Search Previous Year Questions

The correct answer is 330 m/s.

Key Points

• The speed of an ultrasonic wave (or any wave) is determined by the relationship between its frequency and its wavelength, defined by the formula: Speed (v) = Frequency (f) × Wavelength (λ).
• The frequency provided is 33 kHz. In physics calculations, frequencies must be converted to the SI unit Hertz (Hz). Since 1 kHz = 10³ Hz, 33 kHz is equal to 33,000 Hz.
• The wavelength of the wave is given as 0.01 m. This value is already in the standard SI unit (meters) required for the calculation.
• By substituting the known values into the equation: v = 33,000 Hz × 0.01 m.
• The product of these two values results in a wave speed of 330 m/s.

 Additional Information

• Ultrasonic Waves:
• These are longitudinal mechanical waves with frequencies higher than 20,000 Hz (20 kHz), which is the upper threshold of human hearing.
• They possess high energy and shorter wavelengths, allowing them to penetrate deep into materials, making them useful for medical imaging (ultrasonography) and industrial cleaning.
• Animals like bats and dolphins use these waves for echolocation to navigate and hunt.
• Frequency and Period:
• Frequency (f) is the number of cycles per second. It is the reciprocal of the Time Period (T), where f = 1/T.
• Higher frequency waves carry more energy compared to lower frequency waves of the same amplitude.
• Speed of Sound:
• The speed of sound varies depending on the medium (fastest in solids, slowest in gases) and factors like temperature and humidity.
• At 0°C, the speed of sound in dry air is approximately 331 m/s, and at 20°C, it increases to about 343 m/s.

The correct answer is hydrophobic and hydrophilic interactions in water.

Key Points

• Soap molecules are typical surfactants, such as sodium stearate (C₁₇H₃₅COONa), which possess a dual-natured structure consisting of a polar hydrophilic head and a non-polar hydrophobic tail.
• The hydrophilic head is an ionic carboxylate group (COO⁻) that has a strong affinity for water, while the hydrophobic tail is a long hydrocarbon chain that repels water.
• At low concentrations in an aqueous solution, soap behaves as a normal electrolyte; however, when the concentration reaches the Critical Micelle Concentration (CMC), the molecules undergo self-assembly.
• Micelle formation occurs as a result of hydrophobic and hydrophilic interactions, where the hydrophobic tails aggregate in the center of a spherical cluster to minimize contact with water, while the hydrophilic heads face outward to remain in contact with the solvent.
• This spontaneous formation is entropy-driven, primarily due to the hydrophobic effect, which reduces the structured cage of water molecules around the non-polar tails, thereby lowering the system's Gibbs free energy.

 Additional Information

• Analysis of Other Options:
• Hydrogen Bonding: Although water molecules form hydrogen bonds with each other and the soap heads, this interaction alone does not drive the clustering of the long hydrocarbon chains into micelles.
• Electrostatic Attraction: In reality, the anionic heads of soap molecules experience electrostatic repulsion due to identical negative charges; the hydrophobic effect must be strong enough to overcome this repulsion for a micelle to be stable.
• Dispersion Forces: While London dispersion forces occur between the hydrocarbon tails in the core, they are significantly weaker than the hydrophobic interactions that initiate the assembly in water.
• Key Parameters for Micellization:
• Critical Micelle Concentration (CMC): For most soaps, the CMC is approximately 10⁻⁴ to 10⁻³ mol L⁻¹ at room temperature.
• Kraft Temperature (T_k): This is the temperature above which the solubility of the surfactant increases sharply, allowing the formation of associated colloids or micelles.
• Cleansing Action: The hydrophobic core of the micelle traps non-polar grease and oil, while the hydrophilic exterior keeps the entire aggregate suspended in water, enabling the removal of dirt.

The correct answer is 50 m.

Key Points

• In kinematics, a velocity-time graph (v-t graph) illustrates the relationship between an object's velocity and the time elapsed.
• A straight horizontal line parallel to the time axis indicates that the object is moving with a constant velocity, meaning its speed and direction do not change over time.
• The geometric area enclosed between the velocity-time graph and the time axis represents the total distance or displacement covered by the object during that period.
• For a horizontal line at 10 m/s over a duration of 5 seconds, the shape formed under the graph is a rectangle with a height of 10 units and a width of 5 units.
• The mathematical formula for distance in uniform motion is Distance = Velocity × Time.
• By substituting the given values, Distance = 10 m/s × 5 s = 50 m, confirming the total path length covered is 50 meters.

 Additional Information

• Slope of the Graph:
• The slope (gradient) of a velocity-time graph represents the acceleration of the object.
• Since a horizontal line has a zero slope, the acceleration of the object in this scenario is 0 m/s², confirming uniform motion.
• Units and Dimensions:
• The S.I. unit of velocity is meters per second (m/s), and time is measured in seconds (s).
• Multiplying these units results in (m/s) × s = m, which is the standard unit for distance.
• Variable Velocity:
• If the graph were a slanted straight line starting from the origin, it would represent uniformly accelerated motion, where the area would be calculated as a triangle (1/2 × base × height).

The correct answer is Timely irrigation and fertilization.

Key Points

• Irrigation is the artificial application of water to crops at necessary intervals. Timely irrigation is vital to prevent moisture stress, particularly during critical growth stages such as seedling establishment, flowering, and grain filling, which are essential for reaching maximum yield potential.
• Fertilization involves the application of essential nutrients like Nitrogen (N), Phosphorus (P), and Potassium (K). These macronutrients are fundamental for protein synthesis, photosynthesis, and cellular respiration, which directly enhance the biomass production of the crop.
• The synergy between water management and nutrient supply ensures that the plant maintains high turgor pressure and efficient metabolic activity. This leads to vigorous growth and a higher harvest index, contributing significantly to crop yield improvement.

 Additional Information

• Inefficient Practices:
• Monoculture: Planting the same crop year after year leads to soil nutrient depletion and the accumulation of host-specific pests and pathogens, ultimately reducing the overall yield.
• Delayed Harvest: Waiting too long to harvest can cause grain shattering, lodging, and increased exposure to microbial spoilage or pest damage, leading to substantial quantitative losses.
• Pest Neglect: Ignoring signs of infestation or disease allows the problem to cross the economic threshold level (ETL), resulting in severe damage to the photosynthetic surface and decreasing the market value of the produce.

The correct answer is Small, actively dividing cells with thin walls.

Key Points

• Meristematic tissues are defined as a group of young, immature cells that are in a continuous state of division and contribute to the growth of the plant.
• The cells of this tissue are generally small, isodiametric, or spherical in shape, ensuring they can be compactly arranged without intercellular spaces.
• These cells possess thin cellulosic cell walls, which are necessary to allow for rapid expansion and frequent cytokinesis during the growth process.
• They are characterized by dense cytoplasm, a prominent nucleus, and a very high metabolic rate, which is required to support constant cellular replication.
• Unlike permanent tissues, meristematic cells typically lack vacuoles or have very small ones, as their primary function is proliferation rather than storage or structural support.

 Additional Information

• Classification based on Position:
• Apical Meristem: Located at the root tips and shoot tips, these are responsible for primary growth and increasing the length of the plant.
• Lateral Meristem: Found along the longitudinal axis of stems and roots (e.g., vascular cambium and cork cambium), they facilitate secondary growth and increase the girth or thickness.
• Intercalary Meristem: Positioned between permanent tissues, often at the base of nodes or leaves in monocots, assisting in the elongation of internodes.
• Comparison with Permanent Tissues:
• Parenchyma: Often contains large, highly vacuolated cells primarily used for storage of food or water.
• Sclerenchyma: Composed of mature cells with thick, lignified walls that provide mechanical strength but lose the ability to divide.
• Collenchyma: Provides flexibility with cells that have unevenly thickened walls at the corners.

Shortcut Trick

Identify the numerator and denominator as perfect square identities.

Numerator: √(16x⁴ − 72x²y² + 81y⁴) = √(4x² − 9y²)² = 4x² − 9y²

Denominator: √(4x² − 12xy + 9y²) = √(2x − 3y)² = 2x − 3y

Simplified Fraction: (4x² − 9y²) ÷ (2x − 3y) = (2x + 3y)(2x − 3y) ÷ (2x − 3y) = 2x + 3y

Final Expression: (2x + 3y) − (2x − 3y) = 2x + 3y − 2x + 3y = 6y

∴ The correct answer is 6y.

Alternate Method

Given: Expression = [√(16x⁴ − 72x²y² + 81y⁴) ÷ √(4x² − 12xy + 9y²)] − (2x − 3y) and 2x > 3y.

Formula Used: (a − b)² = a² − 2ab + b² and a² − b² = (a − b)(a + b)

⇒ First, simplify the numerator term: 16x⁴ − 72x²y² + 81y⁴ = (4x²)² − 2 × (4x²) × (9y²) + (9y²)² = (4x² − 9y²)²

⇒ Now, simplify the denominator term: 4x² − 12xy + 9y² = (2x)² − 2 × (2x) × (3y) + (3y)² = (2x − 3y)²

⇒ The given expression becomes: [√(4x² − 9y²)² ÷ √(2x − 3y)²] − (2x − 3y)

⇒ Since 2x > 3y, it implies (2x − 3y) is positive. Assuming 2x + 3y is also positive based on provided options.

⇒ [(4x² − 9y²) ÷ (2x − 3y)] − (2x − 3y)

⇒ [((2x − 3y)(2x + 3y)) ÷ (2x − 3y)] − 2x + 3y

⇒ (2x + 3y) − 2x + 3y = 6y

∴ The correct answer is 6y.

Additional Information

Algebraic Identities

Expressions of the form a² − 2ab + b² are perfect squares of (a − b). Identifying these helps in removing square roots during simplification.

Difference of Squares

The identity a² − b² = (a − b)(a + b) is widely used to factorize higher-degree terms and simplify algebraic fractions.

Square Root Property

For any real number 'a', √a² = |a|. The condition 2x > 3y ensures the denominator root is treated as a positive value.

Shortcut Trick

Perform quick prime factorization of 84: 84 = 2 × 2 × 3 × 7.

Identify the distinct prime factors: 2, 3, and 7.

Sum of these factors = 2 + 3 + 7 = 12.

∴ The correct answer is 12.

Alternate Method

Given: The number is 84.

Formula Used: Sum of prime factors = P₁ + P₂ + ... + P_n (where P are distinct prime bases).

⇒ Step 1: Divide 84 by the smallest prime 2: 84 ÷ 2 = 42.

⇒ Step 2: Divide 42 by 2: 42 ÷ 2 = 21.

⇒ Step 3: Divide 21 by 3: 21 ÷ 3 = 7.

⇒ Step 4: Divide 7 by 7: 7 ÷ 7 = 1.

⇒ Prime factors of 84 = 2 × 2 × 3 × 7 = 2² × 3¹ × 7¹.

⇒ Distinct prime factors are 2, 3, and 7.

⇒ Sum = 2 + 3 + 7 = 12.

∴ The correct answer is 12.

Additional Information

Prime Numbers

Numbers that have exactly two factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11, etc.

Distinct Prime Factors

When calculating the sum of prime factors, we generally sum only the unique prime numbers (bases) that divide the given number.

Sum of All Divisors

For a number N = P₁^a × P₂^b, sum of all divisors = [(P₁^a+1 − 1)/(P₁ − 1)] × [(P₂^b+1 − 1)/(P₂ − 1)].

Shortcut Trick

Solve Numerator: (5 × 6) − (36 ÷ 3) + 2 ⇒ 30 − 12 + 2 = 20.

Solve Denominator: (3 × 3) ÷ 9 ⇒ 9 ÷ 9 = 1.

Final Calculation: 20 ÷ 1 = 20.

∴ The correct answer is 20.

Alternate Method

Given:   {[5 of (2 + 4)] − (6² ÷ 3) + 2} / {3 of (5 − 2) ÷ 9}

Formula Used: Apply BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) rule.

Step 1: Simplify the Numerator

⇒ [5 of 6] − {36 ÷ 3} + 2

⇒ 30 − 12 + 2

⇒ 18 + 2 = 20

Step 2: Simplify the Denominator

⇒ 3 of 3 ÷ 9

⇒ 9 ÷ 9 = 1

Step 3: Final Division

⇒ Numerator ÷ Denominator

⇒ 20 ÷ 1 = 20

∴ The correct answer is 20.

Additional Information

BODMAS / VBODMAS Rule

Used to solve multiple operations. V (Vinculum/Bar), B (Brackets), O (Of), D (Division), M (Multiplication), A (Addition), S (Subtraction).

Priority of 'Of'

The operator 'of' represents multiplication but is always calculated before Division and Multiplication.

Order of Brackets

Always prioritize solving from the innermost bracket: Circular () → Curly {} → Square [].

Shortcut Trick

Apply Successive Percentage formula for effective profit: a + b + (ab/100)

Net Profit % = 36 − 25 − (36 × 25)/100 = 11 − 9 = 2%

Given Profit = ₹134 ⇒ 2% of CP = ₹134 ⇒ 1% of CP = ₹67

List Price (Marked Price) = 136% of CP = 136 × 67 = ₹9,112

∴ The correct answer is ₹9,112.

Alternate Method

Given:

Mark-up = 36% above Cost Price (CP)

Discount = 25% on List Price (LP)

Profit = ₹134

Formula Used:

1. LP = CP × (100 + Mark-up%)/100
2. SP = LP × (100 − Discount%)/100
3. Profit = SP − CP

Let the Cost Price (CP) of the fan be ₹100x.

⇒ List Price (LP) = 100x + 36% of 100x = ₹136x

⇒ Discount = 25% of 136x = 136x × (25/100) = ₹34x

⇒ Selling Price (SP) = LP − Discount = 136x − 34x = ₹102x

⇒ Profit = SP − CP = 102x − 100x = 2x

⇒ According to the question: 2x = ₹134

⇒ x = 134 ÷ 2 = ₹67

⇒ List Price = 136x = 136 × 67

⇒ 136 × 67 = 9,112

∴ The correct answer is ₹9,112.

Additional Information

Relationship between CP and MP

The direct ratio of Cost Price to Marked Price is given by: CP/MP = (100 − Discount%)/(100 + Profit%).

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − d₁d₂/100)%.

Mark-up Percentage

It is the percentage by which the Cost Price is raised to decide the Marked Price: Mark-up % = (MP − CP)/CP × 100.

Shortcut Trick

In a right-angled triangle PQR at Q, P + R = 90°, hence complementary relations apply: sec P = cosec R, sin P = cos R, and cos P = sin R.

Substitute these into the expression: 1 + [cosec R (3 cos R − cos R)] ÷ [cosec R (4 sin R − sin R)]

⇒ 1 + (2 cos R) ÷ (3 sin R) = 1 + (2/3) cot R

Given tan R = 1/3 ⇒ cot R = 3.

⇒ 1 + (2/3) × 3 = 1 + 2 = 3.

∴ The correct answer is 3.

Alternate Method

Given: In ΔPQR, ∠Q = 90° and tan R = 1/3.

Let PQ = 1 and QR = 3. Using Pythagoras Theorem:

⇒ PR² = PQ² + QR² = 1² + 3² = 10

⇒ PR = √10

Now, calculate trigonometric ratios for angles P and R:

⇒ sin P = QR/PR = 3/√10 | cos P = PQ/PR = 1/√10 | sec P = √10

⇒ sin R = PQ/PR = 1/√10 | cos R = QR/PR = 3/√10 | cosec R = √10

Substitute these values into the expression 1 + [sec P (3 cos R − sin P)] ÷ [cosec R (4 sin R − cos P)]:

⇒ 1 + [√10 (3 × 3/√10 − 3/√10)] ÷ [√10 (4 × 1/√10 − 1/√10)]

⇒ 1 + [(√10 / √10) × (9 − 3)] ÷ [(√10 / √10) × (4 − 1)]

⇒ 1 + 6 ÷ 3 = 1 + 2 = 3

∴ The correct answer is 3.

Additional Information

Complementary Angle Identities

For any angle θ, sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Identities

Fundamental relations between ratios: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Trigonometric Ratios in Right Triangles

sin θ = Perpendicular/Hypotenuse, cos θ = Base/Hypotenuse, tan θ = Perpendicular/Base.