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Shortcut Trick

Angles subtended by the same arc (Arc AD) are always equal: ∠ABD = ∠ACD = 40°.

Angle formed in a semicircle by the diameter AB is a right angle, meaning ∠ADB = 90°.

In ΔABD, the sum of all interior angles must be 180°.

⇒ ∠BAD = 180° − (90° + 40°) = 50°.

∴ The correct answer is 50°.

Alternate Method

Cyclic Quadrilateral ABCD

A B C D O

40° 40° ?

Given:

⇒ AB is the diameter of the circle.

⇒ ABCD is a cyclic quadrilateral.

⇒ ∠ACD = 40°.

Formulas & Properties Used:

⇒ Angles subtended by the same arc at the circumference are equal.

⇒ The angle subtended by a diameter at any point on the circumference is 90°.

⇒ The sum of the interior angles of a triangle is always 180°.

Calculation:

⇒ Notice that both ∠ACD and ∠ABD are formed by joining the points A and D to the circumference.

⇒ Thus, they are in the same segment of the circle, making them equal.

⇒ ∠ABD = ∠ACD = 40°.

⇒ Since AB is the diameter passing through the center O, the angle it makes at the boundary is 90°.

⇒ Therefore, ∠ADB = 90°.

⇒ Now, in right-angled ΔABD, apply the angle sum property:

⇒ ∠BAD + ∠ABD + ∠ADB = 180°

⇒ ∠BAD + 40° + 90° = 180°

⇒ ∠BAD + 130° = 180°

⇒ ∠BAD = 180° − 130° = 50°.

∴ The correct answer is 50°.

Additional Information

Cyclic Quadrilateral Property

A quadrilateral is called a cyclic quadrilateral if all four of its vertices lie on the circumference of a single circle. The sum of any pair of opposite angles is exactly 180° (e.g., ∠A + ∠C = 180° and ∠B + ∠D = 180°).

Angle in a Semicircle

A foundational circle theorem dictates that the angle subtended by the diameter at any point on the boundary of the circle is always a right angle (90°).

Angles in the Same Segment

When two or more angles are subtended by the exact same arc at any point on the remaining part of the circle, those angles are completely identical in measure.

Shortcut Trick

Substitute sin² A = 1 − cos² A directly into the equation.

⇒ (1 − cos² A) − cos² A = 1/2

⇒ 1 − 2cos² A = 1/2

⇒ 2cos² A = 1/2

⇒ cos² A = 1/4

∴ The correct answer is 1/4.

Alternate Method

Given:

sin² A − cos² A = 1/2

Formula Used:

sin² A + cos² A = 1

Calculations:

⇒ sin² A + cos² A = 1

⇒ sin² A = 1 − cos² A

Substitute the value of sin² A into the given equation:

⇒ (1 − cos² A) − cos² A = 1/2

⇒ 1 − 2cos² A = 1/2

⇒ 2cos² A = 1 − 1/2

⇒ 2cos² A = 1/2

⇒ cos² A = (1/2) × (1/2)

⇒ cos² A = 1/4

∴ The correct answer is 1/4.

Additional Information

Basic Trigonometric Identities

1. sin² θ + cos² θ = 1

2. 1 + tan² θ = sec² θ

3. 1 + cot² θ = cosec² θ

Double Angle Formulas

cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ

Reciprocal Identities

sin θ = 1 ÷ cosec θ | cos θ = 1 ÷ sec θ | tan θ = 1 ÷ cot θ

Shortcut Trick

Directly apply the identity: cosec²A = 1 + cot²A

Square the given value: cot²A = (x + 1/x)² = x² + 1/x² + 2

Substitute to find cosec²A = 1 + x² + 1/x² + 2 = x² + 1/x² + 3

∴ The correct answer is x² + 1/x² + 3.

Alternate Method

Given:

cot A = x + 1/x

Formula Used:

cosec²A − cot²A = 1 ⇒ cosec²A = 1 + cot²A

(a + b)² = a² + b² + 2ab

Calculations:

⇒ cosec²A = 1 + (x + 1/x)²

⇒ cosec²A = 1 + x² + 1/x² + (2 × x × 1/x)

⇒ cosec²A = 1 + x² + 1/x² + 2

⇒ cosec²A = x² + 1/x² + 3

∴ The correct answer is x² + 1/x² + 3.

Additional Information

Pythagorean Trigonometric Identities

sin²A + cos²A = 1, sec²A − tan²A = 1, and cosec²A − cot²A = 1.

Basic Algebraic Expansions

(a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab.

Reciprocal Trigonometric Ratios

sin A = 1 ÷ cosec A | cos A = 1 ÷ sec A | tan A = 1 ÷ cot A.

Shortcut Trick

Observe that each term in the denominator is exactly twice the corresponding term in the numerator.

We know that (2x)³ = 8x³.

By factoring out 2³ = 8 from the denominator, the remaining expression perfectly cancels out the numerator.

Value = 1 ÷ 2³ = 1/8 = 0.125

∴ The correct answer is 0.125.

Alternate Method

Given:

Expression = (0.12³ + 0.06³) ÷ (0.24³ + 0.12³)

Calculations:

⇒ Let's express the denominator terms as multiples of the numerator terms.

⇒ Denominator = (2 × 0.12)³ + (2 × 0.06)³

⇒ Using the exponent rule (a × b)ⁿ = aⁿ × bⁿ:

⇒ Denominator = (2³ × 0.12³) + (2³ × 0.06³)

⇒ Taking 2³ common from both terms:

⇒ Denominator = 8 × (0.12³ + 0.06³)

⇒ Now substitute the denominator back into the given expression:

⇒ Expression = (0.12³ + 0.06³) ÷ [8 × (0.12³ + 0.06³)]

⇒ The term (0.12³ + 0.06³) cancels out entirely from both numerator and denominator.

⇒ Expression = 1 / 8

⇒ Expression = 0.125

∴ The correct answer is 0.125.

Additional Information

Laws of Exponents

When multiplying two terms with the same power, the rule (a × b)ⁿ = aⁿ × bⁿ is applied. This rule simplifies calculations by allowing you to factor out common bases efficiently.

Fraction to Decimal Conversion

It is helpful to remember common fraction conversions to save calculation time during exams: 1/2 = 0.5, 1/4 = 0.25, and 1/8 = 0.125.

Algebraic Identity for Cubes

Sum of cubes identity: a³ + b³ = (a + b)(a² − ab + b²). While not needed in this specific shortcut, it is frequently used to simplify complex polynomial expressions.

Shortcut Trick

Observe the relation between corresponding terms of the numerator and the denominator.

Each term in the numerator is exactly 5 times its corresponding term in the denominator (0.5 = 5 × 0.1, 0.05 = 5 × 0.01, etc.).

Since all terms are cubed, we can safely factor out 5³ from the entire numerator.

The remaining expression perfectly matches the denominator and completely cancels out.

Result = 5³ = 125.

∴ The correct answer is 125.

Alternate Method

Given:

[(0.5)³ + (0.05)³ + (0.005)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

Calculations:

⇒ Let us express the numerator values as direct multiples of the denominator values:

⇒ 0.5 = 5 × 0.1

⇒ 0.05 = 5 × 0.01

⇒ 0.005 = 5 × 0.001

⇒ Substituting these rewritten values into the main expression:

⇒ [(5 × 0.1)³ + (5 × 0.01)³ + (5 × 0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ Expanding the cubes using the rule (a × b)³ = a³ × b³:

⇒ [5³ × (0.1)³ + 5³ × (0.01)³ + 5³ × (0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ Taking 5³ as a common factor from the numerator:

⇒ 5³ × [(0.1)³ + (0.01)³ + (0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ The common bracket term cancels out completely.

⇒ Remaining value = 5³ = 125

∴ The correct answer is 125.

Additional Information

Laws of Exponents

When a product of numbers is raised to a power, the power is smoothly distributed to each multiplier: (a × b)ⁿ = aⁿ × bⁿ.

BODMAS Rule

Simplification of arithmetic expressions always follows the strict sequence: Brackets, Orders (Powers/Roots), Division, Multiplication, Addition, and Subtraction.

Decimal Expansion Tricks

Avoid directly calculating large decimal powers, which consumes time and leads to errors. Always look for common ratios to extract as common factors first.

Shortcut Trick

Directly equate the percentages:

30 × m = 18 × n

m / n = 18 / 30

m / n = 3 / 5

∴ The correct answer is 3 : 5.

Alternate Method

Given:

30% of m = 18% of n

m (3 parts) n (5 parts)

Ratio m : n = 3 : 5

Calculations:

⇒ (30 / 100) × m = (18 / 100) × n

⇒ 30 × m = 18 × n

⇒ m / n = 18 / 30

Dividing the numerator and denominator by their greatest common divisor (6):

⇒ m / n = 3 / 5

⇒ m : n = 3 : 5

∴ The correct answer is 3 : 5.

Additional Information

Compound Ratio

For two given ratios a : b and c : d, the compound ratio is calculated as ac : bd.

Duplicate and Sub-duplicate Ratios

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

Componendo and Dividendo

If two ratios are equal, such that a / b = c / d, then (a + b) / (a − b) = (c + d) / (c − d).

Shortcut Trick

Solve nested square roots step-by-step from the innermost root to the outside.

√289 = 17

√(127 + 17) = √144 = 12

√(388 + 12) = √400 = 20

∴ The correct answer is 20.

Alternate Method

Given:

Expression = √(388 + √(127 + √289))

Calculation:

⇒ First, evaluate the innermost square root: √289 = 17

⇒ Substitute this back into the expression:

⇒ Expression = √(388 + √(127 + 17))

⇒ Simplify the term inside the second square root: 127 + 17 = 144

⇒ Expression = √(388 + √144)

⇒ Evaluate the next square root: √144 = 12

⇒ Substitute this back into the expression:

⇒ Expression = √(388 + 12)

⇒ Expression = √400

⇒ Expression = 20

∴ The correct answer is 20.

Additional Information

Order of Operations for Nested Roots

Always start solving nested square roots from the innermost root and work your way outwards step by step.

Squares of Numbers ending in 3 or 7

Any perfect square that ends in the digit 9 will always have a square root ending in either 3 or 7. For example, 13² = 169 and 17² = 289.

Memorising Perfect Squares

Memorising perfect squares up to 30 helps to significantly speed up simplification problems in competitive exams.

Shortcut Trick For expressions like √a + √b and √c + √d where a + b = c + d, simply check their products ab and cd. Here, the sum under the roots is equal: 6 + 2 = 5 + 3 = 8. Now compare their products: 6 × 2 = 12 and 5 × 3 = 15. Since 12 < 15, the expression with the smaller product is smaller: √6 + √2 < √5 + √3. This means relationship (ii) is correct, while (i) and (iii) are incorrect. ∴ The correct answer is (i) and (iii). Alternate Method Given: Expression A = √6 + √2 Expression B = √5 + √3 Formula Used: (a + b)2 = a2 + b2 + 2ab Calculation: ⇒ Square both expressions to compare their magnitudes easily. ⇒ A2 = (√6 + √2)2 ⇒ A2 = 6 + 2 + 2√(6 × 2) ⇒ A2 = 8 + 2√12 ⇒ B2 = (√5 + √3)2 ⇒ B2 = 5 + 3 + 2√(5 × 3) ⇒ B2 = 8 + 2√15 ⇒ Both squared terms have the same constant sum 8. We only need to compare 2√12 and 2√15. ⇒ Since 12 < 15, we have √12 < √15. ⇒ Therefore, 8 + 2√12 < 8 + 2√15, which means A2 < B2. ⇒ Taking the square root of both sides gives A < B, so √6 + √2 < √5 + √3. ⇒ Thus, relationship (ii) is valid. The question asks to identify the incorrect relationships, which are (i) and (iii). ∴ The correct answer is (i) and (iii). Additional Information Comparison of Surds with Equal Sums When comparing expressions of the form √x + √y and √p + √q, if their sums are equal (x + y = p + q), the greater expression is the one with the larger product of the terms (xy vs pq). Comparison of Surds with Equal Differences When comparing √x − √y and √p − √q, if their differences are equal (x − y = p − q), the greater expression is the one with the smaller product of the terms. Squaring Inequalities For any positive real numbers a and b, if a2 > b2, then it is strictly true that a > b. This property allows converting surds into integer sums for direct and accurate comparison.

Shortcut Trick

Let the marked price (MP) of 1 novel be ₹100.

Cost price (CP) of 1 novel = (MP of 4 novels) ÷ 5 = 400 ÷ 5 = ₹80.

Selling price (SP) of 1 novel after 5% discount on MP = 100 − 5 = ₹95.

Profit on 1 novel = SP − CP = 95 − 80 = ₹15.

Profit % = (15 ÷ 80) × 100 = 1500 ÷ 80 = 18.75%.

∴ The correct answer is 18.75%.

Alternate Method

CP = ₹400 SP = ₹475 MP = ₹500

(For 5 Novels) (For 5 Novels) (For 5 Novels)

Profit = ₹75

Discount = 5%

Given:

⇒ Total purchased = 5 novels

⇒ CP of 5 novels = MP of 4 novels

⇒ Discount offered to customers = 5% on MP

Calculations:

⇒ Let the Marked Price (MP) of 1 novel be ₹100.

⇒ Total MP of 5 novels = 5 × 100 = ₹500.

⇒ Total CP of 5 novels = MP of 4 novels = 4 × 100 = ₹400.

⇒ Total discount on 5 novels = 5% of 500 = (5 ÷ 100) × 500 = ₹25.

⇒ Total SP of 5 novels = Total MP − Total discount = 500 − 25 = ₹475.

⇒ Profit earned = Total SP − Total CP = 475 − 400 = ₹75.

⇒ Profit % = (Profit ÷ Total CP) × 100

⇒ Profit % = (75 ÷ 400) × 100 = 75 ÷ 4 = 18.75%.

∴ The correct answer is 18.75%.

Additional Information

Marked Price and Discount

Discount is an allowance given to the buyer and is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

When two consecutive discounts of a% and b% are given, the single equivalent discount % = a + b − (a × b) ÷ 100.

Dishonest Dealer

If a trader sells goods at the cost price but uses false weights, his profit percentage is given by: (True Weight − False Weight) ÷ False Weight × 100.

Shortcut Trick

Let Total Work = LCM of 28 and 35 = 140 units.

Efficiency of A = 140 ÷ 28 = 5 units/day, Efficiency of B = 140 ÷ 35 = 4 units/day.

Work done by B in the last 17 days alone = 17 × 4 = 68 units.

Work done by A and B together before A left = 140 − 68 = 72 units.

Days A worked before leaving = 72 ÷ (5 + 4) = 72 ÷ 9 = 8 days.

∴ The correct answer is 8 days.

Alternate Method

Given:

Time taken by A alone = 28 days

Time taken by B alone = 35 days

B works alone for the final 17 days to finish the work.

Method Used: LCM Table Method

Calculations:

⇒ Total work to be completed = 140 units

⇒ Combined efficiency of A and B = 5 + 4 = 9 units/day

⇒ Work finished by B alone in the last 17 days = 17 × 4 = 68 units

⇒ Remaining work initially completed by both A and B together = 140 − 68 = 72 units

⇒ Number of days A and B worked together = Remaining Work ÷ Combined Efficiency

⇒ Days they worked together = 72 ÷ 9 = 8 days

⇒ Since A left after working together with B, A worked for exactly 8 days.

∴ The correct answer is 8 days.

Additional Information

Time and Efficiency Relationship

Efficiency is inversely proportional to time. If person A takes x days and person B takes y days, the ratio of their efficiencies is y : x.

Work and Wages

Wages are strictly distributed in proportion to the total work done by each individual. If workers work for the exact same duration, their wages can be directly distributed in the ratio of their respective efficiencies.

Alternate Days Work

When two people work on alternate days, group their combined effort into a 2-day cycle. Find the maximum integer multiple of this cycle that fits within the total work, then calculate the time for the remaining individual days.