)x + mn = 0.

Additional Information

Sum and Product of Roots

For a standard quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

Discriminant and Nature of Roots

The value D = b² − 4ac determines the nature of roots. If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are complex/imaginary.

Quadratic Formula

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

The correct answer is Conductors, alloys, insulators.

Key Points

• Conductors
• Definition: Materials that allow the easy flow of electric current through them.
• Key Characteristics:
• High electrical conductivity
• Low electrical resistance
• Contain free electrons that can move easily within the material.
• Examples:
• Metals (copper, silver, gold, aluminum)
• Graphite
• Saltwater
• Alloys
• Definition: Materials formed by combining two or more metals (or a metal and a non-metal).
• Key Characteristics:
• Often possess properties superior to those of the individual constituent metals.
• Can be designed to have specific electrical, mechanical, and chemical properties.
• Examples:
• Steel (iron and carbon)
• Brass (copper and zinc)
• Bronze (copper and tin)
• Nichrome (nickel and chromium) - used in heating elements
• Insulators
• Definition: Materials that strongly resist the flow of electric current.
• Key Characteristics:
• Low electrical conductivity
• High electrical resistance
• Have tightly bound electrons that cannot move freely.
• Examples:
• Rubber
• Plastic
• Glass
• Wood
• Ceramics

The correct answer is 95% water, 2.5% urea and 2.5% other waste products.

Key Points

• The human excretory system consists of a pair of kidneys, two ureters, a urinary bladder, and a urethra.
• The kidney initiates the formation of urine, and the urethra facilitates urine excretion.
• Humans are ureotelic, and excretion occurs in the form of urea, resulting in less loss of body water.
• Kidneys are reddish-brown, bean-shaped structures. An adult human kidney measures 10-12cm in length, 5-7cm in width, and 2-3cm in thickness, with an average weight of 12-170gm.
• Through nephrons (tiny filtering units), the kidneys remove waste products called urea, as well as extra water and ions from the blood to form urine.
• The nephron is the functional unit of the kidney.
• The pelvis of the kidney is continued as a ureter.
• The ureter is a long and muscular tube.
• The ureter carries urine from the kidneys to the bladder.
• In the pelvic cavity, there is a thin-walled, pear-shaped, white transparent sac called the urinary bladder.
• It temporarily stores the urine.
• The urethra is a membranous tube that arises from the urinary bladder.
• The urethra conducts urine to the exterior.

Thus, the Correct path of urine in our body is Kidney → ureter → urinary bladder → urethra.

Given series:

2 $ 3 @ 4 * 7 8 5 3 # 9 1 > 5 & 4 @ 3 $ # 5 * 6 ! 8 2 & 9 < 4 6 # 3 % 2 = 7 8 * 4 > 7

Condition to be checked:

1. Odd numbers which are immediately preceded by a prime number.
2. Odd numbers which are immediately followed by a symbol.

Prime number → Odd number → Symbol

2 $ 3 @ 4 * 7 8 5 3 # 9 1 > 5 & 4 @ 3 $ # 5 * 6 ! 8 2 & 9 < 4 6 # 3 % 2 = 7 8 * 4 > 7

Thus, there are one such pair (5 3 #) that are following the above conditions.

Hence, "Option 4" is the correct answer.

Mistake Points

Immediately preceded by means immediately before or to come just before.

Immediately followed by means immediately after or to come just after.

Given numbers: Left 202, 323, 266, 182, 315, 285, 193, 217, 231, 186 Right

According to the question, the numbers given are arranged in descending order:

(Left) 202, 323, 266, 182, 315, 285, 193, 217, 231, 186 (Right) ⇒ (Left) 323, 315, 285, 266, 231, 217, 202, 193, 186, 182 (Right)

So, 

The number which is fifth from the right ⇒ (Left) 323, 315, 285, 266, 231, 217, 202, 193, 186, 182 (Right) → 217

The number which is fourth from the left ⇒ (Left) 323, 315, 285, 266, 231, 217, 202, 193, 186, 182 (Right) → 266

Thus, the difference between the number which is fifth from the right and the number which is fourth from the left ⇒ 266 - 217 = 49

Hence, "Option 4" is the correct answer.

Shortcut Trick

Surface Area of sphere = 4πr² = πd²

⇒ 1144 = (22/7) × d²

⇒ d² = (1144 × 7) ÷ 22 = 52 × 7 = 364

⇒ d = √364 = √(4 × 91) = 2√91 cm

∴ The correct answer is 2√91 cm.

Alternate Method

Given: Surface Area of the sphere = 1144 cm², π = 22/7

Formula Used: Surface Area = 4 × π × r² | Diameter (d) = 2 × r

⇒ 4 × (22/7) × r² = 1144

⇒ 88/7 × r² = 1144

⇒ r² = (1144 × 7) ÷ 88

⇒ r² = 13 × 7 = 91

⇒ r = √91 cm

⇒ Diameter (d) = 2 × r = 2 × √91

∴ The correct answer is 2√91 cm.

Additional Information

Volume of Sphere

The volume of a sphere is given by V = (4/3)πr³, where r is the radius.

Hemisphere Properties

For a hemisphere, the Curved Surface Area (CSA) = 2πr² and Total Surface Area (TSA) = 3πr².

Ratio of Surface Area

If the radii of two spheres are in ratio a:b, then their surface areas are in the ratio a²:b².

The logic followed here is:

'BOUTS’ is written as '54%8#'

'ULTRA’ is written as '%38@9'

Here, code for letter 'E' will be a new code other than the given codes.

Therefore, 'BLARE' is written as '539@2'

Hence, "Option 4" is the correct answer.

Given: (left) 573 136 357 442 814 (right)

If 2 is added to the second digit of every number, the numbers become:

• 593 (573: 7+2 = 9)
• 156 (136: 3+2 = 5)
• 377 (357: 5+2 = 7)
• 462 (442: 4+2 = 6)
• 834 (814: 1+2 = 3)

Now, check which of these numbers have a second digit exactly divisible by the third digit:

• 593: 9 is not divisible by 3
• 156: 5 is not divisible by 6
• 377: 7 is divisible by 7
• 462: 6 is divisible by 2
• 834: 3 is divisible by 4

There are three such numbers (377, 462, and 834).

The correct answer is "Option 4".

Shortcut Trick

Combine the coefficients of like terms from all three expressions.

For xy: 7 − 3 − 2 = 2; For yz: 5 + 4 = 9.

For zx: −3 + 9 = 6; For x: 5; For y: −4.

Combine all results: 5x − 4y + 2xy + 9yz + 6zx.

∴ The correct answer is 5x − 4y + 2xy + 9yz + 6zx.

Alternate Method

Given:

Expression 1: 7xy + 5yz − 3zx

Expression 2: 4yz + 9zx − 4y

Expression 3: −3xy + 5x − 2xy

Calculation:

⇒ Simplify Expression 3: −3xy − 2xy + 5x = 5x − 5xy

⇒ Total Sum = (7xy + 5yz − 3zx) + (4yz + 9zx − 4y) + (5x − 5xy)

⇒ Grouping like terms: 5x − 4y + (7xy − 5xy) + (5yz + 4yz) + (−3zx + 9zx)

⇒ Sum = 5x − 4y + 2xy + 9yz + 6zx

Therefore, the correct answer is option (2).

∴ The correct answer is 5x − 4y + 2xy + 9yz + 6zx.

Additional Information

Like Terms

Terms that have the same variables raised to the same powers are called like terms and can be added or subtracted.

Commutative Property of Addition

The sum of algebraic expressions remains the same regardless of the order in which the terms are added (A + B = B + A).

Coefficient

The numerical factor of a term containing variables is known as its coefficient; only these are operated upon during addition of like terms.

The correct answer is Motilal Nehru and Chittaranjan Das .

Key Points

• Indian National Congress (Congress Party) members Motilal Nehru, one of the most well-known lawyers in northern India, and Chitta Ranjan Das, a nationalist politician from Bengal, founded the Swaraj Party in 1923.
• The word "swaraj," which means "self-rule," was used to refer broadly to the campaign for independence from British control and serves as the inspiration for the name of the party.
• The party's main objective was to run in the 1923 elections for the new Central Legislative Assembly and, if successful, to use its position of power to agitate against the British administration in India and disrupt official policies.
• More than 40 Swarajists were elected to the Central Legislative Assembly in 1923, but their representation was never quite enough to stop the British from passing the legislation they sought or thought was necessary to protect internal order in India. The party was dissolved by 1927.

Thus we know that Swaraj Party was founded in 1923 by Motilal Nehru and C.R. Das.

Additional Information

• Chittaranjan Das:
• He was popularly called as Deshbandhu.
• He defended Aurobindo Ghosh in the 1908 Alipore Bomb Case.
• He was the co-founder of the Swarajya Party in 1923 along with Motilal Nehru.
• He was the President of the Gaya session of INC in 1922.
• He was known as the political guru of Subhas Chandra Bose.
• Motilal Nehru:
• 1919 session of the congress was held in Amritsar to condemn the Jallianwala Bagh massacre in the city.
• The 1919 session was held under the presidentship of Motilal Nehru.
• He was the secretary of the Swarajya party formed in the year 1923.

The correct answer is It is a straight line with slope + f, and x intercept + 1 and y intercept - f.

Key Points

• Concave Mirror: Raghu is experimenting with a concave mirror, which can form both real and virtual images depending on the object's position.
• Variables:
• Object Distance (u): The distance of the object from the mirror.
• Image Distance (v): The distance of the image formed by the mirror from the mirror.
• Magnification (m): The ratio of the size of the image to the size of the object (m = v/u).
• Plotting the Graph (u vs. 1/m)
• Purpose: By plotting 'u' (object distance) against '1/m' (inverse of magnification), Raghu aims to analyze the relationship between these variables and potentially determine the focal length (f) of the concave mirror.
• Expected Outcome
• Linear Relationship: If the experiment is conducted accurately, the graph of 'u' versus '1/m' should result in a straight line.
• Slope and Intercept:
• The slope of the line will be equal to the focal length (f) of the concave mirror.
• The intercept on the 'u' axis will be equal to the focal length (f).
• Explanation
• Mirror Formula: The relationship between object distance (u), image distance (v), and focal length (f) of a spherical mirror is given by the mirror formula:
• 1/f = 1/v1 + 1/u

Additional Information

• Magnification: The magnification (m) is related to image distance (v) and object distance (u) as:
• m = -v/u
• Rearranging for 1/m:
• 1/m = -u/v
• Substituting in the Mirror Formula:
• 1/f = 1/v - 1/u
• 1/f = 1/u - 1/m
• Equation of the Line: This equation is in the form of y = mx + c, where:
• y = 1/u
• x = 1/m
• slope (m) = 1/f
• y-intercept (c) = 1/f

Shortcut Trick

Sum of three prime numbers is 40 (an even number).

Sum of three odd numbers is always odd; ∴ one prime number must be 2 (the only even prime).

Remaining sum = 40 − 2 = 38.

Possible pairs of distinct prime numbers whose sum is 38: (7, 31). Product = 2 × 7 × 31 = 434.

∴ The correct answer is 434.

Alternate Method

Given: Sum of three distinct prime numbers (P₁ + P₂ + P₃) = 40.

Logic: Sum of three primes is even ⇒ At least one prime must be even.

⇒ Let P₁ = 2.

⇒ P₂ + P₃ = 40 − 2 = 38.

⇒ Testing prime pairs for sum 38: (3, 35 - No), (5, 33 - No), (7, 31 - Yes).

⇒ Primes are 2, 7, and 31.

⇒ Product = 2 × 7 × 31

⇒ Product = 14 × 31 = 434.

∴ The correct answer is 434.

Additional Information

Even and Odd Prime Logic

The number 2 is the only even prime number. All other prime numbers are odd.

Sum Rules

Odd + Odd + Odd = Odd. Since 40 is even, one of the three primes must be even (2).

Prime Number Property

A prime number has exactly two factors: 1 and itself.

The logic followed here is:

Hence, "Option 2" is the correct answer.

Shortcut Trick

Effective Simple Interest (SI) rate for 4 years = 8% × 4 = 32%.

Effective Compound Interest (CI) rate for 4 years = (1.08⁴ − 1) × 100 ≈ 36.0489%.

Difference in effective rates = 36.0489% − 32% = 4.0489%.

Principal (Sum) = (Actual Difference ÷ Rate Difference) × 100 = (20 ÷ 4.0489) × 100 ≈ ₹493.96.

∴ The correct answer is ₹493.96.

Alternate Method

Given: Rate (R) = 8% p.a., Time (T) = 4 years, CI − SI = ₹20.

Formula Used: Difference = P[(1 + R/100)^T − 1 − (R × T)/100]

⇒ 20 = P[(1 + 8/100)⁴ − 1 − (8 × 4 / 100)]

⇒ 20 = P[(1.08)⁴ − 1 − 0.32]

⇒ 20 = P[1.36048896 − 1.32]

⇒ 20 = P[0.04048896]

⇒ P = 20 ÷ 0.04048896

⇒ P = 493.9619...

∴ The correct answer is ₹493.96.

Additional Information

Difference between CI and SI for 2 Years

The formula is Difference = P × (R/100)².

Difference between CI and SI for 3 Years

The formula is Difference = P × (R/100)² × (3 + R/100).

Effective Rate for Compound Interest

For n years at r% rate, the effective rate is [(1 + r/100)ⁿ − 1] × 100.

The logic followed here is:

Hence, "Option 2" is the correct answer.

The correct answer is Sunil Chhetri.

Key Points

• Sunil Chhetri:
• Sunil Chhetri is associated with football.
• Sunil Chhetri was the captain of the Indian Football Team.
• Since his debut with Mohun Bagan in 2002, Chhetri has scored 157 goals in 361 appearances and remains the top Indian scorer in club football.
• His latest strike now takes his tally to 94 international goals. That’s only third among active players behind Cristiano Ronaldo (128) and Lionel Messi (106).
• He is popularly known as 'Captain Fantastic'.
• He was awarded Arjuna Award in 2011 by the Government of India.
• He was awarded Padma Shri in 2019, India's fourth-highest civilian award.

Additional Information

• Football:
• The first football club in the world ‘Sheffield Football Club’ was founded in the year 1857 in England.
• The first Football club in India was ‘Dalhousie Club’.
• The apex institution of football is the ‘Federation of International Football Association’ (FIFA) was formed on May 21, 1904.
• The headquarters of FIFA is in Paris, France.
• Football was included as a competitive game in Olympic Games officially in 1908.
• India took part in the World Olympic Football Competition in 1948 in London.
• The first World Cup was organised in Uruguay in 1930.
• In India, Indian Football Association (IFA) organises National Football Championship.
• The trophy awarded in this competition is called Santosh Trophy.

Given: Seven people - A, C, G, K, M, R and V are sitting in a straight line facing North.

1) As many people sit to the left of A as to the right of A. 

2. C sits second to the right of A.

3. Only three people sit between C and G. 

4) K is not an immediate neighbour of G. 

5) M sits second to the right of K.

6) R does not sit at any of the extreme ends of the line. 

After positioning R only one position is left which will be occupied by the only person left i.e. V. 

Thus, according to the final arrangement, only one person i.e. G sits between V and R.

Hence, "Option 4" is the correct answer.

The correct answer is 9 ∶ 3 ∶ 3 ∶ 1.

Key Points

• The dihybrid cross between two traits in Mendel's experiments resulted in a phenotypic ratio of 9:3:3:1.
• Explanation:
• Dihybrid Cross: This refers to a cross between two individuals that differ in two observed traits. Mendel studied pea plants, focusing on two traits like seed color (yellow or green) and seed shape (round or wrinkled).
• Phenotypic Ratio: This ratio describes the proportion of offspring displaying different combinations of these two traits.
• 9:3:3:1 Ratio:
• 9: Represents the offspring that exhibit both dominant traits (e.g., yellow and round).
• 3: Represents the offspring that exhibit the dominant trait for one characteristic and the recessive trait for the other (e.g., yellow and wrinkled).
• 3: Represents the offspring that exhibit the recessive trait for the first characteristic and the dominant trait for the second (e.g., green and round).
• 1: Represents the offspring that exhibit both recessive traits (e.g., green and wrinkled).

The correct answer is Toyota Mirai.

Key Points

• Union Minister for Road Transport and Highways Shri Nitin Gadkari launched the world's most advanced technology - developed Green Hydrogen Fuel Cell Electric Vehicle (FCEV) Toyota Mirai along with Union Minister Shri Hardeep Puri, Union Minister Shri RK Singh , Union Minister Shri Mahendra Nath Pandey,  MD Toyota Kirloskar Motor Ltd. Shri Masakazu Yoshimura, VC TKM Ltd. Shri Vikram Kirloskar and officials at New Delhi
• It is an important initiative which will promote clean energy and environmental protection by reducing dependence on fossil fuels and  thereby make India 'Energy Self-reliant' by 2047.
• Fuel Cell Electric Vehicle (FCEV), powered by Hydrogen is one of the best Zero Emission solutions. It is completely environment friendly with no tailpipe emissions other than water.
• Green Hydrogen can be generated from renewable energy and abundantly available biomass. Introduction and adoption of technology to tap into the Green hydrogen’s potential  will play a key role in securing a clean and affordable energy future for India.

The correct answer is Cu.

Key Points

• Mendeleev's periodic table
• ​Dmitri Mendeleev sought to establish a connection between the atomic masses of the elements and their observed chemical and physical properties.
• The physical and chemical properties of the elements are a periodic function of their atomic masses, according to Mendeleev's Periodic Law.
• The periodic table was created by Mendeleev and lists all of the known elements in order of their atomic masses. Groups of elements sharing common properties were arranged in rows. The spaces in between the vertical columns were called periods.

 Additional Information

• Difference between Modern Periodic Table and Mendeleev's Periodic Table:
• The periodic chart of Mendeleev is based on atomic mass whereas the atomic number is the foundation for the modern periodic table.
• About 63 elements can be found on Mendeleev's periodic chart and roughly 118 elements can be found on the modern periodic table.
• Noble gases were not listed in Mendeleev's periodic table (since they were not found at that time), but in the modern periodic table, noble gases are located in a separate group that is designated as group 18.
• Mendeleev's periodic table contained gaps for undiscovered elements, whereas the present periodic table is uniform.
• In Mendeleev's periodic table, transition elements are grouped with other elements, however, in the modern periodic table, they are arranged in separate blocks.

Shortcut Trick For a rectangular pattern with equal rows and columns, the number of cadets must be a perfect square. The total number of cadets is 677. The nearest perfect square less than 677 is 262 = 676. Number of left-out cadets = 677 − 676 = 1. ∴ The correct answer is 1 cadet.

Alternate Method

Given: Total number of NCC cadets = 677 The pattern is a square where the number of rows (x) is equal to the number of columns (x).

Formula Used: Total cadets in the square pattern = x × x = x2 Left-out cadets = Total cadets − Largest perfect square less than or equal to total cadets

Calculations: ⇒ Let us find the square root of 677 using the long division or approximation method. ⇒ We know that 252 = 625 ⇒ We know that 262 = 676 ⇒ We know that 272 = 729 ⇒ Since 676 ≤ 677 < 729, the maximum size of the square formation can have 26 rows and 26 columns. ⇒ Number of cadets in the square pattern = 26 × 26 = 676 ⇒ Left-out cadets = 677 − 676 = 1 ∴ The correct answer is 1 cadet.

Additional Information

Perfect Squares and Formations When objects or people are arranged in a square pattern (equal rows and columns), the total count is always a perfect square (N = x2).

Adding to form a Perfect Square To find the minimum number of units to add to a number N to make it a perfect square, find the next higher perfect square and subtract N from it.

The correct answer is 2020.

Key Points

• Rashtriya Sanskrit Vidyapeetha, Tirupati was elevated to the status of a Central University in 2020.
• This significant development occurred through the passage of the Central Sanskrit Universities Act, 2020 by the Indian Parliament. This Act granted central university status to three institutions:
• Rashtriya Sanskrit Vidyapeetha, Tirupati
• Rashtriya Sanskrit Sansthan, New Delhi
• Shri Lal Bahadur Shastri Rashtriya Sanskrit Vidyapeetha, New Delhi.

Additional Information

• This elevation to Central University status provides several key benefits:
• Enhanced autonomy and financial stability: Central universities receive direct funding from the central government, providing greater financial stability and autonomy in decision-making.
• Increased prestige and recognition: Central university status enhances the institution's reputation and recognition both nationally and internationally.
• Improved research and academic opportunities: The elevated status can facilitate greater access to research grants, collaborations, and academic resources.
• Focus on national importance: As a central university, the institution will play a crucial role in promoting Sanskrit education and research at the national level.

Shortcut Trick

Average age of the 4 boys = 42 ÷ 4 = 10.5 years.

The middle point of the ages (between the 2^nd and 3^rd boy) is 10.5.

Since the interval is 5 years, the gap from the middle to the 2^nd boy is 2.5 years.

Age of 2^nd boy = 10.5 − 2.5 = 8 years.

Age of youngest boy (1^st boy) = 8 − 5 = 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given:

Number of boys = 4

Interval between births = 5 years

Total sum of ages = 42 years

Calculations:

Let the age of the youngest boy be x.

⇒ Ages of the 4 boys are: x, (x + 5), (x + 10), and (x + 15).

⇒ Sum = x + (x + 5) + (x + 10) + (x + 15) = 42

⇒ 4x + 30 = 42

⇒ 4x = 42 − 30

⇒ 4x = 12

⇒ x = 3

∴ The correct answer is 3 years.

Additional Information

Arithmetic Progression (AP) in Ages

When individuals are born at equal intervals, their ages form an AP where the interval is the common difference 'd'.

Average of AP

For an even number of terms, the average is the mean of the two middle terms. Average = Total Sum ÷ Number of terms.

Sum Formula

The sum of n terms of an AP is S_n = (n/2) × [2a + (n−1)d], where 'a' is the first term.

The correct answer is Plants starch resources get used up.

Key Points

• When plants are kept in a dark room for three days, they undergo significant changes due to the lack of sunlight, which is crucial for their survival and growth. Here's what happens:
• Photosynthesis Stops: Plants rely on sunlight to perform photosynthesis, the process of converting light energy into chemical energy (glucose). In the absence of light, photosynthesis ceases. This means the plant can no longer produce its own food.
• Starch Depletion: Plants store food in the form of starch. When photosynthesis stops, the plant starts utilizing the stored starch for energy. Over time, the starch reserves get depleted.
• Chlorophyll Degradation: Chlorophyll, the green pigment responsible for capturing sunlight, starts to degrade in the absence of light. This leads to a loss of green color in the leaves, which may turn yellow or pale.
• Growth Inhibition: Without the energy produced through photosynthesis, the plant cannot synthesize new cells and tissues. This results in stunted growth and development.
• Withering and Death: If the plant is kept in the dark for an extended period, it will eventually exhaust its stored energy reserves and die.

Additional Information

• Other Effects:
• The plant may exhibit weakened stems and leaves.
• It may become more susceptible to diseases and pests.
• Flowering and fruiting may be significantly impacted or cease altogether.

The correct answer is more times per minute than an animal breathing air.

Key Points

• Some fish breathe by gulping water through their mouths. This is how they extract the oxygen they need to survive underwater.
• Water Intake: The fish opens its mouth and takes in a mouthful of water.
• Gill Passage: The water then passes over the gills, which are located on either side of the fish's head.
• Oxygen Extraction: The gills, which are composed of delicate, feathery structures, are richly supplied with blood vessels. As the water flows over the gills, oxygen dissolved in the water diffuses into the bloodstream of the fish.
• Carbon Dioxide Release: Simultaneously, carbon dioxide, a waste product of the fish's metabolism, diffuses from the blood into the water.
• Water Expulsion: Finally, the water, now depleted of oxygen and enriched with carbon dioxide, is expelled from the fish's body through the gill slits.
• This process of water intake, oxygen extraction, and carbon dioxide release is a continuous cycle that allows fish to obtain the oxygen they need for survival.

Additional Information

• Gills are the primary respiratory organs of fish. They are highly specialized structures that maximize the efficiency of gas exchange.
• Countercurrent Exchange: The flow of water over the gills and the flow of blood within the gill filaments are in opposite directions. This countercurrent exchange mechanism ensures that the blood is always exposed to water with a higher oxygen concentration, maximizing the amount of oxygen that can be absorbed.
• Efficiency: Fish gills are incredibly efficient at extracting oxygen from water. This is crucial because the concentration of dissolved oxygen in water is much lower than the concentration of oxygen in air.

The logic followed here is:

Logic: (1st number × 5) + 5 = 2nd number.

Now,

For, 7 : 40

→ (7 × 5) + 5 = 35 + 5 = 40.

For, 9 : 50

→ (9 × 5) + 5 = 45 + 5 = 50.

Similarly, for 12 ∶ ?

→ (12 × 5) + 5 = 60 + 5 = 65.

Hence, the correct answer is "Option 2".

The correct answer is Chhattisgarh.

Key Points

• Chhattisgarh's Dominance: Chhattisgarh emerged as the top coal-producing state in FY 2020-21, surpassing other major coal-producing states like Odisha, Madhya Pradesh, and Jharkhand.
• Abundant Reserves: Chhattisgarh possesses vast coal reserves, particularly in the Korba coalfield, which significantly contributes to its leading position in coal production.
• Mining Infrastructure: The state has a well-developed mining infrastructure, including coal mines, transportation networks, and processing facilities, facilitating efficient coal extraction and transportation.
• Economic Significance: Coal production plays a crucial role in the economy of Chhattisgarh, generating employment and contributing to industrial development.

Given: Amongst six friends A, B, C, D, E and F, each has a different height. 

1) Only one person is shorter than D.

2. Only three people are taller than C.

_ > _ > _ > C > D > _

3) B is taller than F but shorter than E.

4) F is not the shortest.

After positioning B, F and E only one position is left which will be occupied by the only person left i.e. A.

E > B > F > C > D > A

Thus, according to the final arrangement, A is the shortest among all the friends.

Hence, "Option 3" is the correct answer.