RRB ALP CBT-I 2024 Official Paper (Held On: 28 Nov, 2024 Shift 1) (Previous Year Paper)

Shortcut Trick

The area of the largest triangle that can be inscribed in a semi-circle of radius r is always r².

Given radius (r) = 6 cm

⇒ Area = 6²

⇒ Area = 36 cm²

∴ The correct answer is 36 cm².

Alternate Method

Given:

Radius of the semi-circle (r) = 6 cm

Formula Used:

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

For a triangle inscribed in a semi-circle to have the maximum area, its base must be the diameter of the semi-circle and its height must be equal to the radius.

⇒ Base of triangle (b) = Diameter of semi-circle = 2 × r

⇒ Base (b) = 2 × 6 = 12 cm

⇒ Maximum Height (h) = Radius of semi-circle = 6 cm

⇒ Maximum Area = 1/2 × Base × Height

⇒ Area = 1/2 × 12 × 6

⇒ Area = 6 × 6

⇒ Area = 36 cm²

∴ The correct answer is 36 cm².

Additional Information

Angle in a Semi-circle

Any triangle inscribed in a semi-circle with the diameter as one of its sides is always a right-angled triangle (Angle = 90°).

Area and Perimeter of Semi-circle

Area = 1/2 × πr² and Perimeter = πr + 2r (where 2r is the diameter).

Maximum Area of Inscribed Rectangle

The largest rectangle that can be inscribed in a semi-circle of radius r has an area of r².

The logic followed here is:

Hence, the correct answer is "Option 2".

Shortcut Trick

In annual compounding, the amount at the end of a year acts as the principal for the following year.

Interest for the 3^rd year = Amount (3 years) − Amount (2 years) = ₹614.55 − ₹578.40 = ₹36.15

Rate of Interest (R) = (Interest ÷ Principal) × 100

R = (36.15 ÷ 578.40) × 100 = (3615 ÷ 57840) × 100 = 6.25%

∴ The correct answer is 6 ¼%.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹578.40

Amount after 3 years (A₃) = ₹614.55

Formula Used:

A = P(1 + R/100)ⁿ

Calculations:

⇒ A₃ / A₂ = [P(1 + R/100)³] / [P(1 + R/100)²]

⇒ 614.55 / 578.40 = 1 + R/100

⇒ 1 + R/100 = 1.0625

⇒ R/100 = 1.0625 − 1 = 0.0625

⇒ R = 0.0625 × 100 = 6.25%

⇒ R = 6 ¼%

∴ The correct answer is 6 ¼%.

Additional Information

Compound Interest Basics

Compound interest is calculated on the principal plus any accumulated interest. Formula: A = P(1 + r/n)^nt.

Successive Growth

In annual compounding, the growth between any two consecutive years (Year n and Year n+1) is simply the annual rate R% of the amount at Year n.

Comparison with SI

For the first year, if compounding is annual, Simple Interest and Compound Interest on the same principal at the same rate are equal.

The correct answer is a concave lens of suitable power.

Key Points

• Myopia, also known as nearsightedness, is a common vision condition where close objects appear clear, but distant objects are blurry.
• This condition occurs when the eye is too long relative to the focusing power of the cornea and lens of the eye.
• As a result, light rays focus at a point in front of the retina rather than directly on its surface.
• A concave lens (diverging lens) is used to correct myopia by diverging light rays before they enter the eye, allowing them to focus on the retina.
• The power of the concave lens is chosen based on the degree of myopia, ensuring that the light is focused correctly on the retina for clear vision.

Additional Information

• Hyperopia (Farsightedness)
• In hyperopia, distant objects can be seen more clearly than close ones.
• This condition is corrected using a convex lens (converging lens) to focus light correctly on the retina.
• Astigmatism
• Astigmatism is a condition where the cornea or lens is irregularly shaped, causing blurred or distorted vision.
• It can be corrected with cylindrical lenses that compensate for the irregular shape of the cornea or lens.
• Presbyopia
• Presbyopia is an age-related condition where the eye's lens loses its flexibility, making it difficult to focus on close objects.
• This condition is typically corrected with reading glasses or bifocals.

Given: 

Anju's rank from the top = 15^th

Anju's rank from the bottom = 18^th

Total number of students = Anju's rank from the top + Anju's rank from the bottom - 1

(15 + 18) - 1 = 33 - 1 = 32

Hence, The total number of students is "32".

The correct answer is R/2.

Key Points

• The distance between the pole and the focus of a spherical mirror is known as the focal length (f).
• The radius of curvature (R) is the distance between the pole and the center of curvature of the mirror.
• The focal length of a spherical mirror is half of its radius of curvature.
• Mathematically, this relationship is expressed as $f=\frac{R}{2}$
• This applies to both concave and convex mirrors.

Additional Information

• Concave Mirrors
• These mirrors curve inward like a spoon and can form real or virtual images depending on the position of the object.
• They are often used in applications like telescopes and shaving mirrors.
• Convex Mirrors
• These mirrors curve outward and always form virtual, diminished, and upright images.
• They are commonly used as rear-view mirrors in vehicles.
• Mirror Formula
• The mirror equation $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$ relates the focal length (f), image distance (v), and object distance (u) for spherical mirrors.
• This formula is applicable to both concave and convex mirrors.

The correct answer is Medium Range-Microwave Obscurant Chaff Rocket (MR-MOCR)

Key Points

• On June 26, 2024, the Defence Research and Development Organisation (DRDO) handed over the Medium Range-Microwave Obscurant Chaff Rocket (MR-MOCR) to the Indian Navy.
• The ceremony was held in New Delhi.
• DRDO, Indian Navy successfully test vertical launch short range  surface-to-air missile
• The MR-MOCR is a rocket that creates a microwave shield around assets and platforms to obscure radar signals.
• The rocket is fired into space, where it forms a cloud that can shield against threats that use radio frequency seekers.

Additional Information

• DRDO was formed in 1958 from the amalgamation of the then already functioning Technical Development Establishment (TDEs) of the Indian Army and the Directorate of Technical Development & Production (DTDP) with the Defence Science Organisation (DSO).
• Defence Research & Development Organisation (DRDO) is an agency under the Ministry of Defence.
• Headquartered of DRDO in New Delhi,
• DRDO is India's largest research organisation.

The correct answer is Dettol in water.

Key Points

• Dettol is a well-known antiseptic liquid that has a distinctive smell, making it easily identifiable.
• The smell of Dettol is due to the presence of chloroxylenol, which is its active ingredient.
• When Dettol is dissolved in water, its characteristic smell remains, allowing it to be identified by olfactory senses.
• Other solutions like sugar in water, salt in water, and potassium permanganate in water do not have a strong smell, making them harder to identify by smell alone.
• Dettol is commonly used for cleaning and disinfecting surfaces, as well as for first aid purposes due to its antiseptic properties.

Additional Information

• Salt in Water
• Salt (sodium chloride) dissolved in water does not have any significant odor.
• It is commonly used in cooking and food preservation.
• Sugar in Water
• Sugar dissolved in water is sweet to taste but does not have any specific smell.
• It is widely used in beverages and culinary preparations.
• Potassium Permanganate in Water
• Potassium permanganate in water has a pink or purple color, but it does not have a strong smell.
• It is often used as a disinfectant and in water treatment processes.

The logic followed here is: 

Logic: 1^st number ÷ 14  = 2^nd number.

For, 476 : Y

⇒ 476 ÷ 14 = 34.

Similarly, X : 23

⇒ X ÷ 14 = 23, ⇒ X = 23 × 14 = 322.

Thus, X = 322, Y = 34

Hence, the correct answer is "Option1".

Shortcut Trick

Find the LCM of 3, 4, 5, and 6, which is 60.

Divide the given number 2488 by the LCM (60).

2488 ÷ 60 → Remainder = 28.

Least number to add = LCM − Remainder = 60 − 28 = 32.

∴ The correct answer is 32.

Alternate Method

Given: Number = 2488, Divisors = 3, 4, 5, and 6.

Formula: Least number to add = LCM(divisors) − (Number % LCM)

⇒ Step 1: Calculate the Least Common Multiple (LCM) of 3, 4, 5, and 6.

⇒ 3 = 3¹

⇒ 4 = 2²

⇒ 5 = 5¹

⇒ 6 = 2¹ × 3¹

⇒ LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

⇒ Step 2: Perform division to find the current remainder.

⇒ 2488 ÷ 60 = 41 with a remainder of 28 (since 60 × 41 = 2460, 2488 − 2460 = 28).

⇒ Step 3: Find the number needed to reach the next multiple of 60.

⇒ Required Number = Divisor − Remainder = 60 − 28 = 32.

∴ The correct answer is 32.

Additional Information

Divisibility Rule of LCM

A number is divisible by a group of integers if and only if it is divisible by their LCM. For 3, 4, 5, and 6, the combined condition is divisibility by 60.

Addition vs Subtraction for Divisibility

To make a number N divisible by D: subtract the remainder (N % D) or add (D − remainder).

Divisibility by 60

A number is divisible by 60 if it ends in 0 and the remaining digits are divisible by both 3 and 4.

The correct answer is Kolkata Knight Riders

Key Points

• The Kolkata Knight Riders (KKR) won the 17^th edition of the Indian Premier League (IPL) in 2024.
• The KKR beat the Sunrisers Hyderabad by eight wickets in the final match on May 26, 2024.
• The 17^th edition of the IPL was held from March 22 to May 26, 2024.
• The tournament was played across 13 cities in India, with Chennai hosting the opening ceremony and the final.
• The KKR won their third IPL title with this victory.
• Mitchell Starc, a player for the KKR, was named the player of the match.

Additional Information

• The first edition of the Indian Premier League (IPL) cricket tournament was held in the year 2008.
• The IPL is a professional Twenty20 cricket league in India, founded by the Board of Control for Cricket in India (BCCI).
• The league has gained immense popularity and has become one of the most-watched cricket leagues in the world.
• The inaugural season of the IPL featured eight teams representing different cities in India.
• The league format includes a round-robin group stage followed by playoffs and a final match to determine the champion.

Given Equation: 121 C 21 D 11 A 11 B 121 = ?

BODMAS Rule:

Decoding the given information:

Now replacing the signs from left to right using BODMAS Rule, we get:

⇒ 121 + 21 - 11 ÷ 11 × 121

⇒ 121 + 21 - 1 × 121

⇒ 121 + 21 - 121

⇒ 21

Hence, the correct option is "Option 3".

Given: 8 × 2 + 3 − 2 ÷ 6 = ?

According to the question,  ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged

⇒ 8 ÷ 2 - 3 + 2 × 6 = ?

⇒ 4 - 3 + 12 = ?

⇒ 16 - 3 = 13

Hence, the correct answer is "Option 4".

Shortcut Trick

For similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Ratio of Perimeters = Ratio of Sides

75 ÷ 40 = 15 ÷ EF

⇒ EF = (15 × 40) ÷ 75

⇒ EF = 600 ÷ 75 = 8 cm

∴ The correct answer is 8 cm.

Alternate Method

Given:

Perimeter of ▵ABC = 75 cm

Perimeter of ▵DEF = 40 cm

Length of side BC = 15 cm

Formula Used:

If ▵ABC ∼ ▵DEF, then:

Perimeter of ▵ABC ÷ Perimeter of ▵DEF = AB ÷ DE = BC ÷ EF = AC ÷ DF

Calculations:

⇒ Perimeter(▵ABC) ÷ Perimeter(▵DEF) = BC ÷ EF

⇒ 75 ÷ 40 = 15 ÷ EF

⇒ EF = (15 × 40) ÷ 75

⇒ EF = 600 ÷ 75

⇒ EF = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Properties of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides or perimeters: Area₁/Area₂ = (Side₁/Side₂)².

Similarity Criteria

Two triangles are similar if they satisfy the AAA (Angle-Angle-Angle), SSS (Side-Side-Side ratio), or SAS (Side-Angle-Side ratio) criteria.

Linear Ratios

The ratio of any linear components (medians, altitudes, angle bisectors, or circumradii) of two similar triangles is equal to the ratio of their corresponding sides.

The correct answer is Volume of solute / Volume of solution × 100.

Key Points

• Volume by volume percentage (v/v %) is used to express the concentration of a solution when both the solute and the solution are measured in terms of volume.
• The formula for calculating volume by volume percentage is:

Volume by Volume % = (Volume of solute / Volume of solution) × 100

• In this formula:
• Volume of solute = Amount of substance dissolved.
• Volume of solution = Total volume after mixing solute and solvent.
• This method is commonly used for liquid-liquid solutions, such as alcohol in water.
• Option-wise Analysis:
• Mass of solute / Volume of solution × 100 – Used for mass by volume percentage, not v/v percentage.
• Mass of solute / Mass of solution × 100 – Used for mass by mass percentage.
• Volume of solute / Volume of solution × 100 – Correct.
• Volume of solution / Mass of solute × 100 – Incorrect formula.
• Hence, the correct answer is Volume of solute / Volume of solution × 100.

Additional Information

• Mass by Mass Percentage (w/w %):

(Mass of solute / Mass of solution) × 100

• Mass by Volume Percentage (w/v %):

(Mass of solute / Volume of solution) × 100

• Molarity: Number of moles of solute dissolved per litre of solution.
• Solute: The substance that gets dissolved in a solvent.
• Solvent: The substance that dissolves the solute to form a solution.

Important Points

• Example: If 20 mL alcohol is mixed to make 100 mL solution, then:

v/v % = (20 / 100) × 100 = 20%

• Volume percentage is widely used in pharmaceuticals, beverages, perfumes, and chemical industries.

Concept:

Equations of Motion under Gravity:

• The motion of objects under the influence of gravity can be described using the equations of motion.
• For an object thrown vertically upward, the final velocity at the highest point is zero.
• Formula: $v^2 = u^2 + 2as$
• Here, v is the final velocity, u is the initial velocity, a is acceleration (due to gravity), and s is the displacement.
• Acceleration due to gravity (g): The acceleration imparted to objects due to the gravitational force. SI unit: m/s². Dimensional Formula: [LT^-2]. g ≈ 9.8 m/s².

Calculation:

Given,

Mass of the ball, m = 0.5 kg (Not required for calculation)

Maximum height, s = 20 m

Final velocity at the highest point, v = 0 m/s

Acceleration due to gravity, a = -9.8 m/s² (negative as it is upward motion)

Using the equation of motion:

$v^2 = u^2 + 2as$

Substitute the given values:

$0 = u^2 + 2(-9.8)(20)$

⇒ $0 = u^2 - 392$

⇒ $u^2 = 392$

⇒ $u = \sqrt{392}$

⇒ $u ≈ 19.8 , \text{m/s}$

∴ The initial velocity of the ball is approximately 20 m/s.

The correct answer is Janata Dal (S)

Key Points

• H.D. Kumaraswamy is a senior leader of the Janata Dal (Secular) (JD(S)) and has served as the Chief Minister of Karnataka in the past.
• In the 2024 general elections, he contested from the Mandya parliamentary seat in Karnataka and secured a victory.
• Subsequently, he was appointed as the Minister of Heavy Industries in the Union Government.
• This marked a significant milestone for the JD(S), as the party secured representation in the central government, which is relatively rare for a regional party primarily focused on Karnataka.

Additional Information

Shortcut Trick

Valid votes = 80% of 7500 = 6000.

Winner gets 55% of valid votes, so Loser gets 45%.

Majority = (55% − 45%) = 10% of valid votes.

Majority = 10% of 6000 = 600.

∴ The correct answer is 600 votes.

Alternate Method

Given:

Total votes = 7500

Invalid votes = 20%

Winner's share = 55% of valid votes

Calculations:

⇒ Valid votes = 7500 × (100% − 20%)

⇒ Valid votes = 7500 × 80/100 = 6000

⇒ Winner's votes = 55/100 × 6000 = 3300

⇒ Loser's votes = 45/100 × 6000 = 2700

⇒ Majority = Winner's votes − Loser's votes

⇒ Majority = 3300 − 2700 = 600

∴ The correct answer is 600 votes.

Additional Information

Valid Votes Concept

Valid Votes = Total Votes − (Invalid Votes + Votes not cast). All win/loss percentages are usually calculated on valid votes unless stated otherwise.

Majority/Margin

The majority or winning margin is the absolute difference between the votes obtained by the winner and the runner-up.

Percentage Change Base

In multi-step percentage problems, always identify the correct base for each percentage (e.g., 20% of total vs 55% of valid).

The correct answer is Bowman's capsule.

Key Points

• The Bowman's capsule is a cup-shaped structure that encases the glomerulus in the nephron of the kidney.
• It is the first part of the nephron where the filtration of blood takes place to form urine.
• The Bowman's capsule collects the filtrate from the glomerulus and channels it into the proximal convoluted tubule.
• The structure of the Bowman's capsule allows it to efficiently filter blood plasma while preventing the passage of blood cells and large proteins.
• The interaction between the Bowman's capsule and the glomerulus is crucial for the kidney's function of removing waste and excess substances from the bloodstream.

Additional Information

• Proximal Convoluted Tubule
• This is the section of the nephron that follows the Bowman's capsule.
• It is responsible for the reabsorption of water, ions, and nutrients back into the bloodstream.
• Distal Convoluted Tubule
• This is the portion of the nephron that follows the loop of Henle.
• It is involved in the regulation of potassium, sodium, calcium, and pH balance in the body.
• Glomerulus
• A network of capillaries located within the Bowman's capsule.
• It is responsible for the filtration of blood to form the initial urine filtrate.

Shortcut Trick

Winner gets 70%, so the Loser gets 100% − 70% = 30%.

The Majority difference = 70% − 30% = 40%.

According to the question: 40% = 172.

Total valid votes (100%) = (172 ÷ 40) × 100 = 430.

∴ The correct answer is 430.

Alternate Method

Given:

Percentage of votes for the Winner = 70%

Majority of votes = 172

Let the total number of valid votes be V.

⇒ Votes secured by Winner = 70% of V = 0.70V

⇒ Votes secured by Loser = (100% − 70%) of V = 30% of V = 0.30V

⇒ Majority = Votes of Winner − Votes of Loser

⇒ 172 = 0.70V − 0.30V

⇒ 172 = 0.40V

⇒ V = 172 ÷ 0.40

⇒ V = 1720 ÷ 4

⇒ V = 430

Therefore, the total valid votes are 430, which corresponds to option 3.

∴ The correct answer is 430.

Additional Information

Concept of Majority

In a two-candidate election, Majority = (Votes of Winner) − (Votes of Runner-up). If expressed in percentage, Majority % = (Winner %) − (Loser %).

Valid vs. Invalid Votes

Total Votes Polled = Valid Votes + Invalid Votes. Percentages are usually calculated based on valid votes unless stated otherwise.

Winning Margin Formula

Total Valid Votes = (Majority × 100) ÷ (Difference in percentage points between candidates).

The correct answer is xylem, phloem.

Key Points

• Xylem is responsible for the transport of water and minerals from the roots to other parts of the plant.
• Phloem is responsible for the transport of the products of photosynthesis, primarily sugars, from the leaves to other parts of the plant.
• Xylem consists of vessels and tracheids, which are tubular structures that facilitate the upward movement of water and minerals.
• Phloem consists of sieve tubes and companion cells, which work together to transport nutrients throughout the plant.
• The process of water transport in xylem is largely driven by transpiration and root pressure, whereas phloem transport is driven by the pressure-flow mechanism.

Additional Information

• Collenchyma
• It is a type of plant tissue that provides support and flexibility to growing parts of the plant such as stems and leaves.
• Collenchyma cells have thicker cell walls compared to parenchyma cells and are often found under the epidermis.
• Parenchyma
• It is the most common type of plant tissue, involved in various functions including photosynthesis, storage, and tissue repair.
• Parenchyma cells have thin cell walls and large vacuoles, making them versatile in function.
• Sclerenchyma
• It is a type of plant tissue that provides structural support due to its thick, lignified cell walls.
• Sclerenchyma cells are usually dead at maturity and include fibers and sclereids.

Given: Seven people, A, B, C, D, E, F and G are sitting in a row, facing north.

E sits third from the left end. Only three people sit between G and E.

D sits second to the left of C. B is the immediate neighbour of C and F.

Final arrangement, 

Thus, the only four  people sit between A and F.

Hence, the correct answer is "Option 2".

The correct answer is Propanol.

Key Points

• An alcohol is an organic compound with a hydroxyl group (-OH) attached to a carbon atom.
• Propanol, also known as propyl alcohol, has the chemical formula C₃H₇OH.
• Propanol is a three-carbon alcohol, specifically a primary alcohol.
• It is commonly used as a solvent and in the manufacture of other chemicals.
• In contrast, Propanal (an aldehyde), Propanoic (a carboxylic acid), and Propanone (a ketone) do not contain the hydroxyl group characteristic of alcohols.

Additional Information

• Aldehydes
• These compounds have a carbonyl group (C=O) bonded to at least one hydrogen atom.
• Propanal is an example of an aldehyde with the formula CH₃CH₂CHO.
• Carboxylic Acids
• These compounds contain a carboxyl group (-COOH).
• Propanoic acid, also known as propionic acid, has the formula C₂H₅COOH.
• Ketones
• These compounds have a carbonyl group (C=O) bonded to two carbon atoms.
• Propanone, commonly known as acetone, has the formula (CH₃)₂CO.

Given series:

9 # 1 * £ 2 & % 4 @ 6 8 $ 7 & 3 2 5

Condition to be checked:

Numbers which are immediately preceded by a number and also immediately followed by a symbol.

Number → Number → Symbol

Checking the given series:

9 # 1 * £ 2 & % 4 @ 6 8 $ 7 & 3 2 5

Thus, there is one such pattern (6 8 $) that follows the above condition.

Hence, the correct answer "Option 4".

Mistake Points

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

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

The correct answer is It will decrease the concentration of hydronium ions per unit volume.

Key Points

• When an acid is mixed with water, it undergoes ionization and produces hydronium ions (H₃O⁺).
• However, since the acid solution is diluted by adding water, the concentration of hydronium ions per unit volume decreases, even though the total number of hydronium ions may increase.
• This dilution makes the acidic solution less concentrated and increases its pH.
• Hence, the correct answer is Option 4 – It will decrease the concentration of hydronium ions per unit volume.

Additional Information

• Dilution of an acid is an exothermic process, meaning heat is released.
• For safety, acid should always be added to water, and never water to acid, to avoid violent splashing due to the heat evolved.

Important Points

• Adding water to an acid dilutes the solution.
• The concentration of hydronium ions (H₃O⁺) per unit volume decreases.
• Correct Answer: Option 4 – It will decrease the concentration of hydronium ions per unit volume.

The logic followed here is:

FKUZ is related to CORD

For, 'QJKH is related to NNHL

Similarly, for 'JIBP '

So, 'JIBP' will be related to 'GMYT'

Hence, the correct answer is "Option 3".

The correct answer is '6.50%.'

Key Points

• Monetary Policy Committee (MPC) Decision:
• In 2024, the Monetary Policy Committee (MPC) of the Reserve Bank of India (RBI) decided to keep the policy repo rate unchanged at 6.50% under the liquidity adjustment facility (LAF) for the 9^th consecutive time.
• The repo rate is the rate at which the RBI lends short-term funds to commercial banks. Keeping the rate unchanged reflects the RBI's stance on controlling inflation while supporting economic growth.
• By keeping the rate steady, the MPC aims to manage inflationary pressures without dampening economic activity in the country.
• The MPC's decision to keep the repo rate unchanged was made by a majority of 5 out of 6 members.

Additional Information 

Shortcut Trick

Substitute a standard angle θ = 60° into the given equation.

k = cos⁴ 60° − sin⁴ 60° = (1/2)⁴ − (√3/2)⁴ = 1/16 − 9/16 = −8/16 = −1/2.

Now, calculate (1 − k) / (1 + k) = [1 − (−1/2)] / [1 + (−1/2)] = (3/2) / (1/2) = 3.

From the options, only tan² 60° = (√3)² = 3 matches this result.

∴ The correct answer is tan²θ.

Alternate Method

Given: cos⁴ θ − sin⁴ θ = k

Formula Used: a² − b² = (a − b)(a + b) and sin² θ + cos² θ = 1.

⇒ k = (cos² θ − sin² θ)(cos² θ + sin² θ)

⇒ k = (cos² θ − sin² θ) × (1)

⇒ k = cos² θ − sin² θ

⇒ 1 − k = 1 − (cos² θ − sin² θ) = (1 − cos² θ) + sin² θ = sin² θ + sin² θ = 2 sin² θ

⇒ 1 + k = 1 + (cos² θ − sin² θ) = (1 − sin² θ) + cos² θ = cos² θ + cos² θ = 2 cos² θ

⇒ (1 − k) / (1 + k) = (2 sin² θ) / (2 cos² θ)

⇒ (1 − k) / (1 + k) = sin² θ / cos² θ = tan² θ

∴ The correct answer is tan²θ.

Additional Information

Double Angle Identities

The term cos 2θ can be written as cos² θ − sin² θ, 2 cos² θ − 1, or 1 − 2 sin² θ.

Pythagorean Trigonometric Identity

The fundamental identity sin² θ + cos² θ = 1 is used to simplify higher-power trigonometric expressions.

Half-Angle Relationship

The expression (1 − cos 2θ) / (1 + cos 2θ) is a standard identity for tan² θ.

The logic followed here is: 

For, DIJH : KPQO 

For, MRSQ : TYZX 

Similarly, BGHF : ?

Thus, BGHF : INOM

Hence, the correct answer is "Option 2".

Statements:

I. Government authorities of city X are finding different solutions to reduce air pollution in the city during winter - Effect( because, The government taking steps to reduce pollution).

II. City X has seen an acute rise in asthma cases during the last winter - cause( because increased air pollution during winter (an external cause) could have led to a rise in asthma cases).

Thus, Statement I is the effect, and Statement II is the cause.

Hence, the correct answer is "Option 4".

Concept:

 

Principle of Conservation of Energy:

 

• The principle states that energy cannot be created or destroyed, only transformed from one form to another.
• In the context of this problem, the potential energy (PE) at the maximum height will be equal to the initial kinetic energy (KE).

Important terms:

• Potential Energy (PE): Energy possessed by a body due to its position or height.

        SI Unit: Joule (J)

        Dimensional Formula: [M L2 T-2]

• Kinetic Energy (KE): Energy possessed by a body due to its motion.

        SI Unit: Joule (J)

        Dimensional Formula: [M L2 T-2]

 

Calculation:

 

Given:

Mass of the ball (m) = 2 kg

Kinetic energy (KE) = 100 J

We know,

Kinetic Energy = 1/2 mv²

⇒ 100 = 1/2 × 2 × v²

⇒ 100 = v²

⇒ v = √100 = 10 m/s

∴ The velocity of the body will be 10 m/s.

Shortcut Trick

For problems where Cost Price (CP) of 'y' items = Selling Price (SP) of 'x' items:

Gain % = [(y − x) ÷ x] × 100

Gain % = [(400 − 320) ÷ 320] × 100

Gain % = [80 ÷ 320] × 100 = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Selling Price (SP) of 320 mangoes = Cost Price (CP) of 400 mangoes.

Let the CP of 1 mango be ₹1.

⇒ CP of 400 mangoes = ₹400

⇒ SP of 320 mangoes = CP of 400 mangoes = ₹400

Formula Used:

Profit = SP − CP

Profit % = (Profit ÷ CP) × 100

Calculations:

⇒ CP of 320 mangoes = 320 × ₹1 = ₹320

⇒ Profit on 320 mangoes = SP − CP = ₹400 − ₹320 = ₹80

⇒ Gain % = (80 ÷ 320) × 100

⇒ Gain % = (1 ÷ 4) × 100

⇒ Gain % = 25%

∴ The correct answer is 25%.

Additional Information

Profit/Loss on Number of Articles

If the CP of 'A' articles equals the SP of 'B' articles, then the gain or loss percent is [(A − B) ÷ B] × 100. A positive value implies Profit, while a negative value implies Loss.

Relation between CP and SP

SP = CP × [(100 + Gain %) ÷ 100] when there is a profit, and SP = CP × [(100 − Loss %) ÷ 100] when there is a loss.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

The correct answer is Reproduction.

Key Points

• Reproduction is the biological process by which new individual organisms are produced from their parents.
• It is essential for the continuation and stability of species, ensuring that populations can be maintained over generations.
• There are two main types of reproduction: asexual and sexual. Asexual reproduction involves a single organism or cell, while sexual reproduction involves the combination of genetic material from two parents.
• Reproduction allows for the transfer of genetic information from one generation to the next, which is vital for the survival of a species.
• It plays a crucial role in natural selection and evolution, as it introduces genetic variation within a population.

Additional Information

• Mutation
• Mutations are changes in the DNA sequence that can lead to genetic variation within a population.
• They can be beneficial, neutral, or harmful and play a role in the process of evolution.
• Evolution
• Evolution is the process by which different kinds of living organisms develop and diversify from earlier forms over generations.
• It involves mechanisms such as natural selection, mutation, gene flow, and genetic drift.
• Variation
• Variation refers to the differences in the genetic makeup and characteristics within and among populations of organisms.
• It is essential for the process of natural selection and the overall adaptability of a species.

Given:

The value of (0.04)^-1.5 is:

Formula used:

x^-n = 1 / xⁿ

Calculation:

(0.04)^-1.5

⇒ 1 / (0.04)^1.5

⇒ 1 / (0.2)³

⇒ 1 / 0.008

⇒ 125

∴ The correct answer is option (4).

The correct answer is When the initial and final position of the object is the same.

Key Points

• Displacement is defined as the shortest distance between the initial and final positions of an object.
• For displacement to be zero, the object must return to its initial position.
• Distance is a scalar quantity and can be greater than or equal to displacement but never less.
• When the initial and final positions are the same, the path taken does not matter for calculating displacement.
• Displacement takes into account the direction of motion, making it a vector quantity, unlike distance.

Additional Information

• Distance
• It is the total path length traveled by an object, irrespective of the direction.
• Distance is a scalar quantity and only has magnitude.
• Vector Quantity
• Vector quantities have both magnitude and direction.
• Examples include displacement, velocity, and acceleration.
• Scalar Quantity
• Scalar quantities have only magnitude and no direction.
• Examples include distance, speed, and mass.

The correct answer is Yuva Sangam

Key Points

• The Yuva Sangam program organized youth exchange visits across states in 2024, focusing on the '5 Ps'.
• The 5 Ps are: 
• Paryatan: Tourism
• Parampara: Traditions
• Pragati: Development
• Paraspar Sampark: People-to-people connect
• Prodyogiki: Technology
• Yuva Sangam is a program by the Government of India that aims to strengthen connections between people, especially among young people.
• The program involves exposure tours for students and other young people from one state to another.
• The program is part of the Centre's “Ek Bharat Shreshth Bharat” vision, which aims to promote cultural unity and collaboration among India's youth.
• The program provides a platform for young people to meet, interact, and learn from each other.

Additional Information

• National Youth Exchange
• ​The National Youth Exchange Program is an initiative aimed at fostering cultural understanding, unity, and national integration among the youth of India.
• It is typically organized by the Nehru Yuva Kendra Sangathan (NYKS) under the Ministry of Youth Affairs and Sports, Government of India.
• Benefits:
• Builds a sense of unity in diversity among the youth.
• Encourages understanding and appreciation of India’s cultural richness.
• Strengthens leadership skills and enhances teamwork.
• Facilitates networking among participants from different parts of the country.
• India Unites​
• The phrase “India Unites” often symbolizes the idea of national integration and unity in diversity, which aligns with India’s ethos of coexisting harmoniously despite its linguistic, religious, and cultural diversity.
• This concept is reflected in initiatives such as:
• National Integration Day (November 19): Celebrated to promote unity.
• Ek Bharat Shreshtha Bharat Program: A Government of India initiative to enhance cultural exchanges between states and UTs.

The correct answer is 26 July 2024

Key Points

• The Government of India and the Government of the United States of America signed the first ever 'Cultural Property Agreement' to prevent and curb the illicit trafficking of antiquities from India to the USA, today, the 26^th July, 2024 on the sidelines of the 46^th World Heritage Committee, at Bharat Mandapam, New Delhi.
• The agreement was signed by Shri Govind Mohan, Secretary, Ministry of Culture and H.E. Eric Garcetti, Ambassador of the United States of America to India in the august presence of Union Minister of Culture and Tourism, Shri Gajendra Singh Shekhawat.
• The Cultural Property Agreement (CPA) is aligned with the 1970 UNESCO Convention on the Means of Prohibiting and Preventing the Illicit Import, Export and Transfer of Ownership of Cultural Property, to which both countries are States party.
• CPA is another step towards securing India’s rich and diverse cultural heritage and invaluable artifacts of our grand history

Shortcut Trick

In a set of consecutive numbers, the Average is always the Middle Number.

Average (3^rd number) = 3720 ÷ 5 = 744.

The series of even numbers is: 740, 742, 744 (3^rd largest), 746, 748 (largest).

Required Sum = 744 + 748 = 1492.

∴ The correct answer is 1492.

Alternate Method

Given: Sum of 5 consecutive even numbers = 3720.

Let the five consecutive even numbers be x, x + 2, x + 4, x + 6, and x + 8.

⇒ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 3720

⇒ 5x + 20 = 3720

⇒ 5x = 3720 − 20

⇒ 5x = 3700

⇒ x = 3700 ÷ 5 = 740

Calculating required values:

⇒ Third largest number = x + 4 = 740 + 4 = 744

⇒ Largest number = x + 8 = 740 + 8 = 748

⇒ Sum = 744 + 748 = 1492

∴ The correct answer is 1492.

Additional Information

Arithmetic Progression Property

Consecutive even or odd numbers form an Arithmetic Progression (AP) with a common difference (d) of 2. The sum can be calculated using S_n = (n/2) × [2a + (n − 1)d].

Middle Term Shortcut

For an odd number of terms (n) in an AP, the average is always equal to the middle term. Middle Term = Total Sum ÷ n.

Sum of First n Even Numbers

The sum of the first 'n' even natural numbers starting from 2 is given by the formula: n × (n + 1).

Given:

0.292929......

Formula used:

Let x = 0.292929......

Then, 100x = 29.292929......

Calculation:

Let x = 0.292929......

Then, 100x = 29.292929......

⇒ 100x - x = 29.292929...... - 0.292929......

⇒ 99x = 29

⇒ x = 29/99

∴ The correct answer is option 2.

Shortcut Trick

Apply the Volume of Cylinder formula: Volume = πr²h

Substitute values: 5500 = (22/7) × r² × 70

Simplify: 5500 = 220 × r²

⇒ r² = 25 ⇒ Radius (r) = 5 m

Diameter (d) = 2 × Radius = 2 × 5 = 10 m

∴ The correct answer is 10 m.

Alternate Method

Given:

Volume of the cylinder (V) = 5500 m³

Height of the cylinder (h) = 70 m

Value of π = 22/7

Formula Used:

Volume (V) = π × r² × h

Diameter (d) = 2 × r

Calculations:

⇒ Volume = πr²h

⇒ 5500 = (22/7) × r² × 70

⇒ 5500 = 22 × r² × (70 ÷ 7)

⇒ 5500 = 22 × r² × 10

⇒ 5500 = 220 × r²

⇒ r² = 5500 ÷ 220

⇒ r² = 25

⇒ r = √25 = 5 m

⇒ Diameter (d) = 2 × 5

⇒ Diameter = 10 m

∴ The correct answer is 10 m.

Additional Information

Curved Surface Area (CSA)

The area of the curved surface of a cylinder is given by the formula 2πrh.

Total Surface Area (TSA)

The total surface area including the top and bottom circular bases is 2πr(h + r).

Area of Base

The base area of a cylinder is circular and is calculated as πr².

Volume of Cone

The volume of a cone with the same base radius and height is (1/3)πr²h, which is one-third the volume of the cylinder.

The logic followed here is: 

Hence, the correct answer is "Option 3".

Shortcut Trick

Total discount multiplier = 12960 ÷ 18000 = 0.72

First discount multiplier (10%) = 0.90

Second discount multiplier (x) = 0.72 ÷ 0.90 = 0.80

Discount percentage x = (1 − 0.80) × 100 = 20%

∴ The correct answer is 20.

Alternate Method

Given:

Marked Price (MP) = ₹18,000

First Discount = 10%

Final Selling Price (SP) = ₹12,960

Formula Used:

Price after discount = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ Price after 10% discount = 18,000 × (90/100)

⇒ Intermediate Price = 18,000 × 0.9 = ₹16,200

⇒ Now, SP = Intermediate Price × (100 − x)/100

⇒ 12,960 = 16,200 × (100 − x)/100

⇒ 12,960 ÷ 16,200 = (100 − x)/100

⇒ 1296 ÷ 1620 = (100 − x)/100

⇒ 0.8 = (100 − x)/100

⇒ 80 = 100 − x

⇒ x = 100 − 80 = 20

∴ The correct answer is 20.

Additional Information

Successive Discount Formula

A single equivalent discount for two successive discounts a% and b% is given by: (a + b − ab/100)%.

Marked Price and Selling Price

Selling Price = Marked Price × (1 − d₁/100) × (1 − d₂/100) ... × (1 − d_n/100).

Discount vs. Profit

Discount is always calculated on the Marked Price (MP), whereas Profit or Loss is calculated on the Cost Price (CP).

Shortcut Trick

Find the combined sum of the first two expressions by adding coefficients of like terms: (2 − 5)x − 3y + (7 + 4)z = −3x − 3y + 11z.

Subtract this sum from (13x − z) by changing signs of the sum: (13x − z) + (3x + 3y − 11z).

Group and solve: (13 + 3)x + 3y + (−1 − 11)z = 16x + 3y − 12z.

∴ The correct answer is 16x + 3y − 12z.

Alternate Method

Given:

First expression (E₁) = 2x − 3y + 7z

Second expression (E₂) = 4z − 5x

Third expression (E₃) = 13x − z

Calculations:

⇒ Step 1: Calculate the sum (S) of E₁ and E₂.

⇒ S = (2x − 3y + 7z) + (4z − 5x)

⇒ S = (2x − 5x) − 3y + (7z + 4z)

⇒ S = −3x − 3y + 11z

⇒ Step 2: Subtract S from E₃.

⇒ Result = E₃ − S

⇒ Result = (13x − z) − (−3x − 3y + 11z)

⇒ Result = 13x − z + 3x + 3y − 11z

⇒ Result = (13x + 3x) + 3y + (−z − 11z)

⇒ Result = 16x + 3y − 12z

∴ The correct answer is 16x + 3y − 12z.

Additional Information

Like Terms

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

Subtraction of Polynomials

To subtract a polynomial, change the sign of each term in the polynomial being subtracted and then combine the like terms.

Algebraic Sum

The sum is found by grouping the coefficients of identical variables: a×x + b×x = (a + b)×x.

Shortcut Trick

Use the Zig-zag multiplication method for combining ratios.

Arrange the terms: a : b = 3 : 4 and b : c = 2 : 3.

Multiply vertically for a (3 × 2 = 6), diagonally for b (4 × 2 = 8), and vertically for c (4 × 3 = 12).

The resulting ratio is 6 : 8 : 12.

Simplify by dividing all terms by 2 ⇒ 3 : 4 : 6.

∴ The correct answer is 3 : 4 : 6.

Alternate Method

Given:

a : b = 3 : 4

b : c = 2 : 3

Formula Used:

To combine two ratios, the common variable (b) must have the same value in both expressions.

Calculations:

⇒ In ratio a : b, value of b = 4.

⇒ In ratio b : c, value of b = 2.

⇒ To make b equal, multiply the second ratio (b : c) by 2.

⇒ b : c = (2 × 2) : (3 × 2) = 4 : 6.

⇒ Now b = 4 is common to both ratios.

⇒ a : b : c = 3 : 4 : 6.

∴ The correct answer is 3 : 4 : 6.

Additional Information

Compound Ratio

The compound ratio of two ratios a : b and c : d is calculated as (a × c) : (b × d).

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Mean Proportional

If a, b, and c are in continued proportion such that a : b = b : c, then b is the mean proportional where b² = ac.

The correct answer is 10 January 2024 to 18 January 2024

Key Points

• According to available information, the Department for Promotion of Industry and Internal Trade (DPIIT) organised the Startup India Innovation Week 2024 from January 10^th to 18^th, 2024.
• During the Innovation Week 2024, Secretary, DPIIT, Shri Rajesh Kumar Singh delivered the inaugural address in the Startup Seminar: ‘
• The event aimed to promote innovation and entrepreneurship in India by fostering a supportive ecosystem for startups.
• The Theme was "Innovation for Transformation."

Additional Information About the Department for promotion of Industry & Internal Trade:

• ​It is a central government department under the Ministry of Commerce and Industry.
• It was established in the year 1995.
• The department was earlier called the Department of Industrial Policy & Promotion and was renamed as DPIIT in January 2019.
• The role of DPIIT is to promote/accelerate industrial development of the country by facilitating investment in new and upcoming technology, foreign direct investment, and support the balanced development of industries.

The logic followed here is: 

Option 1) XR - CW

Option 2) ZT - PG

Option 3) TN - YS

Option 4) BV - GA

Hence, the correct answer is "Option 2".

Shortcut Trick

The age difference between two people always remains constant (30 years).

Ratio 5 years ago (Anita : Sunita) = 3 : 1

Difference in ratio parts = 3 − 1 = 2 units

2 units → 30 years ⇒ 1 unit = 15 years

Sunita's age 5 years ago = 1 unit = 15 years

Present age of Sunita = 15 + 5 = 20 years

∴ The correct answer is 20 years.

Alternate Method

Given: Age difference = 30 years; 5 years ago, Anita was 3 times as old as Sunita.

Let the present age of Sunita be S years and Anita be A years.

⇒ A − S = 30

⇒ A = S + 30

5 years ago:

⇒ (A − 5) = 3 × (S − 5)

⇒ (S + 30 − 5) = 3S − 15

⇒ S + 25 = 3S − 15

⇒ 3S − S = 25 + 15

⇒ 2S = 40

⇒ S = 20

∴ The correct answer is 20 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same at any point in time, whether in the past, present, or future.

Age Ratio Property

While the absolute difference is constant, the ratio of ages changes every year as both individuals grow older by the same number of years.

General Age Formulation

If the current age is x, then the age n years ago is (x − n) and the age n years hence is (x + n).

The correct answer is 63,000

Key Points

• As per the Union Budget 2024-25, the Pradhan Mantri Janjatiya Unnat Gram Abhiyan aims to improve the socio-economic condition of tribal communities and will cover around 63**,000 villages**.
• Pradhan Mantri Janjatiya Unnat Gram Abhiyan (PMJUGA) is an initiative launched by the Government of India aimed at bringing socio-economic development to tribal communities residing in rural areas.
• The scheme will adopt saturation coverage for tribal families in tribal-majority villages and aspirational districts.
• The Ministry of Tribal Affairs will act as the nodal ministry for the overall monitoring and implementation of the scheme.
• It has various objectives, features which can have a far reaching impact towards the goal of tribal empowerment.

Additional Information

• Objectives of Pradhan Mantri Janjatiya Unnat Gram Abhiyan (PMJUGA)
• Integrated Development: To ensure holistic development of tribal villages by addressing infrastructure gaps and improving basic amenities.
• Livelihood Opportunities: Enhancing livelihood opportunities for tribal communities through skill development, employment generation, and promoting traditional livelihoods.
• Health and Education: Improving access to healthcare facilities and promoting education among tribal populations.
• Cultural Preservation: Preservation and promotion of tribal culture, traditions, and heritage

Given: 'M - N + P ÷ S x T

'M - N + P ÷ S x T - It means that M is brother of N, N is Mother of P, P is father of S, S is the wife of T.

Thus, M is Wife's father's mother's brother of T. 

Hence, the correct answer is "Option 4".

The correct answer is tripled.

Key Points

• The resistance (R) of a conductor is given by the formula: R = ρ(L/A), where:
• ρ is the resistivity of the material (a constant for a given material).
• L is the length of the conductor.
• A is the cross-sectional area of the conductor.
• From the formula, resistance is directly proportional to the length of the conductor. This means that if the length of the conductor increases, its resistance also increases proportionally.
• In this case, when the length (L) is tripled and other factors such as material and cross-sectional area remain constant, the resistance will also be tripled.
• This relationship can be written mathematically as: R₂/R₁ = L₂/L₁, where:
• R₂ and R₁ are the final and initial resistances, respectively.
• L₂ and L₁ are the final and initial lengths, respectively.
• Here, if L₂ = 3L₁, then R₂ = 3R₁, meaning the resistance is tripled.
• This principle is based on the Ohm’s Law and the properties of materials when electricity flows through them.

Additional Information

• Key properties of resistance:
• Resistance depends on the material of the conductor; materials like copper and silver have low resistance compared to other materials.
• It is also affected by temperature. For most conductors, resistance increases with an increase in temperature.
• The cross-sectional area is inversely proportional to resistance; a thicker conductor has lower resistance.
• Resistivity and its significance:
• Resistivity (ρ) is a property of the material and determines how much the material opposes the flow of electric current.
• It is measured in ohm-meters (Ω·m).
• Applications of resistance principles:
• Used in designing electrical circuits and ensuring safety by selecting appropriate materials.
• Helps in manufacturing resistors for various applications in electronic devices.

Given:

Venu starts from Point Y and drives 7 km towards south.

He then takes a left turn, drives 15 km, turns right and drives 38 km.

He then takes a left turn and drives 13 km. He takes a left turn, drives 31 km.

He then turns left, drives 28 km, turns left and drives 13 km to stop at Point Z.

Thus, he drive 27 km towards north in order to reach Point Y again.

Hence, the correct answer is "Option 4".

Statements:

Some ribbons are balloons.

Some balloons are eagles.

The least possible venn diagram is given: 

Conclusions:

(I) All eagles are ribbons - False(because, the is no direct relation between ribbon and eagle is given, so it can be possible only). 

(II) No ribbons are eagles - False(because, the is no direct relation between ribbon and eagle is given, so it can be possible only). 

Thus, Neither conclusion (I) nor (II) follows

Hence, the correct answer is "Option 1".

Given:

Number of cars manufactured in 2012 = 45 (in thousand)

Number of cars manufactured in 2013 = 40 (in thousand)

Formula used:

Total cars manufactured = Cars in 2012 + Cars in 2013

Calculation:

Total cars manufactured = 45 + 40

⇒ Total cars manufactured = 85 (in thousand)

∴ The correct answer is 85 (in thousand).

Statements:

Some trees are flowers.

All flowers are leaves.

The least possible venn diagram is given below: 

Conclusions:

(I) All leaves are trees - False(because, Some flower are trees and all flower is leaves it implies that some leaves are trees, so it can be possible only).

(II) All flowers are trees -  False(because, Some flower are trees, so it can be possible only).

Thus, Neither conclusion (I) nor (II) follows.

Hence, the correct answer is "Option 1".

The correct answer is 10⁻¹⁰ m.

Key Points

• The atomic radius is a measure of the size of an atom, typically the mean or typical distance from the nucleus to the boundary of the surrounding cloud of electrons.
• Hydrogen is the lightest and most abundant chemical element, consisting of only one proton and one electron.
• The atomic radius of a hydrogen atom is approximately 10⁻¹⁰ meters.
• This value is also known as 1 angstrom (Å), a common unit of length used in the field of atomic-scale measurements.
• Understanding the atomic radius is crucial for studying atomic structures, chemical bonding, and molecular interactions.

Additional Information

• Atomic Radii Trends
• Atomic radii generally decrease across a period from left to right due to increasing nuclear charge.
• Atomic radii increase down a group as additional electron shells are added.
• Ionic Radius
• Ionic radius is the radius of an atom's ion.
• Cations are smaller than their parent atoms due to the loss of electron(s) and a resultant increase in nuclear charge.
• Anions are larger than their parent atoms due to the gain of electron(s) leading to increased electron-electron repulsion.
• Van der Waals Radius
• This is a measure of the size of an atom or molecule that includes its electron cloud.
• It is larger than the covalent radius because it accounts for the space occupied by the electron cloud during non-bonded interactions.

The correct answer is Decomposition.

Key Points

• Limestone (CaCO₃) decomposes into quicklime (CaO) and carbon dioxide (CO₂) when heated.
• This type of reaction is known as a thermal decomposition reaction.
• In a decomposition reaction, a single compound breaks down into two or more simpler substances.
• Thermal decomposition usually requires the input of heat energy.
• Quicklime (CaO) is widely used in the construction industry and for making cement.

Additional Information

• Combination Reaction
• Involves two or more substances combining to form a single compound.
• Example: 2H₂ + O₂ → 2H₂O (Formation of water from hydrogen and oxygen).
• Displacement Reaction
• Involves an element displacing another element from a compound.
• Example: Zn + CuSO₄ → ZnSO₄ + Cu (Zinc displaces copper from copper sulfate).
• Double Displacement Reaction
• Involves the exchange of ions between two compounds to form new compounds.
• Example: Na₂SO₄ + BaCl₂ → 2NaCl + BaSO₄ (Sodium sulfate reacts with barium chloride).

Shortcut Trick

List the first 8 prime numbers: 2, 3, 5, 7, 11, 13, 17, and 19.

Calculate the sum: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77.

Perform the division: 77 ÷ 7 = 11.

∴ The correct answer is 11.

Alternate Method

Given: First 8 prime numbers and a divisor of 7.

Formula Used: Result = (Sum of first 8 primes) ÷ 7

⇒ First 8 prime numbers = 2, 3, 5, 7, 11, 13, 17, 19

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19

⇒ Sum = 77

⇒ Required Value = 77 ÷ 7

⇒ Result = 11

∴ The correct answer is 11.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest prime number is 2, which is also the only even prime.

Sum of Primes

There is no general formula to find the sum of the first 'n' prime numbers; they must be listed and summed manually. For the first 10 primes, the sum is 129.

Twin Primes

Twin primes are pairs of prime numbers that differ by exactly 2, such as (3, 5), (5, 7), and (11, 13).

The logic followed here is:

Logic: (1^st number - 5) ÷ 5 = 2^nd number,

For, 430 : Y.

Y = (430 - 5) ÷ 5

Y = 425 ÷ 5 = 85

Y = 85

Similarly, X : 58

X = 58 × 5 + 5

X = 290 + 5 

X = 295.

Thus, X = 295, Y = 85.

Hence, the correct answer is "Option 2".

The correct answer is 82.64 GW.

Key Points

• India's cumulative installed solar power capacity reached 82.64 GW as of 30 April 2024.
• In the financial year 2023-24, India added 15.03 GW of solar power capacity.
• This substantial increase signifies India's continuous progress in renewable energy adoption, particularly solar energy.
• The solar energy sector in India has been rapidly expanding, contributing significantly to the nation's renewable energy targets.
• These figures are in alignment with the goals set by the Indian government to enhance solar power infrastructure and capacity.

Additional Information

• National Solar Mission
• Launched in 2010, it aims to promote the development and use of solar energy for power generation and other uses.
• The mission targets installing 100 GW of grid-connected solar power projects by 2022.
• It includes both solar photovoltaic (PV) and solar thermal technologies.
• The mission is a major part of India's commitment to combat climate change and promote sustainable energy.
• Renewable Energy Targets
• India aims to achieve 175 GW of renewable energy capacity by 2022, which includes 100 GW from solar, 60 GW from wind, 10 GW from biomass, and 5 GW from small hydro.
• The country is working towards an even more ambitious target of 450 GW of renewable energy capacity by 2030.
• These targets are part of India's commitment under the Paris Agreement to reduce carbon emissions.
• Solar Power Advantages
• Solar power is a clean and renewable source of energy, contributing to the reduction of greenhouse gas emissions.
• It helps in reducing dependence on fossil fuels and enhances energy security.
• Solar energy systems have low operating costs and can be installed on various scales, from small rooftop systems to large solar farms.

Shortcut Trick

Total distance covered in the two-way journey = 5 + 5 = 10 km.

Total time taken = 1 hr + (1 hr + 15 min) = 2 hr 15 min = 2.25 hours or 9/4 hours.

Average Speed = Total Distance ÷ Total Time = 10 ÷ (9/4) = 40/9 km/h.

40/9 km/h = 4 4/9 km/h.

∴ The correct answer is 4 4/9 km/h.

Alternate Method

Given:

Distance from home to office = 5 km

Morning time (t₁) = 1 hour

Evening time (t₂) = 1 hour + 15 minutes = 1.25 hours

Formula Used:

Average Speed = Total Distance / Total Time

Calculations:

⇒ Total Distance = 5 km (going) + 5 km (returning) = 10 km

⇒ Total Time = 1 hr + (1 + 15/60) hr

⇒ Total Time = 1 + 1.25 = 2.25 hours

⇒ Converting 2.25 to a fraction: 225/100 = 9/4 hours

⇒ Average Speed = 10 ÷ (9/4)

⇒ Average Speed = (10 × 4) / 9 = 40/9 km/h

⇒ 40/9 = 4 4/9 km/h

∴ The correct answer is 4 4/9 km/h.

Additional Information

Average Speed (Equal Distances)

If an object covers two equal distances with speeds x and y, the average speed is given by (2 × x × y) / (x + y).

Average Speed (Equal Time Intervals)

If an object travels at speed x for time t and speed y for the same time t, the average speed is (x + y) / 2.

Unit Conversion

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

The logic followed here is:

‘MEANS’ is coded as ‘75821’

‘RANGE’ is coded as ‘84592’

‘PRIME’ is coded as ‘64237’

Thus,  code for 'G' is '9'.

Hence, "Option 3" is the correct answer.

The correct answer is maximum at the centre.

Key Points

• The magnetic field strength in a current-carrying circular loop is maximum at the center of the loop.
• At the center of the loop, the magnetic field lines are most concentrated, leading to the highest field strength.
• As you move towards the periphery of the loop, the magnetic field strength decreases.
• The magnetic field strength at the center of a circular loop is given by the formula B = (μ₀I)/(2R), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and R is the radius of the loop.
• This concept is based on the Biot-Savart law which describes the magnetic field generated by a steady current.

Additional Information

• Biot-Savart Law
• This law helps in calculating the magnetic field generated by a steady current.
• It states that the magnetic field at a point is directly proportional to the current and inversely proportional to the distance from the current element.
• Right Hand Rule
• This rule is used to determine the direction of the magnetic field around a current-carrying conductor.
• According to this rule, if you point the thumb of your right hand in the direction of the current, the direction in which your fingers curl gives the direction of the magnetic field.
• Magnetic Field Lines
• These are imaginary lines used to represent the magnetic field.
• The density of these lines indicates the strength of the magnetic field.
• Closer lines represent stronger magnetic fields, while lines further apart indicate weaker fields.

The logic followed here is:

Hence, "Option 4" is the correct answer.

The correct answer is Manu Bhaker and PR Sreejesh

Key Points

• Manu Bhaker and PR Sreejesh were the co-flag bearers for India at the closing ceremony of the 2024 Paris Olympics.
• Manu Bhaker
• A shooter who won two bronze medals at the 2024 Paris Olympics.
• She was the first Indian woman to win a medal in shooting at the Olympics.
• PR Sreejesh
• A goalkeeper for the Indian men's hockey team who won a bronze medal at the 2024 Paris Olympics.
• He is also a hockey legend.

Additional Information

• The host city of the 2024 Summer Olympics is Paris, France.
• The Motto is Made for Sharing.
• Skateboarding, sport climbing, surfing, and breaking have been confirmed as the additional sports proposed for inclusion at the Paris Olympic Games in 2024.
• Los Angeles, California, United States will host the 2028 Summer Olympics.
• The Tokyo Summer Olympic Games will be held from July 23 to August 8, 2021.
• They were originally set to take place from July 24 to August 9, 2020, but were postponed because of the COVID-19 pandemic

The logic followed here is:

Option 1) KM-PJ

Option 2) OQ-TN

Option 3) IK-NH

Option 4) GI-LE

Thus, out of all the options 'GI-LE is the odd one out.

Hence, "Option 4" is the correct answer.

Shortcut Trick

Assume Total Work = LCM(20, 15) = 60 units.

Efficiency of A = 3 units/day and Efficiency of B = 4 units/day.

Work done by B in 6 days = 6 × 4 = 24 units.

Remaining work = 60 − 24 = 36 units.

Time taken by A = 36 ÷ 3 = 12 days.

∴ The correct answer is 12 days.

Alternate Method

Given:

Time taken by A to complete work = 20 days

Time taken by B to complete work = 15 days

B worked alone for = 6 days

Formula Used:

Total Work = Efficiency × Time

Remaining Work = Total Work − Completed Work

Calculations:

⇒ Total Work (LCM of 20 and 15) = 60 units

⇒ Work done by B in 6 days = 6 × 4 = 24 units

⇒ Remaining work to be done by A = 60 − 24 = 36 units

⇒ Time taken by A alone = Remaining Work ÷ Efficiency of A

⇒ Time taken by A = 36 ÷ 3 = 12 days

∴ The correct answer is 12 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of time is x:y, the ratio of efficiency is y:x.

Work and Wages

Wages are always distributed among workers in the ratio of the actual work done by them or their efficiencies (if they work for the same duration).

Fractional Method

Work done by B in 1 day = 1/15. In 6 days, work done = 6/15 = 2/5. Remaining work = 1 − 2/5 = 3/5. Time for A = (3/5) × 20 = 12 days.

Given terms: (Left) 1 $ 7 & 3 4 6 @ 8 Ω % 9 # * £ 2 & 5 (Right)

Required pattern: Symbol - Numbers - Symbol 

​(Left) 1 $ 7 & 3 4 6 @ 8 Ω % 9 # * £ 2 & 5 (Right)

Thus, the only four  pairs follow  the pattern.

Hence, the correct answer is "Option 4".

Given: A, B, C, D, E and F are sitting around a circular table facing the centre.

D sits to the immediate left of F. B is the immediate neighbour of C and F.

A is the immediate neighbour of E and C.

Thus,  the position of D with respect to E is Immediate right of E.

Hence, the correct answer is "Option 4".

Given terms: (Left) * 9 # 1 £ 2 4 & 3 Ω 6 @ 8 $ 7 5 & % (Right)

Required pattern: Symbol - Symbol - Number

 (Left) * 9 # 1 £ 2 4 & 3 Ω 6 @ 8 $ 7 5 & % (Right)

Thus, there is zero pair of given pattern.

Hence, the correct answer is "Option 4".

The correct answer is Amoeba.

Key Points

• Amoeba is a single-celled organism that uses a food vacuole for digestion.
• The food vacuole in Amoeba engulfs food particles through a process called phagocytosis.
• Inside the food vacuole, complex substances are broken down into simpler ones by digestive enzymes.
• The simpler substances then diffuse into the cytoplasm to be used by the cell.
• This process ensures that Amoeba can efficiently obtain nutrients from its environment.

Additional Information

• Plasmodium
• Plasmodium is a parasitic protozoan, known for causing malaria in humans.
• It has a complex life cycle involving both humans and mosquito hosts.
• Spirogyra
• Spirogyra is a filamentous green alga, known for its spiral chloroplasts.
• It reproduces both sexually and asexually through fragmentation and conjugation.
• Lichen
• Lichen is a symbiotic association between a fungus and a photosynthetic partner, usually algae or cyanobacteria.
• They are known for their ability to colonize harsh environments, including bare rocks.

The logic followed here is:

‘PING’ is coded as ‘7953’

‘NOSE’ is coded as ‘4638’

Thus,  code for 'N' is '3'.

Hence, "Option 3" is the correct answer.

The correct answer is Unicellular organisms.

Key Points

• Unicellular organisms are single-celled organisms that perform all necessary functions within one cell.
• Cell division in unicellular organisms, such as bacteria and protozoa, is typically asexual and occurs through processes like binary fission or budding.
• During binary fission, the cell duplicates its genetic material and divides into two equal-sized daughter cells.
• This type of reproduction is efficient and rapid, allowing for quick population growth under favorable conditions.
• Unicellular organisms do not require a mate for reproduction, which simplifies the process and ensures the generation of new individuals.

Additional Information

• Multicellular Organisms
• In multicellular organisms, cell division is used for growth, development, and repair, rather than creating new individuals.
• These organisms reproduce through sexual reproduction, involving the fusion of gametes (sperm and egg cells).
• Mitosis and Meiosis
• Mitosis is a type of cell division resulting in two daughter cells with identical genetic material, used for growth and repair in multicellular organisms.
• Meiosis is a specialized form of cell division that produces gametes with half the genetic material, essential for sexual reproduction.
• Cell Cycle
• The cell cycle includes stages of cell growth, DNA replication, and cell division, ensuring the proper distribution of genetic material to daughter cells.
• Regulation of the cell cycle is crucial for preventing uncontrolled cell division, which can lead to diseases like cancer.

The correct answer is Sodium hydrogen carbonate.

Key Points

• Sodium hydrogen carbonate, also known as baking soda, is commonly used as an ingredient in antacids.
• It helps to neutralize stomach acid, providing relief from heartburn and indigestion.
• Antacids containing sodium hydrogen carbonate are effective in treating acid reflux and peptic ulcers.
• Sodium hydrogen carbonate reacts with hydrochloric acid in the stomach to produce carbon dioxide, water, and salt, reducing acidity.
• It is preferred in antacid formulations because it works quickly to relieve symptoms of hyperacidity.

Additional Information

• Hydrochloric acid
• It is a strong acid produced in the stomach to aid in digestion.
• Excess production of hydrochloric acid can lead to conditions like acid reflux and ulcers.
• Calcium carbonate
• It is another compound often used in antacids to neutralize stomach acid.
• Calcium carbonate is effective in providing longer-lasting relief compared to sodium hydrogen carbonate.
• Magnesium hydroxide
• Commonly known as milk of magnesia, it is used in antacids to neutralize stomach acid and relieve constipation.
• Magnesium hydroxide works by drawing water into the intestines, making it easier to pass stool.

Given:

$\dfrac{0.009}{K} = 0.01$

Formula used:

K = $\dfrac{0.009}{0.01}$

Calculation:

$\dfrac{0.009}{K} = 0.01$

⇒ K = $\dfrac{0.009}{0.01}$

⇒ K = 0.9

∴ The correct answer is option (3).

The correct answer is Neuron.

Key Points

• Neurons are the fundamental units of the brain and nervous system, responsible for receiving sensory input from the external world.
• They conduct information via electrical impulses, known as action potentials, throughout the body.
• Neurons connect with each other to form networks that perform various functions such as motor coordination, sensory perception, and cognitive processes.
• Each neuron consists of a cell body, dendrites, and an axon, which help in the transmission of electrical signals.
• The synapse is the junction between two neurons where information is transmitted from one neuron to another through neurotransmitters.

Additional Information

• Nucleus
• The nucleus is the membrane-bound organelle that contains the cell's chromosomes.
• It acts as the control center of the cell, regulating gene expression and mediating the replication of DNA during the cell cycle.
• Dendrite
• Dendrites are the branched projections of a neuron that receive electrochemical signals from other neural cells.
• They conduct the electrical impulses toward the cell body of the neuron.
• Axon
• The axon is a long, slender projection of a neuron that typically conducts electrical impulses away from the cell body.It is responsible for transmitting information to different neurons, muscles, and glands.