Shortcut Trick

Base of triangle = 4 × (Diagonal of square / √2) = 4 × 7 = 28 cm.

Height of triangle = √(Area of square) = √169 = 13 cm.

Area of triangle = 1/2 × Base × Height = 1/2 × 28 × 13 = 182 sq cm.

∴ The correct answer is 182 sq cm.

Alternate Method

Given:

Diagonal of Square₁ = 7√2 cm

Area of Square₂ = 169 sq cm

Formula Used:

Side of square = Diagonal / √2

Perimeter of square = 4 × Side

Area of square = (Side)²

Area of triangle = (1/2) × base × height

⇒ Side of Square₁ = 7√2 / √2 = 7 cm

⇒ Perimeter of Square₁ = 4 × 7 = 28 cm

⇒ Base (b) of triangle = 28 cm

⇒ Side of Square₂ = √169 = 13 cm

⇒ Height (h) of triangle = 13 cm

⇒ Area of triangle = (1/2) × 28 × 13

⇒ Area = 14 × 13 = 182 sq cm

∴ The correct answer is 182 sq cm.

Additional Information

Square Properties

Relationship between side (a) and diagonal (d): d = a√2. Area = a² = d²/2.

Area of Triangle (Heron's Formula)

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

Equilateral Triangle

Area of an equilateral triangle with side 'a' is (√3 / 4) × a² and height is (√3 / 2) × a.

The logic followed here is:

Hence, "Option 3" is the correct answer.

The correct answer is June

Key Points

• The Public Examinations (Prevention of Unfair Means) Act, 2024, aimed at preventing unfair practices in public examinations and common entrance tests across India, came into effect on June 21, 2024.
• The Act was introduced in the Lok Sabha on February 5, 2024, passed by both houses of Parliament by February 9, 2024, and received presidential assent on February 12, 2024.
• The Act comprehensively defines "unfair means" to include:
• Leaking or attempting to leak question papers or answer keys.
• Assisting candidates during examinations through unauthorized communication or providing solutions.
• Tampering with computer systems or networks related to the examination process.
• Impersonating or facilitating impersonation of candidates.
• Conducting fake examinations or issuing counterfeit documents.
• Altering official records to manipulate merit lists or rankings.
• Penalties:​
• Imprisonment ranging from three to ten years, depending on the severity of the offence.
• Fines up to ₹1 crore for organised malpractices.

Additional Information 

• Issues Related to the Public Examination Bill 2024
• Concerns include the implementation challenges of new technologies, potential state-level resistance to a uniform framework, and ensuring the central regulatory authority functions without bias. Additionally, there are worries about the Bill's impact on the already overburdened judicial system with cases of examination malpractices.
• Discretion to State Governments: Since the said bill is of model draft nature, disparities in its implementation across states are possible. This situation can make its implementation on the ground difficult.
• Creative Loopholes in Sanctions: Provisions like fines on service providers appear to be loose, looking like an eyewash and can easily be dodged by the defaulters.
• Clarity of National Technical Committee: There is no defined guidance on what will constitute the national technical committee and what will be its mandate. This has the potential of creating a committee which might not be of much value in framing security protocols.
• Stringency That Is Legally Challenged: The stringency of measures, including non-bailability and non-compoundability, may run into a legal challenge on the basis of the principles of proportionality and adherence to the principles of natural justice.

Given word: 'ENGLISH'

After applying the specified transformations:

 

Second alphabet from the left end is: M,

Hence, the correct answer is "Option 2".

The correct answer is Yellow.

Key Points

• When white light passes through a prism, it is dispersed into its constituent colours, a phenomenon known as dispersion.
• The seven colours produced are red, orange, yellow, green, blue, indigo, and violet, each having a different wavelength.
• The deviation of light depends on the wavelength; shorter wavelengths (violet, indigo) are deviated more than longer wavelengths (red).
• Among the given options, yellow light has a longer wavelength compared to indigo, blue, and green.
• As a result, yellow light will experience the least deviation out of the provided options when dispersed by a prism.

Additional Information

• Dispersion of Light
• Dispersion occurs because different colours of light travel at different speeds in a medium, leading to different angles of refraction.
• Refractive Index
• The refractive index of a material varies with the wavelength of light, causing the separation of colours in a prism.
• Application in Optics
• Dispersion is utilized in various optical instruments and technologies, such as spectrometers and prisms used in science and industry.

The correct answer is 74.4

Key Points

• According to the World Bank's Women, Business, and the Law (WBL) 2024 report, India scored 74.4 out of 100.0.
• This score indicates that India has a strong legal foundation for women.
• The WBL report is a yearly publication by the World Bank Group that measures the legal rights of women in different countries.
• The WBL 2024 report ranked India 113^th out of 190 countries.​

Additional InformationIndicators Covered:

• Mobility: Measures women’s freedom to travel and move independently.
• Workplace: Evaluates laws prohibiting discrimination in employment.
• Pay: Assesses whether laws mandate equal pay for equal work.
• Marriage: Looks at women’s rights in marriage, including domestic violence laws.
• Parenthood: Reviews support for working mothers, including maternity and parental leave.
• Entrepreneurship: Measures the ease of starting and running businesses as women.
• Assets: Reviews women’s legal rights to own and manage property.
• Pension: Analyzes laws ensuring women have access to retirement benefits.

The correct answer is Phone Pe

Key Points

• In October 2024, ICICI Bank partnered with PhonePe to offer instant credit on UPI for pre-approved customers.
• The credit line is available for up to ₹2 lakh with a repayment period of 45 days
• This facility can be used for purchasing high-ticket items like electronics, travel, and hotel bookings
• It can also be used for bill payments

Additional Information

• ICICI
• ICICI Bank stands for Industrial Credit and Investment Corporation of India.
• ICICI was formed in 1955.
• The objective was to create a development financial institution for providing medium-term and long-term project financing to Indian businesses.
• Headquarters: Mumbai.
• PhonePe
• ​PhonePe is an Indian digital payments and financial services company headquartered in Bengaluru, Karnataka, India
• It was founded in December 2015 by Sameer Nigam, Rahul Chari and Burzin Engineer

The correct answer is Cells.

Key Points

• Cells are the basic structural and functional units of all living organisms.
• They can exist as independent single-celled organisms or as part of a multicellular organism.
• Cells carry out essential processes such as metabolism, energy conversion, and reproduction.
• There are two main types of cells: prokaryotic cells, which do not have a defined nucleus, and eukaryotic cells, which have a defined nucleus and organelles.
• Cells are composed of various organelles that perform specific functions necessary for the cell's survival and proper functioning.

Additional Information

• Tissues
• Tissues are groups of similar cells that work together to perform a specific function.
• There are four basic types of tissues in animals: epithelial, connective, muscle, and nervous tissues.
• Organs
• Organs are structures made up of different types of tissues that work together to perform specific functions within an organism.
• Examples include the heart, lungs, and brain in animals, and roots, stems, and leaves in plants.
• Organ Systems
• Organ systems are groups of organs that work together to perform complex functions necessary for the survival of the organism.
• Examples of organ systems in animals include the circulatory system, respiratory system, and digestive system.

The correct answer is Three days

Key Points

• The Ashtalakshmi Mahotsav 2024 was a three-day cultural festival held from December 6^th to 8^th, 2024, at Bharat Mandapam, Pragati Maidan, New Delhi.
• This inaugural event celebrated the rich cultural heritage and economic potential of Northeast India's eight states—Arunachal Pradesh, Assam, Manipur, Meghalaya, Mizoram, Nagaland, Sikkim, and Tripura—collectively known as the 'Ashtalakshmi'.
• Over 250 artisans and entrepreneurs showcased handicrafts, handlooms, and agri-horticultural products, with a special focus on 34 Geographical Indication (GI)-tagged items
• The fusion of contemporary design with traditional textiles, showcasing Muga silk gowns and Eri silk stoles.

Additional Information

• The first ever Ashtalakshmi Mahotsav aims to offer a dynamic stage for showcasing Northeast India’s vibrant textile sector, tourism opportunities, traditional craftsmanship, and distinctive Geographical Indication (GI) tagged products.
• It will serve as a celebration of the region’s cultural wealth while focusing on its strategic importance in India’s future development.
• The festival has been conceptualized as an annual event that will continue to honor the diversity and cultural richness of Northeast India, propelling the region towards further economic advancement.
• The Mahotsav will also emphasize the significance of infrastructure development in Northeast India and its transformative impact.
• The region’s improved connectivity, industrial growth, and ongoing development in sectors like transport, energy, and digital infrastructure are poised to enhance its strategic role in India's growth story and open new opportunities for trade and business

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

1. Only two people sit to the left of F.

2. F sits second to the right of G.

3. E sits second to the right of F.

4. C sits fourth to the right of B.

5. D sits second to the left of C.

6. A sits an immediate right of C.

Thus, according to the final arrangement, no person sit between B and G.

Hence, "Option 4" is the correct answer.

The correct answer is an acceleration.

Key Points

• When an object falls under gravity, it experiences an acceleration due to the gravitational force exerted by the Earth.
• This acceleration is called gravitational acceleration and is denoted by 'g', which is approximately 9.8 m/s² near the Earth's surface.
• Gravitational acceleration acts on all objects equally, regardless of their masses.
• The direction of the acceleration is towards the center of the Earth.
• During free fall, the only force acting on the object is gravity, making it an ideal example of uniformly accelerated motion.

Additional Information

• Gravitational Force
• Gravitational force is the attractive force between two masses.
• It is one of the four fundamental forces of nature.
• The strength of the gravitational force between two objects depends on their masses and the distance between them.
• Free Fall
• Free fall is the motion of a body where gravity is the only force acting upon it.
• In a vacuum, all objects fall at the same rate regardless of their mass.
• Terminal Velocity
• Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration.
• At terminal velocity, the force of gravity is balanced by the drag force of the medium.

The correct answer is Copper.

Key Points

• When an iron nail is dipped into a copper sulphate solution, a displacement reaction occurs.
• In this reaction, iron (Fe) displaces copper (Cu) from copper sulphate (CuSO₄) solution because iron is more reactive than copper.
• The chemical equation for this reaction is: Fe + CuSO₄ → FeSO₄ + Cu.
• As a result, iron forms iron sulphate (FeSO₄) and copper is displaced from the solution.
• This type of reaction is an example of a single displacement reaction, where a more reactive metal displaces a less reactive metal from its compound.

Additional Information

• Reactivity Series
• The reactivity series is a list of metals arranged in order of their reactivity, from highest to lowest.
• Metals higher in the series can displace metals lower in the series from their compounds in solution.
• Displacement Reactions
• These reactions involve a more reactive element displacing a less reactive element from its compound.
• Displacement reactions are commonly used in metal extraction and refining processes.
• Examples of Displacement Reactions
• Zinc displacing copper from copper sulphate solution: Zn + CuSO₄ → ZnSO₄ + Cu.
• Magnesium displacing hydrogen from hydrochloric acid: Mg + 2HCl → MgCl₂ + H₂.

The correct answer is A straight line.

Key Points

• In uniform motion, an object travels equal distances in equal intervals of time.
• The velocity of the object remains constant throughout the motion.
• The path of an object in uniform motion is a straight line because the direction does not change.
• Uniform motion implies that there is no acceleration involved in the movement of the object.
• This type of motion is an ideal concept often used in physics to simplify complex motion scenarios.

Additional Information

• Non-uniform Motion
• In non-uniform motion, the object travels unequal distances in equal intervals of time.
• This type of motion involves changes in speed and/or direction, leading to acceleration.
• Accelerated Motion
• In accelerated motion, the velocity of the object changes over time.
• This can involve an increase or decrease in speed and can also include changes in direction.
• Circular Motion
• In circular motion, an object moves along the circumference of a circle.
• This type of motion involves a constant change in direction, leading to centripetal acceleration.

The correct answer is Virtual, erect and enlarged.

Key Points

• When an object is placed between the focus (F) and the pole (P) of a concave mirror, the image formed is virtual.
• The image is erect, meaning it is upright and not inverted.
• The image is enlarged, appearing larger than the object.
• This type of image is formed behind the mirror.
• The rays appear to diverge from a point behind the mirror, leading to the virtual image formation.

Additional Information

• Real Image
• A real image is formed when light rays actually converge at a point.
• It is always inverted and can be projected onto a screen.
• Concave Mirror
• A concave mirror is a mirror with a surface that curves inward like a bowl.It can form real and virtual images depending on the object's position relative to the focus and center of curvature.
• Convex Mirror
• A convex mirror has a surface that curves outward.
• It always forms virtual, erect, and diminished images regardless of the object's position.

The logic followed here is:

​Given:

WGN : ZDQ

And,

IUZ : LRC

So, checking each option one by one:

Option 1) PXH: LCD

Option 2) CAT: FXW

Option 3) URL: STI

Option 4) DMV: HHZ

Thus, out of all the options, 'CAT: FXW' follows the same logic as given.

Hence, "Option 2" is the correct answer.

Shortcut Trick

Add the two equations: (3 + 5) Pencils + (5 + 3) Pens = ₹101 + ₹91

8 Pencils + 8 Pens = ₹192

To find the cost of 2 pencils and 2 pens, divide the total by 4:

(8 Pencils + 8 Pens) ÷ 4 = ₹192 ÷ 4 = ₹48

∴ The correct answer is ₹48.

Alternate Method

Given:

Cost of 3 pencils and 5 pens = ₹101

Cost of 5 pencils and 3 pens = ₹91

Let the cost of one pencil be x and one pen be y.

⇒ 3x + 5y = 101 --- (Equation 1)

⇒ 5x + 3y = 91 --- (Equation 2)

Adding both equations:

⇒ (3x + 5x) + (5y + 3y) = 101 + 91

⇒ 8x + 8y = 192

⇒ 8(x + y) = 192

⇒ x + y = 192 ÷ 8 = 24

Now, to find the cost of 2 pencils and 2 pens:

⇒ 2(x + y) = 2 × 24

⇒ 2x + 2y = 48

∴ The correct answer is ₹48.

Additional Information

Symmetric Linear Equations

Equations of the form ax + by = m and bx + ay = n are called symmetric equations. They are best solved by adding and subtracting them to find (x + y) and (x − y) values.

Unitary Cost Calculation

Total Cost ÷ Total Quantity gives the average cost per unit, which helps in finding costs for multiple units of a combined set.

Elimination Method

A general method where you multiply equations by constants to make the coefficients of one variable equal, then subtract to eliminate it.

Shortcut Trick

Using the Alligation Method:

Ratio of Male to Female = 45 : 55 = 9 : 11

Let Female literacy be x%.

(80 − 58) ÷ (58 − x) = 11 ÷ 9

22 ÷ (58 − x) = 11 ÷ 9 ⇒ 2 × 9 = 58 − x

18 = 58 − x ⇒ x = 40%

∴ The correct answer is 40%.

Alternate Method

Given:

Total Population = 100 (Assumption)

Females = 55% of 100 = 55

Males = 100 − 55 = 45

Total Literacy Rate = 58% of 100 = 58

Male Literacy Rate = 80% of 45

Calculations:

⇒ Total Literate people = 100 × (58 ÷ 100) = 58

⇒ Literate Males = 45 × (80 ÷ 100) = 36

⇒ Literate Females = Total Literates − Literate Males

⇒ Literate Females = 58 − 36 = 22

⇒ Female Literacy % = (Literate Females ÷ Total Females) × 100

⇒ Female Literacy % = (22 ÷ 55) × 100

⇒ Female Literacy % = (2 ÷ 5) × 100 = 40%

∴ The correct answer is 40%.

Additional Information

Weighted Average Concept

The total percentage is the weighted average of the subgroups: Average_Total = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Percentage to Fraction Conversion

Fractions like 2/5 = 40%, 1/5 = 20%, and 3/5 = 60% are essential for solving data interpretation and percentage problems quickly.

Base Change Property

Always identify the correct base (denominator). Here, literacy rate for females is calculated with respect to the total female population, not the total population.

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

Conclusions:

(I) Some ants are kites → Follow (As all kites are monkeys and all monkeys are ants. As whole kites comes in monkeys which whole comes in ants, therefore it is true.)

(II) Some monkeys are not kites → Does not follow (As all kites are monkeys. As whole kites comes in monkeys, therefore it is false.)

∴ Here, Only conclusion (I) follows.​

Hence, the correct answer is "Option 3".

The correct answer is Option 2 (1000).

Key Points

• The unit of power is the Watt (W), named after James Watt.
• 1 Kilowatt (KW) is equivalent to 1000 Watts (W).
• The prefix 'kilo-' denotes a factor of 1000.
• Power is the rate at which energy is transferred or converted.
• Common applications of Kilowatts include measuring the power output of engines and the power consumption of electrical appliances.

Additional Information

• Horsepower
• Another unit of power is horsepower (HP), commonly used to measure the power of engines.
• 1 horsepower is approximately equal to 745.7 Watts.
• Megawatt
• 1 Megawatt (MW) is equivalent to 1,000,000 Watts.
• Megawatts are often used to measure the output of power plants or the power consumption of large facilities.
• Electrical Power Measurement
• Electrical power is often measured in Kilowatt-hours (kWh).
• 1 Kilowatt-hour is the amount of energy consumed by a 1000-Watt appliance running for one hour.

Calculation:

Number of people who visited hill station A in 2011 = 777

Number of people who visited hill station B in 2011 = 566

Absolute difference = |777 - 566| = 211

The absolute difference between the number of people who visited hill station A and hill station B in 2011 is 211.

The correct answer is Mohana Singh

Key Points

• Squadron Leader Mohana Singh became the first woman fighter pilot to join the Indian Air Force's (IAF) 18 "Flying Bullets" squadron in 2024.
• The "Flying Bullets" squadron operates the LCA Tejas, India's Indigenous fighter jet
• Mohana Singh is one of the first female fighter pilots in India.
• She was part of the "Tarang Shakti" exercise in Jodhpur, where she flew in the LCA Tejas fighter jet.
• During the exercise, she instructed the Army and Navy Vice Chiefs on the flight.
• The exercise was a message of support for Make in India.

Additional Information

• Avani Chaturvedi
• One of India’s first three women fighter pilots.
• Commissioned in June 2016 in the Indian Air Force (IAF).
• First Indian woman to fly a solo fighter jet (MiG-21 Bison) in 2018.
• Born in Rewa, Madhya Pradesh.
• Graduate from the Indian Air Force Academy, Hyderabad.
• Shivangi Singh
• First woman fighter pilot to join the IAF's Rafale squadron in 2020.
• Part of the prestigious "Golden Arrows" Squadron.
• Hails from Varanasi, Uttar Pradesh.
• Commissioned in 2017 and previously flew MiG-21 Bisons.
• Bhawana Kanth
• One of India’s first three women fighter pilots.
• Commissioned in June 2016 in the IAF.
• Became combat-ready in May 2019, flying MiG-21 Bison.
• Born in Begusarai, Bihar.
• First woman fighter pilot to take part in the Republic Day parade flypast in 2021.

Given:

Anthony starts from Point A and drives 9 km towards West.

He takes a left turn, drives 3 km, turns left again and drives 13 km.

He then takes a right turn and drives 2 km.

He takes a final right turn, drives 4 km and stops at Point P.

Thus, Anthony should drive (3 km + 2 km) 5 km in the north direction in order to reach Point A again.

Hence, "Option 1" is the correct answer.

The correct answer is Six 

Key Points

• India won six medals at the 2024 Paris Olympics, including one silver and five bronze.
• This was India's second-highest medal haul at the Olympics.
• India's medal winners included Manu Bhaker, Swapnil Kusale, Aman Sehrawat, Neeraj Chopra, and Sarabjot Singh.
• Manu Bhaker won India's first medal, a bronze, in the women's 10 m air pistol event.
• She was the first Indian woman to win an Olympic shooting medal.
• Swapnil Kusale won India's third bronze in the men's 50m rifle 3 positions shooting event.
• He was the first Indian to win this event.
• India's men's hockey team also contributed to India's medal haul.

Additional Information

• India competed at the 2024 Summer Olympics in Paris, France, held from 26 July to 11 August 2024.
• The country made its debut at the 1900 Summer Olympics.
• Indian athletes have appeared at every edition of the Summer Olympic Games since 1920 and the Paris Games edition marked India's 26^th appearance at the Summer Olympics.
• India won six medals including a silver and five bronze to be ranked 71^st amongst the 206 NOCs that participated in the Olympics.

The correct answer is Pumice.

Key Points

• Pumice is a correct example of foam in colloids.
• Foam is a type of colloid where gas is dispersed in a solid or liquid medium.
• Pumice forms when gas bubbles are trapped in solidified lava, creating a porous structure.
• This porous nature of pumice makes it light and able to float on water.
• Other examples of foam include whipped cream (gas in liquid) and Styrofoam (gas in solid).

Additional Information

• Mist
• Mist is an example of a colloid where liquid droplets are dispersed in a gas (air).
• It is commonly seen when tiny water droplets are suspended in the air, reducing visibility.
• Jelly
• Jelly is a colloid where a liquid is dispersed in a solid medium.
• It is a semi-solid substance used in various food products.
• Fog
• Fog is similar to mist but consists of larger water droplets dispersed in the air.
• It is a dense layer of water droplets that decreases visibility significantly.

The logic followed here is:

Hence, "Option 4" is the correct answer.

The correct answer is At the growing tips of the stems and roots.

Key Points

• The apical meristematic tissue is located at the growing tips of the stems and roots of a plant.
• It is responsible for the primary growth of the plant, which results in an increase in length.
• These tissues contain actively dividing cells that contribute to the elongation of the plant.
• The apical meristem is crucial for the formation of new organs such as leaves, flowers, and roots.
• It plays a significant role in the development and growth of the plant, ensuring continuous growth at the tips.

Additional Information

• Lateral Meristems
• These are found on the lateral sides of stems and roots.
• Lateral meristems are responsible for the secondary growth, which increases the girth of the plant.
• Intercalary Meristems
• These are located at the base of leaves or internodes.
• They help in the regrowth of grasses and elongation of internodes.
• Root Cap
• A protective structure that covers the tip of a root.
• It protects the apical meristem as the root pushes through the soil.

Shortcut Trick

Total interest percentage = Rate × Time

Net Interest % = 4% × 5 = 20%

Simple Interest = 20% of ₹3,000 = (20 ÷ 100) × 3,000

Simple Interest = ₹600

∴ The correct answer is ₹600.

Alternate Method

Given:

Principal (P) = ₹3,000

Rate (R) = 4% per year

Time (T) = 5 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ SI = (3000 × 4 × 5) ÷ 100

⇒ SI = (3000 × 20) ÷ 100

⇒ SI = 30 × 20

⇒ SI = ₹600

Therefore, the calculated value matches Option (3).

∴ The correct answer is ₹600.

Additional Information

Simple Interest (SI)

Simple interest is calculated only on the initial principal amount. Formula: SI = P × R × T ÷ 100.

Total Amount

The total sum paid at the end of the duration is Principal + Interest. Amount (A) = P + SI = P(1 + RT/100).

Interest Rate Units

The Rate (R) and Time (T) must be in the same time units (e.g., both per year or both per month) before calculation.

Shortcut Trick

Current Sum of ages = 51 + (3 persons × 8 years) = 75 years.

Total ratio parts = 4 + 5 + 6 = 15 units.

15 units = 75 years → 1 unit = 5 years.

Present age of eldest person (6 units) = 6 × 5 = 30 years.

∴ The correct answer is 30 years.

Alternate Method

Given:

Ratio of present ages = 4 : 5 : 6

Sum of ages 8 years ago = 51

Formula Used:

Present Sum = Past Sum + (Number of Persons × Time Gap)

⇒ Let the present ages be 4x, 5x, and 6x.

⇒ Sum of their ages 8 years ago = (4x − 8) + (5x − 8) + (6x − 8) = 51

⇒ 15x − 24 = 51

⇒ 15x = 51 + 24

⇒ 15x = 75

⇒ x = 75 ÷ 15 = 5

⇒ Present age of eldest person = 6x = 6 × 5 = 30 years.

∴ The correct answer is 30 years.

Additional Information

Ratio and Age Relationship

When a ratio a:b is given for present ages, the actual ages are taken as ax and bx. After n years, the ages become (ax + n) and (bx + n).

Sum of Ages over Time

The sum of ages for 'k' persons increases or decreases by (k × n) after or before 'n' years respectively.

Constant Age Difference

The numerical difference between the ages of two individuals remains constant regardless of the number of years passed or gone by.

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

Here, We need to find the symbols that are immediately preceded by an even number and not immediately followed by an odd number.

Required pattern: Even Number - Symbol - Not Odd Number

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

So, there are two such symbols in the series '2 @ 8' and '6 * £'.

Hence, the correct answer is "Option 2".

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

Conclusions:

(I) All bears are parrots → Does not follow (As some parrots are monkeys and some bears are monkeys. So it is possible but not definite.)

(II) Some monkeys are not bears → Does not follow (As some bears are monkeys. So it is possible but not definite.)

∴ Here, Neither conclusion (I) nor (II) follows.​

Hence, the correct answer is "Option 3".

The correct answer is Copper (II) Chloride.

Key Points

• Copper oxide reacts with hydrochloric acid to form copper (II) chloride.
• The chemical equation for the reaction is: CuO + 2HCl → CuCl₂ + H₂O.
• Copper (II) chloride is a compound where copper is in the +2 oxidation state.
• This reaction is a typical acid-base reaction where copper oxide acts as a base.
• The solution formed is usually blue-green due to the presence of copper (II) ions.

Additional Information

• Copper (I) Chloride
• This is a compound where copper is in the +1 oxidation state.
• It has different properties compared to copper (II) chloride, such as color and solubility.
• Oxidation States of Copper
• Copper commonly exhibits +1 and +2 oxidation states in its compounds.
• The +2 oxidation state is more stable in aqueous solutions.
• Acid-Base Reactions
• These reactions involve the transfer of protons (H⁺ ions) from the acid to the base.
• Metal oxides reacting with acids typically form a salt and water.

The correct answer is QK.

 

Key Points

• The work done in moving a charge in an electric field is given by the product of the charge and the potential difference.
• In this case, the charge is +Q and the potential difference is K.
• The formula for work done (W) is: W=Q⋅V
• Here, substituting the given values, W=Q⋅K
• Therefore, the work done in moving the charge +Q across two points with potential difference K is QK

 

Additional Information 

Electric Potential (V)

• Electric potential is the amount of electric potential energy per unit charge at a point in an electric field.
• It is measured in volts (V).

Electric Field (E)

• An electric field is a region around a charged particle where a force would be experienced by other charges.
• It is measured in newtons per coulomb (N/C) or volts per meter (V/m).

Potential Difference (V)

• The potential difference (voltage) between two points is the work done to move a unit charge from one point to another.
• It is a measure of the energy difference per unit charge between two points in an electric field.

The correct answer is New Delhi 

Key Points

• The Fifth Global Standards Symposium (GSS-24) was inaugurated by Mr. Jyotiraditya M. Scindia, Minister of Communications and Development of North Eastern Region, at the Bharat Mandapam, New Delhi  marking a historic event for the Asia-Pacific region.1
• The theme for GSS-24 was “Charting the Next Digital Wave: Emerging Technologies, Innovation, and International Standards”.
• ITU, established in 1865, is the oldest UN-specialized agency for Information and Communication Technologies.
• The focus was on the role of international standards in advancing emerging digital technologies and fostering sustainable development.

Additional Information

• The Global Standards Symposium (GSS) is an important platform established for discussions and coordination in the field of international standards.
• Its main objective is to bring together global policymakers, innovators, and decision-makers to foster cooperation in the field of standards.
• The Past Four GSS Dates, Locations, and Themes:

 

• The GSS-24, held on October 14, 2024, in New Delhi, India, focused on the theme "Charting the Next Digital Wave: Emerging Technologies, Innovation, and International Standards". Over 1,500 global leaders, innovators, and experts participated, discussing the digital revolution and cooperation in the field of standards

Shortcut Trick

In a 'Buy X, Get Y' scheme where Y is the total number of articles received, the Free Articles = Y − X.

Discount % = (Free Articles ÷ Total Articles) × 100

Here, Free articles = 6 − 5 = 1 article.

Total articles received = 6 articles.

Discount % = (1 ÷ 6) × 100 = 16 2/3 %.

∴ The correct answer is 16 2/3 %.

Alternate Method

Given:

Offer = 'Buy 5, Get 6'

Total items received by the customer = 6

Items paid for by the customer = 5

Formula Used:

Discount % = [(Marked Price − Selling Price) ÷ Marked Price] × 100

Let the Marked Price (MP) of each article be ₹1.

⇒ Total MP of 6 articles = 6 × 1 = ₹6.

⇒ Customer pays for 5 articles, so Selling Price (SP) = 5 × 1 = ₹5.

⇒ Discount Amount = MP − SP = 6 − 5 = ₹1.

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (1 ÷ 6) × 100

⇒ Discount % = 16 2/3 %.

∴ The correct answer is 16 2/3 %.

Additional Information

Buy X Get Y Free Scheme

When Y articles are given free on the purchase of X articles, the Discount % = [Y ÷ (X + Y)] × 100.

Relationship between MP and CP

MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %), which helps find the markup over cost price.

Successive Discounts

If two discounts a% and b% are applied, the single equivalent discount = [a + b − (a × b) ÷ 100]%.

Shortcut Trick

For a fixed expenditure of ₹1, Price and Quantity are inversely proportional.

Price Ratio (CP : SP) = 100% : 120% = 5 : 6

Quantity Ratio (CP : SP) = 6 : 5

Given Selling Quantity (5 units) = 10 toffees ⇒ 1 unit = 2

Cost Quantity (6 units) = 6 × 2 = 12 toffees

∴ The correct answer is 12 toffees.

Alternate Method

Given: Selling Price (SP) of 10 toffees = ₹1, Profit = 20%

Formula Used: Cost Price (CP) = SP × [100 / (100 + Gain%)]

⇒ SP of 1 toffee = ₹1/10

⇒ CP of 1 toffee = (1/10) × [100 / (100 + 20)]

⇒ CP of 1 toffee = (1/10) × (100 / 120) = (1/10) × (5 / 6)

⇒ CP of 1 toffee = 1 / 12

⇒ Number of toffees bought for ₹1 = 1 / (1/12) = 12

∴ The correct answer is 12 toffees.

Additional Information

Inverse Relation Rule

When the amount spent is constant, Price × Quantity = Constant. Thus, if price increases by a factor, quantity decreases by the same factor.

Profit/Loss Percentage

The percentage profit or loss is always calculated on the Cost Price (CP) unless stated otherwise. Formula: Profit% = [(SP − CP) / CP] × 100.

Unitary Method in Profit and Loss

It is often easier to find the price or cost of one unit (1 toffee, 1 kg, etc.) to avoid confusion when quantities vary between buying and selling.

Shortcut Trick

Apply the fundamental Trigonometric Identity cos² A = 1 − sin² A in the numerator.

Express the numerator as a product of factors: (1 − sin A)(1 + sin A).

Cancel out the common term (1 − sin A) from both the numerator and denominator.

The resulting simplified expression for y is 1 + sin A.

∴ The correct answer is sin A + 1.

Alternate Method

Given: y = cos² A / (1 − sin A)

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

⇒ y = (1 − sin² A) / (1 − sin A)

⇒ y = ((1 − sin A) × (1 + sin A)) / (1 − sin A)

⇒ y = 1 + sin A

∴ The correct answer is sin A + 1.

Additional Information

Pythagorean Identities

Basic identities in trigonometry include sin² A + cos² A = 1, 1 + tan² A = sec² A, and 1 + cot² A = cosec² A.

Algebraic Simplification

The identity for the difference of two squares, a² − b² = (a − b)(a + b), is frequently used to simplify trigonometric fractions.

Domain of Expression

For a fraction to be defined, the denominator must not be zero. Here, 1 − sin A ≠ 0 implies sin A ≠ 1, which means A ≠ 90°.

The correct answer is the excretory system.

Key Points

• The excretory system is responsible for the removal of nitrogenous waste materials from the body.
• It includes organs such as the kidneys, ureters, bladder, and urethra.
• The kidneys filter blood to remove waste products, which are then excreted as urine.
• Urea, a major nitrogenous waste product, is produced in the liver and excreted by the kidneys.
• The excretory system also helps in maintaining the body's fluid and electrolyte balance.

Additional Information

• Vascular System
• This system consists of vessels that carry blood and lymph through the body.
• It includes arteries, veins, and capillaries.
• Circulatory System
• This system circulates blood through the body, delivering oxygen and nutrients to cells and removing waste products.
• The heart is a central component of the circulatory system.
• Respiratory System
• This system is responsible for the exchange of gases between the body and the environment.
• It includes organs such as the lungs, trachea, and diaphragm.

Statements:

I. Most teachers of school X have left the school and joined school Y.

II. The football team of school X has won the champion’s trophy in the inter-city football championship.

Statement I is the effect - it is possible that the teachers left for reasons like better pay or management issues at school X.

Statement II is the effect - The football team’s victory could be due to their training and performance 

Therefore, both statements are independent effects of different causes and are not directly related to each other.

Hence, Option 2 is correct because both statements are independent effects of different causes.

The logic followed here is:

Hence, "Option 2" is the correct answer.

Given: 36 people are standing in a straight row facing the north.

1. Aman is standing 11^th from the left end.

2. Pratha is standing 12^th from the right end.

So, the number of people standing between Aman and Pratha:

= Total people - (Position of Aman from left + Position of Pratha from right)

= 36 - (11 + 12)

= 36 - 23 = 13.

So, there are '13' people standing between Aman and Pratha.

Hence, the correct answer is "Option 4".

Shortcut Trick

Analyze the prime factorization of the given number:

1083 = 3 × 361 = 3¹ × 19²

For a number to be a perfect square, all prime factors must have even exponents.

Since 3 has an odd exponent (1), it must be multiplied by 3 to make its exponent even (2).

∴ The correct answer is 3.

Alternate Method

Given:

Number = 1083

Formula Used:

A natural number is a perfect square if and only if each of its prime factors can be grouped into equal pairs.

Calculations:

⇒ Resolving the number 1083 into its prime factors:

⇒ 1083 ÷ 3 = 361

⇒ 361 = 19 × 19 = 19²

⇒ So, the prime factorization is: 1083 = 3¹ × 19²

⇒ Here, the prime factor 19 is paired, but the prime factor 3 is not paired.

⇒ To make it a perfect square, we must multiply the number by 3 to complete the pair of 3.

⇒ New number = 1083 × 3 = 3249 = (3 × 19)² = 57² (which is a perfect square)

⇒ Therefore, the smallest natural number to be multiplied is 3.

∴ The correct answer is 3.

Additional Information

Perfect Square Condition

A natural number is a perfect square if and only if all the exponents in its prime factorization are even. For example, N = p₁^2a × p₂^2b is always a perfect square.

Division to Make a Perfect Square

The smallest number by which a given number must be divided to make it a perfect square is the product of all its unpaired prime factors. This value is mathematically identical to the smallest multiplier required.

Square Root of a Large Number

The square root of a perfect square represented in prime factor form N = a^2x × b^2y is calculated as √N = a^x × b^y.

The correct answer is Statement A is true but B is false.

Key Points

• Testosterone is the primary male sex hormone responsible for the development of male reproductive tissues and secondary sexual characteristics such as increased muscle and bone mass, and the growth of body hair.
• Testosterone levels rise significantly during puberty, leading to the changes in appearance seen in boys.
• Progesterone is a female sex hormone involved in the menstrual cycle, pregnancy, and embryogenesis.
• Progesterone does not play a role in the changes seen in boys during puberty.
• Therefore, statement A is correct and statement B is incorrect.

Additional Information

• Estrogen
• Estrogen is the primary female sex hormone responsible for the development and regulation of the female reproductive system and secondary sexual characteristics.
• It plays a crucial role in the menstrual cycle and reproductive system.
• Growth Hormone
• Also known as somatotropin, it stimulates growth, cell reproduction, and cell regeneration in humans and other animals.
• It is important for physical growth and development, especially during puberty.
• Follicle-stimulating hormone (FSH)
• FSH is involved in the regulation of the development, growth, pubertal maturation, and reproductive processes of the body.
• In males, FSH stimulates the production of sperm.

The correct answer is kinetic energy.

Key Points

• Particles of matter are always in a state of motion, which is attributed to their kinetic energy.
• Kinetic energy is the energy that particles possess due to their movement.
• The movement of particles is more pronounced at higher temperatures because the kinetic energy increases with temperature.
• Even in solids, where particles are closely packed, they vibrate in place due to their kinetic energy.
• In liquids and gases, particles move more freely and randomly, demonstrating greater kinetic energy compared to solids.

Additional Information

• Potential Energy
• Potential energy is the stored energy of an object due to its position or state.
• It is not responsible for the movement of particles in matter.
• Intermixing of Particles
• Intermixing of particles, or diffusion, occurs due to the kinetic energy of particles.
• This process is a result of particles moving and spreading out to occupy available space.
• Size of Particles
• The size of particles does not directly influence their movement.
• Kinetic energy is the primary factor affecting the motion of particles, regardless of their size.

Shortcut Trick

To identify a prime number, check its divisibility by prime numbers up to its square root (√n).

For 373, √373 ≈ 19.3. Primes to check: 2, 3, 5, 7, 11, 13, 17, and 19.

Checking 373: It is not divisible by any of these primes (e.g., 373 ÷ 17 = 21.94, 373 ÷ 19 = 19.63).

Since no prime factor exists up to 19.3, 373 is a prime number.

∴ The correct answer is 373.

Alternate Method

Given: Numbers (1) 437, (2) 373, (3) 161, (4) 221.

Formula Used: A Prime Number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Calculations:

⇒ For 161: 161 ÷ 7 = 23 (Divisible by 7, hence Composite)

⇒ For 221: 221 ÷ 13 = 17 (Divisible by 13, hence Composite)

⇒ For 437: 437 ÷ 19 = 23 (Divisible by 19, hence Composite)

⇒ For 373: √373 ≈ 19.3. Primes are 2, 3, 5, 7, 11, 13, 17, 19. None divide 373.

⇒ Therefore, 373 has no factors other than 1 and 373.

∴ The correct answer is 373.

Additional Information

Definition of Prime Number

A prime number is a positive integer greater than 1 that is divisible only by 1 and itself (e.g., 2, 3, 5, 7, 11...).

Composite Number

A composite number has at least one divisor other than 1 and itself. Every composite number can be factored into primes.

Checking for Primes (√n Method)

To determine if n is prime, test its divisibility by all primes p such that p² ≤ n. If no such p divides n, then n is prime.

Co-prime Numbers

Two numbers are co-prime if their Highest Common Factor (HCF) is 1, regardless of whether the numbers themselves are prime.

The logic followed here is:

Option 1) BF-AE

Option 2) DH-CG

Option 3) XB-YC

Option 4) TX-SW

Thus, out of all the options 'XB-YC' is the odd one out.

Hence, "Option 3" is the correct answer.

The logic followed here is:

‘GLOBE’ is coded as ‘84315’

‘GRADE’ is coded as ‘89265’

‘BAKED’ is coded as ‘12756’

Thus,  code for 'R' is '9'.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Option 1) AK - VH

Option 2) YI - TD

Option 3) SC - NX

Option 4) LV - GQ

Thus, out of all the options 'AK - VH' is the odd one out.

Hence, "Option 1" is the correct answer.

Shortcut Trick

Ratio of Speeds (S₁ : S₂) = 10 : (10 + 5) = 10 : 15 = 2 : 3

Since Distance is constant, Time ratio is the inverse: (T₁ : T₂) = 3 : 2

Original Time (3 units) = 2 hours = 120 minutes

Reduction in Time = 3 − 2 = 1 unit ⇒ 120 ÷ 3 = 40 minutes

∴ The correct answer is 40 min.

Alternate Method

Given: Initial speed = 10 km/h, Time = 2 hours, Speed increase = 5 km/h.

Formula: Distance = Speed × Time | Time = Distance ÷ Speed

⇒ Total Distance = 10 km/h × 2 hours = 20 km

⇒ New Speed = 10 + 5 = 15 km/h

⇒ New Time taken = 20 ÷ 15 = 4/3 hours

⇒ Converting to minutes = (4/3) × 60 = 80 minutes

⇒ Original Time = 2 × 60 = 120 minutes

⇒ Time saved (Early reach) = 120 − 80 = 40 minutes

∴ The correct answer is 40 min.

Additional Information

Inverse Proportionality of Speed and Time

When the distance is constant, speed is inversely proportional to time (S ∝ 1/T). If speed increases by a factor, time decreases by the same reciprocal factor.

Unit Conversion

To convert speed from km/h to m/s, multiply by 5/18. To convert hours to minutes, multiply by 60.

Average Speed

For two equal distances covered at speeds x and y, the average speed is given by (2xy) ÷ (x + y).

Shortcut Trick

Identify terms: Let A = x² − y² and B = y² − x².

Required calculation: B − A.

Calculation: (y² − x²) − (x² − y²) = y² − x² − x² + y² = 2y² − 2x².

Factorizing: 2y² − 2x² = −2(x² − y²).

∴ The correct answer is −2(x² − y²).

Alternate Method

Given:

Expression to be subtracted: (x² − y²)

Expression from which to subtract: (y² − x²)

Formula:

Difference = (Term 2) − (Term 1)

Calculations:

⇒ Result = (y² − x²) − (x² − y²)

⇒ Result = y² − x² − x² + y²

⇒ Result = (y² + y²) + (−x² − x²)

⇒ Result = 2y² − 2x²

⇒ Taking −2 as a common factor: −2(x² − y²)

∴ The correct answer is −2(x² − y²).

Additional Information

Rules for Subtracting Polynomials

To subtract one algebraic expression from another, change the sign (from + to − or − to +) of every term in the expression being subtracted and then add the like terms.

Distributive Property of Signs

The negative sign outside a bracket affects all terms inside: −(a − b) = −a + b and −(a + b) = −a − b.

Like Terms

Addition and subtraction are only possible between like terms, which are terms that have the same variables raised to the same powers.

Shortcut Trick

When three numbers are in continued proportion a : x :: x : b, x is the mean proportional.

Direct formula: x = √(a × b)

x = √(48 × 75) = √(3600)

x = 60

∴ The correct answer is 60.

Alternate Method

Given: Proportion 48 : x :: x : 75 and x > 0.

Formula Used: Product of Extremes = Product of Means

⇒ 48 : x = x : 75

⇒ 48 / x = x / 75

⇒ x × x = 48 × 75

⇒ x² = 3600

⇒ x = √3600

⇒ x = 60

∴ The correct answer is 60.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is defined as √(ab).

Third Proportional

If a : b :: b : c, then c is the third proportional to a and b, calculated as c = b² / a.

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional, calculated as d = (b × c) / a.

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

1. Only two people sit to the left of D.

2. E sits second to the right of D.

3) F sits second to the left of D.

4. G sits second to the right of E.

5. A sits second to the left of B.

6. A is not an immediate neighbour of F.

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

Thus, according to the final arrangement, G sits at the extreme right end of the line.

Hence, "Option 3" is the correct answer.

The correct answer is Methane.

Key Points

• Methane is the simplest hydrocarbon with the chemical formula CH₄.
• It is a primary component of natural gas, making it a widely used fuel.
• Methane is a colorless, odorless gas under standard temperature and pressure conditions.
• It is used in residential heating, cooking, and electricity generation.
• Methane has a high energy content per unit of mass, making it an efficient fuel source.

Additional Information

• Ethane
• Ethane is a hydrocarbon with the formula C₂H₆, used as a petrochemical feedstock.
• It is less commonly used as fuel compared to methane.
• Propane
• Propane, with the formula C���H₈, is used as fuel for heating, cooking, and in vehicles.
• It is stored as a liquid in pressurized tanks.
• Butane
• Butane, C₄H₁₀, is used in lighters, portable stoves, and as a propellant in aerosol sprays.
• It is also a component of liquefied petroleum gas (LPG).

The correct answer is ovary, style.

Key Points

• The female reproductive organ of a flower is called the pistil, which is composed of the ovary, style, and stigma.
• The ovary is the swollen bottom part of the pistil, which contains the ovules and is responsible for the development of seeds after fertilization.
• The style is the middle elongated part of the pistil, which connects the ovary to the stigma and serves as a pathway for pollen tubes to reach the ovules.
• The stigma is the terminal part of the pistil that is often sticky, allowing it to capture and hold pollen grains for fertilization.
• The arrangement and structure of the pistil play a crucial role in the reproductive process of flowering plants.

Additional Information

• Pollination
• Pollination is the transfer of pollen from the male anther to the female stigma of a flower.
• It can occur through various agents such as wind, water, insects, birds, and other animals.
• Fertilization
• Fertilization is the fusion of male and female gametes, resulting in the formation of a zygote.
• In flowering plants, fertilization occurs after successful pollination and pollen tube growth.
• Fruit and Seed Development
• After fertilization, the ovary develops into a fruit, and the ovules develop into seeds.
• Fruits protect the seeds and aid in their dispersal to new locations for germination.

The correct answer is Andhra Pradesh

Key Points

• Andhra Pradesh has been awarded the Green Energy Champion Award for 2023-24 at a meeting hosted by the Indian Wind Power Association in Coimbatore
• The award was received by M Kamalakar Babu, Managing Director and Chief Executive Officer of AP Solar Power Corporation Limited, who accepted it on behalf of the State from Lalith Bohra, joint secretary of the Union Ministry of New and Renewable Energy.
• In his acceptance speech, Kamalakar Babu credited Chief Minister N. Chandrababu Naidu for prioritizing renewable energy during his earlier tenure from 2014 to 2019
• This award is a reflection of the state's commitment to sustainable energy and its future goals to continue leading the renewable energy transformation in India.

Additional Information

• India has the fourth-largest installed wind power capacity in the world
• As of 30 September 2024, the total installed wind power capacity was 47.36 gigawatts (GW)
• Wind power capacity is mainly spread across the southern, western, and northwestern states.
• Wind power installations occupy only 2% of the wind farm area facilitating rest of the area for agriculture, plantations, etc
• Wind power plants are also capable to provide fast frequency response in ramping up falling grid frequency.

The correct answer is option 3.

Key Points

• Carbon has the unique ability to form bonds with other atoms of carbon, giving rise to large molecules. This property is called catenation.
• Compounds of carbon having double or triple bonds between their carbon atoms are called unsaturated compounds.
• Compounds of carbon, which are linked by only single bonds between the carbon atoms, are called saturated compounds.
• Carbon compounds can form strong bonds with other elements such as hydrogen, oxygen, nitrogen, and halogens. Thus, statement 3 is incorrect.
• Carbon compounds can exist in various forms, including chains, branched chains, and rings.

Additional Information

• Alkanes
• These are saturated hydrocarbons with single bonds between carbon atoms.
• They follow the general formula C_nH_2n+2.
• Alkenes
• These are unsaturated hydrocarbons with at least one double bond between carbon atoms.
• They follow the general formula C_nH_2n.
• Alkynes
• These are unsaturated hydrocarbons with at least one triple bond between carbon atoms.
• They follow the general formula C_nH_2n-2.

The logic followed here is:

NCPJ is related to PFTO

And,

QAVB is related to SDZG

Similarly,

UNND is related to

Hence, "Option 4" is the correct answer.

Given Equation: 34 D 54 C 18 A 18 B 324 = ?

BODMAS Rule:

Decoding the given information:

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

⇒ 34 - 54 + 18 ÷ 18 × 324

⇒ 34 - 54 + 1 × 324

⇒ 34 - 54 + 324

⇒ 34 - 54 + 324

⇒ 34 + 270

⇒ 304

Hence, the correct option is "Option 4".

Decoding symbols:

Given: N – P # S × T

N – P → N is the brother of P.

P # S → P is the mother of S.

S × T → S is the wife of T.

Thus, N is S's mother's brother or maternal uncle.

Hence, the correct answer is "Option 3".

Given series:

1 * £ 5 $ 8 $ 4 2 & % 6 7 & @ 3 Ω 9

Condition to be checked:

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

Number → Symbol → Symbol

Checking the given series:

1 * £ 5 $ 8 $ 4 2 & % 6 7 & @ 3 Ω 9

Thus there are three such patterns (**1 *** £, 2 & %, 7 & @) that follow the above condition.

Hence, the correct answer "Option 1".

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 Equation: 91 D 13 B 4 A 3 C 2 = ?

BODMAS Rule:

Decoding the given information:

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

⇒ 91 ÷ 13 − 4 + 3 × 2

⇒ 7 − 4 + 3 × 2

⇒ 7 − 4 + 6

⇒ 3 + 6

⇒ 9

Hence, the correct option is "Option 2".

The correct answer is Royal Challengers Bangalore (Bengaluru)

Key Points

• The Royal Challengers Bengaluru (RCB) won the second season of the Women's Premier League (WPL) in 2024.
• The team defeated Delhi Capitals in the final.
• Smriti Mandhana captains the RCB team.
• The matches were held in Bengaluru and Delhi.
• The Board of Control for Cricket in India (BCCI) organizes the league

Additional Information

• The Women's Premier League, also known as WPL, and the Tata WPL for sponsorship reasons, is a women's Twenty20 cricket franchise league in India.
• It is owned and operated by the Board of Control for Cricket in India
• The first season, played in March 2023, saw the Mumbai Indians winning the inaugural title
• The league's structure is based on the structure of the IPL

The logic followed here is:

‘CODE’ is coded as ‘5739’

‘LEGS’ is coded as ‘4256’

Thus,  code for 'E' is '5'.

Hence, "Option 2" is the correct answer.

The correct answer is uniform at all points.

Key Points

• In a long current-carrying solenoid, the magnetic field inside is uniform and parallel to the axis of the solenoid.
• The magnetic field lines inside the solenoid are equally spaced, indicating a constant magnetic field.
• The field outside the solenoid is weak and non-uniform compared to the field inside.
• The strength of the magnetic field inside a solenoid is directly proportional to the number of turns per unit length and the current passing through the solenoid.
• For an ideal long solenoid, the edge effects are negligible, and the magnetic field is considered uniform throughout the length of the solenoid.

Additional Information

• Magnetic Field Lines
• These are imaginary lines used to represent the strength and direction of a magnetic field.They emerge from the north pole and merge at the south pole of a magnet.
• The density of the lines indicates the strength of the magnetic field.
• Electromagnetism
• It is the interaction between electricity and magnetism.
• A current-carrying conductor generates a magnetic field around it.
• Electromagnets are widely used in various applications like motors, transformers, and relays.
• Solenoid Applications
• Solenoids are used in devices that require controlled magnetic fields, such as electromechanical relays and solenoid valves.
• They are also used in magnetic resonance imaging (MRI) machines.
• Solenoids are integral components in many industrial applications, including automation and control systems.

Shortcut Trick

Identify prime numbers between 11 and 30: 13, 17, 19, 23, 29.

Total count of numbers = 5.

Sum of these numbers = 13 + 17 + 19 + 23 + 29 = 101.

Average = Total Sum ÷ Count = 101 ÷ 5 = 20.2.

∴ The correct answer is 20.2.

Alternate Method

Given: Range of numbers between 11 and 30 (both excluded).

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

Step-by-step Calculation:

⇒ Prime numbers in the given range = {13, 17, 19, 23, 29}

⇒ Sum = 13 + 17 + 19 + 23 + 29

⇒ Sum = 30 + 19 + 23 + 29

⇒ Sum = 101

⇒ Number of terms (n) = 5

⇒ Average = 101 ÷ 5

⇒ Average = 20.2

∴ The correct answer is 20.2.

Additional Information

Definition of Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Properties of Averages

If each number in a set is increased by a constant 'k', the average also increases by 'k'.

First 10 Prime Numbers

The sequence of the first ten prime numbers is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

The correct answer is Only c.

Key Points

• The statement "Heterotrophic organisms include plants" is incorrect. Plants are generally autotrophic organisms, meaning they produce their own food through photosynthesis.
• Heterotrophic organisms are those that cannot synthesize their own food and rely on other organisms for nutrition.
• Examples of heterotrophic organisms include animals, fungi, and many bacteria.
• Autotrophic organisms, such as plants, algae, and some bacteria, use light or chemical energy to synthesize food from inorganic substances.
• The other statements (a and b) correctly describe the nutritional modes of organisms.

Additional Information

• Autotrophic Nutrition
• Involves the use of simple inorganic substances to produce food.
• Photosynthesis is a common example, where plants convert carbon dioxide and water into glucose and oxygen using sunlight.
• Heterotrophic Nutrition
• Involves consuming complex organic substances for nutrition.
• Includes various modes such as holozoic, saprophytic, and parasitic nutrition.
• Mixotrophic Nutrition
• Some organisms exhibit both autotrophic and heterotrophic modes of nutrition.
• Examples include certain protists and some plants like Venus flytrap.

Shortcut Trick

A and B finish the work in 5 days, so in 4 days, they complete 4/5 of the total work.

Remaining work = 1 − 4/5 = 1/5.

A completes this 1/5 work in 2 days.

Total time for A alone = 2 × 5 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given: Time taken by (A + B) = 5 days, time worked together = 4 days, time taken by A to finish remaining work = 2 days.

Formula: Total Work = Efficiency × Time

⇒ Let Total Work = LCM(5) = 5 units.

⇒ Efficiency of (A + B) = 5 ÷ 5 = 1 unit/day.

⇒ Work completed by (A + B) in 4 days = 4 × 1 = 4 units.

⇒ Remaining work = 5 − 4 = 1 unit.

⇒ A completes 1 unit in 2 days.

⇒ Efficiency of A = 1 ÷ 2 = 0.5 units/day.

⇒ Time taken by A alone = 5 ÷ 0.5 = 10 days.

∴ The correct answer is 10 days.

Additional Information

Relationship between Time and Efficiency

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

Work and Wages

Wages are generally distributed among workers in the ratio of their efficiencies or the actual work done by each person.

Combined Work Formula

If A can do a piece of work in x days and B in y days, then together they can do it in (x × y) ÷ (x + y) days.

The logic followed here is:

Given: X : 793 :: 679 : Y

So, checking each option one by one:

Option 1) X = 974, Y = 498 → 974 : 793 :: 679 : 498

974 - 793 = 679 - 498

181 = 181 (LHS = RHS)

Option 2) X = 848, Y = 519 → 848 : 793 :: 679 : 519

848 - 793 = 679 - 519

55 ≠ 160 (LHS ≠ RHS)

Option 3) X = 886, Y = 443 → 886 : 793 :: 679 : 443

886 - 793 = 679 - 443

93 ≠ 236 (LHS ≠ RHS)

Option 4) X = 923, Y = 481 → 923 : 793 :: 679 : 481

923 - 793 = 679 - 481

130 ≠ 198 (LHS ≠ RHS)

Hence, "Option 1" is the correct answer.

Shortcut Trick

If each value in a data set is multiplied by a constant k, the average of the set is also multiplied by k.

Here, the Initial Average is 40 and each value is multiplied by 20.

New Average = Initial Average × 20

New Average = 40 × 20 = 800.

∴ The correct answer is 800.

Alternate Method

Given:

Total number of values (n) = 70

Average of these values = 40

Formula Used:

Average = (Sum of all values) ÷ (Total number of values)

⇒ Sum of 70 values = Average × n

⇒ Initial Sum = 40 × 70 = 2800

⇒ If each value is multiplied by 20, the New Sum = (x₁ × 20) + (x₂ × 20) + ... + (x₇₀ × 20)

⇒ New Sum = 20 × (x₁ + x₂ + ... + x₇₀) = 20 × 2800 = 56000

⇒ New Average = New Sum ÷ 70

⇒ New Average = 56000 ÷ 70 = 800

∴ The correct answer is 800.

Additional Information

Addition and Subtraction Rule

If a constant k is added to or subtracted from every value in a set, the new average will be (Old Average + k) or (Old Average − k) respectively.

Division Rule

If every value in a set is divided by a non-zero constant k, the new average will be (Old Average ÷ k).

Combined Average

The average of two groups with n₁ and n₂ elements and averages A₁ and A₂ is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

The correct answer is 40 u.

Key Points

• The atomic mass of an element is the weighted average mass of the atoms in a naturally occurring sample of the element.
• Calcium is a chemical element with the symbol Ca and atomic number 20.
• The atomic mass of calcium is approximately 40 u (atomic mass units).
• Calcium is an essential element for living organisms, particularly in cell physiology.
• It is the fifth most abundant element in the Earth's crust.

Additional Information

• Magnesium
• Magnesium is a chemical element with the symbol Mg and atomic number 12.
• It has an atomic mass of approximately 24.305 u.
• Phosphorus
• Phosphorus is a chemical element with the symbol P and atomic number 15.
• It has an atomic mass of approximately 30.974 u.
• Potassium
• Potassium is a chemical element with the symbol K and atomic number 19.
• It has an atomic mass of approximately 39.098 u.

Shortcut Trick

The ratio of angles J : K : L : M is 2 : 3 : 5 : 6.

Sum of ratio parts = 2 + 3 + 5 + 6 = 16 parts.

Since the total sum of angles is 360°, then 16 parts = 360° → 1 part = 22.5°.

Check co-interior angle sums: J + M = (2 + 6) × 22.5° = 180° and K + L = (3 + 5) × 22.5° = 180°.

When co-interior angles sum to 180°, the sides JK and ML are parallel.

A quadrilateral with only one pair of parallel sides is defined as a Trapezium.

∴ The correct answer is Trapezium.

Alternate Method

Given: The measures of the angles J, K, L, and M of quadrilateral JKLM are in the ratio 2 : 3 : 5 : 6.

Formula Used: Sum of all interior angles of a quadrilateral = 360°.

Calculations:

⇒ Let the angles be 2x, 3x, 5x, and 6x.

⇒ 2x + 3x + 5x + 6x = 360°

⇒ 16x = 360°

⇒ x = 22.5°

⇒ Angle J = 2 × 22.5° = 45°

⇒ Angle K = 3 × 22.5° = 67.5°

⇒ Angle L = 5 × 22.5° = 112.5°

⇒ Angle M = 6 × 22.5° = 135°

⇒ Observe the sum of adjacent angles: J + M = 45° + 135° = 180°.

⇒ Similarly, K + L = 67.5° + 112.5° = 180°.

⇒ Since the sum of co-interior angles is 180°, the side JK is parallel to ML.

⇒ Because only one pair of opposite sides is parallel and the other sides (JM and KL) are not parallel (J + K ≠ 180°), the quadrilateral is a Trapezium.

∴ The correct answer is Trapezium.

Additional Information

Properties of a Trapezium

A Trapezium is a quadrilateral with at least one pair of parallel sides. The angles on the same side of the non-parallel legs are supplementary (sum to 180°).

Sum of Angles in Polygons

For any polygon with n sides, the sum of internal angles is given by (n − 2) × 180°. For a quadrilateral (n=4), the sum is 2 × 180° = 360°.

Parallel Line Properties

If a transversal intersects two lines such that the sum of co-interior angles is 180°, then the two lines are parallel.

Shortcut Trick

For a number to be divisible by 8, the last three digits must be divisible by 8.

The last three digits of 6484a6 are 4a6.

Testing the least value from the options: If a = 0, then 406 ÷ 8 = 50.75 (Not divisible).

Next least value: If a = 1, then 416 ÷ 8 = 52 (Perfectly divisible).

∴ The correct answer is 1.

Alternate Method

Given:

Number: 6484a6

Divisor: 8

Formula Used:

A number is divisible by 8 if the number formed by its last three digits (hundreds, tens, and units place) is exactly divisible by 8.

Calculations:

⇒ The last three digits are 4a6, which can be written as 400 + 10a + 6.

⇒ Since 400 is already divisible by 8 (400 ÷ 8 = 50), we only need to check 10a + 6.

⇒ For a = 0: 10(0) + 6 = 6 (Not divisible by 8).

⇒ For a = 1: 10(1) + 6 = 16 (Divisible by 8: 16 ÷ 8 = 2).

⇒ Since we need the least value, a = 1 is the first digit that satisfies the condition.

∴ The correct answer is 1.

Additional Information

Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4.

Divisibility by 16

A number is divisible by 16 if the last four digits of the number are divisible by 16.

Divisibility by Powers of 2

For any integer n, a number is divisible by 2ⁿ if its last n digits are divisible by 2ⁿ.

Shortcut Trick

Direct formula for Total Surface Area of a closed cylinder: 2πr(r + h)

Substitute values: TSA = 2 × (22/7) × 70 × (70 + 110)

⇒ TSA = 440 × 180

⇒ TSA = 79,200 m²

∴ The correct answer is 79,200 m².

Alternate Method

Given:

Base Radius (r) = 70 m

Height (h) = 110 m

Value of π = 22/7

Formula Used:

Total Surface Area (TSA) = Curved Surface Area + 2 × Base Area

TSA = 2πrh + 2πr² = 2πr(h + r)

Calculations:

⇒ TSA = 2 × (22/7) × 70 × (110 + 70)

⇒ TSA = 2 × 22 × (70 ÷ 7) × 180

⇒ TSA = 44 × 10 × 180

⇒ TSA = 440 × 180

⇒ TSA = 79,200 m²

∴ The correct answer is 79,200 m².

Additional Information

Curved Surface Area (CSA)

The lateral surface area or CSA of a cylinder includes only the side walls and is given by 2πrh.

Volume of a Cylinder

The space occupied by a cylinder is its volume, calculated as Base Area × Height = πr²h.

Hollow Cylinder TSA

For a hollow cylinder with outer radius R and inner radius r, TSA = 2πRh + 2πrh + 2π(R² − r²).

Base Area

Since the base is a circle, the area of one circular base is πr².

The correct answer is Project PARI

Key Points

• ​​The project launched by the Ministry of Culture in 2024 to showcase India's artistic heritage through public art installations is called "Project PARI" (Public Art of India).
• The project aims to enhance the public art scene in India by displaying art installations across various public spaces.
• The initiative is spearheaded by the Ministry of Culture, Government of India.
• Project PARI was launched on the occasion of the 46^th World Heritage Committee Meeting held in New Delhi.

Additional Information

• Project BHARATI
• The Main Focus is the Preservation and promotion of Indian cultural heritage, languages, and traditions.
• The Goal is to digitally document, archive, and showcase India's rich cultural diversity.
• It collaborates with cultural institutions, use of modern technology for virtual exhibitions, and preservation of intangible cultural assets.
• Launched by: Ministry of Culture, Government of India.
• Project KALA
• The main Focus is Promoting Indian art, especially visual arts, through various outreach programs and exhibitions.
• The Goal is to create a digital platform for artists and connect them with global audiences.
• Key Aspects: Virtual art galleries, workshops, and artist grants.
• Launched by: National Gallery of Modern Art (NGMA), Ministry of Culture.
• Project SANSKRITI
• The main Focus is to Revive and support traditional Indian crafts, art forms, and cultural practices.
• The Goal is To provide a sustainable ecosystem for artisans, craftsmen, and traditional performers across the country.
• Key Aspects: Craft fairs, training programs, and employment opportunities for artisans.
• Launched by: Ministry of Textiles, Government of India.

Shortcut Trick

Let the total marks of both students be 100%.

One student secured 56% of the total, so the other secured 100% − 56% = 44%.

Difference in percentage = 56% − 44% = 12%.

Given difference = 9 marks. ∴ 12% ≡ 9 marks → 1% = 0.75 marks.

Marks are: 56 × 0.75 = 42 and 44 × 0.75 = 33.

∴ The correct answer is 42 and 33.

Alternate Method

Given: Difference between marks = 9. Higher marks = 56% of the sum of marks.

Let the marks obtained by the two students be x and (x − 9).

According to the question:

⇒ x = 56% of [x + (x − 9)]

⇒ x = 0.56 × (2x − 9)

⇒ x = 1.12x − 5.04

⇒ 1.12x − x = 5.04

⇒ 0.12x = 5.04

⇒ x = 5.04 ÷ 0.12 = 504 ÷ 12

⇒ x = 42

Marks of the second student = 42 − 9 = 33.

∴ The correct answer is 42 and 33.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert 56% to a fraction: 56/100 = 14/25. This means if the sum is 25 units, the marks are 14 units.

Ratio Method

If Student A gets 56 units and the sum is 100, the ratio of marks A:B is 56:(100 − 56) = 56:44 = 14:11. Difference = 3 units = 9 marks, so 1 unit = 3 marks.

Linear Equations

Always define the variable based on the total sum or the larger value to keep the multiplication factors positive and manageable.

The logic followed here is:

Given: X : 826 :: 723 : Y

So, checking each option one by one:

Option 1) X=593, Y=941 → 593 : 826 :: 723 : 941

593 - 826 = 723 - 941

-233 ≠ -218 (LHS ≠ RHS)

Option 2) X=542, Y=1007 → 542 : 826 :: 723 : 1007

542 - 826 = 723 - 1007

-284 = -284 (LHS = RHS)

Option 3) X=604, Y=1018 → 604 : 826 :: 723 : 1018

604 - 826 = 723 - 1018

-222 ≠ -295 (LHS ≠ RHS)

Option 4) X=518, Y=984 → 518 : 826 :: 723 : 984

518 - 826 = 723 - 984

-308 ≠ -261 (LHS ≠ RHS)

Hence, "Option 2" is the correct answer.