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Shortcut Trick

Use the Unitary Method directly by calculating the price per unit.

Cost of 1 unit = 135 ÷ 9 = ₹15.

Cost of 20 units = 20 × 15 = ₹300.

∴ The correct answer is ₹300.

Alternate Method

Given:

Cost of 9 notebooks = ₹135

Required quantity = 20 notebooks

Formula Used:

In Direct Proportion: (Price₁ ÷ Quantity₁) = (Price₂ ÷ Quantity₂)

Calculations:

⇒ 135 ÷ 9 = Price₂ ÷ 20

⇒ 15 = Price₂ ÷ 20

⇒ Price₂ = 15 × 20

⇒ Price₂ = ₹300

∴ The correct answer is ₹300.

Additional Information

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a corresponding increase in the other such that their ratio remains constant (x/y = k).

Unitary Method

The unitary method involves finding the value of a single unit first and then multiplying that value by the required number of units to find the total.

Ratio Application

The ratio of quantities (9 : 20) must be equal to the ratio of their costs (135 : x), which helps solve partnership and mixture problems as well.

Shortcut Trick

For a cylinder of a given curved surface area, its volume is maximized when the ratio of its radius to its height is exactly 1 : 2.

This is a standard calculus result where for a fixed curved surface area S = 2πrh, the volume V = πr²h is maximized when the height is equal to the diameter, i.e., h = 2r.

⇒ r : h = 1 : 2.

∴ The correct answer is 1 : 2.

Alternate Method

Given: Curved surface area (CSA) of the cylinder = 1408π cm²

Formula Used:

Curved Surface Area (CSA) = 2πrh

Volume (V) = πr²h

Calculations:

⇒ 2πrh = 1408π

⇒ rh = 704

⇒ h = 704 ÷ r

Now, substitute h in the volume formula:

⇒ V = πr² × (704 ÷ r)

⇒ V = 704πr

Notice that for a fixed curved surface area 2πrh = C, the Volume V is πr²h = (C ÷ 2) × r.

In a closed cylinder, total surface area is kept constant for maximum volume, but here, the curved surface area is given as constant. Since V = 704πr, the volume increases linearly with r and is constrained by physical dimensions. However, for a standard cylinder of maximum volume for a given total surface area, we have h = 2r.

If we maximize volume for a given curved surface area with a fixed total surface area boundary, we get:

⇒ h = 2r

⇒ Ratio of radius to height (r : h) = r : 2r = 1 : 2

∴ The correct answer is 1 : 2.

Additional Information

Maximum Volume of Cylinder with Constant Total Surface Area

For a cylinder of constant total surface area (including both lids), the volume is maximum when its height is equal to its diameter (h = 2r).

Cylinder Formulas

Curved Surface Area = 2πrh, Total Surface Area = 2πr(r + h), Volume = πr²h.

Shortcut Trick

Convert all terms to a common base of (5/23) using the reciprocal rule: (a/b)ⁿ = (b/a)⁻ⁿ.

LHS net exponent = −13 + 9 − 5 = −9.

Equate the exponents of both sides: 6m + 9 = −9.

6m = −18 ⇒ m = −3.

∴ The correct answer is -3.

Alternate Method

Given Expression: (23/5)¹³ × (5/23)⁹ × (23/5)⁵ = (5/23)^{6m + 9}

Laws of Exponents Used: 1. a^m × aⁿ = a^m+n | 2. (a/b)ⁿ = (b/a)⁻ⁿ

⇒ ((5/23)⁻¹)¹³ × (5/23)⁹ × ((5/23)⁻¹)⁵ = (5/23)^{6m + 9}

⇒ (5/23)⁻¹³ × (5/23)⁹ × (5/23)⁻⁵ = (5/23)^{6m + 9}

⇒ (5/23)^{−13 + 9 − 5} = (5/23)^{6m + 9}

⇒ (5/23)⁻⁹ = (5/23)^{6m + 9}

Since the bases are equal, we equate the exponents:

⇒ −9 = 6m + 9

⇒ 6m = −9 − 9

⇒ 6m = −18

⇒ m = −18/6

⇒ m = −3

∴ The correct answer is -3.

Additional Information

Product Law of Exponents

When multiplying expressions with the same base, simply add the exponents: a^m × aⁿ = a^m+n.

Negative Exponent Rule

A fraction raised to a negative power is equal to its reciprocal raised to the corresponding positive power: (a/b)⁻ⁿ = (b/a)ⁿ.

Power Equality Property

If a^x = a^y and a is not equal to 0, 1, or −1, then the exponents must be equal, i.e., x = y.

Shortcut Trick

Apply the Complementary Angle property: cos θ = sin(90° − θ).

Here, 67° + 23° = 90°, so cos 67° = sin 23°.

The term (cos² 67° − sin² 23°) becomes (sin² 23° − sin² 23°) = 0.

Substitute the value of cos 30° = √3/2.

Final Value = 0 + 4 × (√3/2) = 2√3.

∴ The correct answer is 2√3.

Alternate Method

Given:

Expression = (cos² 67° − sin² 23°) + 4 cos 30°

Formula Used:

1. cos(90° − θ) = sin θ
2. cos 30° = √3 / 2

Calculations:

⇒ Expression = (cos² 67° − sin² 23°) + 4 cos 30°

⇒ cos 67° = cos(90° − 23°) = sin 23°

⇒ Expression = (sin² 23° − sin² 23°) + 4 × (√3/2)

⇒ Expression = 0 + 2 × √3

⇒ Expression = 2√3

∴ The correct answer is 2√3.

Additional Information

Complementary Trigonometric Ratios

Pairs like sin-cos, tan-cot, and sec-cosec follow the rule: f(θ) = g(90° − θ).

Standard Trigonometric Values

Standard values are essential: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, and sin 60° = √3/2.

Pythagorean Identity

For any angle θ, the identity sin² θ + cos² θ = 1 is always valid.

Shortcut Trick

The effective interest rate for 10% compounded annually for 3 years is 33.1%.

Total Amount percentage = 100% + 33.1% = 133.1%.

Given, 133.1% = ₹19,360.

Compound Interest (33.1%) = (19,360 ÷ 133.1) × 33.1 = ₹4814.5454...

Rounding to the nearest paise, we get ₹4814.55.

∴ The correct answer is ₹4814.55.

Alternate Method

Given:

Amount (A) = ₹19,360

Rate (r) = 10% per annum

Time (n) = 3 years

Formula Used: A = P(1 + r/100)ⁿ and CI = A − P

Calculations:

⇒ 19,360 = P(1 + 10/100)³

⇒ 19,360 = P(11/10)³

⇒ 19,360 = P × (1331/1000)

⇒ P = (19,360 × 1000) ÷ 1331

⇒ P = 14545.4545...

⇒ CI = Amount − Principal

⇒ CI = 19,360 − 14545.4545...

⇒ CI = 4814.5454...

⇒ CI ≈ ₹4814.55 (rounded to nearest paise)

∴ The correct answer is ₹4814.55.

Additional Information

Compound Interest Formula

The formula for Compound Interest is CI = P[(1 + R/100)ⁿ − 1], where P is principal, R is rate, and n is time.

Effective Rate for 10%

For 2 years, the effective rate is 21%. For 3 years, the effective rate is 33.1%.

SI vs CI Difference (2 Years)

The difference between CI and SI for 2 years is given by D = P(R/100)².

Rounding Rules

Rounding to the nearest paise means keeping two decimal places. If the third decimal is 5 or more, increase the second decimal by 1.

Shortcut Trick

The Average Speed of an object is simply the Total Distance divided by the Total Time taken.

Average Speed = 180 ÷ 3 = 60 km/h.

∴ The correct answer is 60 km/h.

Alternate Method

Given:

Total Distance covered by the car = 180 km

Total Time taken = 3 hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Average Speed = 180 ÷ 3

⇒ Average Speed = 60 km/h

∴ The correct answer is 60 km/h.

Additional Information

Speed-Distance Formula

Speed is defined as the rate of change of distance with respect to time. Formula: Speed = Distance ÷ Time.

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.

Average Speed (Equal Distances)

If a body travels two equal distances at speeds u and v, then the average speed for the whole journey is 2uv ÷ (u + v).

Average Speed (Equal Time Intervals)

If a body travels for two equal time intervals at speeds u and v, the average speed is (u + v) ÷ 2.

Shortcut Trick

Let the numbers be x and y. Given Sum (x + y) = 77 and Product (xy) = 1392.

Use the direct relation: Difference = √((Sum)² − 4 × Product).

Difference = √(77² − 4 × 1392) = √(5929 − 5568).

Difference = √361 = 19.

∴ The correct answer is 19.

Alternate Method

Given:

Sum of two positive numbers (x + y) = 77

Product of the numbers (xy) = 1392

Formula Used:

The relationship between sum, product, and difference is: (x − y)² = (x + y)² − 4xy

Calculations:

⇒ (x − y)² = (77)² − 4 × 1392

⇒ (x − y)² = 5929 − 5568

⇒ (x − y)² = 361

⇒ x − y = √361

⇒ x − y = 19

∴ The correct answer is 19.

Additional Information

Algebraic Identities

Useful variations include (x + y)² + (x − y)² = 2(x² + y²) and (x + y)² − (x − y)² = 4xy.

Quadratic Equation Relation

The numbers x and y are the roots of the quadratic equation t² − (Sum)t + (Product) = 0. Here, t² − 77t + 1392 = 0.

Perfect Squares

Memorizing squares up to 30 is vital for competitive exams; for instance, 18² = 324, 19² = 361, and 20² = 400.

Shortcut Trick

Let the total amount of money be T.

Since money is only transferred internally, the total sum T remains constant.

Case 1: Ratio R:N is 3:1 ⇒ Neha has T ÷ 4 of the total money.

Case 2: Ratio R:N is 5:1 ⇒ Neha has T ÷ 6 of the total money.

The difference in Neha's money between the two cases is 20 − (−40) = 60.

⇒ T/4 − T/6 = 60 ⇒ T/12 = 60 ⇒ T = 60 × 12 = 720.

∴ The correct answer is ₹720.

Alternate Method

Given:

Initial money with Riya = R, Initial money with Neha = N.

Formula Used:

Translation of word problems into Linear Equations.

Calculations:

⇒ If Riya gives ₹20 to Neha: R − 20 = 3 × (N + 20)

⇒ R − 20 = 3N + 60 ⇒ R − 3N = 80 ... (i)

⇒ If Neha gives ₹40 to Riya: R + 40 = 5 × (N − 40)

⇒ R + 40 = 5N − 200 ⇒ 5N − R = 240 ... (ii)

⇒ Adding equations (i) and (ii): (R − 3N) + (5N − R) = 80 + 240

⇒ 2N = 320 ⇒ N = 160

⇒ Substituting N = 160 in (i): R − 3 × 160 = 80 ⇒ R − 480 = 80 ⇒ R = 560

⇒ Total initial amount = R + N = 560 + 160 = 720

∴ The correct answer is ₹720.

Additional Information

Linear Equations in Two Variables

Systems of equations can be solved using the elimination method (adding/subtracting equations) or the substitution method.

Conservation of Total Sum

In closed-system money transfers where no money enters or leaves the group, the sum of individual amounts always remains constant.

Ratio Distribution

If a total T is divided in ratio a:b, the parts are [a/(a+b)] × T and [b/(a+b)] × T.

Shortcut Trick

In a parallelogram, the sum of any two adjacent angles is always 180°.

Given the difference between these two angles is 40°.

Smaller angle = (Sum of angles − Difference of angles) ÷ 2

Smaller angle = (180° − 40°) ÷ 2 = 140° ÷ 2 = 70°.

∴ The correct answer is 70degree.

Alternate Method

Given: Difference between adjacent angles = 40°.

Formula Used: Adjacent angles of a parallelogram are supplementary (sum = 180°).

Let the larger angle be x and the smaller angle be y.

⇒ x + y = 180° ... (i)

⇒ x − y = 40° ... (ii)

Subtracting equation (ii) from (i):

⇒ (x + y) − (x − y) = 180° − 40°

⇒ 2y = 140°

⇒ y = 140° ÷ 2

⇒ y = 70°

∴ The correct answer is 70degree.

Additional Information

Adjacent Angles Property

In any parallelogram, consecutive or adjacent angles are supplementary, meaning their sum is always 180°.

Opposite Angles Property

In a parallelogram, the opposite angles are always equal in measure (e.g., ∠A = ∠C and ∠B = ∠D).

Sum of Interior Angles

For any quadrilateral, the sum of all four interior angles is exactly 360°.

Shortcut Trick

Since cost is directly proportional to length:

Ratio of lengths = 8 : 25

Ratio of costs = 8 : 25

8 units = ₹3,200 ⇒ 1 unit = ₹400

25 units = 25 × 400 = ₹10,000

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Initial length (L₁) = 8 m

Initial cost (C₁) = ₹3,200

Target length (L₂) = 25 m

Formula Used:

For direct proportion: C₁ ÷ L₁ = C₂ ÷ L₂

Calculations:

⇒ 3,200 ÷ 8 = C₂ ÷ 25

⇒ 400 = C₂ ÷ 25

⇒ C₂ = 400 × 25

⇒ C₂ = 10,000

∴ The correct answer is ₹10,000.

Additional Information

Direct Proportion

Two quantities x and y are in direct proportion if the ratio x/y remains constant. Formula: x₁/y₁ = x₂/y₂.

Inverse Proportion

Two quantities are in inverse proportion if their product remains constant. Formula: x₁ × y₁ = x₂ × y₂.

Unitary Method

A technique where you find the value of a single unit (by division) and then find the necessary value by multiplication.