Search Previous Year Questions

Shortcut Trick

In any regular polygon, Interior angle + Exterior angle = 180^o.

Given: Interior angle − Exterior angle = 100^o.

2 × Exterior angle = (Sum − Difference) = 180^o − 100^o = 80^o.

⇒ Exterior angle = 40^o.

Number of sides (n) = 360^o ÷ Exterior angle = 360^o ÷ 40^o = 9.

∴ The correct answer is 9.

Alternate Method

Given:

Difference between interior angle (I) and exterior angle (E) of a regular polygon is 100^o.

⇒ I − E = 100^o --- (i)

Formula Used:

1. I + E = 180^o (Linear pair property)
2. Each Exterior angle (E) = 360^o ÷ n, where n is the number of sides.

Calculations:

⇒ We know that I + E = 180^o --- (ii)

⇒ Adding equation (i) and (ii):

⇒ (I − E) + (I + E) = 100^o + 180^o

⇒ 2I = 280^o

⇒ I = 140^o

⇒ Now, substitute I in (ii):

⇒ 140^o + E = 180^o

⇒ E = 180^o − 140^o = 40^o

⇒ Using the formula for number of sides:

⇒ n = 360^o ÷ E

⇒ n = 360^o ÷ 40^o

⇒ n = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is given by (n − 2) × 180^o.

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon, taking one at each vertex, is always 360^o.

Regular Polygon Properties

In a regular polygon, all interior angles are equal and all exterior angles are equal. Each interior angle = [(n − 2) × 180^o] ÷ n.

Shortcut Trick

Ratio of original Cost Price (CP) to Selling Price (SP) is 100 : 121.

New Profit is 13.6%, so the new ratio is 100 : 113.6.

Using the Cross-Product Method with changes (+₹50 for CP and −₹69 for SP):

(100 × 113.6 − 121 × 100) units = (100 × −69 − 113.6 × 50)

−740 units = −6900 − 5680 = −12580 ⇒ 1 unit = ₹17.

∴ The correct answer is ₹1,700.

Alternate Method

Given:

Original Profit = 21%

Increase in CP = ₹50

Decrease in SP = ₹69

New Profit = 13.6%

Formula Used:

SP = CP × (1 + Profit%)

⇒ Let original Cost Price (CP) = x

⇒ Original Selling Price (SP) = x + 21% of x = 1.21x

⇒ New CP = x + 50 and New SP = 1.21x − 69

⇒ According to the question: 1.136 × (x + 50) = 1.21x − 69

⇒ 1.136x + 56.8 = 1.21x − 69

⇒ 1.21x − 1.136x = 56.8 + 69

⇒ 0.074x = 125.8

⇒ x = 125.8 ÷ 0.074 = 1700

∴ The correct answer is ₹1,700.

Additional Information

Profit Percentage

Profit % = (Selling Price − Cost Price) ÷ Cost Price × 100.

Relationship between CP and SP

If an article is sold at a profit of P%, then SP = CP × (100 + P) ÷ 100.

Cross-Product Method

Used when initial ratios and absolute changes are known to find the original values directly.

Shortcut Trick

Test the options to find the smallest positive number.

For 1: Numbers become (3 − 1) = 2 and (5 − 1) = 4.

Third proportional = b² ÷ a = 4² ÷ 2 = 16 ÷ 2 = 8.

The condition is satisfied as 8 is the required third proportional.

∴ The correct answer is 1.

Alternate Method

Given: Initial numbers are 3 and 5. The third proportional to be obtained is 8.

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

Calculations:

⇒ Let x be the smallest positive number to be subtracted.

⇒ First number (a) = 3 − x

⇒ Second number (b) = 5 − x

⇒ Third proportional (c) = 8

⇒ According to the property: (5 − x)² = (3 − x) × 8

⇒ 25 − 10x + x² = 24 − 8x

⇒ x² − 10x + 8x + 25 − 24 = 0

⇒ x² − 2x + 1 = 0

⇒ (x − 1)² = 0

⇒ x − 1 = 0

⇒ x = 1

∴ The correct answer is 1.

Additional Information

Mean Proportional

If b is the mean proportional between a and c, then b = √(a × c) or b² = ac.

Fourth Proportional

If d is the fourth proportional to a, b, and c, then a : b = c : d, which gives d = (b × c) ÷ a.

Continued Proportion

Numbers a, b, c, d are in continued proportion if a/b = b/c = c/d.

Shortcut Trick

Total Work (LCM of 6, 34, 36) = 612 units.

Efficiencies: Eduardo = 102, Abraham = 18, Xavier = 17.

Adjusted Work = Total Work − (Eduardo's 2-day work) + (Abraham's work if he stayed for the last 3 days).

Adjusted Work = 612 − (102 × 2) + (18 × 3) = 612 − 204 + 54 = 462 units.

Total time = Adjusted Work ÷ (Combined Efficiency of Abraham and Xavier) = 462 ÷ (18 + 17) = 13.2 days.

∴ The correct answer is 13.2 days.

Alternate Method

Given: Time taken by Eduardo = 6 days, Abraham = 34 days, and Xavier = 36 days.

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

⇒ Let the total number of days taken to finish the work be T.

⇒ Work done by Eduardo (for 2 days) = 102 × 2 = 204 units.

⇒ Work done by Abraham (for T − 3 days) = 18 × (T − 3) units.

⇒ Work done by Xavier (for T days) = 17 × T units.

⇒ Total Work = (Work by Eduardo) + (Work by Abraham) + (Work by Xavier)

⇒ 612 = 204 + 18(T − 3) + 17T

⇒ 612 = 204 + 18T − 54 + 17T

⇒ 612 = 150 + 35T

⇒ 35T = 612 − 150 ⇒ 35T = 462

⇒ T = 462 ÷ 35 = 13.2 days.

∴ The correct answer is 13.2 days.

Additional Information

Time and Efficiency Relation

Efficiency is inversely proportional to the time taken to complete a task. Work = Efficiency × Time.

Leaving Before Completion

If a person leaves 'n' days before completion, adding their 'n' days' potential work to the total work allows you to treat them as having worked until the end.

LCM Method for Total Work

Assuming Total Work as the LCM of given time durations helps in calculating efficiencies as whole numbers, simplifying arithmetic steps.

Shortcut Trick

Time taken: Patna Express = 9h, Delhi Express = 11h. Let Distance = 99 units (LCM).

Speeds: Patna Express = 11 units/h, Delhi Express = 9 units/h.

Distance covered by Patna Express before 2:00 PM (4 hours) = 11 × 4 = 44 units.

Meeting time from 2:00 PM = Remaining distance ÷ Relative speed = (99 − 44) ÷ (11 + 9) = 55 ÷ 20 = 2 hours 45 min.

∴ The correct answer is 04:45PM.

Alternate Method

Given:

Patna Express (Delhi to Patna): 10:00 AM to 07:00 PM = 9 hours.

Delhi Express (Patna to Delhi): 02:00 PM to 01:00 AM = 11 hours.

Formula Used:

Time = Distance ÷ Relative Speed

⇒ Let total distance = 99 km.

⇒ Speed of Patna Express = 99 ÷ 9 = 11 km/h.

⇒ Speed of Delhi Express = 99 ÷ 11 = 9 km/h.

⇒ Patna Express distance from 10:00 AM to 02:00 PM = 11 × 4 = 44 km.

⇒ Distance between trains at 02:00 PM = 99 − 44 = 55 km.

⇒ Relative Speed = 11 + 9 = 20 km/h.

⇒ Time taken to meet = 55 ÷ 20 = 2.75 hours.

⇒ 2.75 hours = 2 hours + (0.75 × 60) min = 2 hours 45 min.

⇒ Meeting time = 02:00 PM + 2 hours 45 min = 04:45 PM.

∴ The correct answer is 04:45PM.

Additional Information

Relative Speed in Opposite Directions

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

Standard Meeting Time Logic

To find the meeting time, first calculate the distance covered by the early-starting train to bring both trains to the same starting time, then divide the remaining distance by the relative speed.

Average Speed for Same Distance

If a journey is covered at speed S₁ and returned at speed S₂, the Average Speed is (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

Average Speed = (Total Distance) ÷ (Total Time)

Total Distance = 6 + 18 = 24 km

Total Time = (6 ÷ 6) + (18 ÷ 9) = 1 + 2 = 3 hours

Average Speed = 24 ÷ 3 = 8 km/h

∴ The correct answer is 8 km/h.

Alternate Method

Given:

Distance covered while walking (d₁) = 6 km

Speed while walking (s₁) = 6 km/h

Distance covered while running (d₂) = 18 km

Speed while running (s₂) = 9 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

⇒ Time taken for walking (t₁) = 6 ÷ 6 = 1 hour

⇒ Time taken for running (t₂) = 18 ÷ 9 = 2 hours

⇒ Total Distance = 6 + 18 = 24 km

⇒ Total Time = 1 + 2 = 3 hours

⇒ Average Speed = 24 ÷ 3 = 8 km/h

∴ The correct answer is 8 km/h.

Additional Information

Average Speed (Equal Distances)

If a person travels equal distances at speeds u and v, the average speed is given by the harmonic mean: 2uv ÷ (u + v).

Unit Conversion

To convert speed from km/h to m/s, multiply by 5/18. To convert from m/s to km/h, multiply by 18/5.

Relative Speed

When two objects move in the same direction, relative speed = |u − v|. When they move in opposite directions, relative speed = u + v.

Shortcut Trick

Diameter of the sphere = 28 cm → Radius (r) = 14 cm.

Area of one circle with r = 14 cm is πr² = 616 cm².

Total Surface Area of a sphere = 4 × πr² = 4 × 616 = 2464 cm².

∴ The correct answer is 2464 cm².

Alternate Method

Given:

Diameter of the spherical tank (d) = 28 cm

Value of π = 22/7

Formula Used:

Radius of sphere (r) = d ÷ 2

Total Surface Area of sphere = 4 × π × r²

Calculations:

⇒ Radius (r) = 28 ÷ 2 = 14 cm

⇒ Total Surface Area = 4 × (22/7) × 14 × 14

⇒ Total Surface Area = 4 × 22 × 2 × 14

⇒ Total Surface Area = 88 × 28

⇒ Total Surface Area = 2464 cm²

∴ The correct answer is 2464 cm².

Additional Information

Volume of a Sphere

The total space enclosed inside a sphere is given by the formula: Volume = (4/3) × π × r³.

Curved Surface Area of a Hemisphere

For a hemisphere of radius r, the curved surface area is exactly half of the sphere's surface area: CSA = 2 × π × r².

Total Surface Area of a Hemisphere

For a solid hemisphere, the total surface area includes the curved surface and the flat circular base: TSA = 3 × π × r².

Shortcut Trick

Use the direct trigonometric identity: sec²θ = 1 + tan²θ.

Substitute tan θ = 2 into the formula: sec²θ = 1 + (2)² = 1 + 4 = 5.

Taking square root on both sides: sec θ = √5.

Since θ is in the first quadrant, sec θ must be positive.

Thus, the value matches Option (3).

∴ The correct answer is √5.

Alternate Method

Given: tan θ = 2 and θ is in the first quadrant.

Formula Used: In a right-angled triangle, tan θ = Perpendicular / Base and sec θ = Hypotenuse / Base. By Pythagoras Theorem: H² = P² + B².

⇒ tan θ = 2 / 1 → Perpendicular (P) = 2 and Base (B) = 1

⇒ Hypotenuse (H) = √(P² + B²)

⇒ H = √(2² + 1²) = √(4 + 1) = √5

⇒ sec θ = Hypotenuse / Base = √5 / 1 = √5

Since the angle is in the first quadrant, all trigonometric ratios are positive, confirming the result matches Option (3).

∴ The correct answer is √5.

Additional Information

Pythagorean Identities

There are three fundamental identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Quadrant Rules (ASTC)

In the First Quadrant (0 to 90°), all trigonometric ratios are positive. In the Second, only sin and cosec are positive.

Reciprocal Relations

Trigonometric functions are related by reciprocals: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ.

Shortcut Trick

For a constant Rupee amount, Quantity₁ × SP%₁ = Quantity₂ × SP%₂

Here, Quantity₁ = 5, Loss = 34% (SP%₁ = 66%), Gain = 10% (SP%₂ = 110%)

⇒ 5 × 66 = x × 110

⇒ 330 = 110 × x

⇒ x = 3

∴ The correct answer is 3.

Alternate Method

Given:

Selling Price (SP₁) of 5 lemons = ₹1

Loss = 34% | Target Gain = 10%

Formula Used:

Cost Price (CP) = SP ÷ (1 − Loss%) | New SP = CP × (1 + Gain%)

Calculations:

⇒ SP of 1 lemon = ₹1 ÷ 5 = ₹0.20

⇒ Since this is at 34% loss, 66% of CP = ₹0.20

⇒ CP of 1 lemon = 0.20 ÷ 0.66 = ₹20/66 = ₹10/33

⇒ To gain 10%, New SP = 110% of (₹10/33)

⇒ New SP per lemon = (110 ÷ 100) × (10 ÷ 33) = (11 ÷ 10) × (10 ÷ 33) = 1 ÷ 3

⇒ New SP of 1 lemon = ₹1/3

⇒ Number of lemons sold for ₹1 = 1 ÷ (1/3) = 3

∴ The correct answer is 3.

Additional Information

Inverse Variation in Profit and Loss

When the total amount received remains the same, the quantity and the selling price percentage (100 ± P/L%) are inversely proportional. n₁ × %SP₁ = n₂ × %SP₂.

Successive Percentage Change

If a value is changed by a% and then by b%, the net change is given by the formula [a + b + (a × b) ÷ 100]%.

Relation between CP and SP

Cost Price is always considered as the base (100%). Profit increases the percentage above 100, while Loss decreases it below 100.

Shortcut Trick

For a hollow cylinder, Total Surface Area = 2π(R + r)(h + R − r).

Given: Outer radius (R) = 10 cm, Inner radius (r) = 8 cm, Height (h) = 200 cm.

TSA = 2 × (22/7) × (10 + 8) × (200 + 10 − 8)

TSA = 2 × (22/7) × 18 × 202 = 159984 ÷ 7 ≈ 22854.85 cm².

∴ The correct answer is 22854.85 cm².

Alternate Method

Given:

⇒ Outer radius (R) = 10 cm

⇒ Inner radius (r) = 8 cm

⇒ Length (h) = 2 m = 2 × 100 = 200 cm

Formula Used:

⇒ Total Surface Area (TSA) = Outer CSA + Inner CSA + 2 × Area of base rings

⇒ TSA = 2πRh + 2πrh + 2π(R² − r²)

Calculations:

⇒ TSA = 2 × (22/7) × [ (10 × 200) + (8 × 200) + (10² − 8²) ]

⇒ TSA = (44/7) × [ 2000 + 1600 + (100 − 64) ]

⇒ TSA = (44/7) × [ 3600 + 36 ]

⇒ TSA = (44 × 3636) ÷ 7

⇒ TSA = 159984 ÷ 7

⇒ TSA ≈ 22854.857 cm²

∴ The correct answer is 22854.85 cm².

Additional Information

Volume of Hollow Cylinder

The volume of the material in a hollow cylinder is given by V = π × (R² − r²) × h.

Curved Surface Area (CSA)

The total curved surface area is the sum of the inner and outer surface areas: CSA = 2πh(R + r).

Cross-sectional Area of the Ring

The area of the base ring (annulus) at each end of the pipe is π × (R² − r²).