Search Previous Year Questions

The correct answer is - 50 cm.

Key Points

• When two or more thin lenses are placed in physical contact with each other coaxially, the total optical power (P) of the combination is equal to the algebraic sum of the individual powers of the lenses.
• The formula for the equivalent power is expressed as P = P₁ + P₂. In this specific case, the given powers are P₁ = + 4 D and P₂ = − 6 D.
• By substituting these values into the formula, the net power of the lens combination is calculated as P = (+ 4) + (− 6) = − 2 D.
• The focal length (f) of a lens system is mathematically defined as the reciprocal of its power. When the power is measured in Dioptres (D), the focal length is initially calculated in meters (m) using the relation f = 1 / P.
• For the resultant power of − 2 D, the focal length in meters is f = 1 / (− 2) = − 0.5 m.
• To convert the focal length into centimeters (cm) as required by the question, the value in meters must be multiplied by 100, leading to − 0.5 × 100 = − 50 cm.

 Additional Information

• Nature of Lens and Power:
• A convex lens is known as a converging lens and is assigned a positive power because it brings parallel rays of light to a real focus.
• A concave lens is known as a diverging lens and is assigned a negative power because it causes parallel rays of light to spread out from a virtual focus.
• Since the resultant power in this calculation is negative (− 2 D), the entire lens combination behaves effectively as a single concave (diverging) lens with a focal length of 50 cm.
• The S.I. unit of power of a lens is the Dioptre (D), where 1 D is equivalent to the power of a lens whose focal length is exactly 1 meter.

The correct answer is 100 J.

Key Points

• Kinetic Energy (K.E.) is the energy an object possesses due to its motion. It is mathematically expressed by the formula \( K = \frac{1}{2}mv^{2} \), where m represents the mass of the object and v represents its velocity.
• Based on the initial conditions, the object has a velocity (v₁) of 5 m/s and a Kinetic Energy (K₁) of 25 J. By rearranging the formula to solve for mass: 25 = 1/2 × m × 5² → 25 = 1/2 × m × 25, which gives a mass (m) of 2 kg.
• When the velocity (v₂) is doubled to 10 m/s, the new kinetic energy is calculated using the same mass: K₂ = 1/2 × 2 × 10² = 1 × 100 = 100 J.
• A fundamental property of this relationship is that Kinetic Energy is directly proportional to the square of the velocity (K ∝ v²). Since the velocity increased by a factor of 2 (from 5 to 10), the kinetic energy must increase by a factor of 2² = 4. Therefore, 25 J × 4 = 100 J.

 Additional Information

• Nature of Kinetic Energy:
• It is a scalar quantity, meaning it has magnitude but no specific direction.
• The SI unit is the Joule (J), defined as kg⋅m²/s².
• It can never be negative because mass is positive and the square of velocity is always non-negative.
• Work-Energy Theorem: It states that the net work done on an object results in a change in its kinetic energy (W = ΔK).
• Momentum Relationship: Kinetic energy can also be expressed in terms of momentum (p) as \( K = \frac{p^{2}}{2m} \).

The correct answer is Virtual, erect, and highly diminished at the focus.

Key Points

• A concave lens is known as a diverging lens because it bends parallel light rays outward after they pass through it.
• When an object is positioned at infinity, the rays of light hitting the lens are parallel to the principal axis.
• After refraction, these rays diverge and do not meet on the other side; however, when produced backward, they appear to meet at the principal focus (F₁).
• The image formed is virtual because the light rays do not actually intersect, and erect because it maintains the same vertical orientation as the object.
• Since the rays appear to converge at a single point (the focus), the image is considered highly diminished or point-sized.

 Additional Information

• Image Characteristics of Concave Lenses:
• Regardless of the position of the object, a concave lens always forms a virtual, erect, and diminished image.
• The magnification (m) for a concave lens is always positive and less than 1.
• These lenses are used to correct Myopia (near-sightedness) as they help diverge light before it reaches the eye.
• Lens Formula and Sign Convention:
• The relationship between focal length (f), object distance (u), and image distance (v) is given by the formula: \( 1/f = 1/v - 1/u \).
• According to the Cartesian sign convention, the focal length of a concave lens is always negative.
• A convex lens, by contrast, can form both real and virtual images depending on the distance of the object from the optical center.

The correct answer is They occupy the top trophic level in food chains.

Key Points

• The phenomenon responsible for this occurrence is Biological Magnification (or Biomagnification), which refers to the progressive increase in the concentration of harmful chemicals at each higher trophic level.
• Many agricultural chemicals, such as pesticides (e.g., DDT) and industrial wastes, are non-biodegradable, meaning they cannot be broken down by biological processes or excreted by organisms.
• These chemicals enter the food chain at the producer level (plants) via soil or water and are transferred to consumers as they feed on the organisms from the level below.
• Since humans generally occupy the topmost trophic level in most food chains, they consume a large volume of biomass from lower levels, leading to the maximum accumulation of these toxins in their bodies over time.

 Additional Information

• Trophic Levels:
• These represent the functional steps in a food chain or food web, beginning with autotrophs (producers) and ending with apex predators.
• While energy decreases by 90 percent at each step (the 10 percent law), non-biodegradable substances do not dissipate, causing their concentration to spike as the total biomass of the level decreases.
• Bioaccumulation vs. Biomagnification:
• Bioaccumulation refers to the buildup of a substance within a single organism's body over its lifetime.
• Biomagnification refers to the increase in concentration as the substance moves through the entire ecosystem from one level to the next.
• Health Impacts: High concentrations of heavy metals and synthetic chemicals in humans can lead to carcinogenic effects, reproductive issues, and damage to the nervous system.

The correct answer is Change in motion.

Key Points

• An unbalanced force is defined as a situation where the net force acting on an object is non-zero, meaning the individual forces do not cancel each other out.
• According to Newton's Second Law of Motion, the acceleration of an object is produced when an unbalanced force acts upon it; this is mathematically represented as Force = mass × acceleration.
• Because acceleration involves a change in velocity, the application of an unbalanced force must result in a change in motion, which can manifest as a change in speed, a change in direction, or both.
• If an object is stationary, an unbalanced force will cause it to start moving; if it is already moving, the force will cause it to speed up, slow down, or turn.
• Unlike balanced forces, which maintain an object's current state of equilibrium, unbalanced forces are the primary cause of mechanical work and kinetic energy variation.

 Additional Information

• Balanced Forces:
• These occur when the resultant force is exactly zero Newtons.
• An object under balanced forces follows Newton's First Law, remaining at rest or continuing at a constant velocity.
• Mass:
• In Newtonian physics, mass is a measure of inertia and remains constant regardless of the external forces applied.
• Forces affect the motion of the mass but do not increase or decrease the quantity of matter itself.
• Inertia:
• This is the inherent property of matter that resists changes in its state of motion.
• The magnitude of the unbalanced force required to change motion is directly proportional to the inertia (mass) of the object.

Shortcut Trick

For any regular polygon, the sum of an interior angle and an exterior angle at a vertex is always 180°.

Exterior angle = 180° − 156° = 24°

Since the polygon is regular, all its exterior angles are equal.

Sum of 7 exterior angles = 7 × 24° = 168°

∴ The correct answer is 168°.

Alternate Method

Given:

Interior angle of the regular polygon = 156°

Formula Used:

1. Exterior Angle = 180° − Interior Angle
2. Sum of 'k' exterior angles = k × Exterior Angle (for a regular polygon)

Calculations:

⇒ One Exterior angle = 180° − 156°

⇒ One Exterior angle = 24°

⇒ In a regular polygon, all exterior angles are of the same measure.

⇒ Measure of 7 exterior angles = 7 × 24°

⇒ Required Sum = 168°

∴ The correct answer is 168°.

Additional Information

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon, regardless of the number of sides, is always 360°.

Number of Sides

The number of sides (n) of a regular polygon can be calculated using the formula: n = 360° ÷ Exterior Angle. Here, n = 360 ÷ 24 = 15 sides.

Interior Angle Sum

The sum of all interior angles of an n-sided polygon is given by (n − 2) × 180°.

Shortcut Trick

Solve using VBODMAS priority: Brackets → Of → Division → Addition → Subtraction.

Inner part: (1/2 ÷ (1/2 of 1/2) + 1/2) = (1/2 ÷ 1/4 + 1/2) = (2 + 1/2) = 2.5.

Outer part: (1/2 of 2.5) − 0.5 = 1.25 − 0.5 = 0.75.

∴ The correct answer is 0.75.

Alternate Method

Given Expression: (1/2) of ((1/2) ÷ (1/2) of (1/2) + (1/2)) − (1/2)

Method: Applying the BODMAS rule to simplify the operations step by step.

⇒ 1/2 of (1/2 ÷ (1/2 × 1/2) + 1/2) − 1/2

⇒ 1/2 of (1/2 ÷ 1/4 + 1/2) − 1/2

⇒ 1/2 of (1/2 × 4 + 1/2) − 1/2

⇒ 1/2 of (2 + 1/2) − 1/2

⇒ 1/2 × 5/2 − 1/2

⇒ 5/4 − 1/2

⇒ 1.25 − 0.5 = 0.75

∴ The correct answer is 0.75.

Additional Information

VBODMAS Rule

It stands for Vinculum, Bracket, Of, Division, Multiplication, Addition, and Subtraction. It defines the specific sequence of operations to solve any mathematical expression.

Priority of 'Of' Operator

The 'Of' operator represents multiplication but is prioritized higher than Division and Multiplication in the order of operations.

Fraction Division

To divide a number by a fraction, we multiply the number by the reciprocal of that fraction (e.g., a ÷ b/c = a × c/b).

Shortcut Trick

Radius (r) = Diameter ÷ 2 = 20 ÷ 2 = 10 cm.

Volume = πr²h = (22 ÷ 7) × 10 × 10 × 42.

Since 42 is a multiple of 7, the calculation becomes: 22 × 100 × 6.

Volume = 132 × 100 = 13200 cm³.

∴ The correct answer is 13200 cm³.

Alternate Method

Given:

Diameter of the cylinder (d) = 20 cm

Height of the cylinder (h) = 42 cm

π = 22 ÷ 7

Formula Used:

Volume of Cylinder = πr²h

Where r is the radius and h is the height.

Calculations:

⇒ Radius (r) = Diameter ÷ 2

⇒ Radius (r) = 20 ÷ 2 = 10 cm

⇒ Volume = (22 ÷ 7) × (10)² × 42

⇒ Volume = (22 ÷ 7) × 100 × 42

⇒ Volume = 22 × 100 × (42 ÷ 7)

⇒ Volume = 22 × 100 × 6

⇒ Volume = 132 × 100

⇒ Volume = 13200 cm³

∴ The correct answer is 13200 cm³.

Additional Information

Curved Surface Area (CSA) of Cylinder

The area of the side surface of the cylinder is given by the formula: CSA = 2 × π × r × h.

Total Surface Area (TSA) of Cylinder

The sum of the curved surface area and the areas of the two circular bases: TSA = 2πr(r + h).

Volume of a Hollow Cylinder

If R is the external radius and r is the internal radius, then Volume = π × (R² − r²) × h.

Shortcut Trick

Break the expression into four independent segments using the BODMAS rule.

Segment 1: 96 ÷ (4 × 4) = 96 ÷ 16 = 6

Segment 2: 36 ÷ (9 × 3) = 36 ÷ 27 = 4/3

Segment 3: (14 × 2 ÷ 4) = 28 ÷ 4 = 7

Segment 4: 18 ÷ (3 × 3) = 18 ÷ 9 = 2

Total = (6 + 4/3) + (7 + 2) = 15 + 4/3 = 49/3

∴ The correct answer is 49/3.

Alternate Method

Given Expression: [{(96 ÷ (4 × 4)) + (36 ÷ (9 × 3))} + {(14 × 2 ÷ 4) + (18 ÷ (3 × 3))}]

Method Used: BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction).

⇒ [{(96 ÷ 16) + (36 ÷ 27)} + {(28 ÷ 4) + (18 ÷ 9)}]

⇒ [{6 + 4/3} + {7 + 2}]

⇒ [{(18 + 4) / 3} + 9]

⇒ [22/3 + 9]

⇒ (22 + 27) / 3

⇒ 49/3

∴ The correct answer is 49/3.

Additional Information

The BODMAS/VBODMAS Rule

Always solve in the order: Vinculum (Bar), Brackets (), Of, Division, Multiplication, Addition, and Subtraction.

Types of Brackets

Standard order for solving nested brackets is: Small Brackets ( ) → Curly Brackets { } → Square Brackets [ ].

Fraction Simplification

To add a whole number and a fraction like x + a/b, use the formula (x × b + a) / b.

Shortcut Trick

Apply the successive percentage change formula: [x + y + (xy ÷ 100)]%

Where x = +30% (increase) and y = −35% (decrease).

Net Change = 30 − 35 + [(30 × −35) ÷ 100]

Net Change = −5 − 10.5 = −15.5%

Negative sign indicates a percentage decrease.

∴ The correct answer is 15.5% decrease.

Alternate Method

Given: Initial tourists increase = 30%; Subsequent monsoon decrease = 35%.

⇒ Let the initial number of tourists be 100.

⇒ Number of tourists after summer increase = 100 + (30% of 100) = 130.

⇒ Number of tourists after monsoon drop = 130 − (35% of 130).

⇒ Monsoon drop amount = (35 ÷ 100) × 130 = 45.5.

⇒ Final tourists count = 130 − 45.5 = 84.5.

⇒ Overall percentage change = 100 − 84.5 = 15.5%.

∴ The correct answer is 15.5% decrease.

Additional Information

Effective Percentage Change

For two successive changes of a% and b%, the net change is given by the formula [a + b + (ab ÷ 100)]%.

Multiplier Method

The final value can be calculated as Initial Value × (1 + r₁ ÷ 100) × (1 + r₂ ÷ 100). In this case: 1 × 1.30 × 0.65 = 0.845.

Profit and Loss Context

This same logic applies to successive discounts or a markup followed by a discount in Profit and Loss problems.