The logic followed here is :-

(First term + 2)² = Second term

In 6 : 64 → (6 + 2)² = 64

In 11 : 169 → (11 + 2)² = 169

Similarly,

In 8 : 100 → (8 + 2)² = 100

Hence, the correct answer is "100".

The correct answer is Konkani.

Key Points

• Konkani is an Indo-Aryan language spoken predominantly in the Konkan region of India, which includes Goa, Maharashtra, and Karnataka.
• It is the official language of the Indian state of Goa and is also recognized as a minority language in the states of Karnataka and Maharashtra.
• Konkani has several dialects, including Goan Konkani, Mangalorean Konkani, and Malvani Konkani.
• The script used to write Konkani varies depending on the region and community.
• In Goa, it is written in the Devanagari script, while in Maharashtra and Karnataka, it is written in the Kannada script.

 Additional Information

• Kokborok is a Tibeto-Burman language.
• It is spoken in the Indian state of Tripura and parts of Bangladesh.
• Khamti is a Tai language.
• It is spoken in parts of Myanmar and India's Arunachal Pradesh state.
• Tamang is a Tibeto-Burman language.
• It is spoken in Nepal, India, and Bhutan.

Shortcut Trick

Identify complementary angles: 12° + 78° = 90° and 23° + 67° = 90°.

Since cos² θ + cos² (90° − θ) = 1 and sin² θ + sin² (90° − θ) = 1.

Numerator = cos² 12° + cos² (90° − 12°) = cos² 12° + sin² 12° = 1.

Denominator = 2(sin² 78° + sin² (90° − 78°)) = 2(sin² 78° + cos² 78°) = 2(1) = 2.

Value = 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given: Expr = (cos² 12° + cos² 23°) / [2(sin² 78° + sin² 67°)]

Formula Used: cos(90° − θ) = sin θ | sin(90° − θ) = cos θ | sin² θ + cos² θ = 1

⇒ Numerator: cos² 12° + cos² 23°

⇒ Using cos θ = sin(90° − θ), cos 23° = sin(90° − 23°) = sin 67°

⇒ Numerator = cos² 12° + sin² 67° (Alternatively, use cos² 23° = sin² 67°)

⇒ Let's convert denominator: sin² 78° = cos² (90° − 78°) = cos² 12°

⇒ sin² 67° = cos² (90° − 67°) = cos² 23°

⇒ Expression = (cos² 12° + cos² 23°) / [2(cos² 12° + cos² 23°)]

⇒ Expression = 1 / 2

∴ The correct answer is 1/2.

Additional Information

Trigonometric Identities

Fundamental identity: sin² θ + cos² θ = 1 for any angle θ.

Complementary Angle Rules

If A + B = 90°, then sin A = cos B, tan A = cot B, and sec A = cosec B.

Square Sum for Complementary Angles

If A + B = 90°, then sin² A + sin² B = 1 and cos² A + cos² B = 1.

Shortcut Trick

Subtract the remainders from the numbers: (1876 − 1), (12503 − 3), (6877 − 2)

Numbers to find HCF for: 1875, 12500, 6875

Since 12500 and 1875 both end in multiples of 125, the HCF must be a multiple of 125.

Check options: Only 625 is a factor of 1875 (1875 = 625 × 3) and 12500 (12500 = 625 × 20).

∴ The correct answer is 625.

Alternate Method

Given: Numbers 1876, 12503, 6877 and remainders 1, 3, 2.

Formula: Greatest number = HCF(Number₁ − Remainder₁, Number₂ − Remainder₂, Number₃ − Remainder₃)

⇒ First number = 1876 − 1 = 1875

⇒ Second number = 12503 − 3 = 12500

⇒ Third number = 6877 − 2 = 6875

⇒ Prime Factorization of 1875 = 3 × 5⁴

⇒ Prime Factorization of 12500 = 2² × 5⁵

⇒ Prime Factorization of 6875 = 11 × 5⁴

⇒ HCF = 5⁴ = 625

∴ The correct answer is 625.

Additional Information

Highest Common Factor (HCF)

The largest number that divides two or more numbers without leaving a remainder. Also known as Greatest Common Divisor (GCD).

HCF with Remainders

To find the greatest number that divides x, y, and z leaving remainders a, b, and c, calculate HCF(x−a, y−b, z−c).

Common Remainder Property

If the remainder is the same but unknown, the greatest number is HCF(|x−y|, |y−z|, |z−x|).

The correct answer is Ge**.**

Key Points

• Period 3 of the Modern Periodic Table includes elements from sodium (Na) to argon (Ar).
• S and Al are all located before Ar in the same period.
• Ge does NOT belong to period 3 of the Modern Periodic Table.
• It belongs to period 4.

Additional Information

• Ar is the last element in period 3 and has an electron configuration of 1s2 2s2 2p6 3s2 3p6.
• Ge is a metalloid element with properties between those of metals and nonmetals, and is used in electronic devices.
• S is a nonmetal element that is essential for life and commonly found in minerals and fuels.
• Al is a metal element that is the third most abundant element in the Earth's crust and is commonly used in construction materials.
• Knowing the correct period of an element can help predict its properties and reactivity, as elements within the same period share similar electronic configurations.

The correct answer is Jyotirao Phule.

Key Points

• Jyotirao Phule was a social reformer, activist, and writer who lived in the 19th century in Maharashtra, India.
• He founded the Satyashodhak Samaj in 1873 with the aim of uplifting the lower castes and women through education and social reforms.
• The Satyashodhak Samaj was a movement that challenged the caste system and advocated for social equality and justice.
• Jyotirao Phule was also a strong advocate of women's education and established the first school for girls in Pune in 1848.

Additional Information

• Swami Dayanand Saraswati was a Hindu religious leader.
• He founded the Arya Samaj, a movement that aimed to reform Hinduism and promote Vedic teachings.
• Narayana Guru was a social reformer and spiritual leader from Kerala.
• He advocated for social equality and religious tolerance.
• Ishwarchandra Vidyasagar was a Bengali polymath.
• He worked towards women's education and social reforms in the 19th century.

The correct answer is Hanukkah.

Key Points

• Hanukkah is a Jewish festival that lasts for eight days and nights.
• It commemorates the rededication of the Second Temple in Jerusalem after it was reclaimed from the Seleucid Empire in the 2nd century BCE.
• The festival is also known as the Festival of Lights because of the tradition of lighting a special candelabrum called a menorah.
• Other customs associated with Hanukkah include playing with dreidels (spinning tops) and eating foods fried in oil, such as latkes (potato pancakes) and sufganiyot (jelly donuts).

Additional Information

• Nauroj is a Persian New Year celebration.
• It takes place on the spring equinox.
• Anthurium is a type of tropical flower often used in floral arrangements.
• Hornbill is a type of bird found in Africa and Asia.

Shortcut Trick

To expand (3a + 4b + 5c)², identify the coefficients: x = 3, y = 4, z = 5.

Squared terms: (3a)² = 9a², (4b)² = 16b², (5c)² = 25c².

Product terms: 2 × (3 × 4)ab = 24ab, 2 × (4 × 5)bc = 40bc, 2 × (5 × 3)ac = 30ac.

Summing all results gives the final expansion directly.

∴ The correct answer is 9a² + 16b² + 25c² + 24ab + 40bc + 30ac.

Alternate Method

Given: The expression to expand is (3a + 4b + 5c)².

Formula Used: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

⇒ Let x = 3a, y = 4b, and z = 5c.

⇒ Expansion = (3a)² + (4b)² + (5c)² + 2(3a)(4b) + 2(4b)(5c) + 2(5c)(3a)

⇒ Expansion = 9a² + 16b² + 25c² + 2(12ab) + 2(20bc) + 2(15ac)

⇒ Expansion = 9a² + 16b² + 25c² + 24ab + 40bc + 30ac

∴ The correct answer is 9a² + 16b² + 25c² + 24ab + 40bc + 30ac.

Additional Information

Algebraic Identity for Trinomial Square

The square of a trinomial is given by (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Sign Variations in Expansion

If a term is negative, the signs of its products change: (a + b − c)² = a² + b² + c² + 2ab − 2bc − 2ca.

Sum of Squares Formula

The formula can be rearranged to find the sum of squares: a² + b² + c² = (a + b + c)² − 2(ab + bc + ca).

Shortcut Trick

Simplify first purchase: 12 bats + 18 balls = 2100 ⇒ 2 bats + 3 balls = 350 (Divide by 6).

Multiply this by 4 ⇒ 8 bats + 12 balls = 1400.

Compare with second purchase: 8 bats + 22 balls = 1900.

Difference in price ⇒ (22 − 12) balls = 1900 − 1400 ⇒ 10 balls = 500 ⇒ 1 ball = 50.

Substitute to find bat cost ⇒ 2 bats + 3(50) = 350 ⇒ 2 bats = 200 ⇒ 1 bat = 100.

∴ The correct answer is Rs. 100.

Alternate Method

Given:

Total cost of 12 bats and 18 balls = Rs. 2,100

Total cost of 8 bats and 22 balls = Rs. 1,900

Calculations:

Let the cost of one bat be x and the cost of one ball be y.

⇒ 12x + 18y = 2100 ... (i)

⇒ 8x + 22y = 1900 ... (ii)

Divide equation (i) by 6:

⇒ 2x + 3y = 350 ... (iii)

Multiply equation (iii) by 4 to align coefficients of x with equation (ii):

⇒ 8x + 12y = 1400 ... (iv)

Subtract equation (iv) from equation (ii):

⇒ (8x + 22y) − (8x + 12y) = 1900 − 1400

⇒ 10y = 500 ⇒ y = 50

Substitute y = 50 in equation (iii):

⇒ 2x + 3(50) = 350

⇒ 2x + 150 = 350 ⇒ 2x = 200

⇒ x = 100

The cost of a bat is Rs. 100.

∴ The correct answer is Rs. 100.

Additional Information

Elimination Method

A process of solving a system of equations by multiplying equations by suitable factors so that the coefficients of one variable become equal, then adding or subtracting to eliminate that variable.

Linear Equations in Two Variables

Equations of the form Ax + By = C, where A, B, and C are constants. A pair of such equations can be solved to find the unique values of x and y.

Substitution Method

Expressing one variable in terms of the other from one equation (e.g., x = (C − By)/A) and substituting it into the second equation to solve for the unknown.

Shortcut Trick

Area = 2916 m²

Side = √2916

Since 50² = 2500 and 60² = 3600, the side is between 50 and 60.

The last digit of 2916 is 6, so the square root must end in 4 or 6.

Checking 54²: 54 × 54 = 2916.

∴ The correct answer is 54 m.

Alternate Method

Given:

Area of the square plot = 2916 m²

Formula Used:

Area of a square = (Side)²

Calculations:

⇒ Side² = 2916

⇒ Side = √2916

⇒ Side = √(2 × 2 × 3 × 3 × 3 × 3 × 3 × 3)

⇒ Side = 2 × 3 × 3 × 3

⇒ Side = 54 m

∴ The correct answer is 54 m.

Additional Information

Perimeter of a Square

The perimeter is the total length of the boundary, calculated as 4 × Side.

Diagonal of a Square

The diagonal of a square with side 's' is given by s√2.

Area in terms of Diagonal

The area can also be expressed as (Diagonal)² ÷ 2.

The correct answer is option 4 ) power

 

Power is a physical quantity that measures the rate at which work is done or energy is transferred or converted per unit of time.

$P=\frac{W}{t}$

Key Points

• In (SI), the standard unit of power (P) is the watt (W).
• Watt (W) which is defined as one joule of energy per second.
• Power can be used to describe the amount of work done, energy consumed, or energy transferred per unit of time.
• Power can be expressed in various forms, including mechanical power, electrical power, and thermal power.

Additional Information  There are several different units for measuring power, depending on the context and the system of measurement being used.

 

The unit of power, the "watt," (W) is named after James Watt, a Scottish engineer and inventor.

The correct answer is 2018.

Key Points

• Shaktikanta Das was appointed as the Reserve Bank of India's Governor in 2018.
• Shaktikanta Das succeeded Urjit Patel as the 25th Governor of RBI.
• Prior to his appointment as the Governor, Shaktikanta Das served as the Economic Affairs Secretary in the Ministry of Finance.
• He has also served as a member of the 15th Finance Commission and India's Sherpa to G20.

Additional Information 

• The Reserve Bank of India is both the nation's central bank and the regulatory agency in charge of overseeing the country's banking industry.
• The Indian government's Ministry of Finance is the owner of it.
• It is in charge of managing the Indian rupee's supply, issuing it, and keeping it in circulation.
• It was founded in 1935, Kolkata.

Given codes are:-

Here, code tu is used for word high.

Hence, the correct answer is "tu".

The correct answer is Option - 2

Key Points 

• Baking Powder is a mixture of a carbonate or bicarbonate and a weak acid (edible ).
• Baking powder is used to increase the volume and lighten the texture of baked goods.
• It works by releasing CO₂ gas into a batter or dough through an Acid-Base Reaction causing bubbles in the wet mixture to expand.

Important Points

• The base and acid are prevented from reacting prematurely by the inclusion of a buffer such as Cornstarch.
• Baking powder is used instead of Yeast for end-products where fermentation products would be undesirable where the batter lacks the elastic structure to hold gas bubbles for more than a few minutes, and to speed the production of baked goods.
• Because carbon dioxide is released at a faster rate through the acid-base reaction than through fermentation, breads made by chemical leavening are called Quick breads.
• The introduction of baking powder was revolutionary in minimizing the time and labor required to make breadstuffs. It led to the creation of new types of cakes, cookies, biscuits, and other baked goods.

Additional Information

• Most commercially available baking powders are made up of Sodium Bicarbonate (NaHCO₃) and one or more edible acids.
• Sodium Bicarbonate is also called baking soda.
• How Baking Powder Works-
• Acid-base reaction -
• NaHCO₃ + H⁺ → Na⁺ + CO₂ + H₂O
• The first to create a form of baking powder was an English chemist and food manufacturer Alfred Bird in 1843.
• Difference between Baking soda and Baking Powder.

Given number: 986 432 631 387 267

Arranging all numbers in ascending order

267 387 432 631 986

The third greatest number is 432 and the second digit of the third greatest number is 3

The smallest number is 267 and the third digit of the smallest number is 7.

So, the sum of the third greatest number and the third digit of the smallest number is = 3 + 7 = 10 

Hence, the answer is "10".

The correct answer is Articles 32 and 226.

Key Points

• The Constitution of India provides for the writ jurisdiction of the Supreme Court and High Courts under Articles 32 and 226 respectively.
• A writ is a formal written order issued by a court to a person or entity, either to perform a specific act or to refrain from doing a particular act.
• The writs that can be issued by the Supreme Court and High Courts include habeas corpus, quo warranto, mandamus, certiorari, and prohibition.

Additional Information

• Habeas corpus is a writ that can be used to secure the release of a person who has been unlawfully detained.
• Quo warranto is a writ that can be used to challenge the authority of a person holding public office.
• Mandamus is a writ that can be used to compel a public official or government agency to perform a specific duty.
• Certiorari is a writ that can be used to review the decision of a lower court or tribunal.
• Prohibition is a writ that can be used to prevent a court or tribunal from exceeding its jurisdiction.

Calculation:

Decrease in the sales of the brand S from the year 2019 to 2020.

401 - 255 = 146

% decreased = 146/401 × 100 = 36.40%.

∴ Option 2 is the correct answer.

The logic followed here is:

MODERN : RONMED

 GERMAN : AENGMR

Similarly,

 JUNIOR : OURJIN

Hence, the correct answer is "OURJIN".

The correct answer is Option 4.

Key Points 

• Adipose tissue which is known as body fat, is found below the skin and between the internal organs.
• It is a connective tissue that extends throughout your body.
• It is found under the skin (subcutaneous fat), between internal organs (visceral fat), and even in the inner cavities of bones (bone marrow adipose tissue) and in breast tissue.

Important Points

• Adipose tissue consists of lipid-rich cells called adipocytes.
• The adipocytes are storage reservoirs that store fat globules in a structural network of fibers.
• Apart from storage, it also serves as a cushion and insulator to protect our bodies.
• **Types of Adipocyte-**There are three types of Adipocytes.

Function of Adipose tissue-

Additional InformationDifferent types of tissue-

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The correct answer is Prasant Chandra Mahalanobis.

Key Points

• Prasant Chandra Mahalanobis was a renowned Indian statistician who was born in 1893 and passed away in 1972.
• He was a founder of the Indian Statistical Institute and made significant contributions to the field of statistics, including the development of the Mahalanobis distance measure.
• Mahalanobis was also an important figure in India's economic development, serving as the chief architect of the Second Five Year Plan, which aimed to promote industrialization and self-sufficiency in the country.

Additional Information

• Jagdish Natwarlal Bhagwati was an Indian-American economist who made significant contributions to the field of international trade theory.
• ​He was a professor at Columbia University and served as an economic advisor to several governments around the world.
• Prof. VKRV Rao was an Indian economist and educationist who played a key role in the development of higher education in India.
• He was the founder of the Institute of Social and Economic Change in Bangalore.
• VR Gadgil was an Indian politician and freedom fighter who was a member of the Indian National Congress.
• He served as a member of parliament and held several ministerial positions in the government.

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

Conclusion:

I) Some chairs are pens. Does not follows (There is no relation between chairs and pens)

II) Some tables are inks. Follows (In the above figure shows there is a relation between tables and pens)

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

The correct answer is virtual and erect.

Key Points

• The power of the lens (P) is given as +2.0 D. Since the power is positive, the lens is a convex (converging) lens.
• The focal length (f) is calculated using: f = 1/P = 1/2 = 0.5 m = 50 cm.
• The object is placed at a distance of 10 cm, which is less than the focal length (50 cm), i.e., the object lies between the optical centre and the principal focus of the convex lens.
• When an object is placed within the focal length of a convex lens, the image formed is virtual, erect, and magnified on the same side of the lens as the object.
• Using the lens formula: 1/f = 1/v − 1/u, where f = +50 cm and u = –10 cm, we get 1/50 = 1/v + 1/10, which gives v = –12.5 cm. The negative sign indicates that the image is virtual.
• Hence, the image formed is virtual and erect.

Additional Information

• Power of a Lens:
• P = 1/f (where f is in metres).
• The SI unit of power is the dioptre (D).
• Convex Lens:
• Has positive focal length.
• Converges parallel rays of light to the principal focus.
• Image Formation by a Convex Lens:
• Beyond 2F: Real, inverted, diminished.
• At 2F: Real, inverted, same size.
• Between F and 2F: Real, inverted, magnified.
• At F: Image formed at infinity.
• Within F: Virtual, erect, magnified.
• Lens Sign Convention:
• Object distance (u) is taken as negative.
• Virtual image distance (v) is negative.
• Focal length of a convex lens is positive.
• Applications of Convex Lens:
• Used in magnifying glasses, cameras, microscopes, telescopes, and for correcting hypermetropia (long-sightedness).

Shortcut Trick

Identify the corresponding vertices by observing the markings on the sides.

Vertex Q is common to both triangles. Side QR (single tick) corresponds to side QP (single tick).

Side QS (double tick) corresponds to side QT (double tick).

Following the order of vertices in ΔQRS: Q → R (single tick) → S (double tick).

The matching order in the second triangle is Q → P (single tick) → T (double tick).

∴ The correct answer is ΔQPT.

Alternate Method

Given: In the figure, side QR = QP (indicated by single tick) and side QS = QT (indicated by double tick).

Formula Used: SAS (Side-Angle-Side) Congruence Criterion states that if two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle, the triangles are congruent.

In ΔQRS and ΔQPT:

⇒ QR = QP (Given by single tick markings)

⇒ ∠RQS = ∠PQT (Vertically opposite angles are always equal)

⇒ QS = QT (Given by double tick markings)

⇒ By SAS Congruence Criterion, ΔQRS ≅ ΔQPT.

∴ The correct answer is ΔQPT.

Additional Information

SAS Congruence Rule

Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

CPCT (Corresponding Parts of Congruent Triangles)

Once two triangles are proved congruent, all their remaining corresponding parts (angles and sides) are also equal.

Vertically Opposite Angles

When two straight lines intersect at a point, the angles formed opposite to each other at the vertex are always equal.

The correct answer is Abhijeet Kunte.

Key Points

• Abhijeet Kunte is an Indian chess player who has represented the country in several international events.
• He has won several national and international tournaments, including the Commonwealth Chess Championship in 2003.
• Kunte has also been awarded the Arjuna Award in 2002, which is India's second-highest sporting honour.

Additional Information

• Dhyan Chand Award is India's highest award for lifetime achievement in sports and games, given by the Ministry of Youth Affairs and Sports.
• The award is named after the legendary hockey player Dhyan Chand, who won three Olympic gold medals for India in 1928, 1932, and 1936.
• Pritam Siwach is a boxer who won a gold medal in the 2019 National Championships.
• Sandhya Gurung is a football player who has represented India in the South Asian Games.
• Radhakrishnan Nair is a coach who has trained several Indian badminton players.

Shortcut Trick

Net percentage change = x − y − (xy/100) = 10 − 10 − (10 × 10)/100 = −1%.

Successive change of 10% increase and 10% decrease results in a net 1% decrease.

99% of initial population = 4,73,220.

100% of initial population = (4,73,220 ÷ 99) × 100 = 4,78,000.

∴ The correct answer is 4,78,000.

Alternate Method

Given:

First year increase = 10%

Second year decrease = 10%

Final population = 4,73,220

Let the population at the beginning of the first year be P.

⇒ P × (1 + 10/100) × (1 − 10/100) = 4,73,220

⇒ P × (110/100) × (90/100) = 4,73,220

⇒ P × (11/10) × (9/10) = 4,73,220

⇒ P × (99/100) = 4,73,220

⇒ P = (4,73,220 × 100) ÷ 99

⇒ P = 4,780 × 100

⇒ P = 4,78,000

∴ The correct answer is 4,78,000.

Additional Information

Successive Percentage Change

When a value changes by x% and then y%, the net change is given by [x + y + (xy/100)]%.

Same Percent Increase and Decrease

If a value is increased by x% and then decreased by x%, there is always a net decrease of (x²/100)%.

Population Formula

Population after n years = P(1 ± R/100)ⁿ, where P is the initial population and R is the rate of growth/decay.

Given Statement:

N < L > J < K > I = H ≤ G

Conclusion:

I) J = G → False (As J < K > I = H ≤ G → no clear relation between J and G can be determined)

II) K > H → True (As K > I = H → There is the relation between K and H can be determined)

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

The correct answer is Chess.

Key Points

• Priyanka Nutakki is an Indian chess player.
• In 2022, she was awarded the FIDE title of Woman Grandmaster (WGM).
• Chess is a board game played between two players on a checkered board.
• It is a popular game worldwide and is recognized as a sport by the International Olympic Committee.

Additional Information

• Hockey is a team sport played on a field with a stick and a ball.
• It is a popular sport in countries like India, Pakistan, and Australia.
• Badminton is a racquet sport played on a court with a shuttlecock.
• It is a popular sport in countries like China, Indonesia, and India.
• Wrestling is a combat sport involving grappling techniques.
• It is a popular sport in countries like the United States, Russia, and Iran.

Shortcut Trick

To double a sum at Simple Interest, use the formula: Time (T) = 100 ÷ Rate of Interest (R).

Given R = 6.25% (since 6 ¹/₄ = 6.25).

T = 100 ÷ 6.25 = 16 years.

∴ The correct answer is 16 years.

Alternate Method

Given Principal (P) = P

Given Amount (A) = 2P (as the sum gets doubled)

Given Rate of Interest (R) = 6 ¹/₄% = 25/4% per annum.

⇒ Simple Interest (SI) = Amount − Principal = 2P − P = P

⇒ Formula: SI = (P × R × T) ÷ 100

⇒ P = (P × (25 ÷ 4) × T) ÷ 100

⇒ 1 = (25 × T) ÷ 400

⇒ 25 × T = 400

⇒ T = 400 ÷ 25

⇒ T = 16 years.

∴ The correct answer is 16 years.

Additional Information

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is the Principal, R is the Rate per annum, and T is Time in years.

Amount in SI

The total Amount (A) is the sum of Principal and Simple Interest: A = P + SI = P × [1 + (R × T) ÷ 100].

Shortcut for n-times sum

If a sum becomes 'n' times itself in T years at SI, then R = [(n − 1) × 100] ÷ T.

(Left) R # A $ G H K % T B M & N Y ÷  X B / L Q @ D (Right)

1. Second symbol and third letter from the left end 

R # A $ G H K % T B M & N Y ÷  X B / L Q 

In the above series second symbol and third letter is $ G 

2. The fifth and six terms from the right end

R # A $ G H K % T B M & N Y ÷  X B / L Q @ D

In the above series the fifth and six terms from the right end is B /

If The second symbol and third letter from the left end of series are combined with the fifth and six terms from the right end of the series, then we get $ G / B.

Hence, the correct answer is "$ G / B".

The correct answer is Maharashtra.

Key Points

• According to the 2011 Census data, the concentration of Buddhism is the largest in Maharashtra.
• Maharashtra has the highest number of Buddhists in India, with over 6 million people following the religion.
• The state is home to important Buddhist pilgrimage sites such as Ajanta and Ellora caves, and the famous Buddhist stupa at Sanchi.
• While Uttarakhand also has a significant Buddhist population, it is much smaller than Maharashtra's.
• Goa and Andhra Pradesh have relatively small Buddhist populations compared to other states in India.

Additional Information

• Buddhism is an ancient religion that originated in India over 2,500 years ago and spread across Asia and beyond.
• It is based on the teachings of Siddhartha Gautama, who is also known as the Buddha or the enlightened one.
• Today, Buddhism is one of the major world religions, with millions of followers around the globe.

Shortcut Trick

When two articles are sold at the same Selling Price (SP), one at a gain of x% and another at a loss of x%, there is always a net loss.

Net loss % = x² ÷ 100 = 15² ÷ 100 = 225 ÷ 100 = 2.25%.

Total Selling Price = 20,000 × 2 = Rs. 40,000.

Loss Amount = [Total SP ÷ (100 − Loss%)] × Loss% = [40,000 ÷ (100 − 2.25)] × 2.25.

Loss Amount = (40,000 ÷ 97.75) × 2.25 ≈ Rs. 921.

∴ The correct answer is Rs. 921 loss.

Alternate Method

Given: SP₁ = SP₂ = Rs. 20,000 | Gain = 15% | Loss = 15%. Formulas: CP = [SP × 100] ÷ [100 ± Profit/Loss%] | Total Loss = Total CP − Total SP.

⇒ CP of 1st article = (20,000 × 100) ÷ 115 ≈ Rs. 17,391.30. ⇒ CP of 2nd article = (20,000 × 100) ÷ 85 ≈ Rs. 23,529.41. ⇒ Total CP = 17,391.30 ± 23,529.41 = Rs. 40,920.71. ⇒ Total SP = 20,000 × 2 = Rs. 40,000. ⇒ Overall Loss = 40,920.71 − 40,000 = Rs. 920.71 ≈ Rs. 921.

∴ The correct answer is Rs. 921 loss.

Additional Information

Net Percentage Change

When a value is increased by x% and then decreased by x%, the net change is always a decrease of (x²/100)%.

Equal Selling Price Case

If SP of two items is same, and profit% on one equals loss% on other, net loss = (Common Profit or Loss % / 10)².

The correct answer is Option 2.

Key Points 

• Pole (P/O): Geometric center of the spherical surface of the mirror.

• Centre of curvature (C): Centre of the sphere of which the mirror is a part.

• Focus point: The point where incident rays parallel to the principal axis meet or seem to be met.

• For a small spherical mirror, the principal focus ‘F’ lies exactly at the midpoint of the pole P and the center of curvature C.

• Therefore, the focal length of a spherical mirror (concave or convex) is equal to half of its radius of curvature.

• Or, The distance between the pole and the center of the curvature is twice the distance between the pole and the focus.

• This means if R = distance between the center of the curvature and the pole (Radius of curvature), f = distance between the focus point and the center of curvature (focal length).

• Then **R = 2f.**​

Additional Information 

• Some other useful terms -
• Principal Axis -  the straight line passing through the center of curvature C and pole P of a spherical mirror, produced on both sides.
• Aperture - The portion of a mirror from which the reflection of light actually takes place is called the aperture of the mirror. It is also called the linear aperture of the mirror. The aperture of a spherical mirror is denoted by the diameter of its reflecting surface.
• Types of spherical mirror -
• There are two types of spherical mirrors -
• 1. Convex Spherical Mirror: whose inner (cave type) surface is silvered and reflection takes place at the outer (convex) surface.
• 2. Concave Spherical Mirror: whose outer bulged surface is silvered and reflection takes place from the inner hollow (cave type) surface.

Uses of the concave and concave mirror -

The correct answer is Assam.

Key Points

• Assam has the oldest operating oil refinery in India since 1901, which is Digboi Refinery.
• It was established by the Assam Oil Company, which was later taken over by the Indian Oil Corporation in 1981.
• The refinery has a processing capacity of 0.65 million metric tonnes per annum and produces various petroleum products like petrol, diesel, kerosene, and LPG.
• Digboi Refinery is also known for its historical significance as it played a vital role during World War II and supplied fuel for the Allied forces.

Additional Information

• Madhya Pradesh has three oil refineries, namely Bina Refinery, Bharat Oman Refinery, and Ratnagiri Refinery.
• West Bengal has two oil refineries, namely Haldia Refinery and Barauni Refinery.
• Mumbai has two oil refineries, namely Mumbai Refinery and Bina Refinery.

From the given information: 

Ashish is moving 1k straight from his home but there is no direction is given in which direction he is moving. We can start by taking any direction. So we are starting with east direction.

After that he turned left and moved 3 km straight.

He then turned left and travelled 4 km, from there he turned left and moved 1 km.

So according to the question, if the final position of Ashish is in west direction then the starting position was in south direction.

Hence, the correct answer is "South".

The correct answer is 0.7.

Key Points

• Marginal Propensity to Consume (MPC) is the increase in consumer spending when disposable income increases by one unit.
• In the given saving function, the formula is S = -30 + 0.3Y, where S is savings and Y is income.
• Therefore, the consumption function will be C = Y - S.
• To calculate MPC, we need to take the derivative of the consumption function with respect to income, which will be MPC = dC/dY.
• Here, dC/dY = 1 - dS/dY.
• In the given saving function, dS/dY = 0.3, so MPC will be 1 - 0.3, which is equal to 0.7.
• Hence, the correct answer is 0.7.

Additional Information

• Based on income level, MPC fluctuates.
• MPC usually decreases with increasing income.
• The Keynesian multiplier, which characterizes the impact of more investment or government expenditure as an economic stimulus, has MPC as its primary factor.
• MPC cannot be more than 1, which means that a consumer cannot spend more than the increase in income.
• The marginal propensity to save is the decrease in savings when disposable income increases by one unit.
• 0 is the case where there is no increase in consumer spending with an increase in disposable income, which is highly unlikely.

The correct answer is decreasing marginal utility.

Key Points

• Marginal utility refers to the additional satisfaction or value that a consumer derives from consuming an additional unit of a product.
• Decreasing marginal utility means that as a consumer consumes more and more units of a product, the additional satisfaction or value derived from consuming each additional unit decreases.
• This happens because the consumer's needs and wants are limited, and as they consume more of a product, their needs and wants related to that product are increasingly satisfied, leading to a decrease in marginal utility.
• This concept is important for businesses to understand because it helps them determine the optimal quantity of a product to produce and sell, as well as the optimal price to charge for it.

Additional Information

• Non-satiation refers to the concept that consumers always want more of a good and will never be fully satisfied.
• It is the opposite of decreasing marginal utility.
• Increasing marginal utility refers to the concept that as a consumer consumes more units of a product, the additional satisfaction or value derived from consuming each additional unit increases.
• This concept is rare and not commonly observed in real-world scenarios.

The given series is as follows:

 (Left) 734 462 397 354 881 (Right)

 The numbers after arranging in ascending order are as follows

 (Left) 354 397 462 734 881 (Right)

So we can see that after rearranging the number only 881 remain's unchanged.

Hence, the correct answer is "option 1".

The correct answer is 2025-26.

Key Points

• The Pradhan Mantri Poshan Shakti Nirman (PM POSHAN) scheme aims to improve the nutritional status of children in the age group of 6 months to 6 years, adolescent girls, pregnant women, and lactating mothers.
• The scheme was launched in March 2018 and is implemented by the Ministry of Women and Child Development.
• The scheme provides financial assistance to States/UTs for the implementation of various interventions such as supplementary nutrition, counselling, and growth monitoring.
• The Government of India has approved the PM POSHAN for providing one hot cooked meal in Government and Government - aided Schools from 2021-22 to 2025-26.

Additional Information

• The national midday meal program in schools, often known as the midday meal scheme, was superseded by the PM-POSHAN initiative.
• It also includes pre-primary students from balvatikas, or youngsters in the age range of three to five.
• Supplementary nourishment is provided by the initiative to children in high-anemia prevalence areas and aspirational districts.
• The central government recently asked union territory authorities and state governments to look into the potential of implementing millets under the PM POSHAN Scheme.

Shortcut Trick

Diagonal of a cube = side × √3

Given, 10√3 = side × √3 ⇒ side = 10 cm

Total Surface Area (TSA) = 6 × side² = 6 × 10² = 600 cm²

∴ The correct answer is 600 cm².

Alternate Method

Given:

Longest diagonal of cube = 10√3 cm

Formula Used:

1. Length of diagonal of a cube = a√3 (where 'a' is the side)
2. Total Surface Area (TSA) of a cube = 6a²

Calculations:

⇒ a√3 = 10√3

⇒ a = 10 cm

⇒ TSA = 6 × (10)²

⇒ TSA = 6 × 100

⇒ TSA = 600 cm²

∴ The correct answer is 600 cm².

Additional Information

Volume of Cube

The space occupied by the cube is given by a³.

Lateral Surface Area (LSA)

The area of the four side faces of a cube is 4a².

Diagonal of a Face

The diagonal of any single square face of the cube is a√2.

Shortcut Trick

The diameter of the largest circle in a rectangle is equal to the shorter side of the rectangle.

Here, shorter side = 7 m ⇒ Diameter (d) = 7 m.

Circumference = π × d = (22/7) × 7 = 22 m.

∴ The correct answer is 22 m.

Alternate Method

Given: Length (L) = 12 m, Breadth (B) = 7 m.

Formula Used: Diameter of largest inscribed circle = min(Length, Breadth); Circumference = 2πr.

⇒ For a circle to fit inside a 12 m × 7 m rectangle, its diameter cannot exceed the smaller dimension.

⇒ Diameter (d) = 7 m.

⇒ Radius (r) = d/2 = 7/2 m.

⇒ Circumference = 2 × (22/7) × (7/2).

⇒ Circumference = 22 m.

∴ The correct answer is 22 m.

Additional Information

Inscribed Circle in a Square

For a square of side 'a', the diameter of the inscribed circle is exactly 'a' and the area is πa²/4.

Rectangle Properties

The largest circle's radius is half the width (shorter side) of the rectangle.

Circle Mensuration

Area of circle = πr² and Circumference = 2πr = πd.

The correct answer is Jharkhand.

Key Points

• Jharkhand has the largest coal reserves in India, accounting for about 17% of the country’s total coal reserves.

Additional Information

• The coal reserves in Chhattisgarh contribute significantly to the state's revenue and power generation capacity.
• Coal mining activities in the state are managed by public and private sector companies, including South Eastern Coalfields Limited (SECL).
• Chhattisgarh's coalfields support a large number of mining and industrial activities, with long-term plans to expand coal-based energy production.
• Along with Jharkhand, Odisha, and West Bengal, Chhattisgarh forms the core of India's coal-producing belt, critical to the country's energy security.

The correct answer is Chloroplasts.

Key Points 

• Chlorophyll is a green pigment that is responsible for photosynthesis in plants.
• Chloroplasts are organelles found in plant cells that contain chlorophyll and are responsible for photosynthesis.
• The manufacture of active substances such vitamins, lipids, amino acids, phytohormones, nucleotides, and secondary metabolites is carried out by chloroplasts.

 Additional Information

• Ribosomes are involved in protein synthesis.
• The messenger RNA (mRNA) sequence is read by the ribosome, which then converts the genetic code into a specific string of amino acids. These amino acids develop into long chains and fold to form proteins.
• Mitochondria are responsible for cellular respiration.
• Adenosine triphosphate, a little molecule, stores the chemical energy generated by the mitochondria.
• Lysosomes are responsible for breaking down waste material.
• They disintegrate extraneous or worn-out cell components.
• They could be applied to eliminate bacterial and viral invaders.

The correct answer is O.

Key Points

• The element that can form both a double bond and a single bond with a carbon atom is oxygen (O).
• Oxygen can form a double bond with carbon by sharing two electrons and a single bond by sharing one electron.
• Fluorine (F) cannot form a single bond with carbon as it requires only one electron to complete its octet and carbon requires four electrons to complete its octet.
• Therefore, fluorine can only form a single bond with carbon.
• Chlorine (Cl) and bromine (Br) can form a single bond with carbon but cannot form a double bond as they require two electrons to complete their octet and carbon requires four electrons to complete its octet.
• In addition to oxygen, nitrogen (N) can also form both a double bond and a single bond with carbon.
• However, nitrogen requires three electrons to complete its octet and carbon requires four electrons to complete its octet.
• Therefore, nitrogen can only form a triple bond or a single bond with carbon.

Additional Information

• It is important to note that the ability of an element to form a double or single bond with carbon depends on the number of valence electrons it has and its electronegativity.
• Fluorine (F) is the most electronegative element and can form only a single bond with carbon.
• Chlorine (Cl) and bromine (Br) have lower electronegativity than oxygen (O) and nitrogen (N) and can form a single bond with carbon.
• Oxygen (O) and nitrogen (N) are both highly electronegative elements and can form both single and double bonds with carbon.

The correct answer is Canada

Key Points

• The Constitution of India is a federal constitution, but it also has some unitary features.
• This feature is known as a Quasi Federal Government system.
• The Constitution of the United States of America is the oldest written national constitution still in use.
• It was adopted on September 17, 1787.
• The United States of America is a federal republic, with power divided between the federal government and the states.
• This division of powers is similar to the system in India.
• The United States of America also has a written Constitution, with a Bill of Rights that protects individual rights and freedoms.

Additional Information

• The United Kingdom is a unitary state, with power concentrated in the central government.
• The UK does not have a written constitution, and its system of government is based on parliamentary sovereignty.
• Canada 
• The Canadian Constitution Act of 1867 established the federal system of government in Canada.
• South Africa is a unitary state, with power concentrated in the central government.
• The Constitution of South Africa was adopted in 1996, and it established a federal system of government with power divided between the national government and the provinces.

The given series is as follows:

 (Left) 485 746 379 137 254 (Right)

 The numbers after arranging in ascending order are as follows

 (Left) 137 254 379 485 746 (Right)

So we can see that after rearranging the number only 379 remain's unchanged.

Hence, the correct answer is "379".

The following seating arrangement is as follows:

There are 20 students of a class standing in a straight row facing north.

1) Shilpa is 7th from the right end

2) Nikhil is 18th from the left end

So the final arrangement,

So as per the arrangement 3 students are sitting between Shilpa and Nikhil.

Hence, the correct answer is "3".

Joint Forest Management (JFM) is a collaborative approach to forest management that involves local communities and government forest departments.

Key Points

• It is based on the principle that local communities have a vested interest in the sustainable management of their forests, and that they should be given a say in how their forests are managed.
• JFM is an example of a participatory approach to development.
• Participatory approaches emphasize the importance of involving local communities in the planning and implementation of development projects.
• This can lead to more sustainable and effective development outcomes.

Key Points

• Single-man dominance is an example of a top-down approach, in which decisions are made by a single individual or small group of individuals.
• Economic equality is an example of a distributive justice approach, which focuses on the fair distribution of resources.
• Political equality is an example of a procedural justice approach, which focuses on ensuring that everyone has a fair opportunity to participate in decision-making processes.

Hence, Joint forest management is an example of participatory approach.

The correct answer is C6H12O6.

Key Points

• Glucose is represented by C6H12O6.
• Glucose is a simple sugar and the most important carbohydrate in the body.
• Glucose is a monosaccharide, which means it is a single sugar molecule.
• It is a primary source of energy for the body and is used by every cell.
• Glucose is commonly found in foods such as fruits, vegetables, and grains.
• When glucose levels in the body are too high, it can lead to health problems such as diabetes.

Additional Information

• C2H5OH is the formula for ethanol, which is a type of alcohol.
• Ethanol is a colorless, volatile, flammable liquid with a strong flavor and odor similar to wine.
• CH3COOH is the formula for acetic acid, which is the main component of vinegar.
• The distinctive smell of vinegar is caused by the fermentation byproduct known as acetic acid.

The correct answer is Uttar Pradesh.

Key Points

• As the top-ranked state in Category I, Uttar Pradesh announced that it has won the "E-Panchayat Puraskar 2021."
• Assam and Chhattisgarh came in second and third, respectively.
• Tamil Nadu and Odisha came in third, according to the government.

Additional Information

• e-Panchayat Puraskar is an annual award scheme launched by the Ministry of Panchayati Raj to recognize the best performing Panchayats in the country.
• Category I includes Panchayats with a population of up to 5,000 and Category II includes Panchayats with a population of more than 5,000.
• Haryana, and Rajasthan were also among the states that received awards in different categories of e-Panchayat Puraskar 2021.
• The award scheme also aims to encourage and motivate Panchayats to improve their functioning and services for citizens.

The given logic here is as follows:

The difference of numbers follows the prime number series.

Hence, the correct answer is "118".

Prime number series: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43............

The correct answer is IIT Kanpur.

Key Points

• As per a news report of February 2022, IIT Kanpur developed a biodegradable nanoparticle to protect crops from fungal and bacterial infections.
• A novel biodegradable carbenoid metabolite based on nanoparticles has been developed by researchers.
• It can assist farmers in preventing bacterial and fungal diseases in their crops and serve as an alternative to chemical-based insecticides.
• IIT Kanpur is a public technical and research university located in Uttar Pradesh.

Additional Information

• IIT Guwahati is a premier institute of engineering and technology located in Assam.
• IIT Kharagpur is a public technical and research university located in West Bengal.
• The technical university known as IIT Roorkee is situated in Roorkee, Uttarakhand, India.

Given:-

Six people → U, V, W, X, Y and Z are sitting in a row facing north.

1. Z is sitting second to the right of V ans immediate left of V.

                                     

                             

2. Only one person is sitting between U and V.

Case 1 → Not possible as position at which U could be placed is already occupied by Z.

Case 2 → Not possible as only position is empty to the left of V and to the right of V the position is already occupied by Z.

Case 3:-

3. W is not an immediate neighbour of U.

4. The only position left will be occupied by X.

Hence, the correct answer is "Z".

Shortcut Trick

Distance = [ (S₁ × S₂) ÷ (S₁ + S₂) ] × Total Time

⇒ Distance = [ (3 × 5) ÷ (3 + 5) ] × 4

⇒ Distance = (15 ÷ 8) × 4 = 15 ÷ 2 = 7.5 km

∴ The correct answer is 7.5 km.

Alternate Method

Given:

Speed (Going) = 3 km/h

Speed (Returning) = 5 km/h

Total Time = 4 hours

Formula Used:

Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

Total Distance (Round Trip) = Average Speed × Total Time

Calculations:

⇒ Average Speed = (2 × 3 × 5) ÷ (3 + 5) = 30 ÷ 8 = 3.75 km/h

⇒ Total Distance (covered in both ways) = 3.75 × 4 = 15 km

⇒ Distance between Home and School = Total Distance ÷ 2

⇒ Required Distance = 15 ÷ 2 = 7.5 km

∴ The correct answer is 7.5 km.

Additional Information

Average Speed for Constant Distance

When a person travels the same distance at two different speeds S₁ and S₂, the average speed is (2S₁S₂) ÷ (S₁ + S₂).

Speed and Time Relation

For a constant distance, speed is inversely proportional to time (S₁ ÷ S₂ = T₂ ÷ T₁).

Distance Formula

Distance = Speed × Time. Ensure units are consistent (e.g., km/h with hours or m/s with seconds).

The correct answer is Tungsten.

Key Points

• Tungsten has a very high melting point which makes it ideal for use in the heating element of a bulb.
• Tungsten is frequently utilized in heavy metal alloys that are used to make cutting tools, such as high speed steel.
• It is also utilized to create wear-resistant coatings in what are known as "superalloys."
• Known to exist among the strongest materials are tungsten and diamond.

Additional Information

• Mercury is a liquid metal at room temperature and is used in fluorescent bulbs.
• Another name for it is quicksilver.
• Silicon is a semiconductor material.
• It is a tetravalent metalloid that is a hard, brittle crystalline solid with a metallic sheen that is blue-grey.
• Copper is a good conductor of electricity and is used in electrical wiring and circuits.
• It is an extremely high thermal and electrical conductivity metal that is ductile, soft, and malleable.

Given:-

45 × 15 ÷ 40 - 30 + 5

If '+' and '-' are interchanged and also '×' and '÷' are interchanged then

New equation → 45 ÷ 15 × 40 + 30 - 5

Now, By applying BODMAS Rule, we get

45 ÷ 15 × 40 + 30 - 5

3 × 40 + 30 - 5

120 + 30 - 5

150 - 5 = 145

Hence, the correct answer is "145".

The least possible Venn diagram for the given statements is as follows:-

Conclusions:

(I) All pencils are bottles → Follows (As all pencils are bags and all bags are bottles so, it is definite that all pencils are bottles.)

(II) Some bags are notebooks → Follows (As some notebooks are pencils and all pencils are bags so, it is definite that some bags are notebooks.)

Hence, the correct answer is "Both conclusions I and II follows".

Given number:-

Left 354982 Right

New number after arranging each digit of the given number in ascending order:-

Left 234589 Right

Digits whose position will remain unchanged as compared to that in original number → 4 and 8

Hence, the correct answer is "Two".

Given:-

1. Shyam starts from Point A and drives 15 km towards the north.

2. He then takes a right turn, drives 5 km, turns right and drives 20 km.

3. He then takes a right turn and drives 10 km.

4. He takes a final right turn, drives 5 km and stops at point P.

5. Distance and direction towards which shyam has to travel to reach point A from point P in a straight line.

Hence, the correct answer is "5 km towards the east".

Let the number of girls be ' x '.

And the number of boys be ' x + 10 '.

Given → Total Students = 50.

So, (x) + (x + 10) = 50.

2x + 10 = 50

2x = 40

x = 20.

So, the number of girls (x) is 20.

And the number of boys (x + 10) is 30.

Hence, the correct answer is " 30 ".

Shortcut Trick

For right-angled triangles, the specific congruence criterion is RHS (Right angle-Hypotenuse-Side).

Two right triangles are congruent if the hypotenuse and one side of one triangle are equal to the corresponding parts of the other.

While other criteria like SAS, ASA, and SSS also apply, RHS is the unique condition specifically defined for right triangles.

∴ The correct answer is When their hypotenuses are equal and one of the sides of one of these triangles is equal in length to the corresponding side of the other triangle.

Alternate Method

Given: Two triangles with one angle equal to 90°.

Property Used: RHS Congruence Rule.

B C A Q R P Hypotenuse Hypotenuse Side

Shortcut Trick

For a continued proportion a : b :: b : c, the middle term is the mean proportional.

Trick logic: y = (18 × 18) ÷ 6

⇒ y = 18 × 3 = 54

∴ The correct answer is 54.

Alternate Method

Given: 6 : 18 :: 18 : y

Formula Used: Product of Extremes = Product of Means

⇒ 6 × y = 18 × 18

⇒ 6y = 324

⇒ y = 324 ÷ 6

⇒ y = 54

∴ The correct answer is 54.

Additional Information

Third Proportional

In a continued proportion a : b :: b : c, the term c is called the third proportional to a and b, where c = b² ÷ a.

Mean Proportional

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

Fourth Proportional

In the proportion a : b :: c : d, d is the fourth proportional and is calculated as (b × c) ÷ a.

The correct answer is sunlight.

Key Points

• When silver chloride (AgCl) is exposed to sunlight, it undergoes photodecomposition and forms silver metal and chlorine gas.
• This reaction causes the white AgCl to turn grey.
• AgCl is a white, insoluble, and inert compound that is commonly used in photography and silver plating.
• It is also used in the production of specialty glass, ceramics, and catalysts.
• CO₂ and O₂ do not have any significant effect on AgCl.
• Water can cause AgCl to dissolve and form a colorless solution.

Additional Information

• CO₂ is a colorless and odorless gas that is present in the atmosphere.
• It is a greenhouse gas that contributes to global warming.
• It is also used in the production of carbonated beverages, dry ice, and fire extinguishers.
• O₂ is a colorless and odorless gas that is essential for the survival of most living organisms.
• It is used in the production of steel, welding, and medical applications.
• Sunlight is a form of electromagnetic radiation that is emitted by the sun.
• It contains a range of wavelengths, including ultraviolet (UV) radiation.
• UV radiation can cause photodecomposition of certain compounds, such as AgCl.

Calculation:

$\frac{2+ \sqrt3}{2- \sqrt3}+ \frac{2- \sqrt3}{2+ \sqrt3}+ 11$

By Rationalization of the $\frac{2+ \sqrt3}{2- \sqrt3}+ \frac{2- \sqrt3}{2+ \sqrt3}$

⇒ $\frac{(2+ \sqrt3)(2+ \sqrt3)}{4- 3}$ + $\frac{(2- \sqrt3)(2- \sqrt3)}{4- 3}$ + 11

⇒ $4 + 3 + 4\sqrt{3}$ + $4 + 3 - 4\sqrt{3}$ + 11

⇒ 14 + 11

⇒ 25

The square root of 25 is 5.

∴ Option 4 is the correct answer.

The correct answer is Lack of O₂.

Key Points

• Our muscles need oxygen to function properly.
• Lack of oxygen can cause muscle cramps.
• Oxygen helps in the process of energy production in our muscles.
• Without enough oxygen, this process becomes less efficient, leading to muscle fatigue and cramping.
• Lack of oxygen can also cause a buildup of lactic acid in the muscles, which can contribute to cramping.

Additional Information

• Oxygen is also important for removing waste products from the muscles.
• Without enough oxygen, waste products can build up and cause muscle cramps.
• Carbon dioxide is a waste product of cellular respiration and has no direct impact on muscle cramps.
• Excess carbon dioxide can lead to respiratory acidosis, which can cause muscle weakness and fatigue, but not necessarily cramps.
• Excess oxygen is not harmful to the body, but it may not necessarily prevent muscle cramps.

The correct answer is m = v/u.

Key Points

• Magnification is the ratio of the size of the image to the size of the object.
• It is a dimensionless quantity.
• Image distance (v) is the distance between the lens and the image formed by the lens.
• It is measured in meters.
• Object distance (u) is the distance between the lens and the object being viewed.
• It is also measured in meters.
• The correct relation between m, v, and u for a lens is given by the formula m = v/u.
• This means that magnification is directly proportional to the object distance and inversely proportional to the image distance.

Additional Information

• Magnification can be positive or negative depending on the orientation of the image formed by the lens.
• If the image is upright and virtual, magnification is positive.
• If the image is inverted and real, magnification is negative.
• The focal length of a lens is the distance between the lens and the point at which the light rays converge or diverge.
• It is denoted by the symbol f.
• The formula for the focal length of a lens is given by 1/f = 1/v - 1/u.
• This formula is used to determine the focal length of a lens if the image distance and object distance are known.
• The unit of measurement for focal length is meters.

Shortcut Trick

For any quadratic equation of the form ax² + bx + c = 0, calculate the discriminant (D = b² − 4ac).

In x² + 7x − 60 = 0, we have a = 1, b = 7, and c = −60.

D = 7² − 4 × 1 × (−60) = 49 + 240 = 289.

Since the discriminant D > 0, the equation has two real and unequal roots.

∴ The correct answer is two real and unequal roots.

Alternate Method

Given:

The quadratic equation is x² + 7x − 60 = 0.

Formula Used:

The nature of roots is determined by the discriminant (D):

D = b² − 4ac

1. If D > 0, roots are real and unequal.
2. If D = 0, roots are real and equal.
3. If D < 0, roots are imaginary (no real roots).

Calculations:

⇒ Comparing x² + 7x − 60 = 0 with ax² + bx + c = 0:

⇒ a = 1, b = 7, c = −60

⇒ D = (7)² − 4 × 1 × (−60)

⇒ D = 49 + 240

⇒ D = 289

⇒ Since 289 > 0, the value of the discriminant is positive.

⇒ Therefore, the equation has two real and unequal roots.

∴ The correct answer is two real and unequal roots.

Additional Information

Quadratic Formula

The roots of ax² + bx + c = 0 are given by x = [−b ± √(b² − 4ac)] ÷ 2a.

Sum and Product of Roots

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

Nature of Roots Table

D > 0 and a perfect square: Roots are rational and unequal. D > 0 and not a perfect square: Roots are irrational and unequal.

Complex Roots

If D < 0, roots are complex conjugates of each other, often written in the form p ± iq.

The logic used here is → 6 is added to each letter to get the diagonally opposite letter.

KHADI:

DRINK:

Similarly,

INLET:

Hence, the correct answer is " ZKRTO ".

The correct answer is Kerala.

Key Points

• India’s first Graphene Innovation Centre set up in Kerala.
• Graphene is a form of carbon that has unique properties such as high thermal and electrical conductivity, mechanical strength, and flexibility.
• It is considered a wonder material with potential applications in various fields such as electronics, energy, and medicine.
• India is one of the leading countries in graphene research and has several research institutions and startups working on graphene-related projects.
• The centre is a collaboration between the Telangana government and the Indian Institute of Technology (IIT) Hyderabad.
• It aims to promote research and development of graphene-based technologies and products.

Additional Information

• Kerala, Gujarat, and Karnataka are among the states in India that have active research and development activities in graphene.
• Kerala has a Graphene Research Centre at the Cochin University of Science and Technology.
• Gujarat has a Graphene Centre at the Gujarat Council on Science and Technology.
• Karnataka has several institutions and startups working on graphene research, including the Indian Institute of Science (IISc) and the startup Log 9 Materials.

The correct answer is blue-green.

Key Points

• Copper (II) chloride (CuCl₂) is a compound that is greenish-brown in solid form.
• However, when it is dissolved in water, it forms a blue-green solution.
• The blue-green colour of the solution is due to the presence of Cu2+ ions, which absorb light in the orange and red regions of the spectrum and appear blue-green to the human eye.

Additional Information

• Copper (II) chloride is commonly used as a reagent in chemistry experiments and is also used in the production of pigments, dyes, and wood preservatives.
• Copper (I) chloride can produce a red or green colour in solution depending on the concentration and conditions.
• Cobalt (II) chloride produces a pink/red colour in solution.
• Nickel (II) chloride produces a yellow-green colour in solution.

Shortcut Trick

Identify the Common Factors among the given numbers: 450, 1500, and 1650.

Find the Difference between the closest numbers: 1650 − 1500 = 150.

Check if 150 divides all numbers: 450 ÷ 150 = 3, 1500 ÷ 150 = 10, 1650 ÷ 150 = 11.

Since 150 divides all three numbers, it is the Highest Common Factor (HCF).

∴ The correct answer is 150.

Alternate Method

Given: Numbers 450, 1500, and 1650.

Formula Used: Prime Factorization Method.

Calculations:

⇒ Prime factors of 450 = 2 × 3 × 3 × 5 × 5 = 2¹ × 3² × 5²

⇒ Prime factors of 1500 = 2 × 2 × 3 × 5 × 5 × 5 = 2² × 3¹ × 5³

⇒ Prime factors of 1650 = 2 × 3 × 5 × 5 × 11 = 2¹ × 3¹ × 5² × 11¹

⇒ HCF = Product of the lowest power of all common prime factors.

⇒ HCF = 2¹ × 3¹ × 5² = 2 × 3 × 25 = 150.

∴ The correct answer is 150.

Additional Information

HCF (Highest Common Factor)

The largest positive integer that divides each of the integers without a remainder. Also known as GCD (Greatest Common Divisor).

Difference Method for HCF

The HCF of any two or more numbers is always a factor of the difference between those numbers.

Relation between HCF and LCM

For two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a,b) × LCM(a,b).

The correct answer is is negative.

Key Points

• The height of the image of an object below principal axis of a spherical mirror is negative.
• This is because the image formed by a spherical mirror below the principal axis is virtual and inverted.
• As a result, the height of the image will be negative.
• The position of the object will affect the location and size of the image, but it does not determine the height of the image.
• The height of the image will vary depending on where it is located with respect to the principal axis, and if the image is located below the principal axis, the height will be negative.

Additional Information

• The height of the image will vary depending on where it is located with respect to the principal axis.
• The position of the object will affect the location and size of the image, but it does not determine the height of the image.
• If the image is located above the principal axis, the height will be positive.
• Spherical mirrors are mirrors that have a curved surface, with the shape of a sphere.
• The principal axis is an imaginary line that passes through the center of the sphere and the vertex of the mirror.
• The height of the image refers to the distance between the top of the image and the principal axis, or between the bottom of the image and the principal axis.
• The height of the object refers to the distance between the top of the object and the principal axis, or between the bottom of the object and the principal axis.

The logic used here is → Numbers are multiplied by 2 to get the alternate numbers of the series.

Hence, the correct answer is " 56 ".

Shortcut Trick

Identify the nature of roots by calculating the discriminant (D = b² − 4ac).

For the equation 3x² − 4√3 x + 4 = 0, D = (−4√3)² − 4(3)(4).

D = (16 × 3) − 48 = 48 − 48 = 0.

Since D = 0, the equation must have two equal real roots.

∴ The correct answer is two equal real roots.

Alternate Method

Given quadratic equation: 3x² − 4√3 x + 4 = 0

The standard form of a quadratic equation is ax² + bx + c = 0.

By comparison, we find the coefficients:

a = 3

b = −4√3

c = 4

The nature of roots is determined by the discriminant (D) formula:

D = b² − 4ac

⇒ D = (−4√3)² − 4 × 3 × 4

⇒ D = (16 × 3) − 48

⇒ D = 48 − 48

⇒ D = 0

When the discriminant of a quadratic equation is equal to zero, the roots are real and equal.

∴ The correct answer is two equal real roots.

Additional Information

Nature of Roots of Quadratic Equation

The value D = b² − 4ac is called the discriminant. It decides whether the roots are real, distinct, or imaginary.

Distinct Real Roots

If D > 0, the roots are real and distinct (unequal).

Imaginary Roots

If D < 0, the roots are imaginary (no real roots exist for the equation).

Perfect Square Discriminant

If D > 0 and is a perfect square, the roots are rational and distinct (provided a, b, c are rational).

The correct answer is A traditional way of collecting water.

Key Points

• A 'bawri' is a traditional way of collecting water.
• It is a stepwell, which is a type of well or pond that was used for accessing groundwater in ancient times.
• Stepwells were constructed by digging into the ground and building steps leading down to the water level.
• They were commonly found in the arid regions of India and were used for both irrigation and drinking water.
• Stepwells were also used for social gatherings and religious ceremonies.

Additional Information

• A narrow bore tube-well is a modern method of accessing groundwater.
• It involves drilling a narrow borehole into the ground and installing a tube or pipe to bring the water to the surface.
• This method is commonly used in areas where groundwater is available at a deep level.
• An underground tank is a storage facility for water.
• It is used to store water for later use, particularly in areas where water is scarce.
• A canal is a man-made waterway that is used to transport water from one place to another.
• Canals are commonly used for irrigation, particularly in agricultural areas.

The correct answer is 5.6.

Key Points

• Acid rain is caused by the release of sulfur dioxide and nitrogen oxides into the atmosphere, which react with water, oxygen, and other chemicals to form sulfuric acid, nitric acid, and other pollutants.
• The pH scale ranges from 0 to 14, with 7 being neutral.
• Anything below 7 is considered acidic, and anything above 7 is considered basic.
• The pH of acid rain is typically between 4.2 and 5.6, which is significantly more acidic than normal rainwater (pH 5.6-6.5).
• Acid rain can have harmful effects on the environment, including damaging crops and forests, making bodies of water more acidic, and causing respiratory problems in humans.

Additional Information

• A pH value of 8.6 is actually considered basic, not acidic.
• Water with a pH of 8.6 is slightly alkaline and can be found in some natural springs and wells.
• A pH value of 6.6 is still slightly acidic, but less so than acid rain.
• This pH level can be found in some natural bodies of water and is generally safe for aquatic life.
• A pH value of 7.6 is neutral, not acidic.
• This pH level can be found in most tap water and is safe for human consumption.

The correct answer is A coil carrying conductor experiences force in the region of non-uniform magnetic field.

Key Points

• An electric motor works on the principle of converting electrical energy into mechanical energy.
• It consists of a coil of wire (conductor) placed in a magnetic field.
• When an electric current is passed through the coil, it experiences a force due to the interaction of the magnetic field and the current-carrying conductor.
• This force causes the coil to rotate, which in turn rotates the shaft of the motor and produces mechanical energy.
• This phenomenon is known as electromagnetic induction, which is the basis of the working of electric motors and generators.

Additional Information

• The induced voltage is directly correlated with the wire's number of coils or twists.
• The voltage generated increases with the number of turns.
• The induced voltage changes as the magnetic field changes.
• The conductor can be moved in the magnetic field or the magnetic field can be moved around the conductor to accomplish this.
• The electromagnetic induction concept underlies the operation of AC generators.
• Electrical transformers operate on the principle of electromagnetic induction.
• The electromagnetic induction principle underpins the magnetic flow meter.

The correct answer is source of a constant magnetic field.

Key Points

• A solenoid is a coil of wire, and when current flows through it, a magnetic field is generated.
• The magnetic field inside a solenoid is uniform and strong, making it useful for applications such as electromagnets and magnetic locks.
• The strength of the magnetic field can be increased by increasing the current or the number of turns in the coil.
• A solenoid is a type of electromagnet that consists of a coil of wire wrapped around a cylindrical core.
• The magnetic field produced by a solenoid is proportional to the current flowing through the coil and the number of turns in the coil.
• Solenoids are commonly used in a variety of applications, including door locks, valves, relays, and motors.

Additional Information

• A solenoid carrying current does not emit light on its own.
• However, it can be used in conjunction with other components such as LEDs or incandescent bulbs to create a light source.
• If the solenoid is not constructed properly or if there is a defect in the coil, it may produce a non-uniform magnetic field.
• A solenoid can be used to convert electrical energy into mechanical energy.
• When current flows through the coil, it creates a magnetic field that can be used to move a plunger or other mechanical component.

The correct answer is Empowers them to decide about child bearing.

Key Points

• Formal health education provides knowledge about sexually transmitted diseases, which is important for women to protect themselves and their partners from infections.
• It also provides information about nutritional needs of children, which is important for women to ensure that their children grow up healthy and strong.
• Formal health education helps to reduce child and maternal deaths by teaching women about the importance of prenatal care, safe delivery practices, and postnatal care for both the mother and the child.
• Empowering women to decide about child bearing is also an important aspect of formal health education, as it enables them to make informed choices about their reproductive health and family planning.

Additional Information

• Sexually transmitted diseases (STDs) are infections that are passed from one person to another through sexual contact.
• Formal health education can provide information on how to prevent and treat STDs, as well as how to reduce the risk of transmission.
• Nutritional needs of children vary depending on their age, gender, and activity level.
• Formal health education can provide information on how to meet these needs through a balanced diet and healthy eating habits.
• Child and maternal deaths are a major public health concern, particularly in low-income countries.
• ​Formal health education can help to address this issue by improving access to healthcare services and promoting healthy behaviors.
• Empowering women to make decisions about child bearing can have a positive impact on their health and well-being, as well as that of their families.
• This can include access to family planning services, education on contraception methods, and support for safe and healthy pregnancies.

Shortcut Trick

For two pipes filling a tank in x and y hours, the time taken together is (x × y) / (x + y).

Required time = (25 × 35) / (25 + 35)

Required time = 875 / 60

Divide by 5 = 175 / 12

∴ The correct answer is 175/12 hours.

Alternate Method

Given: Time for Pipe 1 = 25 hours, Pipe 2 = 35 hours.

Formula Used: Total Work = LCM of individual times; Efficiency = Total Work ÷ Time.

⇒ Combined Efficiency = 7 + 5 = 12 units/hour

⇒ Time taken = Total Work ÷ Combined Efficiency

⇒ Time taken = 175 / 12 hours

∴ The correct answer is 175/12 hours.

Additional Information

Pipes & Cisterns Concept

If an inlet pipe fills a tank in A hours and an outlet pipe empties it in B hours, net work per hour = (1/A) − (1/B).

Efficiency and Time

Efficiency is inversely proportional to time. If the ratio of time is 5:7, the ratio of efficiency is 7:5.

Shortcut Trick

Using the Net Deviation method from the mean (56):

Deviations: (50−56), (63−56), (62−56), (64−56), (48−56), (51−56), (55−56), (48−56)

Sum of deviations = −6 + 7 + 6 + 8 − 8 − 5 − 1 − 8 = −7

Since the total net deviation must be zero: (x − 56) + (−7) = 0 ⇒ x = 56 + 7 = 63

∴ The correct answer is 63 kg.

Alternate Method

Given:

Weights of 9 members = 50, 63, x, 62, 64, 48, 51, 55, 48 kg

Mean weight = 56 kg

Formula Used:

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

Calculations:

⇒ Sum of weights = 56 × 9

⇒ 50 + 63 + x + 62 + 64 + 48 + 51 + 55 + 48 = 504

⇒ 441 + x = 504

⇒ x = 504 − 441

⇒ x = 63

∴ The correct answer is 63 kg.

Additional Information

Arithmetic Mean

The average of a set of numerical values, calculated by adding them together and dividing by the count of terms: Σx ÷ n.

Properties of Mean

If a constant k is added to or subtracted from every term, the new mean will be (Old Mean ± k).

Median of Data

The middle value when the data is arranged in ascending order. For 9 terms, it is the (9+1)÷2 = 5^th term.