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Shortcut Trick

Convert each surd into its simplest form by extracting perfect square factors.

√98 = 7√2 | √32 = 4√2 | √50 = 5√2

Expression = 7√2 − 4√2 + 5√2 = 8√2

∴ The correct answer is 8√2.

Alternate Method

Given:

The expression to simplify is √98 − √32 + √50

Method Used:

Prime factorisation method to simplify the square roots.

Calculations:

⇒ √98 = √(2 × 49) = √(2 × 7 × 7) = 7√2

⇒ √32 = √(2 × 16) = √(2 × 4 × 4) = 4√2

⇒ √50 = √(2 × 25) = √(2 × 5 × 5) = 5√2

Substituting these simplified values into the original expression:

⇒ 7√2 − 4√2 + 5√2

⇒ (7 − 4 + 5)√2

⇒ (3 + 5)√2

⇒ 8√2

∴ The correct answer is 8√2.

Additional Information

Properties of Surds

For any positive numbers a and b, √(a × b) = √a × √b. This property allows us to separate perfect squares from non-perfect squares under the root.

Addition and Subtraction of Surds

Only like surds (surds having the exact same number under the root) can be added or subtracted directly. For example, x√a + y√a = (x + y)√a.

BODMAS Rule

Always follow the order of operations. Roots (Orders) are simplified first, followed by addition and subtraction from left to right as they have the same priority level.

Shortcut Trick When two circles touch each other internally, they share exactly one point of contact. A common tangent is a straight line that touches both circles without crossing inside them. Since one circle is inside the other, any line touching the smaller circle elsewhere will cut through the larger circle. Thus, a tangent can only be drawn exactly at their single shared point of contact. ∴ The correct answer is 1. Alternate Method

C₁ C₂ P Common Tangent

Given: Two circles are touching each other internally. Concept Used: ⇒ If two circles touch internally, the distance between their centers is equal to the difference of their radii: C1C2 = R − r. ⇒ Because the smaller circle lies completely inside the larger circle, no transverse common tangents can be drawn across them. ⇒ The circles share only one common point, which is their point of contact (let's call it P). ⇒ A common tangent must pass through point P and be perpendicular to the straight line joining the centers C1 and C2. ⇒ Because there is only one point of contact, exactly one common tangent exists. ∴ The correct answer is 1. Additional Information Circles completely non-intersecting and outside each other Distance between centers > R + r. There are exactly 4 common tangents (2 direct and 2 transverse). Circles touching externally Distance between centers = R + r. There are exactly 3 common tangents (2 direct and 1 transverse at the point of contact). Circles intersecting at two points R − r < Distance between centers < R + r. There are exactly 2 common tangents (both are direct). One circle completely inside another Distance between centers < R − r. There are 0 common tangents.

Shortcut Trick

The perpendicular from the center bisects the chord, forming a right-angled triangle with the radius.

Half of the chord length = √(Radius² − Distance²)

Half chord = √(26² − 10²) = √(676 − 100) = √576 = 24 cm.

Total length of chord AB = 2 × 24 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Radius of the circle (OA) = 26 cm

Distance of the chord from the center (OM) = 10 cm

Formula Used:

The perpendicular drawn from the center of a circle to a chord bisects the chord.

By Pythagoras theorem: Hypotenuse² = Base² + Perpendicular²

O

A B

M 10 cm

26 cm

Right Angled Triangle OMA

Calculations:

⇒ In right-angled triangle OMA,

⇒ OA² = OM² + AM²

⇒ 26² = 10² + AM²

⇒ 676 = 100 + AM²

⇒ AM² = 576

⇒ AM = √576 = 24 cm

⇒ Since OM bisects the chord AB, Length of AB = 2 × AM

⇒ AB = 2 × 24 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Perpendicular from Center to Chord

The perpendicular drawn from the center of a circle to any chord always bisects the chord into two exactly equal halves.

Equal Chords Property

Chords that are equal in length are always equidistant from the center of the circle, which is why chord CD is identically 10 cm from the center.

Pythagorean Triplets

The values (10, 24, 26) form a standard Pythagorean triplet. It is easily derived by multiplying the base triplet (5, 12, 13) by 2.

Shortcut Trick

Area of a segment = r² × [(πθ ÷ 360°) − (sinθ ÷ 2)]

⇒ Area = 6² × [(π × 60° ÷ 360°) − (sin60° ÷ 2)]

⇒ Area = 36 × [(π ÷ 6) − (√3 ÷ 4)]

⇒ Area = (6π − 9√3) sq. cm

∴ The correct answer is (6π−9√3) sq. cm.

Alternate Method

 60° O A B 6 cm 6 cm Segment

Given:

Radius of the circle (r) = 6 cm

Central angle of the sector (θ) = 60°

Formula Used:

Area of sector = (θ ÷ 360°) × πr²

Area of equilateral triangle = (√3 ÷ 4) × side²

Calculations:

⇒ Area of sector OAB = (60° ÷ 360°) × π × 6²

⇒ Area of sector OAB = (1 ÷ 6) × 36π = 6π sq. cm

⇒ Since two sides (OA and OB) are equal radii and the included angle is 60°, the remaining angles are also 60°. Thus, ▵OAB is an equilateral triangle with side 6 cm.

⇒ Area of triangle OAB = (√3 ÷ 4) × 6²

⇒ Area of triangle OAB = (√3 ÷ 4) × 36 = 9√3 sq. cm

⇒ Area of corresponding segment = Area of sector − Area of triangle

⇒ Area of corresponding segment = (6π − 9√3) sq. cm

∴ The correct answer is (6π−9√3) sq. cm.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ is directly given by the formula A = (θ ÷ 360°) × πr².

Length of an Arc

The length of an arc corresponding to a central angle θ can be calculated using the formula L = (θ ÷ 360°) × 2πr.

Area of Triangle in a Sector

The area of the triangle formed by two respective radii and their corresponding chord is obtained generally by A = (1 ÷ 2) × r² × sinθ.

Shortcut Trick

To convert a pure recurring decimal to a fraction, write the repeating digits over the same number of 9s.

Here, the repeating sequence is 12, which has two digits.

Fraction = 12 / 99

Simplifying by dividing both numerator and denominator by 3 gives 4 / 33.

∴ The correct answer is 4/33.

Alternate Method

Given:

Decimal value = 0.121212...

Method Used:

Let the repeating decimal be x. Multiply it by 10ⁿ (where n is the number of repeating digits) and subtract the original x to solve for the fraction.

Calculation:

⇒ Let x = 0.121212... — (Equation 1)

⇒ Multiply by 100 since two digits (12) are repeating.

⇒ 100x = 12.121212... — (Equation 2)

⇒ Subtracting Equation 1 from Equation 2:

⇒ 100x − x = 12.121212... − 0.121212...

⇒ 99x = 12

⇒ x = 12 / 99

⇒ Dividing numerator and denominator by 3:

⇒ x = 4 / 33

∴ The correct answer is 4/33.

Additional Information

Pure Recurring Decimal

A decimal in which all digits after the decimal point repeat continuously (e.g., 0.333...). It is converted by placing the repeating part over an equal number of 9s.

Mixed Recurring Decimal

A decimal where some digits after the decimal point do not repeat, followed by a repeating sequence (e.g., 0.12333...).

Converting Mixed Recurring Decimal

Fraction = (Total number formed after decimal − Non-repeating part) ÷ (Number of 9s equal to repeating digits followed by 0s equal to non-repeating digits).

Calculation:

(3.2 - 1$\frac35$) + (2$\frac14$ ÷ 0.5)

⇒ (3.2 - $8\over5$) + ($9\over4$ ÷ 0.5)

⇒ (3.2 - 1.6) + (2.25 ÷ 0.5)

⇒ 1.6 + 4.5

⇒ 6.1

Hence the correct answer is option (2).

Shortcut Trick

Marketing department's share = ₹36,000 × [5 ÷ (4 + 5 + 3)] = ₹36,000 × (5/12) = ₹15,000

Remaining amount after deductions = ₹15,000 − ₹3,000 − ₹1,500 = ₹10,500

Ratio of top performers = 5 : 4, so the smaller share gets 4 parts out of 9.

Smaller share = ₹10,500 × (4/9) = ₹4,666.67

∴ The correct answer is ₹4,666.67.

Alternate Method

Total: 36k 5/12 Mktg: 15k - 4.5k Rem: 10.5k 5 parts 4 parts Smaller Share = 4/9 × 10500 = 4666.67

Given:

Total bonus pool = ₹36,000

Bonus ratio for Sales, Marketing, and Operations = 4 : 5 : 3

Marketing deductions = ₹3,000 (team building) and ₹1,500 (intern)

Top performers' allocation ratio = 5 : 4

Calculations:

⇒ Total initial ratio parts = 4 + 5 + 3 = 12 parts

⇒ Marketing department's share = (5 ÷ 12) × 36,000 = ₹15,000

⇒ Total fixed deductions = 3,000 + 1,500 = ₹4,500

⇒ Remaining amount for top performers = 15,000 − 4,500 = ₹10,500

⇒ The remaining amount is split into 5 + 4 = 9 parts.

⇒ The smaller share corresponds to the 4 parts ratio.

⇒ Value of the smaller share = (4 ÷ 9) × 10,500

⇒ Value = 42,000 ÷ 9 = ₹4,666.66...

∴ The correct answer is ₹4,666.67.

Additional Information

Ratio Distribution Rule

When a quantity Q is divided in the ratio a : b : c, the share of the first part is calculated directly as [a ÷ (a+b+c)] × Q.

Sequential Allocations

When solving hierarchical distribution problems, calculate the pool for each subsequent stage by subtracting any fixed amounts before applying the new ratio constraints.

Proportions in Partnerships

If profits, revenues, or bonuses are distributed among groups, the respective shares behave similarly to partnership accounts, where total allocation strictly bounds internal distributions.

Shortcut Trick

For continued fractions, always calculate from the bottommost nested denominator upwards.

Bottommost expression = 2 + 1/2 = 5/2

Middle expression = 2 + 1 / (5/2) = 2 + 2/5 = 12/5

Final top expression = 1 + 1 / (12/5) = 1 + 5/12 = 17/12

∴ The correct answer is 17/12.

Alternate Method

Given:

x = 1 + 1 / (2 + 1 / (2 + 1/2))

Calculations:

⇒ First, we simplify the innermost fraction at the bottom: 2 + 1/2

⇒ 2 + 1/2 = (4 + 1) / 2 = 5/2

⇒ Substitute this back into the expression:

⇒ x = 1 + 1 / (2 + 1 / (5/2))

⇒ Taking the reciprocal: 1 / (5/2) = 2/5

⇒ x = 1 + 1 / (2 + 2/5)

⇒ Now, simplify the new bottom fraction: 2 + 2/5

⇒ 2 + 2/5 = (10 + 2) / 5 = 12/5

⇒ Substitute this back into the expression:

⇒ x = 1 + 1 / (12/5)

⇒ Taking the reciprocal: 1 / (12/5) = 5/12

⇒ x = 1 + 5/12

⇒ x = (12 + 5) / 12 = 17/12

∴ The correct answer is 17/12.

Additional Information

Continued Fractions

A continued fraction is evaluated by working from the bottommost nested fraction upwards to the main fraction.

Reciprocal Rule

The reciprocal of a fraction a/b is b/a. This property is used repeatedly when resolving 1 ÷ (a/b).

BODMAS Rule

For nested expressions, calculations must always resolve the deepest levels first according to the standard order of operations.

Shortcut Trick

Profit ratio is directly proportional to Equivalent Capital (Capital × Time).

Kapil's equivalent capital = (10 × 6) + (6 × 6) = 60 + 36 = 96 units

Manish's equivalent capital = 8 × 9 = 72 units

Profit Ratio = 96 : 72 = 4 : 3

∴ The correct answer is 4:3.

Alternate Method

Given:

Kapil's initial investment = ₹1,00,000

Manish's investment after 3 months = ₹80,000

Kapil withdrew ₹40,000 after 3 more months (i.e., at the end of 6 months).

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

₹1,00,000 × 6m

₹60,000 × 6m

Kapil

₹80,000 × 9m

Manish

0m 3m 6m 12m

Profit Ratio = (100k × 6 + 60k × 6) : (80k × 9) = 960 : 720 = 4:3

Calculations:

⇒ Kapil invested ₹1,00,000 for the first 6 months.

⇒ After 6 months, his capital becomes = ₹1,00,000 − ₹40,000 = ₹60,000.

⇒ Kapil invested ₹60,000 for the remaining 6 months of the year.

⇒ Kapil's equivalent capital = (1,00,000 × 6) + (60,000 × 6)

⇒ Kapil's equivalent capital = 6,00,000 + 3,60,000 = 9,60,000

⇒ Manish joined after 3 months, so his money was invested for (12 − 3) = 9 months.

⇒ Manish's equivalent capital = 80,000 × 9 = 7,20,000

⇒ Profit Ratio = Kapil's equivalent capital : Manish's equivalent capital

⇒ Profit Ratio = 9,60,000 : 7,20,000

⇒ Profit Ratio = 96 : 72 = 4 : 3

∴ The correct answer is 4:3.

Additional Information

Compound Partnership

When partners invest for different time periods or change their capital mid-way, profit is distributed based on the product of Capital and Time (Equivalent Capital).

Simple Partnership

When all partners invest their capital for the exact same time period, the profit is divided purely in the ratio of their invested capitals.

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or commission from the profit before the remaining profit is divided among all partners based on investment.

Shortcut Trick

Profit ratio of partners = Ratio of their (Capital × Time).

A's Profit : B's Profit = (60000 × 10) : (90000 × 8) = 600000 : 720000 = 5 : 6

Total profit ratio parts = 5 + 6 = 11 units, which corresponds to the total profit of ₹40,000.

A's share = (5 ÷ 11) × 40000 = 200000 ÷ 11 ≈ ₹18,181.82

∴ The correct answer is ₹18,181.82.

Alternate Method

Given:

A's investment = ₹60,000 | Time period = 10 months

B's investment = ₹90,000 | Time period = 8 months

Total Profit = ₹40,000

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Partner A₹60,000× 10 months= 600,000Partner B₹90,000× 8 months= 720,000Profit Ratio → 600k : 720k = 5 : 6Total Profit = ₹40,000

Calculations:

⇒ Profit Ratio of A to B = (60000 × 10) : (90000 × 8)

⇒ Profit Ratio = 600000 : 720000

⇒ Simplifying the ratio by dividing by 120000, we get 5 : 6

⇒ Total parts in the profit ratio = 5 + 6 = 11

⇒ A's Share = (A's ratio ÷ Total ratio) × Total Profit

⇒ A's Share = (5 ÷ 11) × 40000

⇒ A's Share = 200000 ÷ 11 ≈ 18181.8181...

∴ The correct answer is ₹18,181.82.

Additional Information

Compound Partnership

When partners invest different amounts for different time periods, the profit ratio is calculated as the product of their investment and time: (Capital₁ × Time₁) : (Capital₂ × Time₂).

Simple Partnership

If all partners invest their capital for the exact same period of time, their profit is shared precisely in the ratio of their initial investments.

Active vs Sleeping Partner

An active partner manages the business and usually receives a fixed salary or a certain percentage of the total profit before the remaining profit is distributed based on the normal capital ratio. A sleeping partner only invests the capital.

Finding the Time Ratio

If the profit ratio and the investment capital ratio are both known, the time ratio can be directly found using: Time Ratio = Profit Ratio ÷ Capital Ratio.