Search Previous Year Questions

Shortcut Trick

Total deer in the reserve = 200.

Since 10% of 200 = 20.

Then, 30% of 200 = 3 × 20 = 60.

∴ The correct answer is 60 deer.

Alternate Method

Given:

Total number of deer = 200

Percentage of deer tagged with GPS collars = 30%

Formula Used:

Value = (Percentage ÷ 100) × Total Value

Calculations:

⇒ Number of tagged deer = (30 ÷ 100) × 200

⇒ Number of tagged deer = 30 × (200 ÷ 100)

⇒ Number of tagged deer = 30 × 2

⇒ Number of tagged deer = 60

∴ The correct answer is 60 deer.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide by 100. Example: 30% = 30/100 = 3/10.

Finding Percentage of a Number

To find x% of y, the formula is (x × y) ÷ 100. This is a fundamental concept in data interpretation and commercial arithmetic.

Complementary Percentage

If 30% are tagged, then the remaining (100% − 30%) = 70% are untagged. Untagged deer = 0.70 × 200 = 140.

Shortcut Trick

Time taken by Van 1 = 4h and Van 2 = 3.5h. Let total distance = 28 units (LCM of 4, 3.5).

Speeds: V₁ = 28 ÷ 4 = 7 units/h, V₂ = 28 ÷ 3.5 = 8 units/h.

Distance covered by Van 1 by 7:00 a.m. (in 2 hours) = 7 × 2 = 14 units.

Remaining distance at 7:00 a.m. = 28 − 14 = 14 units. Relative speed = 7 + 8 = 15 units/h.

Meeting time = 7:00 a.m. + (14 ÷ 15) × 60 min = 7:00 a.m. + 56 min = 7:56 a.m.

∴ The correct answer is 7:56 a.m.

Alternate Method

Given:

Van 1 (Jaipur to Agra): Departure = 5:00 a.m., Arrival = 9:00 a.m.

Van 2 (Agra to Jaipur): Departure = 7:00 a.m., Arrival = 10:30 a.m.

Formula Used:

Distance = Speed × Time | Relative Speed (Opposite) = S₁ + S₂ | Time = Distance ÷ Speed

⇒ Time taken by Van 1 = 9:00 − 5:00 = 4 hours

⇒ Time taken by Van 2 = 10:30 − 7:00 = 3.5 hours = 7/2 hours

⇒ Let total distance (D) = LCM(4, 3.5) = 28 km

⇒ Speed of Van 1 (S₁) = 28 ÷ 4 = 7 km/h

⇒ Speed of Van 2 (S₂) = 28 ÷ 3.5 = 8 km/h

⇒ Distance covered by Van 1 from 5:00 a.m. to 7:00 a.m. = 7 × 2 = 14 km

⇒ Remaining distance between vans at 7:00 a.m. = 28 − 14 = 14 km

⇒ Relative speed (opposite direction) = 7 + 8 = 15 km/h

⇒ Time to meet after 7:00 a.m. = 14 ÷ 15 hours

⇒ Converting to minutes = (14/15) × 60 = 56 minutes

⇒ Meeting Time = 7:00 a.m. + 56 minutes = 7:56 a.m.

∴ The correct answer is 7:56 a.m.

Additional Information

Relative Speed Concept

When two objects move toward each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Meeting Time with Different Starts

Always calculate the lead distance covered by the first object until the second object starts moving before applying relative speed.

Time Conversion Shortcut

To convert hours into minutes, multiply by 60. For example, 1/15 hour = 4 minutes; 14/15 hour = 56 minutes.

Shortcut Trick

In a 30°−60°−90° triangle, the ratio of the sides opposite to the angles is 1 : √3 : 2.

Side opposite to 30° (Height) = 1 unit = 50 m.

Side opposite to 60° (Distance) = √3 units = 50√3 m.

∴ The correct answer is 50√3 m.

Alternate Method

Given: Height of the tower (AB) = 50 m, Angle of elevation (∠ACB) = 30°.

Formula: tan θ = Perpendicular ÷ Base

Let the distance of the man from the foot of the tower be x.

⇒ In right-angled triangle ABC, tan 30° = AB ÷ BC

⇒ 1 ÷ √3 = 50 ÷ x

⇒ x = 50 × √3

⇒ x = 50√3 m

∴ The correct answer is 50√3 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when the object is above the horizontal level.

Trigonometric Ratios for Standard Angles

tan 30° = 1 ÷ √3 | tan 45° = 1 | tan 60° = √3.

Angle of Depression

The angle formed by the line of sight with the horizontal when the object is below the horizontal level. Numerically, Angle of Elevation = Angle of Depression.

Shortcut Trick

For a 30° angle in a right triangle, the ratio of Height : Hypotenuse is 1 : 2.

Total Tree Height = Standing Height + Broken (Hypotenuse) part = 1 + 2 = 3 units.

Given, 3 units = 20 m ⇒ 1 unit = 20/3 m.

Height of the broken point is 1 unit = 20/3 m.

∴ The correct answer is 20/3 m.

Alternate Method

Given:

Total length of the tree = 20 m

Angle of elevation (θ) = 30°

Let the height of the break point from the ground be 'h' meters.

Length of the broken part (Hypotenuse) = (20 − h) meters.

Formula Used:

sin θ = Perpendicular ÷ Hypotenuse

Calculations:

⇒ sin 30° = h ÷ (20 − h)

⇒ 1/2 = h ÷ (20 − h)

⇒ 20 − h = 2h

⇒ 3h = 20

⇒ h = 20/3 m

∴ The correct answer is 20/3 m.

Additional Information

Trigonometric Ratios

For standard angles: sin 30° = 1/2, sin 45° = 1/√2, and sin 60° = √3/2.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when looking up at an object.

Right Triangle Ratios

In a 30°−60°−90° triangle, the sides are always in the ratio 1 : √3 : 2.

Shortcut Trick

Convert successive discounts into a single multiplier ratio: 20% = 1/5 and 25% = 1/4.

The Marked Price (MP) to Selling Price (SP) ratio is: (5 × 4) : (4 × 3) = 20 : 12 = 5 : 3.

Given final Selling Price (3 units) = ₹5,40,000.

Value of 1 unit = 5,40,000 ÷ 3 = ₹1,80,000.

Marked Price (5 units) = 5 × 1,80,000 = ₹9,00,000.

∴ The correct answer is Rs. 900000.

Alternate Method

Given:

Final Selling Price of the car = Rs. 5,40,000

First Discount (D₁) = 20% | Second Discount (D₂) = 25%

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − D₁)/100 × (100 − D₂)/100

Calculations:

⇒ 5,40,000 = MP × (100 − 20)/100 × (100 − 25)/100

⇒ 5,40,000 = MP × (80/100) × (75/100)

⇒ 5,40,000 = MP × (4/5) × (3/4)

⇒ 5,40,000 = MP × (3/5)

⇒ MP = (5,40,000 × 5) ÷ 3

⇒ MP = 1,80,000 × 5

⇒ MP = 9,00,000

∴ The correct answer is Rs. 900000.

Additional Information

Single Equivalent Discount

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Marked Price vs Selling Price

Marked Price is the list price before discount, while Selling Price = MP × (1 − d/100) where d is the discount percentage.

Effective Multiplier Method

If multiple discounts are applied, the final SP = MP × m₁ × m₂... where m is the fraction (100−d)/100.

Shortcut Trick

Use the direct ratio: CP : MP = (100 − Discount%) : (100 + Gain%).

CP : MP = (100 − 34) : (100 + 10) = 66 : 110.

Simplified Ratio: CP : MP = 3 : 5.

According to the question, 3 units = Rs. 6000 → 1 unit = Rs. 2000.

Marked Price (5 units) = 5 × 2000 = Rs. 10,000.

∴ The correct answer is Rs. 10,000.

Alternate Method

Given:

Cost Price (CP) = Rs. 6000

Gain Percentage = 10%

Discount Percentage = 34%

Formula Used:

Selling Price (SP) = CP × (100 + Gain%)/100

Marked Price (MP) = SP × 100 / (100 − Discount%)

Calculations:

⇒ SP = 6000 × (110 / 100)

⇒ SP = 6000 × 1.1 = Rs. 6600

⇒ MP = 6600 × 100 / (100 − 34)

⇒ MP = 6600 × 100 / 66

⇒ MP = 100 × 100 = Rs. 10,000

∴ The correct answer is Rs. 10,000.

Additional Information

Direct CP and MP Relation

The formula MP/CP = (100 + Gain%)/(100 − Discount%) is extremely useful for exams to find any one unknown variable out of the four.

Discount and Profit Bases

Always remember that Discount is calculated on the Marked Price, while Profit or Loss is always calculated on the Cost Price.

Successive Discounts

If two successive discounts x% and y% are given, the single equivalent discount is [x + y − (x × y)/100]%.

Shortcut Trick

Total work remains constant, so use M₁D₁ = M₂D₂.

46 × 16 = 32 × Required Days

Required Days = (46 × 16) ÷ 32

Required Days = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Alternate Method

Given:

Initial count of men (M₁) = 46

Initial time taken (D₁) = 16 days

New count of men (M₂) = 32

Formula Used:

When the amount of work is the same, we apply the inverse variation rule:

M₁ × D₁ = M₂ × D₂

Calculations:

⇒ 46 × 16 = 32 × D₂

⇒ 736 = 32 × D₂

⇒ D₂ = 736 ÷ 32

⇒ D₂ = 23

∴ The correct answer is 23 days.

Additional Information

General Chain Rule Formula

The complete relation for work is (M₁×D₁×H₁)/W₁ = (M₂×D₂×H₂)/W₂, where H is hours and W is units of work.

Concept of Man-Days

Total work is often measured in man-days, which is the product of the number of men and the number of days they work.

Inverse Proportion Rule

If Work is constant, Men and Days are inversely proportional. If the number of men decreases by a certain ratio, the days must increase by the reciprocal of that ratio.

Shortcut Trick

The perpendicular distance from the center to chord AB is OC = √(13² − 12²) = 5 cm.

Given CF = 5 + 4√5 cm and CF = OC + OF (for parallel chords on opposite sides), then OF = 4√5 cm.

Half the length of chord DE is FD = √(13² − (4√5)²) = √(169 − 80) = √89 cm.

Full length of chord DE = 2 × √89 = 2√89 cm.

∴ The correct answer is 2√89 cm.

Alternate Method

Given:

Radius (R) = 13 cm

Chord AB = 24 cm, OC ⊥ AB

Distance CF = (5 + 4√5) cm, OF ⊥ DE

Formula Used:

Length of perpendicular from center to chord = √(Radius² − (Chord/2)²)

Calculations:

⇒ AC = CB = AB ÷ 2 = 24 ÷ 2 = 12 cm

⇒ In right triangle OCB, OC² + CB² = OB²

⇒ OC² + 12² = 13²

⇒ OC² + 144 = 169

⇒ OC² = 25 ⇒ OC = 5 cm

⇒ Since OC and OF are perpendicular to parallel chords, CF = OC + OF

⇒ 5 + 4√5 = 5 + OF

⇒ OF = 4√5 cm

⇒ In right triangle OFD, OF² + FD² = OD² (where OD = Radius = 13)

⇒ (4√5)² + FD² = 13²

⇒ 80 + FD² = 169

⇒ FD² = 169 − 80 = 89

⇒ FD = √89 cm

⇒ Length of chord DE = 2 × FD = 2√89 cm

∴ The correct answer is 2√89 cm.

Additional Information

Perpendicular from the Center

A perpendicular drawn from the center of a circle to a chord bisects the chord into two equal segments.

Distance between Parallel Chords

The distance between two parallel chords on opposite sides of the center is the sum of their perpendicular distances from the center (d = d₁ + d₂).

Chord Length Formula

The length of a chord L at a distance d from the center of a circle with radius R is given by L = 2√(R² − d²).

Shortcut Trick

Identify the last digit of the area: 5, so the side must end in 5.

Estimate the range: 180² = 32,400 and 190² = 36,100.

Since 34,225 lies between 32,400 and 36,100, the side must be 185.

Verification: 18 × 19 = 342, then append 25 → 34,225.

∴ The correct answer is 185 units.

Alternate Method

Given: Area of the square = 34,225 square units.

Formula Used: Area of square = (side)²

⇒ side² = 34,225

⇒ side = √34,225

⇒ side = √(25 × 1369)

⇒ side = 5 × √1369

⇒ side = 5 × 37

⇒ side = 185 units

∴ The correct answer is 185 units.

Additional Information

Perimeter of a Square

The total boundary length of a square is given by the formula: Perimeter = 4 × side.

Diagonal of a Square

The straight line distance between opposite corners is: Diagonal = side × √2.

Area using Diagonal

If the diagonal (d) is known, the area can be calculated as: Area = d² ÷ 2.

Shortcut Trick

Length of tangents from an external point are equal ⇒ PR = PS = 14 cm.

Radii of the same circle are equal ⇒ OR = OS = 10 cm.

Perimeter of quadrilateral ORPS = Sum of all four sides.

Perimeter = 2 × (Radius + Tangent Length) = 2 × (10 + 14) = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Radius of circle (OR = OS) = 10 cm

Length of tangent PR = 14 cm

Formula Used:

Perimeter of a quadrilateral = Sum of the lengths of its four sides

Tangent property: Tangents drawn from an external point to a circle are equal in length.

Calculations:

⇒ In quadrilateral ORPS, the sides are OR, RP, PS, and SO.

⇒ PR and PS are tangents from point P ⇒ PR = PS = 14 cm.

⇒ OR and OS are radii of the circle ⇒ OR = OS = 10 cm.

⇒ Perimeter of ORPS = OR + RP + PS + SO

⇒ Perimeter = 10 + 14 + 14 + 10

⇒ Perimeter = 24 + 24 = 48 cm.

∴ The correct answer is 48 cm.

Additional Information

Radius and Tangent Relationship

The radius of a circle is always perpendicular to the tangent at the point of contact (∠ORP = ∠OSP = 90°).

Cyclic Quadrilateral Property

Quadrilateral ORPS is a cyclic quadrilateral because the sum of opposite angles ∠ORP + ∠OSP = 180°, which implies ∠ROS + ∠RPS = 180°.

Line Joining Center and External Point

The line OP bisects the angle between the two tangents (∠RPS) and the angle between the two radii (∠ROS).