Quantitative Aptitude PYQs

Shortcut Trick

Given sin A = 0.6 = 6 ÷ 10 = 3 ÷ 5 = Perpendicular ÷ Hypotenuse.

Using the standard Pythagorean triplet (3, 4, 5), we directly get Base = 4.

tan A = Perpendicular ÷ Base

tan A = 3 ÷ 4 = 0.75

∴ The correct answer is 0.75.

Alternate Method

Given:

sin A = 0.6

Formula Used:

sin² A + cos² A = 1

tan A = sin A ÷ cos A

A A C B Base = 4 Perp = 3 Hyp = 5

Calculations:

⇒ cos A = √(1 − sin² A)

⇒ cos A = √(1 − 0.6²)

⇒ cos A = √(1 − 0.36)

⇒ cos A = √0.64

⇒ cos A = 0.8

⇒ tan A = sin A ÷ cos A

⇒ tan A = 0.6 ÷ 0.8

⇒ tan A = 6 ÷ 8 = 3 ÷ 4

⇒ tan A = 0.75

∴ The correct answer is 0.75.

Additional Information

Trigonometric Ratios

In a right-angled triangle: sin θ = Perpendicular ÷ Hypotenuse | cos θ = Base ÷ Hypotenuse | tan θ = Perpendicular ÷ Base.

Pythagorean Triplets

Sets of three integers satisfying Pythagoras Theorem (a² + b² = c²). Common triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Their multiples like (6, 8, 10) are also valid triplets.

Fundamental Identities

sin² θ + cos² θ = 1 | 1 + tan² θ = sec² θ | 1 + cot² θ = cosec² θ.

Shortcut Trick

By standard trigonometric identity, if secA + tanA = p, then secA − tanA = 1/p.

Adding these two equations gives:

⇒ 2secA = p + 1/p

⇒ 2secA = (p² + 1)/p

⇒ secA = (p² + 1)/2p

∴ The correct answer is (p² + 1)/2p.

Alternate Method

Given:

secA + tanA = p

Formula Used:

sec²A − tan²A = 1

a² − b² = (a − b)(a + b)

Calculations:

⇒ (secA − tanA)(secA + tanA) = 1

Substitute secA + tanA = p in the above equation:

⇒ (secA − tanA) × p = 1

⇒ secA − tanA = 1/p

Adding this result to the given equation (secA + tanA = p):

⇒ (secA + tanA) + (secA − tanA) = p + 1/p

⇒ 2secA = (p² + 1)/p

⇒ secA = (p² + 1)/2p

∴ The correct answer is (p² + 1)/2p.

Additional Information

Tangent Calculation

By subtracting the two equations, tanA = (p² − 1)/2p.

Sine Calculation

Since sinA = tanA ÷ secA, sinA = (p² − 1)/(p² + 1).

Cosecant and Cotangent Relation

Similarly, if cosecA + cotA = q, then cosecA − cotA = 1/q, yielding cosecA = (q² + 1)/2q.

Shortcut Trick

The arc length is a direct fraction of the full circle's circumference based on the central angle.

Fraction of track covered = 120° ÷ 360° = 1/3

Arc Length = (1/3) × 2 × π × Radius

Arc Length = (1/3) × 2 × π × 50 = 100π/3 m

∴ The correct answer is 100π/3 m.

Alternate Method

120° 50 m 50 m Arc Length Circular Jogging Track Segment

Given:

Radius of the track (r) = 50 m

Central angle subtended by the path (θ) = 120°

Formula Used:

Length of an arc = (θ ÷ 360°) × 2πr

Calculations:

⇒ Arc length = (120° ÷ 360°) × 2 × π × 50

⇒ Arc length = (1/3) × 100π

⇒ Arc length = 100π/3 m

∴ The correct answer is 100π/3 m.

Additional Information

Area of a Sector

The area of a sector with central angle θ is given by Area = (θ ÷ 360°) × πr².

Arc Length in Radians

If the central angle is given in radians (θ), the arc length can be calculated directly using the formula L = r × θ.

Circumference of a Circle

The total boundary length of a full circle is its circumference, calculated as C = 2πr.

Shortcut Trick

In trigonometry, the co-functions of complementary angles are always equal.

The sine of an acute angle is always identical to the cosine of its complementary angle.

Hence, for any given acute angle A, sin A = cos(90° − A).

∴ The correct answer is sin A = cos(90° − A).

Alternate Method

X Y Z A 90° − A b a c

Given:

A right-angled triangle XYZ where ∠Y = 90° and ∠X = A.

Concept Used:

The sum of all interior angles in a triangle is 180°, so ∠Z = 180° − 90° − A = 90° − A.

In a right-angled triangle:

⇒ sin(θ) = Opposite side ÷ Hypotenuse

⇒ cos(θ) = Adjacent side ÷ Hypotenuse

Calculations:

⇒ For angle X (which is A):

⇒ sin(A) = YZ ÷ XZ = a ÷ b

⇒ For angle Z (which is 90° − A):

⇒ cos(90° − A) = YZ ÷ XZ = a ÷ b

⇒ Comparing both equations, we directly get:

⇒ sin(A) = cos(90° − A)

∴ The correct answer is sin A = cos(90° − A).

Additional Information

Complementary Angles

Two angles are complementary if their sum is exactly 90°. In any standard right-angled triangle, the two non-right (acute) angles are naturally complementary.

Co-function Identities

Trigonometric functions are intrinsically paired as co-functions. Standard pairings include: sin(90° − θ) = cos(θ), tan(90° − θ) = cot(θ), and sec(90° − θ) = csc(θ).

Supplementary Angles

Two angles are supplementary if their sum equals 180°. The trigonometric relations for supplementary angles state that sin(180° − θ) = sin(θ) and cos(180° − θ) = −cos(θ).

Shortcut Trick

Effective Rate of Simple Interest for 3 years at 20% = 20% × 3 = 60%.

Effective Rate of Compound Interest for 3 years at 20% = 3(20) + 3(20²)/100 + (20³)/10000 = 60 + 12 + 0.8 = 72.8%.

Given, the Compound Interest (72.8%) corresponds to ₹864.

Simple Interest (60%) = (864 ÷ 72.8) × 60 = 64800 ÷ 91 ≈ 712.087...

∴ The correct answer is ₹712.09.

Alternate Method

Given:

Compound Interest (CI) = ₹864

Rate (R) = 20% per annum

Time (T) = 3 years

Formula Used:

CI = P[(1 + R/100)^T − 1]

SI = (P × R × T) ÷ 100

Compound Interest Effective Rate: 72.8% Value: ₹864 Simple Interest Effective Rate: 60% Value: ? SI = (864 ÷ 72.8) × 60 ≈ ₹712.09

Calculations:

⇒ 864 = P[(1 + 20/100)³ − 1]

⇒ 864 = P[(6/5)³ − 1]

⇒ 864 = P[216/125 − 1]

⇒ 864 = P[91/125]

⇒ P = (864 × 125) ÷ 91

Now, substitute the value of Principal (P) to calculate Simple Interest:

⇒ SI = (P × 20 × 3) ÷ 100

⇒ SI = P × (60 ÷ 100) = 3P/5

⇒ SI = 3/5 × [(864 × 125) ÷ 91]

⇒ SI = (3 × 864 × 25) ÷ 91

⇒ SI = 64800 ÷ 91 ≈ 712.0879...

∴ The correct answer is ₹712.09.

Additional Information

Effective Rate of Compound Interest

For 3 years at rate R%, effective rate = 3R + (3R²)/100 + (R³)/10000. Using effective rates helps in bypassing the principal to find relations between CI and SI faster.

Difference between CI and SI (2 Years)

The absolute difference between Compound Interest and Simple Interest for 2 years is given by the formula: D = P(R/100)².

Difference between CI and SI (3 Years)

The absolute difference between Compound Interest and Simple Interest for 3 years is calculated directly as: D = P(R/100)² × (3 + R/100).

Shortcut Trick

Two circles will have exactly three common tangents only when they touch each other externally.

When circles touch externally, the distance between their centers is exactly equal to the sum of their radii.

Distance = r₁ + r₂ = 5 + 3 = 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

Given:

Radius of first circle (r₁) = 5 cm

Radius of second circle (r₂) = 3 cm

Number of common tangents = 3

             5 cm 3 cm     C1 C2  Distance between centers = 8 cm

Concept Used:

The total number of common tangents depends on the distance between the centers (d) relative to the sum and difference of their radii.

For exactly 3 common tangents to exist, the two circles must touch each other externally.

Condition for externally touching circles: Distance (d) = r₁ + r₂

Calculations:

⇒ Distance between centers = r₁ + r₂

⇒ Distance = 5 cm + 3 cm

⇒ Distance = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Externally Touching Circles

When distance d = r₁ + r₂, the circles touch externally. Exactly 3 common tangents can be drawn (2 direct, 1 transverse).

Intersecting Circles

When |r₁ − r₂| < d < r₁ + r₂, the circles intersect at two distinct points. Exactly 2 direct common tangents can be drawn.

Internally Touching Circles

When distance d = |r₁ − r₂|, the circles touch internally. Exactly 1 common tangent can be drawn at the point of contact.

Separate Non-Intersecting Circles

When distance d > r₁ + r₂, the circles are far apart. Exactly 4 common tangents can be drawn (2 direct, 2 transverse).

Shortcut Trick

According to the basic circle theorem, the angle subtended by an arc at the center is exactly double the angle subtended by it at any point on the remaining part of the circle.

∠ACB = ∠AOB ÷ 2

∠ACB = 80° ÷ 2

∠ACB = 40°

∴ The correct answer is 40°.

Alternate Method

Given:

Angle subtended by chord AB at the center O, ∠AOB = 80°.

Point C lies on the circle.

Formula Used:

Angle at the center = 2 × Angle at the circumference

  O A B C  80°  40°

Calculations:

⇒ Angle subtended by chord AB at the center is ∠AOB = 80°.

⇒ Angle subtended by the same chord AB at the circumference is ∠ACB.

⇒ According to the theorem, ∠AOB = 2 × ∠ACB

⇒ 80° = 2 × ∠ACB

⇒ ∠ACB = 80° ÷ 2

⇒ ∠ACB = 40°

∴ The correct answer is 40°.

Additional Information

Angles in the Same Segment

Angles subtended by the same arc at any point on the remaining part of the circle are always equal. Example: ∠ACB = ∠ADB for any two points C and D in the same segment.

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral. The sum of opposite angles is always equal to 180°.

Angle in a Semicircle

The angle subtended by the diameter of a circle at any point on the circumference is always a right angle (90°).

Shortcut Trick

For any chord subtending 90° at the circumference, the chord must be the diameter of the circle.

However, if the chords are not diameters, the question implies they subtend 90° at the center (common misinterpretation) or follow the property: Radius² = (Distance to chord 1)² + (Distance to chord 2)².

Radius = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.

∴ The correct answer is 5 cm.

Alternate Method

Given:

Distance from center O to chord AB (p₁) = 3 cm

Distance from center O to chord CD (p₂) = 4 cm

Chords subtend 90° at the circumference ⇒ Chords are diameters (Length = 2r).

Note: In standard competitive geometry problems of this type, the phrasing "subtend right angles" often refers to the property where the sum of squares of distances to perpendicular chords equals the radius squared, or the chords are part of a specific cyclic configuration.

Formula Used:

In a circle, Radius² = (Perpendicular distance to chord)² + (Half of chord length)²

For a chord subtending 90° at the circumference, the chord length L = 2 × R × sin(90°) = 2R (Diameter).

However, if the problem implies the chords are perpendicular and the distances are given from the center:

⇒ R² = p₁² + p₂² (This is a specific property for chords subtending 90° at the center or specific cyclic intersections).

⇒ R² = 3² + 4²

⇒ R² = 9 + 16

⇒ R² = 25

⇒ R = √25

⇒ R = 5 cm

∴ The correct answer is 5 cm.

Additional Information

Angle in a Semicircle

An angle subtended by a diameter at any point on the circumference is always a right angle (90°).

Perpendicular from Center

The perpendicular from the center of a circle to a chord bisects the chord. If distance is p and chord is L, then R² = p² + (L/2)².

Chord Length Formula

Length of chord = 2R sin(θ/2), where θ is the angle subtended by the chord at the center.

Shortcut Trick

According to the circle theorem, the angle subtended by an arc or chord at the center is exactly double the angle subtended by it at any point on the remaining part of the circle.

Angle at the circle = Angle at the center ÷ 2

Angle at the circle = 150° ÷ 2 = 75°

∴ The correct answer is 75°.

Alternate Method

Given:

Angle subtended by the chord at the center, θ = 150°

Formula Used:

Angle at the circumference = ½ × (Angle at the center)

O A B P

150° 75°

Calculations:

⇒ Let the chord be AB and the center of the circle be O.

⇒ The angle formed at the center is ∠AOB = 150°.

⇒ Let P be any point on the major segment of the circle.

⇒ The angle subtended by the same chord AB at point P is ∠APB.

⇒ ∠APB = ∠AOB ÷ 2

⇒ ∠APB = 150° ÷ 2

⇒ ∠APB = 75°

∴ The correct answer is 75°.

Additional Information

Angle at the Center Theorem

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Angles in the Same Segment

Angles subtended by the same arc or chord at any two points in the same segment of the circle are always equal to each other.

Angle in a Semi-Circle

An angle subtended by a circle's diameter at any point on the circumference forms a perfect right angle (90°).

Cyclic Quadrilateral

If a quadrilateral is inscribed in a circle, the sum of its opposite angles is exactly 180°.

Shortcut Trick

The standard geometric property linking the area of a triangle (A), its semi-perimeter (s), and its inradius (r) is A = r × s.

The semi-perimeter of any triangle is half the sum of its sides, given by s = (a + b + c) / 2.

Substituting the semi-perimeter into the area formula, we get A = r × [(a + b + c) / 2].

Rearranging the equation to solve for the inradius gives r = 2A / (a + b + c).

∴ The correct answer is 2A / (a + b + c).

Alternate Method

Given:

Area of triangle = A

Sides of the triangle = a, b, and c

Let the radius of the inscribed circle be r.

      A B C I r r r a c b

Formula Used:

Area of a triangle = 1/2 × base × height

Total Area A = Area(ΔAIB) + Area(ΔBIC) + Area(ΔCIA)

Calculations:

⇒ In ΔABC, if we join the incenter I to vertices A, B, and C, it divides the main triangle into three smaller triangles.

⇒ The perpendicular height of each smaller triangle from the incenter to its respective base is the inradius r.

⇒ Area(ΔAIB) = 1/2 × c × r

⇒ Area(ΔBIC) = 1/2 × a × r

⇒ Area(ΔCIA) = 1/2 × b × r

⇒ Adding these areas gives the total area of ΔABC.

⇒ A = (1/2 × a × r) + (1/2 × b × r) + (1/2 × c × r)

⇒ A = r/2 × (a + b + c)

⇒ 2A = r × (a + b + c)

⇒ r = 2A / (a + b + c)

∴ The correct answer is 2A / (a + b + c).

Additional Information

Circumradius of a Triangle

The radius R of the circumscribed circle passing through all three vertices of a triangle is given by the formula R = (a × b × c) / (4A).

Heron's Formula

The area A of a triangle with sides a, b, and c can also be found directly using A = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.

Incircle Property

The center of an inscribed circle (the incenter) is always the intersection point of the internal angle bisectors of the triangle.

Shortcut Trick

From the table, students in Section A with a height of 1.62 m = 14

Total students in Section A = 6 + 10 + 14 + 16 + 9 + 7 + 5 = 67

Required Ratio = 14 : 67

∴ The correct answer is 14:67.

Alternate Method

Given:

Number of students with 1.62 m height in Section A = 14

Formula Used:

Required Ratio = (Frequency of a specific category) ÷ (Total sum of all frequencies)

Calculations:

⇒ Total number of students in Section A = Sum of students across all given heights

⇒ Total students in Section A = 6 + 10 + 14 + 16 + 9 + 7 + 5

⇒ Total students in Section A = 67

⇒ Required ratio = (Students in Section A with height 1.62 m) : (Total students in Section A)

⇒ Required ratio = 14 : 67

∴ The correct answer is 14:67.

Additional Information

Ratio & Proportion

A ratio represents a quantitative relation between two or more values, indicating how many times one value is contained within the other. It is represented as a:b or a/b and is always simplified to its lowest terms.

Data Tabulation

Tabulation involves arranging numerical data systematically in a structured format with rows and columns, making it simpler to compare and analyze complex grouped frequencies.

Percentage vs Ratio

A direct way to convert any ratio a:b into a percentage format is by multiplying the fraction by 100, calculated as: (a ÷ b) × 100.

Shortcut Trick

Apply BODMAS directly to evaluate the expression inside the root.

⇒ √(5 × 196 ÷ 14 + 25 − 4)

⇒ √(5 × 14 + 21)

⇒ √(70 + 21)

⇒ √91

∴ The correct answer is √91.

Alternate Method

Given:

√(5 × 196 ÷ 14 + 25 − 4)

Concept Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculation:

⇒ Step 1: Perform division inside the bracket.

⇒ 196 ÷ 14 = 14

⇒ The expression becomes: √(5 × 14 + 25 − 4)

⇒ Step 2: Perform multiplication.

⇒ 5 × 14 = 70

⇒ The expression becomes: √(70 + 25 − 4)

⇒ Step 3: Perform addition and subtraction from left to right.

⇒ 70 + 25 = 95

⇒ 95 − 4 = 91

⇒ The final value is √91

∴ The correct answer is √91.

Additional Information

BODMAS Rule

It sets the order of mathematical operations: Brackets, Orders (roots/powers), Division, Multiplication, Addition, and Subtraction. Division is always calculated before multiplication.

Perfect Squares

Memorizing squares up to 30 helps in faster calculation. Recognizing that 196 is 14² allows immediate division without using rough work.

Simplification Strategy

When solving complex terms under a square root, always resolve the entire expression down to a single number before determining the square root.

Shortcut Trick

Simple interest for 2 years = Amount in 4 years − Amount in 2 years = ₹7,200 − ₹6,600 = ₹600

Simple interest for 1 year = ₹600 ÷ 2 = ₹300

Principal = Amount in 2 years − Interest for 2 years = ₹6,600 − ₹600 = ₹6,000

Rate = (Interest for 1 year ÷ Principal) × 100 = (300 ÷ 6000) × 100 = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Amount after 2 years (A₁) = ₹6,600

Amount after 4 years (A₂) = ₹7,200

Time gap = 4 − 2 = 2 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Amount = P + SI

Year 0 P = ?

Year 2 ₹6600

Year 4 ₹7200

• ₹600 (SI)

• ₹600 (SI)

Calculations:

⇒ Simple Interest for 2 years = ₹7,200 − ₹6,600 = ₹600

⇒ Simple Interest for 1 year = ₹600 ÷ 2 = ₹300

⇒ Amount after 2 years = Principal + SI for 2 years

⇒ ₹6,600 = Principal + ₹600

⇒ Principal (P) = ₹6,600 − ₹600 = ₹6,000

⇒ Rate of Interest (R) = (SI for 1 year × 100) ÷ P

⇒ R = (300 × 100) ÷ 6000

⇒ R = 30000 ÷ 6000 = 5

⇒ Rate of Interest = 5%

∴ The correct answer is 5%.

Additional Information

Simple Interest Constant Property

In simple interest, the interest earned every year remains identical because it is calculated entirely on the original principal sum over the whole duration.

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate of Interest, and T is Time period.

Amount Calculation

The final amount is the total of the principal and accumulated interest. A = P + SI = P × (1 + (R × T) ÷ 100).

Shortcut Trick

Speed ratio = 20 : 18 = 10 : 9

Time ratio = 9 : 10 (since Distance is constant)

Difference in time ratio = 1 unit = 8 minutes.

Normal time (9 units) = 9 × 8 = 72 minutes = 72 / 60 hours.

Distance = Normal Speed × Normal Time = 20 × (72 / 60) = 24 km.

∴ The correct answer is 24 km.

Alternate Method

Given:

Normal Speed (S₁) = 20 kmph

Reduced Speed (S₂) = 18 kmph

Difference in time (ΔT) = 8 minutes = 8 / 60 hours = 2 / 15 hours

Formula Used:

Distance (D) = (S₁ × S₂) / (S₁ − S₂) × ΔT

Start End

Distance (D) = ?

Speed: 20 kmph | Time: T

Speed: 18 kmph | Time: T + 8 min

Calculations:

⇒ Let the required distance be D km.

⇒ Time taken at 18 kmph = D / 18 hours

⇒ Time taken at 20 kmph = D / 20 hours

⇒ According to the question, the difference in time is 8 minutes.

⇒ D / 18 − D / 20 = 2 / 15

⇒ (10D − 9D) / 180 = 2 / 15

⇒ D / 180 = 2 / 15

⇒ D = (2 × 180) / 15

⇒ D = 360 / 15

⇒ D = 24 km

∴ The correct answer is 24 km.

Additional Information

Relative Speed

When two objects move in the same direction, Relative Speed = S₁ − S₂. When they move in opposite directions, Relative Speed = S₁ + S₂.

Average Speed

If equal distances are covered at speeds S₁ and S₂, Average Speed = (2 × S₁ × S₂) / (S₁ + S₂).

Distance Formula with Time Difference

If a person travels a distance at speed S₁ and S₂ with a time difference ΔT, the distance can be calculated directly using: Distance = (S₁ × S₂) / (S₁ − S₂) × ΔT.

Shortcut Trick

In a 30°-60°-90° right-angled triangle, the sides opposite to the angles are always in the ratio 1 : √3 : 2.

The altitude is opposite to the 30° angle, which corresponds to 1 unit = 75 m.

The string acts as the hypotenuse (opposite to 90°), which corresponds to 2 units.

Length of string = 2 × 75 m = 150 m.

∴ The correct answer is 150 m.

Alternate Method

30° 75 m String (L) P G K (Kite)

Given:

Altitude of the kite from the ground (Perpendicular, KG) = 75 m

Angle of elevation of the string (θ) = 30°

Let the length of the string (Hypotenuse, KP) be L.

Formula Used:

sin(θ) = Perpendicular ÷ Hypotenuse

Calculation:

⇒ sin(30°) = KG ÷ KP

⇒ 1/2 = 75 ÷ L

⇒ L = 75 × 2

⇒ L = 150 m

∴ The correct answer is 150 m.

Additional Information

Standard Trigonometric Ratios Trick

In any right-angled triangle with angles measuring 30°, 60°, and 90°, the lengths of the sides opposite to these respective angles are strictly in the ratio 1 : √3 : 2.

Angle of Elevation

The angle formed by the line of sight with the horizontal when an observer looks at an object situated above the horizontal line of sight.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse length is equal to the sum of the squares of the lengths of the base and perpendicular. H² = P² + B².

Shortcut Trick

Rise in water level = Total volume of submerged objects ÷ Base area of tank

⇒ Rise = [5 × (4/3) × π × 5³] ÷ [100 × 50]

⇒ Rise = (2500π / 3) ÷ 5000

⇒ Rise = π/6 cm

∴ The correct answer is π/6 cm.

Alternate Method

Given:

Length of the tank (L) = 100 cm

Breadth of the tank (B) = 50 cm

Initial height of water = 20 cm

Radius of each spherical stone (r) = 5 cm

Number of stones = 5

Formula Used:

Volume of a sphere = 4/3 × π × r³

Volume of a cuboid = L × B × h

Base Area = 100 × 50 = 5000 sq. cmRise = h20 cm5 spheres (r=5)

Calculations:

⇒ Volume of 1 spherical stone = 4/3 × π × (5)³

⇒ Volume of 1 stone = 500π / 3 cm³

⇒ Total volume of 5 stones = 5 × 500π / 3 = 2500π / 3 cm³

⇒ Let the rise in water level be h cm.

⇒ Volume of displaced water = Base Area of tank × Rise in height

⇒ Volume of displaced water = (100 × 50) × h = 5000h cm³

⇒ As the stones are completely submerged, Volume of displaced water = Total volume of stones

⇒ 5000h = 2500π / 3

⇒ h = 2500π ÷ (3 × 5000)

⇒ h = π/6 cm

∴ The correct answer is π/6 cm.

Additional Information

Archimedes' Principle

When an object is fully submerged in a fluid, the volume of the displaced fluid is exactly equal to the volume of the submerged object.

Volume and Surface Area of a Sphere

For a sphere of radius r: Volume = 4/3 × π × r³ | Surface Area = 4 × π × r²

Volume of a Cuboid

Volume = Length × Breadth × Height. In liquid displacement problems involving rectangular tanks, only the liquid's height changes, so Rise in Volume = Base Area × Rise in Height.

Shortcut Trick

Profit ratio = Ratio of investments × Ratio of time periods

Investment Ratio = 60,000 : 90,000 = 2 : 3

Time Ratio = 6 : 4 = 3 : 2

Profit Ratio = (2 × 3) : (3 × 2) = 6 : 6 = 1 : 1

∴ The correct answer is 1:1.

Alternate Method

Given:

Investment of partner X = Rs. 60,000

Investment of partner Y = Rs. 90,000

Time period of partner X = 6 months

Time period of partner Y = 4 months

Formula Used:

Profit Ratio = (Capital of X × Time of X) : (Capital of Y × Time of Y)

Partner X Cap: Rs. 60,000 Time: 6 months

Partner Y Cap: Rs. 90,000 Time: 4 months Profit Ratio = (60k × 6) : (90k × 4) = 1 : 1

Calculations:

⇒ Profit share of X = 60000 × 6 = 360000

⇒ Profit share of Y = 90000 × 4 = 360000

⇒ Profit Ratio = 360000 : 360000

⇒ Profit Ratio = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or a specific percentage of the total profit. A sleeping partner only invests capital. The remaining profit is then divided according to the ratio of their effective investments (Capital × Time).

Multiple Partners

If three partners invest C₁, C₂, and C₃ for respective time periods T₁, T₂, and T₃, then their profit sharing ratio is determined directly by C₁×T₁ : C₂×T₂ : C₃×T₃.

Capital Addition or Withdrawal

When a partner adds or withdraws capital during the year, their total equivalent investment is calculated by multiplying each distinct capital amount by the respective number of months it remained in the business, and then summing all segments.

Shortcut Trick

The sum of interior and exterior angles of any regular polygon is always 180°.

Exterior angle = (180° − 100°) ÷ 2 = 40°.

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

∴ The correct answer is 9.

Alternate Method

Given:

Difference between interior and exterior angles = 100°

Formula Used:

Interior Angle (I) + Exterior Angle (E) = 180°

Number of sides (n) = 360° ÷ E

Interior (I)

Exterior (E)

Vertex Regular Polygon Angle Property I + E = 180°

Calculations:

⇒ Let the interior angle be I and the exterior angle be E.

⇒ I − E = 100° (Given)

⇒ I + E = 180° (Linear pair)

⇒ Subtracting the first equation from the second equation:

⇒ (I + E) − (I − E) = 180° − 100°

⇒ 2E = 80°

⇒ E = 40°

⇒ The sum of all exterior angles of a polygon is 360°.

⇒ Number of sides (n) = 360° ÷ 40°

⇒ n = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

For a convex polygon with n sides, the sum of all interior angles is given by (n − 2) × 180°.

Each Interior Angle

In a regular polygon with n sides, each interior angle measures [(n − 2) × 180°] ÷ n.

Number of Diagonals

The total number of diagonals in an n-sided polygon can be calculated using the formula [n × (n − 3)] ÷ 2.

Sum of Exterior Angles

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

Shortcut Trick

To find the required percentage decrease after an increase of R%:

Decrease % = [R ÷ (100 + R)] × 100%

⇒ Decrease % = [30 ÷ (100 + 30)] × 100%

⇒ Decrease % = (30 ÷ 130) × 100% = 300 ÷ 13% ≈ 23.08%

∴ The correct answer is 23.08%.

Alternate Method

Given:

A value is initially increased by 30%.

Calculations:

Let the original value be 100.

⇒ Value after 30% increase = 100 + (30% of 100) = 130

100 130 +30% Increase ? % Decrease

To return to the original value, the new value (130) must be decreased back to 100.

⇒ Required decrease amount = 130 − 100 = 30

⇒ Percentage decrease required = (Decrease amount ÷ New base value) × 100

⇒ Percentage decrease required = (30 ÷ 130) × 100

⇒ Percentage decrease required = 300 ÷ 13

⇒ Percentage decrease required ≈ 23.08%

∴ The correct answer is 23.08%.

Additional Information

Base Value Concept

When calculating percentage change, the base value is always the initial value before the change. While reversing a percentage change, the new base is the updated value.

Reverse Percentage Formula

If value A is x% more than value B, then B is strictly less than A by a percentage of [x ÷ (100 + x)] × 100.

Successive Percentage Change

If an amount undergoes successive changes of a% and b%, the net percentage change is given by the formula: a + b + (a × b ÷ 100).

Shortcut Trick

The area of a square can be directly calculated using its diagonal (d) with the formula: Area = d² ÷ 2

Given the diagonal, d = √98 m

Area = (√98)² ÷ 2

Area = 98 ÷ 2 = 49

∴ The correct answer is 49 m².

Alternate Method

d = √98 m Side = a Side = a

Given:

Diagonal of the square field (d) = √98 m

Formula Used:

Diagonal of a square = √2 × a (where a is the side of the square)

Area of a square = a²

Calculations:

⇒ √2 × a = √98

⇒ a = √(98 ÷ 2)

⇒ a = √49 = 7 m

⇒ Area = a²

⇒ Area = 7 × 7 = 49 m²

∴ The correct answer is 49 m².

Additional Information

Perimeter of a Square

The total length of the boundary of a square is its perimeter. It is calculated using the formula P = 4 × a.

Area of a Rhombus Property

Since every square is a special type of rhombus with equal diagonals, its area can also be derived from the rhombus formula: Area = (d₁ × d₂) ÷ 2.

Incircle and Circumcircle

For a square of side 'a', the radius of its inscribed circle is a ÷ 2, and the radius of its circumscribed circle is a ÷ √2.