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Shortcut Trick

Net Discount = a + b − (a × b) ÷ 100

Here, a = 40 and b = 60.

Net Discount = 40 + 60 − (40 × 60) ÷ 100

Net Discount = 100 − 2400 ÷ 100 = 100 − 24 = 76%.

∴ The correct answer is 76%.

Alternate Method

Given: First discount = 40%, Second discount = 60%.

Formulas: Equivalent Discount = 100 − [100 × (100 − d₁)/100 × (100 − d₂)/100]

⇒ Let the Marked Price (MP) be 100.

⇒ After 40% discount, Price = 100 − 40 = 60.

⇒ After 60% discount on 60, Price = 60 − (60 × 60 ÷ 100) = 60 − 36 = 24.

⇒ Total Discount = MP − Final SP = 100 − 24 = 76.

⇒ Equivalent Discount% = (76 ÷ 100) × 100 = 76%.

∴ The correct answer is 76%.

Additional Information

Successive Discounts

When a series of discounts are applied one after another on the reducing balance, they are called successive discounts. The total effective discount is always less than the simple sum of individual discounts.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP × (100 − D%)/100.

The logic followed here is:

For, ASD ∶ EWH

For, FGH ∶ JKL

Similarly, for JKL ∶ NOP

Hence, the correct answer is "Option 2".

The logic followed here is:

Rank of Radhika from the Top = 18^th

Rank of Radhika from the Bottom = 32^th

So,

Total number of the students = Rank from the Top + Rank from the Bottom - 1

= 18 + 32 - 1

= 50 - 1

= 49.

Hence, the correct answer is "Option 1".

The correct answer is 10^th.

Key Points

• India ranks 10^th in plant diversity in the world as on December 2010.
• India is one of the 17 mega-diverse countries in the world, with a rich variety of flora and fauna.
• It is home to about 8% of the world's total plant species.
• The plant diversity in India is distributed across various biomes, including tropical rainforests, dry deciduous forests, alpine meadows, and mangroves.
• This diversity is influenced by factors such as climate, topography, and soil type.
• The National Biodiversity Act of 2002 and the National Biodiversity Authority of India are some of the initiatives taken by the government to conserve and sustainably use the country's biodiversity.

Additional Information

• The top three countries in plant diversity are Brazil, Colombia, and China, with around 55,000, 50,000, and 45,000 plant species, respectively.
• An international plant science journal called Plant Diversity (formerly known as Plant Diversity and Resources) publishes significant original research as well as review papers that enhance our knowledge of the historical and contemporary distribution of plants and aid in the creation of more phylogenetically accurate taxonomic classifications.
• The community of plant systematic and evolutionary biologists will find fresh information on or insights into evolutionary processes and mechanisms presented in it.

Given:

Susheela started her morning walk and noted that her shadow is at her back.

After some time she turned left and then again turned right.

Finally she turned right and started running.

So, Susheela was running in the 'South' direction.

Hence, the correct answer is "Option 4".

Shortcut Trick

Given sec θ = 5/3, identify the Pythagorean triplet (3, 4, 5).

Here, Hypotenuse (H) = 5, Base (B) = 3, ∴ Perpendicular (P) = √(5² − 3²) = 4.

Calculate required ratios: tan θ = 4/3, cos θ = 3/5, sin θ = 4/5, sec θ = 5/3.

Substitute: [3(4/3) − 5(3/5)] / [5(4/5) − 3(5/3)] = [4 − 3] / [4 − 5] = 1 / (−1) = −1.

∴ The correct answer is -1.

Alternate Method

Given: sec θ = 5/3 (H/B)

Formula Used: sin θ = P/H, cos θ = B/H, tan θ = P/B

⇒ H = 5, B = 3

⇒ P = √(H² − B²) = √(5² − 3²) = √16 = 4

⇒ Numerator = 3 × (4/3) − 5 × (3/5) = 4 − 3 = 1

⇒ Denominator = 5 × (4/5) − 3 × (5/3) = 4 − 5 = −1

⇒ Expression = 1 / (−1) = −1

∴ The correct answer is -1.

Additional Information

Trigonometric Ratios

Reciprocal relations: sin θ = 1/cosec θ, cos θ = 1/sec θ, and tan θ = 1/cot θ.

Pythagorean Identity

The fundamental identity: sin²θ + cos²θ = 1. Also, 1 + tan²θ = sec²θ.

Pythagorean Triplets

Common triplets used in exams include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

The correct answer is two-tier.

Key Points

• The 73^rd Amendment of the Constitution was passed in 1992 to provide constitutional status to the Panchayati Raj Institutions (PRIs) and to strengthen the local self-governance in rural areas of India.
• The Amendment provides for a three-tier structure of local governance in states with a population of more than 20 lakhs, while states with a population of less than 20 lakhs can have a minimum two-tier structure.

Additional Information

• The two-tier structure includes the Gram Panchayats at the village level and the Panchayat Samiti at the block level, while the three-tier structure includes the Zilla Parishad at the district level as the third tier.
• The purpose of the Amendment is to empower the local communities and to ensure their participation in decision-making, planning and implementation of developmental programs.
• Union rule, on the other hand, refers to the central government's control over the state governments, which is not related to the local governance structure.

Shortcut Trick

Arrange the wages in ascending order: 5000, 6000, 8000, 8500, 9300, 9500.

Since the number of observations (n = 6) is even, the median is the average of the two middle terms.

Middle terms = 8000 and 8500.

Median = (8000 + 8500) ÷ 2 = 16500 ÷ 2 = 8250.

∴ The correct answer is ₹8,250.

Alternate Method

Given: Wages = {5000, 6000, 8000, 8500, 9300, 9500}

Formula: For an even number of observations (n), Median = [ (n/2)^th term + (n/2 + 1)^th term ] ÷ 2

⇒ Number of observations (n) = 6

⇒ n/2 = 3^rd term = 8000

⇒ (n/2 + 1) = 4^th term = 8500

⇒ Median = (8000 + 8500) ÷ 2

⇒ Median = 16500 ÷ 2

⇒ Median = 8250

∴ The correct answer is ₹8,250.

Additional Information

Median for Odd Data Set

If the number of observations (n) is odd, the median is the exact middle value located at the (n + 1)/2^th position.

Empirical Relationship

For a moderately skewed distribution, 3 × Median = Mode + 2 × Mean. This helps find any one measure when the other two are known.

Impact of Outliers

The median is a measure of central tendency that is not affected by extremely large or small values (outliers), unlike the arithmetic mean.

Shortcut Trick

Let the Father's age be F and Son's age be S.

Given: F + 2S = 34 and 1.5F + S = 45.

To eliminate S, multiply the second equation by 2: 3F + 2S = 90.

Subtracting the first equation: (3F + 2S) − (F + 2S) = 90 − 34 ⇒ 2F = 56.

Dividing by 2 gives the Father's age: F = 28 years.

∴ The correct answer is 28 years.

Alternate Method

Given:

Case 1: F + 2S = 34 years

Case 2: 1.5F + S = 45 years

Calculations:

⇒ Let Father's age be x and Son's age be y.

⇒ x + 2y = 34 --- (i)

⇒ 1.5x + y = 45 --- (ii)

⇒ From (ii), y = 45 − 1.5x

⇒ Substitute value of y in (i): x + 2(45 − 1.5x) = 34

⇒ x + 90 − 3x = 34

⇒ −2x = 34 − 90

⇒ −2x = −56

⇒ x = 56 ÷ 2

⇒ x = 28

∴ The correct answer is 28 years.

Additional Information

Linear Equations in Two Variables

These are equations of the form ax + by + c = 0, which can be solved using Substitution, Elimination, or Cross-multiplication methods.

Age Problems

In age-related problems, if the current age is x, then n years ago the age was (x − n) and n years later the age will be (x + n).

Method of Elimination

This involves multiplying equations by suitable constants to make the coefficients of one variable numericaly equal, then adding or subtracting to eliminate that variable.

Shortcut Trick

If x and y are two numbers, their third proportional is given by y² ÷ x.

Here, x = (a + b) and y = (a + b)².

Third Proportional = [(a + b)²]² ÷ (a + b) = (a + b)⁴ ÷ (a + b) = (a + b)³.

∴ The correct answer is (a + b)³.

Alternate Method

Given: First term (A) = (a + b), Second term (B) = (a + b)².

Formula Used: If c is the third proportional to a and b, then a : b = b : c, which implies c = b²/a.

⇒ Let the third proportional be x.

⇒ (a + b) : (a + b)² :: (a + b)² : x

⇒ (a + b) / (a + b)² = (a + b)² / x

⇒ x = [(a + b)² × (a + b)²] / (a + b)

⇒ x = (a + b)⁴ / (a + b)

⇒ x = (a + b)³

∴ The correct answer is (a + b)³.

Additional Information

Mean Proportional

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

Fourth Proportional

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

Continued Proportion

Three numbers a, b, c are said to be in continued proportion if a/b = b/c, where b² = ac.