RRB Group D Previous Year Paper (Held on: 22 Sept 2022 Shift 3)

The correct answer is urban local bodies

Key Points

• The Twelfth Schedule of the Indian Constitution is related to the powers, authority, and responsibilities of urban local bodies, specifically municipalities.
• The schedule was added to the Constitution by the 74th Amendment Act of 1992, and came into effect on June 1, 1993.
• The Twelfth Schedule gives municipalities the power to: Plan urban areas, Regulate land use, Supply water, Ensure public health, and Provide welfare.
• The Constitution also provides for three types of municipalities: 
• Nagar Panchayat
• Municipal Council
• Municipal Corporation
• The members of a municipality are usually elected directly.
• The schedules of the Indian Constitution are additional provisions that provide detailed information about various aspects of the constitution.
• There are 18 list

Additional Information The List of 18 Functional Items under the 12th schedule is:

1. Urban planning including town planning.
2. Regulation of land use and construction of buildings.
3. Planning for economic and social development.
4. Roads and bridges.
5. Water supply for domestic, industrial and commercial purposes.
6. Public health, sanitation conservancy and solid waste management,
7. Fire services.
8. Urban forestry, protection of the environment and promotion of ecological aspects.
9. Safeguarding the interests of weaker sections of society, including the handicapped and mentally retarded.
10. Slum improvement and up-gradation.
11. Urban poverty alleviation.
12. Provision of urban amenities and facilities such as parks, gardens, playgrounds.
13. Promotion of cultural, educational and aesthetic aspects.
14. Burials and burial grounds, cremations, cremation grounds and electric crematoriums,
15. Cattle ponds, prevention of cruelty to animals.
16. Vital statistics including registration of births and deaths.
17. Public amenities including street lighting, parking lots, bus stops and public conveniences.
18. Regulation of slaughterhouses and tanneries.

Shortcut Trick

Total frequency N = 5 + 7 + 4 + 8 + 2 = 26.

Median class is determined by the cumulative frequency just greater than or equal to N ÷ 2.

N ÷ 2 = 26 ÷ 2 = 13.

Cumulative frequencies (CF): 5 → 12 → 16.

The 13^th term falls in the group with CF 16, which corresponds to x = 3.

∴ The correct answer is 3.

Alternate Method

Given:

x values: 1, 2, 3, 4, 5

f (frequency): 5, 7, 4, 8, 2

Formula Used:

If the total number of observations (N) is even, Median = [ (N ÷ 2)^th term + (N ÷ 2 + 1)^th term ] ÷ 2.

Calculations:

⇒ Let us calculate the Cumulative Frequency (CF) for each class:

⇒ For x = 1, f = 5 → CF = 5

⇒ For x = 2, f = 7 → CF = 5 + 7 = 12

⇒ For x = 3, f = 4 → CF = 12 + 4 = 16

⇒ For x = 4, f = 8 → CF = 16 + 8 = 24

⇒ For x = 5, f = 2 → CF = 24 + 2 = 26

⇒ Total frequency N = 26, which is an even number.

⇒ Position 1 = N ÷ 2 = 13^th observation.

⇒ Position 2 = (N ÷ 2) + 1 = 14^th observation.

⇒ The CF just greater than 12 is 16, meaning both the 13^th and 14^th observations fall under x = 3.

⇒ Median = (3 + 3) ÷ 2 = 3

∴ The correct answer is 3.

Additional Information

Mean

Mean is the arithmetic average of a distribution, calculated as Mean = ∑(f × x) ÷ ∑f.

Mode

Mode is the value that appears most frequently in a data set. In this distribution, x = 4 has the highest frequency (8), so the Mode is 4.

Empirical Relationship

For moderately skewed distributions, the empirical relationship between mean, median, and mode is given by: Mode = 3 × Median − 2 × Mean.

Shortcut Trick

Divisibility rule of 9: The sum of digits must be divisible by 9.

Sum of digits = 4 + a + 5 + 1 + 2 + 4 + 3 + 5 + 6 + 7 + 8 + 9 + 7 + 3 + 4 = 68 + a.

The next multiple of 9 after 68 is 72.

⇒ 68 + a = 72 ⇒ a = 4.

∴ The correct answer is 4.

Alternate Method

Given: A 15-digit number 4a5124356789734 is divisible by 9.

Formula: A number is divisible by 9 if the sum of its digits is a multiple of 9.

⇒ Sum = 4 + a + 5 + 1 + 2 + 4 + 3 + 5 + 6 + 7 + 8 + 9 + 7 + 3 + 4

⇒ Sum = 68 + a

For the number to be divisible by 9, (68 + a) must be divisible by 9.

⇒ Multiples of 9 near 68 are 63, 72, 81...

⇒ Since 'a' is a single digit, 68 + a = 72

⇒ a = 72 − 68

⇒ a = 4

∴ The correct answer is 4.

Additional Information

Divisibility by 3

Similar to 9, a number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility by 11

The difference between the sum of digits at odd places and sum of digits at even places must be 0 or a multiple of 11.

Digital Root

The single-digit sum of all digits in a number is called its digital root; for numbers divisible by 9, the digital root is always 9.

Given: Amongst six people, P, Q, R, S, T and U, each has a different weight.

1. R is heavier than T.

R > T

2. U is lighter than P.

P > U

3. Q is lighter than T but heavier than S.

T > Q > S

4. S is heavier than P.

T > Q > S > P

Thus, final arrangement:

R > T > Q > S > P > U

So, R is the heaviest amongst them.

Hence, the correct answer is "R".

The correct answer is 2022.

Key Points

• The Criminal Procedure (Identification) Bill passed in Parliament in 2022.
• It takes the place of the Identification of Prisoners Act, 1920, a statute from the colonial era that permits police to measure individuals who are accused, detained, or awaiting trial in criminal situations.
• It gives the police legal permission to collect bodily and biological samples from both criminal suspects and prisoners.
• The Law Commission of India's 87th Report reviewed this statute in 1980 and made several recommendations for changes. These recommendations were added in the bill.

Additional Information

• On December 3, 2021, the Central Vigilance Commission (CVC) (Amendment) Bill, 2021 was presented to the Lok Sabha. 
• The Central Vigilance Commission (Amendment) Ordinance, 2021 is superseded by it.
• The CVC Act of 2003 is intended to be amended by the bill.  A Central Vigilance Commission is to be established in accordance with the 2003 Act to investigate claims of offenses against the Prevention of Corruption Act, 1988.
• On December 20, 2021, the Election Laws (Amendment) Bill, 2021 was presented to the Lok Sabha. 
• In order to achieve specific electoral reforms, the Bill alters the Representation of the People Acts of 1950 and 1951.
• The 1950 Act specifies voter criteria, electoral roll compilation, seat distribution, and boundary delimitation for elections.
• The 1951 Act governs election administration, as well as election-related offenses and disputes.
• The Farm Laws Repeal Bill, 2021 was introduced in Lok Sabha on November 29, 2021 by the Minister of Agriculture and Farmers’ Welfare, Mr. Narendra Singh Tomar.
• The three agriculture laws that Parliament passed in September 2020 are repealed by the Bill.
• These include the Farmers' Produce Trade and Commerce (Promotion and Facilitation) Act, 2020, the Farmers (Empowerment and Protection) Agreement on Price Assurance and Farm Services Act, 2020, and the Essential Commodities (Amendment) Act, 2020.
• These laws were passed in order to:
• (i) establish a legal framework for contract farming;
• (ii) make it easier for farmers to trade their produce without obstacles outside of the markets that are alerted by the various state Agricultural Produce Marketing Committee (APMC) laws; and
• (iii) control the supply of specific foods (like cereals, pulses, and onions) only in extreme cases, like famine, war, or sharp price increases.

Shortcut Trick For equations to have real roots, the Discriminant (D) must be D ≥ 0. From x2 + kx + 64 = 0 ⇒ k2 ≥ 256 ⇒ k ≥ 16. From x2 − 8x + k = 0 ⇒ 64 − 4k ≥ 0 ⇒ k ≤ 16. Comparing both conditions, the only possible positive value for k is 16. ∴ The correct answer is 16. Alternate Method Given: Equation 1: x2 + kx + 64 = 0 Equation 2: x2 − 8x + k = 0 Formula Used: For a quadratic equation ax2 + bx + c = 0, the roots are real if the Discriminant (D) = b2 − 4ac ≥ 0. Calculations: ⇒ For equation x2 + kx + 64 = 0 to have real roots: ⇒ k2 − 4 × 1 × 64 ≥ 0 ⇒ k2 − 256 ≥ 0 ⇒ k2 ≥ 256 ⇒ k ≥ 16 (considering positive value of k) ⇒ For equation x2 − 8x + k = 0 to have real roots: ⇒ (−8)2 − 4 × 1 × k ≥ 0 ⇒ 64 − 4k ≥ 0 ⇒ 4k ≤ 64 ⇒ k ≤ 16 ⇒ To satisfy both conditions k ≥ 16 and k ≤ 16 simultaneously, the value of k must be exactly 16. ∴ The correct answer is 16. Additional Information Nature of Roots based on Discriminant (D) If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary. Sum and Product of Roots For ax2 + bx + c = 0, the sum of roots (α + β) = −b/a and the product of roots (αβ) = c/a. Quadratic Formula The values of x are calculated as x = (−b ± √D) / 2a, where D is the discriminant.

The correct answer is θ = β

Key Points

• The ray of light is refracted inside the equilateral prism and travels parallel to its base.
• For this condition, the angle of incidence (θ) and the angle of emergence (β) are equal due to the symmetry of the prism and the path of light.
• All angles of the prism are 60°.
• The refracted ray traveling parallel to the base ensures that the angles of incidence and emergence are symmetric.
• At both the entry and exit points, the angles of incidence and refraction are equal, leading to θ = β
• When the refracted ray is parallel to the base, the deviation angles inside the prism ensure that the light exits symmetrically.

Additional Information

• Prism
• A prism is a type of polyhedron that has two polygonal bases which are parallel to each other.
• In the field of optics, a prism is known as a transparent optical element with flat, polished surfaces that can refract light.
• The two polygonal bases are connected by lateral faces.
• These lateral faces are typically rectangular, but can sometimes be parallelograms.
• Prism Formulas are generally defined as,
• Surface Area of a Prism = 2 × Base Area + Lateral Surface Area
• Volume of a Prism = Base Area × Height

The correct answer is Cilia sweep mucus from the lungs and airway towards the nose.

Key Points

• Role of Cilia in the Respiratory Tract:
• Cilia are tiny, hair-like structures lining the respiratory tract, especially the trachea and bronchi.
• Their primary function is to move mucus and trapped particles (like dust, bacteria, and viruses) upward toward the throat and nose.
• This process is part of the mucociliary escalator, which helps keep the airways clear and prevents infections.
• Cilia are mobile, tiny, finger-like projections on the surface of airway cells.

Additional Information

• Respiratory System​
• In the respiratory process, the exchange of gases takes place in the lungs and alveoli.
• As a result of respiratory exhalation, air reaches the various alveoli of the lungs.
• A dense network of blood cells is present around the vasculature.
• The transport of oxygen is mainly by the red pigment hemoglobin found in the blood.
• Gaseous exchange in the lungs is called external respiration.
• The trachea is an important part of the respiratory system that connects the nose and mouth to the lungs.
• The larynx is a respiratory organ present in the throat of humans and other mammalian organisms that helps in speaking.
• The act of transporting oxygen and carbon dioxide from the lungs to the body cells and back to the lungs is called the transport of gases

Shortcut Trick

Selling Price can be calculated directly by applying the Discount on the Marked Price.

Multiplier SP = MP × (100 − Discount%) ÷ 100

SP = 4,500 × (100 − 4.2) ÷ 100

SP = 4,500 × 95.8 ÷ 100

SP = 45 × 95.8 = 4,311

∴ The correct answer is ₹4,311.00.

Alternate Method

Given → Marked Price (MP) = ₹4,500, Discount% = 4.2% Formulas: Discount = (Discount% ÷ 100) × MP | SP = MP − Discount

⇒ Discount Amount = (4.2 ÷ 100) × 4,500 ⇒ Discount Amount = 4.2 × 45 = 189 ⇒ Selling Price (SP) = 4,500 − 189 ⇒ Selling Price (SP) = 4,311

∴ The correct answer is ₹4,311.00.

Additional Information

Marked Price & Discount

The Marked Price is the price listed on the label, and Discount is always calculated on this value to find the Selling Price.

Profit and Loss Relation

Profit is realized when the Selling Price is higher than the Cost Price, even after allowing a discount on the Marked Price.

Given number → 6513742678

If 1 is added to each odd digit and 2 is subtracted from each even digit then,

New number → 4624820486

The digits that will appear more than once in the new number thus formed are:-

4624820486

Hence, the correct answer is "4, 2, 8, and 6".

Therefore, the correct answer is option 1 (Four).

The correct answer is 2 ∶ 27.

Key Points

Resistance (R) is given by the formula:

 $R = \frac{\rho . l }{A}$LA

Where, $\rho$ is the resistivity, $l$ is the length, and $A$ is the cross-sectional area.

Given that the resistances of the two wires are equal:

$R_1 = R_2$

$\frac{\rho_1 . l_1}{A_1} = \frac{\rho_2 . l_2}{A_2}$

From the problem,

$\frac{A_1}{A_2} = \frac{1}{9}$A1A2=19A1A2=19A1A2=19 and $\frac{l_1}{l_2} = \frac{3}{2}$L1L2=32L1L2=32L1L2=32

Substituting these values, we get:

$\frac{\rho_1 . l_1}{l_2} = \frac{\rho_2 . A_1}{A_2}$

$\frac{\rho_1 . 3}{2} = \frac{\rho_2 . 1}{9}$

Solving this, we get

$\frac{\rho_1}{\rho_2} = \frac{2}{27}$

Hence, the ratio of their resistivities is 2 ∶ 27.

Additional Information 

Addition in Series $R_{eq}$ = R1 + R2 + R3.

Addition in Parallel  $\frac{1}{{{R_{eq}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}}$.

The correct answer is Deepawali

Key Points

• Diwali is not **related to agriculture, while Lohri, Pongal, and Baisakhi are all harvest festivals.**​
• Diwali is the Hindu festival of lights.
• It is one of the most important and popular festivals in India, and is also celebrated by Hindus around the world.
• Diwali symbolizes the victory of light over darkness, good over evil, and knowledge over ignorance.
• Diwali is celebrated on the darkest night of the Hindu lunar month Kartika, which typically falls between mid-October and mid-November in the Gregorian calendar.
• The festival lasts for five days, and each day has its own special significance.
• It is not a harvest festival celebrated.

Additional Information

• Baisakhi festival
• Baisakhi is a major event for the farmers as it commemorates the harvest time of the Rabi season.
• Wheat crops could be seen growing and swinging all across Punjab at this time of year.
• Sikhs hold kirtans, visit local Gurdwaras, and community fairs, hold Nagar kirtan processions, raise the Nishan Sahib flag, and gather to socialize and share festive foods.
• Lohri:
• Lohri, the bonfire harvest festival of Punjab celebrated in the month of Magh, is symbolic of new beginnings.
• The festival is celebrated every year on January 13 with much enthusiasm and zeal.
• This festival denotes the harvesting of the Rabi crops and hence all the farmers get together in order to thank god for giving them such a wonderful harvest.
• The rituals related to Lohri symbolize the attachment of the people with Mother Nature.
• Pongal
• ​A major harvest festival celebrated in Tamil Nadu, India, that typically takes place in mid-January.
• Pongal is dedicated to the Sun God, Surya, and includes rituals, feasting, and cultural activities.
• A traditional bull-taming sport called Jalli Kattu is also part of the festivities.

Each of six teachers, J, K, L, M, N and O, have their classes on six different days of a week starting from Monday and ending on Saturday of the same week, but not necessarily in the same order.

1. Only two teachers have classes before M's class.

2. J has the class on Tuesday, and O has the class on Saturday.

3. Only one teacher has the class after the class of K.

4. N's class is not between the classes of K and M.

Thus the final arrangement:

So, N has the class on Monday.

Hence, the correct answer is "N".

The correct answer is the Heating of calcium carbonate

Key Points

• Heating of calcium carbonate is not an example of a combination reaction
• A combination reaction occurs when two or more reactants combine to form a single product.
• However, when calcium carbonate is heated, it undergoes thermal decomposition to form calcium oxide and carbon dioxide.
• Examples of combination reactions
• Formation of water from hydrogen gas and oxygen gas
• Formation of calcium oxide by the combination of calcium and oxygen
• Formation of calcium hydroxide by adding water to quicklime

Additional Information

• Combination reaction
• ​A combination reaction is when two or more reactants combine to form a single product.
• The three types of combination reactions are between two elements, two compounds, or between an element and a compound.
• Displacement reaction:
• The reaction in which an ion (or an atom) in a compound is replaced by an ion (or an atom) of another element is known as a displacement reaction.
• It is denoted by X + YZ  → XZ+Y
• Fe (s) + CuSO4 (aq) → FeSO4 (aq) + Cu (s)
• In the above reaction, iron is more reactive than copper, it displaces copper, resulting in the formation of ferrous sulphate.
• Fe2O3 (s) + 2Al (s) → Al2O3 (s) + 2F3 (s)
• In the above reaction, aluminium is more reactive than iron, it displaces iron, resulting in the formation of aluminium oxide.
• Decomposition reaction:
• In a Decomposition reaction, a compound breaks down into two or more components, in which one component must be in the elemental state.
• Decomposition reaction is the opposite reaction of the combination reaction.
• CaCO3 (s) → CaO (s) + CO2 (g)

The correct answer is The principal focus.

Key Points

• When light rays from an object are parallel to the principal axis of a concave mirror, they converge after reflection and meet at the principal focus.
• This is a fundamental property of concave mirrors:
• Parallel rays to the principal axis always converge at the focus after reflection.
• Thus, the image will be formed at the principal focus, and it will be real, inverted, and highly diminished.

​

Additional Information

• Spherical mirrors are of two types -
• Convex mirror, and
• Concave mirror.
• The image formed by the spherical mirrors depends on the position of the object.
• Convex mirror: If the outer surface of the spherical mirror is the reflecting surface then it is called a convex mirror. It is also called the diverging mirror.
• The nature of the image formed by a convex mirror does not depend on the distance of the object from the mirror. These images are always virtual, smaller than the object, and situated behind the mirror.
• Concave mirror: If the inner surface of the spherical mirror is the reflecting surface, then it is called a concave mirror. It is also called a focusing mirror/converging mirror.
• The size of the image produced by these mirrors can be larger or smaller than the object, depending upon the distance of the object from the mirror.
• The concave mirror can form both real as well as virtual images of any object.
• Plane mirrors are the most commonly used mirrors. Lateral inversion is seen in the plane mirror.

Important Points

• Formation of images for different positions of the object by a concave mirror:

The correct answer is Impulse

Key Points

• The term for buying that occurs when a consumer experiences a sudden, powerful urge to buy something immediately is called "impulse buying".
• Impulse buying refers to a spontaneous purchase made without prior planning or consideration
• Emotions or appealing displays often trigger it at the moment.
• Impulse buying:
• Unplanned: The decision to buy is made on the spot, not part of a pre-planned shopping list.
• Emotional influence: Emotions like excitement, desire, or even stress can drive impulse purchases.
• Immediate gratification: The consumer feels a strong urge to buy something right away.

Additional Information To reduce dissonance, consumers may engage in a number of activities, such as:

• Seeking out information about the product from other sources, such as reviews or word-of-mouth.
• Comparing prices between brands.
• Reading product specifications.
• Talking to friends or family about their experiences with the product.
• By engaging in these activities, the consumer hopes to confirm their decision and reduce any feelings of uncertainty or doubt.

The correct answer is Part IV A(Article 51 A)

Key Points

• The Part and Article of the Constitution of India that is related to Fundamental Duties is Part IV A (Article 51 A).
• Article 51A (a):-
• It outlines the fundamental duty of every citizen to abide by the Constitution
• It respects its ideals and institutions, the National Flag and the National Anthem.
• The National Anthem is a symbol of national unity and integrity and must be respected by all citizens.
• The 42nd Amendment Act of 1976 added the Fundamental Duties of citizens to the Constitution.
• The duties are described as a moral obligation of citizens to promote patriotism and protect India's cohesion
• One of the Fundamental Duties is the duty of parents or guardians to provide educational opportunities to their children between the ages of six and fourteen.
• This duty was added by the 86th Amendment Act of 2002.

Additional Information

• List of Fundamental Duties of India
• It shall be the duty of every citizen of India —
• To abide by the Constitution and respect its ideals and institutions, the National Flag and the National Anthem;
• To cherish and follow the noble ideals which inspired our national struggle for freedom;
• To uphold and protect the sovereignty, unity, and integrity of India.
• To defend the country and render national service when called upon to do so;
• To promote harmony and the spirit of common brotherhood amongst all the people of India transcending religious, linguistic and regional or sectional diversities; to renounce practices derogatory to the dignity of women;
• To value and preserve the rich heritage of our composite culture;
• To protect and improve the natural environment including forests, lakes, rivers, wildlife and to have compassion for living creatures;
• To develop the scientific temper, humanism, and the spirit of inquiry and reform;
• To safeguard public property and to abjure violence;
• To strive towards excellence in all spheres of individual and collective activity so that the nation constantly rises to higher levels of endeavor and achievement;
• Who is a parent or guardian, to provide opportunities for education to his child, or as the case may be, ward between the age of six to fourteen years.

The correct answer is Ravi Kumar

Key Points

• Ravi Kumar Dahiya:- 
• ​Ravi Kumar Dahiya is an Indian freestyle wrestler who won a silver medal in the men's 57 kg freestyle wrestling at the 2020 Summer Olympics.
• He is the first Indian wrestler to win a medal at the Olympic Games in over two decades.
• Ravi Kumar Dahiya has been awarded the "Major Dhyan Chand Khel Ratna Award 2021" for Wrestling. (Hence option 4 is correct)
• Dahiya is also a two-time Asian champion and a Commonwealth Games gold medalist.
• He is widely regarded as one of the best wrestlers in the world in his weight category.

Additional Information

• Sumit Antil​
• Sumit Antil is an Indian Paralympian and javelin thrower.
• He won a gold medal in the men's javelin throw F64 category at the 2020 Summer Paralympics.
• He holds the current world record, having thrown 68.55 meters in the Paralympic final.
• Sumit Antil was born on 6 July 1998 in Khewda, Sonipat, Haryana, India.
• He lost his left leg below the knee after he was involved in a motorbike accident in 2015 at the age of 17.
• Lovlina Borgohain
• Lovlina Borgohain is an Indian boxer who plays in the 69Kg category.
• She is 23 years old and hails from the Golaghat district of Assam.
• She won a bronze medal in the women's welterweight category at Tokyo Olympics 2021.
• With this win, she became the third boxer to win a medal at the Olympics for India after Vijender Singh and MC Mary Kom's bronze at the 2008 and 2012 Olympics respectively.
• Krishna Nagar​
• Krishna Nagar won the gold medal in badminton at the Tokyo Paralympics 2020.
• This achievement marked a significant moment in India's performance at the Paralympic Games.
• Krishna Nagar competed in the Men's Singles SH6 category.
• His victory added to India's growing success in para-badminton on the international stage.

The logic followed here is:

'MANNER' is written as 'OETVOD'

'BEHAVE' is written as 'DINIFQ'

Similarly, 'POLITE' will be written as:

Hence, the correct answer is "RSRQDQ".

The correct answer is 20 m.

Key Points

Step 1: Calculate the Required Magnification

Given

• Length of mature whale $h_i= -30m$
• Length of baby whale $h_o= 10m$

The magnification required is: $m= \frac{h_i}{h_o}= \frac{-30}{10}=-3$

This means the image needs to be three times the size of the object.

Step 2: Apply Magnification Formula for Mirrors

The magnification (m) is also related to the object distance (p) and the image distance (q) by: $m=-\frac{q}{p}$

We know $m = -3$, So $\frac{q}{p}= -3$

$q=-3p$

Step 3: Use the Mirror Formula

The mirror formula is: $\frac{1}{f}= \frac{1}{p}+\frac{1}{q}$, where f is the focal length of the mirror.

Given the focal length of the mirror $f=-15m$, substitute $q=-3p$ into the mirror formula:

$\frac{1}{-15}=\frac{1}{-p}+ \frac{1}{-3p}$

$-\frac{1}{15}=\frac{ -3-1}{3p}$

$\frac{1}{15}=\frac{4}{3p}$

$p=20$

The baby whale should be placed 20.0 m in front of the concave mirror to produce a real image three times the size of the baby whale, making it appear as long as a mature whale.

Additional Information

• Concave Mirror:
• A concave mirror is a spherical mirror with its reflecting surface curved inward, resembling a portion of the interior of a sphere.
• It can form real and virtual images depending on the position of the object relative to the focal point and center of curvature.
• When an object is placed beyond the center of curvature, the image formed is real, inverted, and diminished.
• When an object is placed between the focal point and the center of curvature, the image formed is real, inverted, and magnified.
• Focal Length (f):
• The focal length of a mirror is the distance between the mirror's surface and its focal point, where parallel rays of light either converge or appear to diverge after reflecting off the mirror.
• For a concave mirror, the focal length is positive, indicating that the focal point is on the same side as the reflecting surface.
• Mirror Formula:
• The mirror formula relates the object distance (u), the image distance (v), and the focal length (f) of a spherical mirror:
• $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
• Magnification (m):
• Magnification is the ratio of the height of the image (h') to the height of the object (h), which can also be expressed in terms of distances:
• $m=\frac{h'}{h}=-\frac{v}{u}$

Shortcut Trick

Ratio of Surface Area S_s : S_c = 1 : 1

4πr² = 6a² ⇒ r/a = √(6/4π) = √(3/2π)

Ratio of Volumes = (4/3)πr³ : a³ = (4/3)π(r/a)³ : 1

V_s/V_c = (4π/3) × (3/2π) × √(3/2π) = 2 × √(3/2π) = √(12/2π) = √(6/π)

∴ The correct answer is √6 : √π.

Alternate Method

Given: Surface Area of Sphere = Surface Area of Cube

Formula Used: Surface Area of Sphere = 4πr²; Surface Area of Cube = 6a²; Volume of Sphere = (4/3)πr³; Volume of Cube = a³

⇒ 4πr² = 6a²

⇒ r²/a² = 6/(4π) = 3/(2π)

⇒ r/a = √(3/2π)

⇒ Volume Ratio = [(4/3)πr³] / a³

⇒ Ratio = (4π/3) × (r/a)³

⇒ Ratio = (4π/3) × (r/a)² × (r/a)

⇒ Ratio = (4π/3) × (3/2π) × √(3/2π)

⇒ Ratio = 2 × √3 / √(2π) = √(4 × 3) / √(2π) = √12 / √(2π)

⇒ Ratio = √(12 / 2π) = √(6/π) = √6 : √π

∴ The correct answer is √6 : √π.

Additional Information

Mensuration of Sphere

For a Sphere of radius r, Volume = (4/3)πr³ and Total Surface Area = 4πr².

Mensuration of Cube

For a Cube of side a, Volume = a³ and Total Surface Area = 6a².

Isoperimetric Inequality

For a fixed Surface Area, a Sphere encloses the maximum Volume among all solids.

Given:

Each vowel in the word LOCKER is changed to the letter preceding it in the English alphabetical order and each consonant is changed to the letter following it in the English alphabetical order.

So, letter D will appear twice in the group of letters thus formed.

Hence, the correct answer is "D".

Given: Eight friends, Hari, Ganpati, Falak, Ekta, Janak, Rohit, Mangesh and Tony, are sitting around a round table, facing the centre.

1. Falak is sitting second to the left of Tony, who is sitting third to the left of Hari.

2) Ekta is sitting second to the left of Hari.

3. Ganpati is sitting fourth to the right of Tony.

4. Janak is sitting third to the right of Mangesh, who is not an immediate neighbour of either Falak or Tony.

Thus, the final arrangement:

So, Rohit is second to the right of Janak.

Hence, the correct answer is "Rohit".

The correct answer is a diatomic molecule

Key Points

• Hydrogen exists as a diatomic molecule (H₂) at standard temperature and pressure, unlike other alkali metals.
• It has an electronic configuration as 1s¹ (similar to alkali metals).
• Henry Moseley observed that there is a relationship between wavelength and atomic number.
• Accordingly Mendeleev's periodic law was modified and the atomic number was adopted as the basis of the modern periodic table.
• Atomic no. gives the number of protons in nucleus and the no. of electrons in orbit around the nucleus.
• This number keeps on increasing by one when we move from one elements to others in a row.
• The properties of elements could be predicted with more precision when they are arranged as per their increasing atomic no.
• Additionally the modern periodic table take cares of 3 limitation of Mendeleev periodic table.

Additional Information

• Limitations of Mendeleev periodic table are as follows:
• Electronic configuration of H resemble with that of alkali metals but it existed as a diatomic molecules just like halogen hence no fixed position could be given to H.
• Mendeleev could not assigned location to isotopes as they were discovered long after Mendeleev periodic table.
• Atomic masses do not increase in a regular manner, so it was not possible to predict how many elements could be discovered between 2 elements.

The correct answer is Adi Sankaracharya

Key Points

• The doctrine of Advaita or Monism was given by Adi Sankaracharya
• The philosophy of Advaita means ‘non-dualism, it means the soul is no different from Brahman.
• He used to say to people that “Atman is Brahman”.
• He started a Hindu revivalist movement in the ninth century to give a new orientation to Hinduism.
• He was born in Kaladi in Kerela.
• The doctrine of Advaita was too abstract to appeal to the common man.

Additional Information

The correct answer is Nuclear energy

Key Points

• Nuclear energy is not a non-conventional source of energy, but rather a conventional energy source.
• Non-conventional energy 
• It is renewable energy which is used by human beings for a long time.
• This type of energy is continuously produced in nature and is inexhaustible.
• Non-conventional sources of energy include solar energy, wind, and biomass.
• These sources do not produce any pollution in the environment.
• Examples are Wood, Solar energy, wind energy, hydropower, tidal energy, geothermal energy, Biomass Energy, etc.
• Geothermal Energy - It Utilizes heat from beneath the Earth's surface.
• Solar Energy - It Harnesses energy from the Sun.
• Tidal Energy- It Uses the kinetic energy of tides.

Additional Information A conventional source of energy is a natural resource that is available in nature and is available in limited quantities.

• They are also known as a non-renewable source of energy as once depleted, they cannot be re-generated easily.
• They are formed from decay matter over hundreds and millions of years.
• Examples are Oil, Coal, petroleum, natural gas, nuclear fuels, hydel power, etc.

Shortcut Trick

Identify the first and last multiples of 6 in the range (18, 78).

First number (a) = 24 | Last number (l) = 72

Number of terms (n) = [(Last − First) ÷ Common Difference] + 1

n = [(72 − 24) ÷ 6] + 1 = [48 ÷ 6] + 1 = 8 + 1 = 9

∴ The correct answer is 9.

Alternate Method

Given: Range = (18, 78) excluded, Divisor = 6

Formula Used: a_n = a + (n − 1)d

⇒ The natural numbers divisible by 6 between 18 and 78 are 24, 30, 36, ..., 72.

⇒ This is an Arithmetic Progression (AP) where:

⇒ First term (a) = 24

⇒ Common difference (d) = 6

⇒ Last term (a_n) = 72

⇒ 72 = 24 + (n − 1) × 6

⇒ 72 − 24 = (n − 1) × 6

⇒ 48 = (n − 1) × 6

⇒ n − 1 = 48 ÷ 6 = 8

⇒ n = 8 + 1 = 9

∴ The correct answer is 9.

Additional Information

Arithmetic Progression (AP)

A sequence where the difference between consecutive terms is constant. The n^th term is a_n = a + (n − 1)d.

Divisibility Rule for 6

A number is divisible by 6 if it is simultaneously divisible by both 2 (even) and 3 (sum of digits is divisible by 3).

Sum of n terms in AP

The sum S_n is given by (n ÷ 2) × (a + l), where a is the first term and l is the last term.

The correct answer is Sustainable management of natural resources is an easy task.

Key Points

• Sustainable management of natural resources is a complex challenge, facing numerous obstacles:
• Competing interests: Balancing economic development, environmental protection, and social needs can be difficult, leading to conflicts.
• Limited knowledge: Fully understanding the long-term impacts of resource use and developing effective solutions requires ongoing research and innovation.
• Lack of enforcement: Sustainable practices may require regulations and incentives, which can be challenging to implement and enforce effectively.
• Therefore, while acknowledging individual interests and the reality of finite resources, managing them sustainably involves addressing multifaceted challenges, making it far from an easy task.

Additional Information

• Sustainable resource management entails preserving natural resources so they can be used effectively and not abused for personal gain.
• Natural resource management:
• It is imperative that natural resource management be done for the benefit of present and future generations.
• It entails caring for the environment of marine life and biodiversity as well as the management of resources including coal, water, soil, land, animals, and plants.
• Considering the requirements of the next generation in order to ensure that they can profit from natural resources as well.
• It enables us to leave them in a better world.

The correct answer is 2.88 J

Key Points

• A 12 V battery is connected across a 250 Ω resistance.
• We need to calculate the heat produced in 5 seconds.
• Given:
• Voltage (V) = 12 V
• Resistance (R) = 250 Ω
• Time (t) = 5 s
• Calculation:
• The heat produced (H) can be calculated using Joule's law of heating, which is:
• H = (V² / R) * t
• H = (12² / 250) * 5
• H = (144 / 250) * 5
• H = 0.576 * 5
• H = 2.88 J
• Therefore, the heat produced in 5 seconds is 2.88 Joules.

Additional Information

• Joule's law of heating​
• According to the Joule's law of heating, heat produced when a current 'I' flows through a material of resistance 'R' for a period of time 'T' is I2RT.
• The Joule's law of heating states that heat produced in a resistor is directly proportional to the square of the current flowing through a resistor, directly proportional to the resistance of the given resistor and directly proportional to the time for which the currents flows through the resistor.
• Therefore H=I2Rt, where H is Heat, I is current, R is Resistance, t is time.

Shortcut Trick

Remaining Savings % = 100% − 84% = 16%

Given: 16% = ₹8,000

Monthly Income (100%) = (8000 ÷ 16) × 100

Income = 500 × 100 = ₹50,000

∴ The correct answer is ₹50,000.

Alternate Method

Given:

Expenditure = 84% of income

Monthly Savings = ₹8,000

Calculations:

⇒ Let Total Income be ₹X.

⇒ Savings % = 100% − 84% = 16%

⇒ 16% of X = 8,000

⇒ (16 ÷ 100) × X = 8,000

⇒ X = (8,000 × 100) ÷ 16

⇒ X = 500 × 100

⇒ X = 50,000

∴ The correct answer is ₹50,000.

Additional Information

Income Formula

Income is always the sum of Expenditure and Savings (Income = Expenditure + Savings).

Percentage Base

In income-based problems, the Total Income is always considered as the base value or 100%.

Successive Percentage Change

If income increases by a% and then b%, effective change = [a + b + (a × b)/100]%.

Shortcut Trick

Check through options: If Son age = 30 years.

Then Father age = 80 − 30 = 50 years.

Condition: 2 × Son age = 2 × 30 = 60 years.

Difference = 60 − 50 = 10 years (Matches condition).

∴ The correct answer is 30 years.

Alternate Method

Let Father age be F and Son age be S.

Given: F + S = 80 ⇒ F = 80 − S --- (i)

According to problem: 2S = F + 10 --- (ii)

Substitute (i) in (ii):

⇒ 2S = (80 − S) + 10

⇒ 2S = 90 − S

⇒ 2S + S = 90

⇒ 3S = 90

⇒ S = 30 years

∴ The correct answer is 30 years.

Additional Information

Linear Equations in Two Variables

Problems involving two unknowns can be solved using substitution or elimination methods to find individual values.

Age Relation Constants

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

Ratio of Ages

If the ratio of ages is given as a:b, we assume the ages to be ax and bx to set up the equation.

Given,

Statement:

Sale up to 70% on all clothing items in the store and online.

Conclusions:

I. All clothing items are 70% off during the sale period - False (Because, the statement mentions "up to 70%," which means some items may be discounted less than 70%, but not necessarily all items are at 70% off).

II. There are some clothing items that are not on sale - False (Because, the statement indicates that all clothing items have a discount of up to 70%, which implies all items are on sale, though the discount percentage might vary).

So, Neither conclusion I nor II follows.

Hence, the correct answer is "Option 1".

Given:

Sales volumes for 2017 (in thousands):

S1 = 80, S2 = 75, S3 = 95, S4 = 85, S5 = 75

Formula used:

Average = $\frac{\text{Sum of sales volumes}}{\text{Number of shops}}$

Calculations:

Sum of sales volumes = 80 + 75 + 95 + 85 + 75

⇒ Sum = 410

Number of shops = 5

⇒ Average = $\frac{410}{5}$

⇒ Average = 82

∴ The average sales volume of Philips bulbs for 2017 is 82,000.

The correct answer is Zero

Key Points

• Rotating the magnet about its own axis, while it is aligned with the solenoid's axis, does not change the magnetic flux through the solenoid.
• This is because the orientation of the magnetic field lines relative to the solenoid does not change only the magnet is being spun around its axis.
• The magnetic field strength and direction at any point in the solenoid remain constant (Zero) during the rotation of the magnet.
• If the magnet were moved differently or oriented differently, you would get a varying magnetic flux and an induced current.

Additional Information

• Solenoid
• A solenoid is an instrument consists of copper coiling over a cylinder designed to create a strong magnetic field inside the coil because of the flow of current through the coil.
• By wrapping the same wire many times around a cylinder, the magnetic field due to the flow of current can become quite strong.
• Hence we can say that the strength of the magnetic field will change as a current through coil or number of turn’s changes.
• Magnetic field strength is independent of the diameter of the cylinder of a solenoid.
• Strength of the magnetic field in a solenoid will be directly proportional to the number of turns and amount of current flowing through a wire and will be inversely proportional to its length.
• Thus it is given by $B = \frac{{{\mu _0}NI}}{l}$

Shortcut Trick Given: CP of 63 articles = SP of 54 articles. Ratio of CP : SP = 54 : 63 = 6 : 7. Profit = SP − CP = 7 − 6 = 1 unit. Gain% = (Profit ÷ CP) × 100 = (1 ÷ 6) × 100 = 16 2/3%. Since SP > CP, there is a gain. ∴ The correct answer is 16 2/3% gain.

Alternate Method Given: CP × 63 = SP × 54 Formulas: Profit% = ((SP − CP) ÷ CP) × 100 | SP = CP × (100 + P%) ÷ 100

⇒ Let the CP of one article be Re. 1. ⇒ Total CP of 63 articles = Rs. 63. ⇒ Total SP of 54 articles = Total CP of 63 articles = Rs. 63. ⇒ SP of one article = 63 ÷ 54 = 7/6. ⇒ Profit per article = 7/6 − 1 = 1/6. ⇒ Gain% = (1/6 ÷ 1) × 100 = 16 2/3%. ∴ The correct answer is 16 2/3% gain.

Additional Information Cost Price vs Selling Price Relationship When the CP of 'x' articles equals the SP of 'y' articles, Gain/Loss% = (x − y) ÷ y × 100. If positive, it is a gain; if negative, it is a loss. Profit and Loss Basics Profit occurs when SP > CP. Loss occurs when CP > SP. Both are always calculated on the Cost Price (CP) unless stated otherwise.

The logic followed here is:

Given Sequence: L G _ A L L _ F B L _ G F _ L L G F _ L

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

Option 1) FFCCD → L G F A L / L F F B L / C G F C L / L G F D L

Option 2) FLGDC → L G F A L / L L F B L / G G F D L / L G F C L

Option 3) FCGLD → L G F A L / L C F B L / G G F L L / L G F D L

Option 4) FGLCD → L G F A L / L G F B L / L G F C L / L G F D L

Here, 4th letter is increased by 1.

A + 1 = B; B + 1 = C; C + 1 = D; 

Hence, the correct answer is "FGLCD".

The correct answer is the atmosphere is less dense at high altitudes

Key Points

• Breathing is harder at high altitudes compared to sea level
• It is because the atmosphere is less dense at high altitudes.
• This means that the air is thinner, and there is less oxygen available for the body to breathe in
• There is a decrease in atmospheric pressure as well as the oxygen content of the atmosphere resulting in a lower partial pressure of oxygen than that of carbon dioxide resulting in difficulty in breathing.

Additional Information

• Altitude Sickness
• At sea level, there are around 21% more oxygen molecules than usual.
• While the concentration does not change with the increase in altitude but the amount of oxygen molecules per breath and the atmospheric pressure decreases.
• At 12,000 feet (3,658 metres), there are around 40% less oxygen molecules in each breath.
• Even while at rest, your breathing rate needs to increase for the body to receive enough oxygen.
• Continued exposure to higher altitudes without enough acclimatisation might result in very serious, even fatal illnesses.

Given: Five friends named Kishore, Jayan, Lahari, Manyaki and Nalini live on five different floors of the same building.

1. Kishore lives on odd numbered floor but not on third floor and Nalini lives on the bottommost floor.

2. Manyaki and Jayan lives on even numbered floors but Manyaki doesn't lives above Jayan.

3. Lahari lives on the floor immediately above Manyaki.

So, Lahari lives on the third floor.

Hence, the correct answer is "Lahari".

Given:

5432.5678 + 435.692 + 29.54 + 6.3 + 0.4321

Formula used:

Sum of the given numbers

Calculation:

5432.5678 + 435.692 + 29.54 + 6.3 + 0.4321

⇒ 5432.5678 + 435.692 = 5868.2598

⇒ 5868.2598 + 29.54 = 5897.7998

⇒ 5897.7998 + 6.3 = 5904.0998

⇒ 5904.0998 + 0.4321 = 5904.5319

∴ The correct answer is option (4).

Given series: + 25 @ 36 # 29 37 % 14 - 7

Symbols have a prime number as an immediate neighbour on either of its sides.

Condition need to be check:

Symbol (Either) → Prime number → Symbol (Or)

• 25 @ 36 # 29 37 % 14 - 7

So, 3 symbols have a prime number as an immediate neighbour on either of its sides.

Hence, the correct answer is "3".

According to the given information:

From starting point he ran 11 km towards West, turned right and ran for 12 km and turned right again and ran 11 km and then turned right again and ran 9 km.

Distance between present position and starting point = 12 - 9 = 3 km.

So, he has to run 3 km now in order to reach the starting point from his present position to complete the marathon.

Hence, the correct answer is "3 km".

Given Statements:

Some investors are urbans.

All stores are urbans.

Some cottages are stores.

The possible Venn diagram is as follows,

Conclusions:

I. Some cottages are investors. - False (Because, there is no direct relation given between cottages and investors so, it can be possible but not definite)

II. Some investors are stores. - False (Because, there is no direct relation given between investors and stores so, it can be possible but not definite)

III. Some cottages are urbans. - True (Because, Some cottages are stores and All stores are urbans. So, Some cottages are also urbans)

Hence, the correct answer is "Only conclusion III follows".

Given: 

Option 1) 13 + 7 × 3 = 47

After interchanging the numbers and signs:

⇒ 13 × 3 + 7 = 47

⇒ 39 + 7 = 47

⇒ 46 = 47

Option 2) 16 × 3 + 7 = 39

After interchanging the numbers and signs:

⇒ 16 + 7 × 3 = 39

⇒ 16 + 21 = 39

⇒ 37 = 39

Option 3) 14 + 3 × 7 = 35

After interchanging the numbers and signs:

⇒ 14 × 7 + 3 = 35

⇒ 98 + 3 = 35

⇒ 101 = 35

Option 4) 3 + 7 × 9 = 30

After interchanging the numbers and signs:

⇒ 7 × 3 + 9 = 30

⇒ 21 + 9 = 30

⇒ 30 = 30

Hence, the correct answer is "3 + 7 × 9 = 30".

Statements:

Some cisterns are ponds.

All ponds are streets.

Some streets are cisterns.

The possible Venn diagram is as follows,

Conclusions:

I. Some streets are ponds. - True (Because, All ponds are streets given. So, Some streets are ponds it is definite)

II. All cisterns are streets. - False (Because, Some streets are cisterns given. So, All cisterns are streets. It is not definite)

III. All ponds are cisterns. - False (Because, Some cisterns are ponds. So, All cisterns are streets. It is not definite)

IV. No street is a cistern. - False (Because, Some streets are cisterns given. So, No street is a cistern. It is not definite)

Hence, the correct answer is "Only conclusion I follows".

The correct answer is Line A - it was growing but now it is not growing.

Key Points

• Line A indicates that the size of the ozone hole was increasing previously.
• The size has stabilized and is no longer expanding.
• This implies that the efforts to reduce the factors causing the ozone depletion have been effective.
• Monitoring the size of the ozone hole is crucial for understanding the effectiveness of environmental policies.

Additional Information

• Ozone Hole:
• The ozone hole refers to the region over Antarctica where the concentration of ozone in the stratosphere is significantly reduced during certain times of the year.
• This phenomenon was first observed in the late 1970s and has been a major environmental concern ever since.
• Causes of Ozone Depletion:
• Chlorofluorocarbons (CFCs), halons, and other ozone-depleting substances (ODS) are the primary causes of ozone depletion.
• These substances release chlorine and bromine atoms upon exposure to ultraviolet (UV) light, which then break down ozone molecules.
• Montreal Protocol:
• The Montreal Protocol is an international treaty adopted in 1987 to phase out the production and consumption of ozone-depleting substances.
• It has been successful in reducing the emissions of ODS and has contributed to the stabilization of the ozone layer.
• Impact of Ozone Depletion:
• Ozone depletion leads to higher levels of harmful UV radiation reaching the Earth's surface.
• This can cause increased cases of skin cancer, cataracts, and other health issues, as well as negatively affect ecosystems and wildlife.

Given series:

• V ! ^ C F A * B Y S & Y T G $ V # Y ! $ C ^ A C $ Y N 

Condition to be checked:

Symbols which are immediately preceded by a symbol and immediately followed by a letter.

Symbol → Symbol → Letter

• V ! ^ C F A * B Y S & Y T G $ V # Y ! $ C ^ A C $ Y N

Thus, there is two such pattern (! ^ C and ! $ C) that are following the above conditions.

Hence,"2" is the correct answer.

Mistake Points

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

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

The correct answer is Khelo India

Key Points

• In the Union Sports Budget for the financial year 2022-23, the biggest beneficiary was the Khelo India Scheme.
• The Ministry has furnished an Expenditure Finance Committee (EFC) memorandum to the Ministry of Finance for extension/continuation of the Khelo India Scheme from 2021-22 to 2025- 26,
• A number of Rs 8750 crore has been estimated as a financial implication of the new Khelo India Scheme (2021-22 to 2025-26) in the EFC memorandum furnished to the Ministry of Finance.
• An amount of Rs 657.71 crore has been allocated in the Budget Estimate (B.E.) for the year 2022-23 under the Khelo India Scheme.
• This information was given by former Minister of State for Youth Affairs & Sports Shri Kiren Rijiju.

Additional Information

• National Service Scheme
• ​NSS aims to develop the personality and leadership qualities of students through community service.
• It encourages youth to engage in social service activities and helps them become responsible citizens.
• It primarily targets students in universities, colleges, and schools.
• Volunteers engage in community-based activities, often related to education, health, sanitation, and environmental awareness.
• Urban Sports Infrastructure Scheme
• ​This scheme aims to develop and upgrade sports infrastructure in urban areas to promote sports and physical activities among citizens, particularly the youth.
• The scheme provides financial assistance to state and Union Territory governments for creating and improving sports facilities such as sports complexes, stadiums, and playgrounds in urban areas.
• National Sports Talent Search Scheme (NSTSS)
• ​​​NSTSS aims to identify and nurture young talent in various sports disciplines across India
• It ensures that the country produces world-class athletes for international competitions.
• Under the scheme, talented athletes are provided with scholarships and advanced training under expert coaches.
• The focus is on identifying talent early and providing them with the necessary resources for their development.

The correct answer is aqueous

Key Points

• The notation (aq) in a chemical equation represents  Aqueous.
• (aq) stands for "aqueous," which means the substance is dissolved in water to form a solution.
• For example, in the equation:
• NaCl (aq)→Na+(aq)+Cl−(aq)
• Sodium chloride is dissolved in water, and the ions dissociate in the aqueous medium.

Additional Information

• Chemical equations are symbolic representations of chemical reactions.
• They provide information about the reactants, products, and their respective quantities.
• Balancing chemical equations is essential to satisfy the law of conservation of mass.
• Different states of matter are indicated to provide a clear understanding of the reaction conditions.

The correct answer is placenta

Key Points

• The placenta is the organ that supplies nutrients to a fetus from the mother
• The human placenta is called a haemochorial type.
• It is in direct contact with the fetus.
• The placenta forms a thick membranous envelope all over the fetus which gives it protection during the gestation period.
• The blood and another kind of supply are facilitated by the chorionic villi.

Additional Information

• Zygote:
• The zygote is a single cell formed by the fusion of a sperm cell and an egg cell during fertilization.
• It represents the first stage of development in sexually reproducing organisms
• It eventually develops into an embryo through successive cell divisions.
• Endosperm:
• The endosperm is a tissue found in the seeds of flowering plants, providing nutrition to the developing embryo.
• It forms after the fertilization process in plants, specifically as a result of double fertilization, where one sperm nucleus fuses with two polar nuclei in the ovule.
• Cervix:
• The cervix is a part of the female reproductive system in mammals.
• It is the lower part of the uterus that connects the uterus to the vagina
• It plays a key role during childbirth by dilating to allow the baby to pass through.

Given:

Number of girls in colleges D and E:

D = 3000, E = 3250

Total girls in colleges A, B, and C:

A = 2500, B = 3000, C = 2000

Formula used:

Percentage = $\frac{\text{Total girls in D and E}}{\text{Total girls in A, B, and C}}\times 100$

Calculations:

Total girls in D and E = 3000 + 3250

⇒ 6250

Total girls in A, B, and C = 2500 + 3000 + 2000

⇒ 7500

Percentage = $\frac{6250}{7500} \times 100$

⇒ $\frac{6250 \times 100}{7500}$

⇒ 83.33%

∴ The percentage of girls in colleges D and E is 83.33%.

According to the given information:

1. Jyoti covered a distance of 3 km in a particular direction and then turned to her right.
2. She further travelled 1 km and there after she turned to the left and walked 5 km.
3. And then finally she turned to right and walked in the north direction.

So, she was facing in West direction when she started her journey.

Hence, the correct answer is "West".

The logic followed here is:

Hence, the correct answer is "225".

Shortcut Trick

Let θ = 45°.

Substitute sin 45° = 1/√2 and cos 45° = 1/√2, so sin² θ = 1/2 and cos² θ = 1/2.

Also, sec² θ = 2 and cosec² θ = 2.

Expression = (1/(2 − 1/2) + 1/(2 − 1/2)) × (1/2 × 1/2) = (2/3 + 2/3) × (1/4) = 4/3 × 1/4 = 1/3.

Now test option (4): (1 − 1/4)/(2 − 1/4) = (3/4)/(7/4) = 3/7 (Incorrect). Test option (1): (1 − 1/4)/(2 + 1/4) = (3/4)/(9/4) = 3/9 = 1/3 (Correct).

∴ The correct answer is (1 − cos² θ sin² θ)/(2 + cos² θ sin² θ).

Alternate Method

Given:

The expression is $\left(\frac{1}{\sec ^2 \theta-\cos ^2 \theta}+\frac{1}{\operatorname{cosec}^2 \theta-\sin ^2 \theta}\right)\left(\sin ^2 \theta \cos ^2 \theta\right)$

Formula Used:

⇒ sec² θ = 1/cos² θ

⇒ cosec² θ = 1/sin² θ

Calculations:

⇒ sec² θ − cos² θ = (1 − cos⁴ θ)/cos² θ = ((1 − cos² θ)(1 + cos² θ))/cos² θ = (sin² θ(1 + cos² θ))/cos² θ

⇒ Therefore, 1/(sec² θ − cos² θ) = cos² θ/(sin² θ(1 + cos² θ))

⇒ Similarly, 1/(cosec² θ − sin² θ) = sin² θ/(cos² θ(1 + sin² θ))

⇒ Sum = [cos⁴ θ(1 + sin² θ) + sin⁴ θ(1 + cos² θ)] / [sin² θ cos² θ (1 + sin² θ)(1 + cos² θ)]

⇒ Numerator of the sum = cos⁴ θ + sin⁴ θ + sin² θ cos² θ (sin² θ + cos² θ) = (sin² θ + cos² θ)² − 2 sin² θ cos² θ + sin² θ cos² θ = 1 − sin² θ cos² θ

⇒ Denominator of the sum = sin² θ cos² θ (1 + sin² θ + cos² θ + sin² θ cos² θ) = sin² θ cos² θ (2 + sin² θ cos² θ)

⇒ Multiplying by (sin² θ cos² θ) simplifies the entire expression to: (1 − sin² θ cos² θ)/(2 + sin² θ cos² θ)

∴ The correct answer is (1 − cos² θ sin² θ)/(2 + cos² θ sin² θ).

Additional Information

Trigonometric Identities

Recall that sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and cosec² θ − cot² θ = 1.

Algebraic Expressions in Trigonometry

Expressing sin⁴ θ + cos⁴ θ as (sin² θ + cos² θ)² − 2 sin² θ cos² θ = 1 − 2 sin² θ cos² θ is extremely useful for simplifying higher power trigonometric equations.

Given Statements:

All gadgets are electronics.

Some electronics are wirelesses.

Some wirelesses are toys.

The possible Venn diagram is as follows,

Conclusions:

I. Some gadgets are toys. - False (Because, there is no direct relation between gadgets and toys so it can be possible but not possible)

II. Some electronics are gadgets. - True (Because, All gadgets are electronics. So Some electronics are gadgets, it is definite)

Hence, the correct answer is "Only conclusion II follows".

Shortcut Trick

For compounded half-yearly, the Rate becomes 12% ÷ 2 = 6% and the Time becomes 1 × 2 = 2 periods.

Effective Rate for 2 periods = 6 + 6 + (6 × 6) ÷ 100 = 12.36%.

Interest = 12.36% of 80,000 = 12.36 × 800 = ₹9,888.

Total Amount = 80,000 + 9,888 = ₹89,888.

∴ The correct answer is ₹89,888.

Alternate Method

Given:

Principal (P) = ₹80,000

Rate (R) = 12% per annum

Time (T) = 1 year

Compounding = Half-yearly

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Where r = Rate per compounding period and n = number of compounding periods.

Calculations:

⇒ Adjusted Rate (r) = 12% ÷ 2 = 6% per half-year

⇒ Number of periods (n) = 1 × 2 = 2

⇒ A = 80,000 × (1 + 6/100)²

⇒ A = 80,000 × (106/100) × (106/100)

⇒ A = 80,000 × 1.06 × 1.06

⇒ A = 80,000 × 1.1236

⇒ A = 89,888

∴ The correct answer is ₹89,888.

Additional Information

Compounding Frequency

When interest is compounded quarterly, the rate is divided by 4 and the time is multiplied by 4.

Effective Rate of Interest

For two periods, the effective rate is calculated as: x + y + (xy/100), which helps in avoiding long power calculations.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years (or 2 periods) is P(r/100)².

Shortcut Trick

Ratio of Keyboard cost : Mouse cost = 7 : 1

Total ratio parts = 7 + 1 = 8 units

8 units = ₹3,600 → 1 unit = 3,600 ÷ 8 = ₹450

Keyboard cost = 7 units = 7 × ₹450 = ₹3,150

∴ The correct answer is ₹3,150.

Alternate Method

Given:

Total cost of keyboard and mouse = ₹3,600

Cost of keyboard is 7 times the cost of the mouse.

Formula Used:

Total Cost = Keyboard Cost + Mouse Cost

Calculations:

⇒ Let the cost of the mouse be ₹x.

⇒ The cost of the keyboard = ₹7x.

⇒ According to the question, x + 7x = 3,600

⇒ 8x = 3,600

⇒ x = 3,600 ÷ 8

⇒ x = 450

⇒ Cost of the keyboard = 7x = 7 × 450 = ₹3,150

∴ The correct answer is ₹3,150.

Additional Information

Ratio and Proportion

A ratio compares two quantities. If A:B = x:y, then A = [x/(x+y)] × Total and B = [y/(x+y)] × Total.

Unitary Method

Find the value of a single unit first by dividing the total value by the total number of units, then multiply by the required number of units.

Percentage Calculation

To find what percentage the keyboard is of the total cost: (7/8) × 100 = 87.5%.

The correct answer is 1.58

Key Points

• The refractive index (n) of a medium can be determined using the equation n = c/v,
• Where
• v is the speed of light in the medium.

c is the speed of light in a vacuum

• The dispersion of light in a prism depends on the wavelength; shorter wavelengths bend more.
• However, without specific details on the prism's material or angles, we can't directly calculate n from the given wavelength (380 nm).
• The correct option,1.58, is provided without a direct calculation path, implying it might be a given property of the material for that wavelength.

Additional Information Refraction:

• When light rays pass through one medium to another medium, the band from their original path. This phenomenon is called the refraction of light.
• When it passes through a rarer to denser medium, it band towards normal.
• When it passes from denser to rarer medium, it's band away from the normal.

Reflection: 

• When light falls on the surface and get back to the same medium then it is called reflection.
• As light rays remain in the same medium, hence its wavelength remains unchanged.

The correct answer is Copper

Key Points

• Copper (Cu) is the metal that forms a green coating due to corrosion when exposed to air.
• This happens because copper reacts with moisture and gases in the air,
• It forms a mixture of copper carbonate and copper hydroxide.
• To clean tarnished copper vessels, you can use lemon or tamarind juice to restore their characteristic lustre

Additional Information

• Corrosion
• Metals when kept open react with oxygen and moisture present in the air.
• This process is called corrosion.
• The metal undercover chemically reacts with the oxygen in the atmosphere and forms an oxide on the surface of the metal.
• Rust formation over Iron when it is kept in presence of moist air is an example of corrosion.
• It leads to loss of strength and shine of metal.

The logic followed here is:

Logic: (1st number)³ + 10 = 2nd number.

For, 2 : 18

⇒ (2)³ + 10 = 8 + 10 = 18.

For, 4 : 74

⇒ (4)³ + 10 = 64 + 10 = 74.

Similarly, for 9 : ?

⇒ (9)³ + 10 = 729 + 10 = 739.

Hence, the correct answer is "Option 3".

The correct answer is The President.

Key Points

• The President is the head of the State and the Union Executive (Article 52).
• Elected by an Electoral College consisting of MPs and MLAs through proportional representation by single transferable vote (Article 54).
• Holds office for 5 years and can be re-elected (Article 56).
• Exercises executive, legislative, judicial, and emergency powers with the aid and advice of the Council of Ministers (Articles 72, 123, 356).
• The Prime Minister is the head of the Union Executive of the Government of India
• The Prime Minister is appointed by the President.
• Other ministers shall be appointed by the President on the advice of the Prime Minister.
• He is also the head of the council of ministers.
• Article 74 to Article 78 of Part V deals with the Prime Minister and Council of Ministers.
• He is the chairman of the cabinet and presides over the meetings of the Cabinet.
• The prime minister is the leader of the lower house of Parliament.
• He is the chief spokesman of the Central Government and also in foreign relations.
• He acts as the chief advisor to the President of India.
• The Prime Minister submits his resignation to the president of India.
• He is the head of the Nuclear Command Authority, NITI Aayog, Appointments Committee of the Cabinet, Department of Atomic Energy, Department of Space and Ministry of Personnel, Public Grievances and Pensions.
• He communicates all decisions of the Cabinet related to the administration of the affairs of the Union and proposals for legislation to the President and therefore acts as the link between President and cabinet.
• The death or resignation of the Prime Minister automatically leads to the dissolution of the Council of Ministers.
• To become a prime minister of India, he/she should have completed 30 years of age if he is a member of the Rajya Sabha or 25 years of age if he is a member of the Lok Sabha.

Additional Information

• The Union executive consists of the President, the Vice-President, and the Council of Ministers with the Prime Minister as the head to aid and advise the President.
• The Union Government has three organs – the Executive, the Legislature, and the Judiciary.
• Chief Justice of India does not form a part of the Union executive.
• The speaker of Lok Sabha is a member of Parliament but is not a part of the Union Executive.

Confusion Points

• The President of India is the Head of the State and Supreme Commander of the armed forces whereas PM is the head of the government and does not represent the state.

Shortcut Trick

Use the direct algebraic identity: (a − b)³ + (a + b)³ = 2a³ + 6ab².

Here, substitute a = 7x and b = 1.

⇒ 2(7x)³ + 6(7x)(1)²

⇒ 2(343x³) + 42x

⇒ 686x³ + 42x

∴ The correct answer is 686x³ + 42x.

Alternate Method

Given:

The expression is (7x − 1)³ + (7x + 1)³.

Formula Used:

(a + b)³ = a³ + b³ + 3a²b + 3ab²

(a − b)³ = a³ − b³ − 3a²b + 3ab²

Calculations:

⇒ Let us expand both terms individually using the formulas.

⇒ (7x − 1)³ = (7x)³ − 1³ − 3(7x)²(1) + 3(7x)(1)²

⇒ (7x − 1)³ = 343x³ − 1 − 147x² + 21x

⇒ (7x + 1)³ = (7x)³ + 1³ + 3(7x)²(1) + 3(7x)(1)²

⇒ (7x + 1)³ = 343x³ + 1 + 147x² + 21x

⇒ Now, add the two expanded forms together:

⇒ (343x³ − 1 − 147x² + 21x) + (343x³ + 1 + 147x² + 21x)

⇒ The terms with opposite signs cancel out (−1 and +1, −147x² and +147x²).

⇒ = (343x³ + 343x³) + (21x + 21x)

⇒ = 686x³ + 42x

∴ The correct answer is 686x³ + 42x.

Additional Information

Sum and Difference of Cubes

Basic algebraic identities are: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Algebraic Identities for Squares

Similarly, for squares: (a + b)² + (a − b)² = 2(a² + b²) and (a + b)² − (a − b)² = 4ab.

Cubes Subtraction Trick

Similar to the sum of cubes, the difference expression can be simplified directly as: (a + b)³ − (a − b)³ = 2b³ + 6a²b.

The logic followed here is:

'SCOPE' is written as 'JZNMX'

'GAMUT' is written as 'VBPHI'

Similarly, 'RANGE' will be written as:

Hence, the correct answer is "KBOVX".

According to the given information:

1. Anu lives on the sixth floor.

2. Only two people live between Anu and Shubham.

3. Only 4 people live above Aditya.

4. Sakshi lives on the topmost floor.

5) Diksha lives immediately below Sakshi.

6. Ravi lives immediately above Deep.

Thus, final arrangement:

So, Deep lives on floor 1.

Hence, the correct answer is "1".

The correct answer is 56.

Key Points

• Newland assumed that only 56 elements exist in nature.
• In 1866, John Newlands, an English scientist, arranged the then-known elements in order of increasing atomic mass.
• He started with the element with the lowest atomic mass (hydrogen) and ended with Thorium which was the 56th element.
• He found that every eighth element has the same properties as the first.
• He compared it to the octave found in music.
• That's why he called it the 'law of octaves'.
• This is known as 'Newlands' Law of Octaves'.
• The properties of lithium and sodium were found to be similar in Newlands' octave.

Additional Information

• Law of Octaves was applicable only up to calcium, as after calcium every eighth element did not possess properties similar to that of the first.
• It was assumed by Newlands that only 56 elements existed in nature and no more elements would be discovered in the future.
• But, later on, several new elements were discovered, whose properties did not fit into the Law of Octaves.
• In 1869, just five years after John Newlands put forward his Law of Octaves, a Russian chemist called Dmitri Mendeleev published a periodic table.
• Mendeleev also arranged the elements known at the time in order of relative atomic mass, but he did some other things that made his table much more successful.

The correct answer is Basic

Key Points

• Washing soda (sodium carbonate, Na₂CO₃) is a basic compound.
• When dissolved in water, it produces hydroxide ions (OH⁻), making the solution alkaline.
• This basic nature is why it is used in cleaning agents to neutralize acidic substances and remove grease or oil.
• Washing soda is a water-softening agent because it reacts with the calcium and magnesium ions in hard water to form insoluble precipitates, thereby removing hardness from the water.
• This property enhances its effectiveness in cleaning.

Additional InformationSome common chemical compounds with their common names are:

Shortcut Trick

Total jackfruits now = 18 + 2 = 20.

Total decrease in weight = 20 × 20 g = 400 g = 0.4 kg.

Total weight of 2 new jackfruits = (2 × Old Mean) − Total decrease = (2 × 7.2) − 0.4 = 14 kg.

Weight of each new jackfruit = 14 ÷ 2 = 7 kg.

∴ The correct answer is 7.00 kg.

Alternate Method

Given:

Initial number of jackfruits = 18

Initial mean weight = 7.2 kg

Decrease in mean = 20 g = 0.02 kg

Calculations:

⇒ Total weight of 18 jackfruits = 18 × 7.2 = 129.6 kg

⇒ Total number of jackfruits after addition = 18 + 2 = 20

⇒ New mean weight = 7.2 − 0.02 = 7.18 kg

⇒ Total weight of 20 jackfruits = 20 × 7.18 = 143.6 kg

⇒ Total weight of 2 added jackfruits = 143.6 − 129.6 = 14.0 kg

⇒ Weight of each added jackfruit = 14.0 ÷ 2 = 7.00 kg

∴ The correct answer is 7.00 kg.

Additional Information

Arithmetic Mean formula

The mean is calculated as the sum of all observations divided by the total number of observations: Mean = (Sum of values) ÷ (Total number of items).

Effect of New Entry on Mean

If the added value is less than the original mean, the new mean will decrease. If the added value is more, the mean will increase.

Unit Conversion

Always ensure all units are the same before calculation. In this case, 20 g = 20 ÷ 1000 = 0.02 kg.

Shortcut Trick

Initial ratio of Boys : Girls = 1 : 1. To match the future Boys part, use 2 : 2.

Final ratio after 8 girls left = 2 : 1.

Difference in Girls parts = 2 − 1 = 1 unit.

Given 1 unit = 8 girls, initial total parts = 2 + 2 = 4 units.

Initial total = 4 × 8 = 32.

∴ The correct answer is 32.

Alternate Method

Given: Initial Boys = Girls. After 8 girls left, Boys = 2 × Girls.

⇒ Let the initial number of boys and girls be x each.

⇒ Number of girls after 8 left = x − 8.

⇒ According to the problem: x = 2 × (x − 8).

⇒ x = 2x − 16.

⇒ 2x − x = 16.

⇒ x = 16.

⇒ Initial total number of students = x + x = 16 + 16 = 32.

∴ The correct answer is 32.

Additional Information

Ratio Balancing

When one part of a ratio is changed and the other remains constant, multiply the original ratio to make the constant parts equal in both states.

Linear Equations in One Variable

Standard approach for solving word problems by translating text into algebraic expressions like ax + b = c.

Constant Quantity Concept

In problems where only one group changes (e.g., girls leaving), the other group (boys) acts as a fixed reference point for calculation.

Given word:  'DANCER' 

So, there is one such pair i.e. AC.

Hence, the correct answer is "Option (3)".

Shortcut Trick

Convert the equation to standard form Ax² + Bx + C = 0 → x² − 4ax + 3a² = 0.

Identify the coefficients: A = 1, B = −4a, C = 3a².

Apply the sum of roots property: Sum = −B/A.

Sum = −(−4a) / 1 = 4a.

∴ The correct answer is 4a.

Alternate Method

Given: x² + 3a² = 4ax

Method: Standard Quadratic Root Relations

⇒ Rearranging the quadratic equation: x² − 4ax + 3a² = 0

⇒ Comparing with Ax² + Bx + C = 0, we get:

⇒ Coefficient A = 1

⇒ Coefficient B = −4a

⇒ Coefficient C = 3a²

⇒ Sum of roots (α + β) = −B / A

⇒ Sum of roots = −(−4a) / 1

⇒ Sum of roots = 4a

∴ The correct answer is 4a.

Additional Information

Product of Roots

For any quadratic equation Ax² + Bx + C = 0, the product of roots (α × β) is given by C/A. In this case, product = 3a² / 1 = 3a².

Discriminant and Nature of Roots

The value D = B² − 4AC determines the nature of roots. Here, D = (−4a)² − 4(1)(3a²) = 16a² − 12a² = 4a². Since D ≥ 0, the roots are real.

Factorization Method

The equation x² − 4ax + 3a² = 0 can be factored as (x − 3a)(x − a) = 0. The individual roots are x = 3a and x = a, and their sum is 3a + a = 4a.

Shortcut Trick For a mean proportional relationship a : x :: x : b, the value of x is √(a × b). x = √(2.5 × 4.9) = √(12.25) Since 3.52 = 12.25, the value of x is 3.5. ∴ The correct answer is 3.5. Alternate Method Given: 2.5 : x :: x : 4.9 and x > 0. Formula Used: Product of Extremes = Product of Means (ad = bc).

⇒ 2.5 / x = x / 4.9 ⇒ x2 = 2.5 × 4.9 ⇒ x2 = 12.25 ⇒ x = √12.25 ⇒ x = 3.5 ∴ The correct answer is 3.5. Additional Information Mean Proportional The mean proportional between two numbers 'a' and 'b' is the value 'x' such that a:x = x:b, calculated as x = √(ab). Third Proportional In the ratio a:b = b:c, 'c' is called the third proportional and is calculated as c = b2 / a. Fourth Proportional In the proportion a:b = c:d, 'd' is the fourth proportional to a, b, and c, calculated as d = (b × c) / a.

The correct answer is bleaching action

Key Points

• Bleaching Action is not used in Baking soda.
• Baking soda (sodium bicarbonate) is a mild alkaline compound with various uses:
• Antacids:
• It neutralizes excess stomach acid, providing relief from heartburn and indigestion.
• Fire extinguishers:
• It reacts with acids to produce carbon dioxide, which helps extinguish fires.
• Baking powder:
• A key ingredient in baking powder, where it reacts with acids to release carbon dioxide, causing baked goods to rise.

Additional Information

• Baking soda was first manufactured by French chemist Nicolas Leblanc.
• It was first manufactured in 1971.
• The common name of Sodium bicarbonate is Baking soda.
• Sodium bicarbonate is non flammable.
• Baking soda is basic in nature.
• Uses of Sodium bicarbonate are following:-
• It acts as an antacid that is used to treat stomach
• It is used in fire extinguishers due to the formation of soapy foam
• It acts as a pesticide
• It neutralizes the effect of acid as a neutralizer
• In the stomach, Reduces the acidity

The logic followed here is:

Hence, the correct answer is "148".