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Shortcut Trick

Product of diagonals = 2 × Area = 2 × 600 = 1200.

Identify two numbers with product 1200 and difference 10: 40 and 30.

Semi-diagonals are 20 cm and 15 cm. Using Pythagorean triplet (15, 20, 25), the side is 25 cm.

Perimeter = 4 × 25 = 100 cm.

∴ The correct answer is 100 cm.

Alternate Method

Given: Area of Rhombus = 600 cm², difference between diagonals = 10 cm.

Formula Used: Area = (1/2) × d₁ × d₂; Side (s) = √[(d₁/2)² + (d₂/2)²]; Perimeter = 4s.

Let the smaller diagonal be d cm. Then the longer diagonal is (d + 10) cm.

⇒ Area = (1/2) × d × (d + 10) = 600

⇒ d² + 10d = 1200

⇒ d² + 10d − 1200 = 0

⇒ d² + 40d − 30d − 1200 = 0

⇒ d(d + 40) − 30(d + 40) = 0

⇒ (d − 30)(d + 40) = 0

⇒ d = 30 cm (Since length cannot be negative)

⇒ d₁ = 30 cm and d₂ = 30 + 10 = 40 cm

⇒ Side of rhombus (s) = √[(30/2)² + (40/2)²]

⇒ s = √[15² + 20²] = √[225 + 400]

⇒ s = √625 = 25 cm

⇒ Perimeter = 4 × 25 = 100 cm

∴ The correct answer is 100 cm.

Additional Information

Properties of a Rhombus

All sides are equal, and diagonals bisect each other at 90°. Opposite angles are equal, and adjacent angles are supplementary (sum = 180°).

Relation Between Side and Diagonals

The sum of the squares of the diagonals is equal to four times the square of the side: d₁² + d₂² = 4s².

Area of Rhombus

Area can also be calculated using base and height: Area = Base × Height, or using trigonometry: Area = s² × sin(θ).

Shortcut Trick

Break down the denominator terms: 2√12 = 4√3, √32 = 4√2, and √50 = 5√2.

Combine like terms: (5√3 − 4√3) + (5√2 − 4√2) = √3 + √2.

Factorize the numerator: 3 + √6 = √3 × √3 + √3 × √2 = √3(√3 + √2).

The expression simplifies to: √3(√3 + √2) ÷ (√3 + √2) = √3.

∴ The correct answer is √3.

Alternate Method

Given Expression: (3 + √6) ÷ (5√3 − 2√12 − √32 + √50)

Step 1: Simplify the Denominator

⇒ Denominator = 5√3 − 2√(4 × 3) − √(16 × 2) + √(25 × 2)

⇒ Denominator = 5√3 − 2(2√3) − 4√2 + 5√2

⇒ Denominator = 5√3 − 4√3 − 4√2 + 5√2

⇒ Denominator = (5 − 4)√3 + (5 − 4)√2 = √3 + √2

Step 2: Factorize the Numerator

⇒ Numerator = 3 + √6

⇒ Numerator = (√3 × √3) + (√3 × √2)

⇒ Numerator = √3(√3 + √2)

Step 3: Simplify the Ratio

⇒ Result = [√3(√3 + √2)] ÷ (√3 + √2)

⇒ Result = √3

∴ The correct answer is √3.

Additional Information

Simplification of Surds

To simplify a surd, extract the largest perfect square factor from under the radical sign, such as √12 = √(4 × 3) = 2√3.

Addition and Subtraction of Surds

Only like surds (those with the same irrational part) can be added or subtracted together, e.g., 5√2 + 3√2 = 8√2.

Conjugate Surds

The conjugate of (√a + √b) is (√a − √b). Multiplying a surd by its conjugate results in a rational number: (√a + √b)(√a − √b) = a − b.

Shortcut Trick

Given sin A = 5/7, then cosec A = 7/5, cos² A = 1 − (5/7)² = 24/49, and sec A = 7/√24 = 7/2√6.

Numerator = 5(7/5) − 7(5/7) = 7 − 5 = 2.

Denominator = 2√6(7/2√6) + 7(24/49) = 7 + 24/7 = 73/7.

Final value = 2 ÷ (73/7) = 14/73.

∴ The correct answer is 14/73.

Alternate Method

Given:

sin A = 5/7

Formula Used:

sin A = Opposite/Hypotenuse | cos² A = 1 − sin² A | sec A = 1/cos A | cosec A = 1/sin A

⇒ Base = √(Hypotenuse² − Opposite²) = √(7² − 5²) = √(49 − 25) = √24 = 2√6

⇒ cosec A = 7/5

⇒ sec A = 7/2√6

⇒ cos² A = (2√6/7)² = 24/49

⇒ Numerator = 5 × (7/5) − 7 × (5/7) = 7 − 5 = 2

⇒ Denominator = 2√6 × (7/2√6) + 7 × (24/49) = 7 + 24/7 = (49 + 24)/7 = 73/7

⇒ Required Value = 2 ÷ (73/7) = 14/73

∴ The correct answer is 14/73.

Additional Information

Trigonometric Identities

The fundamental identity is sin² A + cos² A = 1, from which we derive sec² A − tan² A = 1 and cosec² A − cot² A = 1.

Reciprocal Ratios

The primary ratios and their reciprocals are: cosec A = 1/sin A, sec A = 1/cos A, and cot A = 1/tan A.

Double Angle Formula

cos 2A can be expressed in terms of sin and cos as: cos 2A = cos² A − sin² A = 2cos² A − 1 = 1 − 2sin² A.

Shortcut Trick

A number is divisible by 88 if it is divisible by both its co-prime factors 8 and 11.

Quick check for 8: The last three digits must be divisible by 8.

For 1058, the last three digits are 058, which is not divisible by 8 (58 ÷ 8 = 7.25).

Therefore, 1058 cannot be divisible by 88.

∴ The correct answer is 1058.

Alternate Method

Given: The numbers are 1232, 968, 1144, and 1058.

Formula: Divisibility by 88 ⇒ Divisibility by 8 AND 11.

⇒ For 1232: Last 3 digits 232 ÷ 8 = 29 (Yes); Difference of sums (1+3) − (2+2) = 0 (Yes). Divisible by 88.

⇒ For 968: Last 3 digits 968 ÷ 8 = 121 (Yes); Difference of sums (9+8) − 6 = 11 (Yes). Divisible by 88.

⇒ For 1144: Last 3 digits 144 ÷ 8 = 18 (Yes); Difference of sums (1+4) − (1+4) = 0 (Yes). Divisible by 88.

⇒ For 1058: Last 3 digits 058 ÷ 8 = 7.25 (No). Difference of sums (1+5) − (0+8) = −2 (No).

⇒ Since 1058 fails both tests, it is not divisible by 88.

∴ The correct answer is 1058.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits are divisible by 8 or if the last three digits are 000.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and the sum of digits at even positions is either 0 or a multiple of 11.

Composite Divisibility

If a number is divisible by two co-prime numbers a and b, then it must also be divisible by their product a × b.

Shortcut Trick

Compare the ratios: Present (5:7) and After 4 years (6:8).

The difference between parts is equal: (6 − 5) = 1 unit and (8 − 7) = 1 unit.

This 1 unit difference corresponds to the time gap of 4 years.

⇒ 1 unit = 4 years.

⇒ Younger sister's present age = 5 units = 5 × 4 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given:

Ratio of present ages = 5 : 7

Ratio after 4 years = 6 : 8

Calculation:

Let the present ages be 5x and 7x.

⇒ (5x + 4) / (7x + 4) = 6 / 8

⇒ (5x + 4) / (7x + 4) = 3 / 4

⇒ 4 × (5x + 4) = 3 × (7x + 4)

⇒ 20x + 16 = 21x + 12

⇒ 21x − 20x = 16 − 12

⇒ x = 4

⇒ Present age of younger sister = 5x = 5 × 4 = 20 years.

∴ The correct answer is 20 years.

Additional Information

Ratio and Age Problems

When the difference between the ratio parts is same for both subjects, the change in value is directly proportional to the change in ratio units.

Equating Differences

If the difference between ratios is not equal, multiply the ratios by the difference of the other ratio to make the unit gap constant.

Age Gap Property

The difference between the ages of two people remains constant throughout their lives, regardless of the number of years passed.

Shortcut Trick

The age difference between the mother and daughter is 36 − 12 = 24 years.

Since the age difference remains constant, when the mother is twice the daughter's age (ratio 2:1), the difference of 1 unit must equal 24 years.

Therefore, the daughter's age at that time will be 1 unit = 24 years.

The number of years required is 24 − 12 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given:

Current age of Mother = 36 years

Current age of Daughter = 12 years

Let the required number of years be x.

After x years:

Mother's age = 36 + x

Daughter's age = 12 + x

According to the condition (Mother = 2 × Daughter):

⇒ 36 + x = 2 × (12 + x)

⇒ 36 + x = 24 + 2x

⇒ 36 − 24 = 2x − x

⇒ x = 12

∴ The correct answer is 12 years.

Additional Information

Constant Age Difference

The difference between the ages of two people always remains the same, regardless of how many years pass (Difference = Age₁ − Age₂).

Ratio Method for Ages

If the ratio of ages changes from a:b to c:d over time, the difference between the ratio parts (scaled appropriately) must correspond to the constant age difference.

Age after 'n' years

If the current age is A, then the age after n years is A + n, and the age n years ago was A − n.

Shortcut Trick

If the sum of three numbers is zero (a + b + c = 0), then a³ + b³ + c³ = 3abc.

Here, a = 21, b = −2, and c = −19.

Checking the sum: 21 + (−2) + (−19) = 21 − 21 = 0.

Therefore, the value is 3 × 21 × (−2) × (−19) = 63 × 38 = 2394.

∴ The correct answer is 2394.

Alternate Method

Given: Expression 21³ + (−2)³ + (−19)³

Calculation:

⇒ Calculate 21³ = 21 × 21 × 21 = 9261

⇒ Calculate (−2)³ = (−2) × (−2) × (−2) = −8

⇒ Calculate (−19)³ = (−19) × (−19) × (−19) = −6859

⇒ Sum = 9261 + (−8) + (−6859)

⇒ Sum = 9261 − 8 − 6859

⇒ Sum = 9261 − 6867 = 2394

∴ The correct answer is 2394.

Additional Information

Algebraic Identity for Cubes

The fundamental identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

The Conditional Property

When (a + b + c) = 0, the entire expression (a + b + c)(a² + b² + c² − ab − bc − ca) becomes zero, which simplifies the equation to a³ + b³ + c³ = 3abc.

Cubes of Negative Numbers

The cube of a negative number is always negative, i.e., (−x)³ = −(x³).

Shortcut Trick

Use the weighted average formula where Time acts as the weight.

Average Speed = (S₁t₁ + S₂t₂ + S₃t₃) ÷ (t₁ + t₂ + t₃)

Average Speed = (52 × 3 + 65 × 4 + 76 × 3) ÷ 10

Average Speed = (156 + 260 + 228) ÷ 10 = 644 ÷ 10 = 64.4 km/hr.

∴ The correct answer is 64.4 km/hr.

Alternate Method

Given:

Time spent by Arshdeep = 3 hr, Speed = 52 km/hr

Time spent by Sandeep = 4 hr, Speed = 65 km/hr

Time spent by Avtar = 3 hr, Speed = 76 km/hr

Formula Used:

Distance = Speed × Time

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Distance covered by Arshdeep = 52 × 3 = 156 km

⇒ Distance covered by Sandeep = 65 × 4 = 260 km

⇒ Distance covered by Avtar = 76 × 3 = 228 km

⇒ Total Distance = 156 + 260 + 228 = 644 km

⇒ Total Time = 3 + 4 + 3 = 10 hr

⇒ Average Speed = 644 ÷ 10 = 64.4 km/hr

∴ The correct answer is 64.4 km/hr.

Additional Information

Average Speed for Equal Distance

When a person travels two equal distances at speeds x and y, the average speed is given by 2xy ÷ (x + y).

Average Speed for Equal Time

If a person travels at speeds x and y for equal intervals of time, the average speed is simply the arithmetic mean: (x + y) ÷ 2.

Relative Speed

Speed of one object with respect to another. For opposite directions, add speeds (S₁ + S₂); for the same direction, subtract them (S₁ − S₂).

Shortcut Trick

Compare pairs using Cross-Multiplication: a c/d if (a × d) > (b × c).

1. 7/11 and 9/14 → 7×14 = 98, 9×11 = 99 ∴ 9:14 is larger.
2. 9/14 and 11/18 → 9×18 = 162, 11×14 = 154 ∴ 9:14 is larger.
3. 9/14 and 13/21 → 9×21 = 189, 13×14 = 182 ∴ 9:14 is larger.

∴ The correct answer is 9 : 14.

Alternate Method

Given: Ratios are 7 : 11, 9 : 14, 11 : 18, and 13 : 21.

Formula Used: Convert each ratio into its Decimal Equivalent for direct comparison.

Calculations:

⇒ 7 ÷ 11 ≈ 0.6363

⇒ 9 ÷ 14 ≈ 0.6428

⇒ 11 ÷ 18 ≈ 0.6111

⇒ 13 ÷ 21 ≈ 0.6190

⇒ Comparing the values: 0.6428 > 0.6363 > 0.6190 > 0.6111

⇒ Therefore, 9 : 14 is the greatest ratio.

∴ The correct answer is 9 : 14.

Additional Information

Comparison of Fractions

For any two fractions a/b and c/d, if ad > bc, then a c/d. If ad < bc, then a/b < c/d.

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Compounded Ratio

The compounded ratio of (a : b) and (c : d) is calculated as (a × c) : (b × d).

Shortcut Trick

Use the Successive Multiplying Factor method for effective discounts.

Net Multiplier = (100 − 40)% × (100 − 20)% = 0.6 × 0.8 = 0.48

Final Price = 2200 × 0.48 = 22 × 48 = 1056

∴ The correct answer is ₹1056.

Alternate Method

Given: Marked Price (MP) = ₹2,200 | First Discount = 40% | Additional Discount (Credit Card) = 20%

Formula Used: Selling Price (SP) = MP × (1 − d₁/100) × (1 − d₂/100)

⇒ Price after 40% discount = 2200 − (40% of 2200)

⇒ Price = 2200 − 880 = ₹1320

⇒ Price after 20% additional discount = 1320 − (20% of 1320)

⇒ Price = 1320 − 264 = ₹1056

∴ The correct answer is ₹1056.

Additional Information

Net Successive Discount Formula

For two successive discounts of a% and b%, the Equivalent Single Discount = (a + b − ab/100)%.

Marked Price (MP) and Selling Price (SP)

The price printed on the label is MP. The actual price after deducting the Discount from MP is the SP.

Successive Percentage Change

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). Use '+' for increase and '−' for decrease (discount).