Shortcut Trick

The area of a general quadrilateral equals half the product of its diagonal and the sum of perpendiculars dropped on it.

⇒ Area = 1/2 × Diagonal × (Sum of perpendiculars)

⇒ Area = 1/2 × 70 × (18 + 20)

⇒ Area = 35 × 38 = 1330

∴ The correct answer is 1330 cm².

Alternate Method

Given:

Diagonal BD = 70 cm

Perpendicular AL = 18 cm

Perpendicular CM = 20 cm

Formula Used:

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

Area of Quadrilateral ABCD = Area of ΔABD + Area of ΔBCD

Calculation:

⇒ Area of ΔABD = 1/2 × BD × AL

⇒ Area of ΔABD = 1/2 × 70 × 18 = 630 cm²

⇒ Area of ΔBCD = 1/2 × BD × CM

⇒ Area of ΔBCD = 1/2 × 70 × 20 = 700 cm²

⇒ Total Area = 630 + 700 = 1330 cm²

∴ The correct answer is 1330 cm².

Additional Information

Area of Rhombus

The area is calculated as half the product of its diagonals: Area = 1/2 × d₁ × d₂.

Area of Trapezium

Calculated as half the sum of parallel sides multiplied by the distance between them: Area = 1/2 × (a + b) × h.

Cyclic Quadrilateral

A quadrilateral whose vertices all lie on a single circle. Its area can be found using Brahmagupta's formula: Area = √((s−a)(s−b)(s−c)(s−d)), where s is the semi-perimeter.

Shortcut Trick

Total capacity = LCM(5.6, 7) = 28 units.

Efficiency: Pipe A = +5 units/hr (emptying), Pipe B = −4 units/hr (filling).

Net work in 1 cycle (2 hours) = 5 − 4 = 1 unit emptied.

Safe zone before Pipe A's final turn = 28 − 5 = 23 units.

Time for 23 units = 23 × 2 = 46 hours, remaining 5 units emptied by A in 1 hour.

∴ The correct answer is 47 hours.

Alternate Method

Given:

Time taken by outlet pipe A to empty the tank = 5.6 hours.

Time taken by inlet pipe B to fill the tank = 7 hours.

Pipes are opened alternatively for 1 hour each, starting with A.

Calculations:

⇒ Let emptying the tank be positive work (+), and filling be negative work (−).

⇒ Work done in the 1^st hour by A = 5 units.

⇒ Work done in the 2^nd hour by B = −4 units.

⇒ Total work done in 1 cycle (2 hours) = 5 − 4 = 1 unit.

⇒ Since A can empty 5 units in its turn, we calculate full cycles for 28 − 5 = 23 units.

⇒ Time required to empty 23 units = 23 cycles × 2 hours/cycle = 46 hours.

⇒ Remaining work = 28 − 23 = 5 units.

⇒ In the 47^th hour, Pipe A will empty the remaining 5 units in exactly 1 hour.

⇒ Total time taken = 46 + 1 = 47 hours.

∴ The correct answer is 47 hours.

Additional Information

Pipes and Cisterns

Inlet pipes add to the volume (+ efficiency) and outlet pipes remove volume (− efficiency). The total work is assumed as the LCM of the given time periods.

Alternate Work Concept

When positive and negative work happen alternatively, always subtract the positive work of the final step from the total work to avoid overcounting cycles, similar to the monkey climbing a greased pole puzzle.

Efficiency vs Time

Efficiency is inversely proportional to time taken. Total Work = Efficiency × Time.

Shortcut Trick

To find the ratio of the extremes (c : a) directly, multiply the given ratios and invert.

⇒ a / c = (a / b) × (b / c)

⇒ a / c = (2 / 3) × (5 / 7) = 10 / 21

⇒ c / a = 21 / 10

∴ The correct answer is 21 : 10.

Alternate Method

Given:

⇒ a : b = 2 : 3

⇒ b : c = 5 : 7

Calculations:

⇒ To combine the ratios, make the common term 'b' equal in both.

⇒ The value of b is 3 in the first ratio and 5 in the second.

⇒ Multiply the first ratio by 5 and the second ratio by 3 (LCM of 3 and 5 is 15).

⇒ a : b = (2 × 5) : (3 × 5) = 10 : 15

⇒ b : c = (5 × 3) : (7 × 3) = 15 : 21

⇒ Combining them, we get a : b : c = 10 : 15 : 21

⇒ Therefore, the ratio c : a = 21 : 10

∴ The correct answer is 21 : 10.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is called a compound ratio. For a : b and c : d, it is ac : bd.

Duplicate and Sub-duplicate Ratio

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

Invertendo and Alternendo

For a proportion a : b = c : d, invertendo gives b : a = d : c, and alternendo gives a : c = b : d.

Shortcut Trick

Since ABC forms a straight line, ∠OBD = 180° − 120° = 60°.

Because OB and OD are radii, ΔOBD is equilateral, making ∠BOD = 60°.

The tangent CD is perpendicular to radius OD, so ∠ODC = 90°.

In right-angled ΔODC, tan(60°) = CD ÷ OD ⇒ √3 = 7√3 ÷ R.

Solving this gives R = 7 cm. The diameter is 2 × 7 = 14 cm.

∴ The correct answer is 14 cm.

Alternate Method

O A B C D

120° 60° 7√3 R

Given:

⇒ CD is tangent to the circle at point D.

⇒ CD = 7√3 cm

⇒ ∠DBC = 120°

Formula Used:

⇒ In a right-angled triangle, tan(θ) = Perpendicular ÷ Base

⇒ Angles forming a linear pair add up to 180°

Calculations:

⇒ Since point B lies on the straight line AC, ∠OBD and ∠DBC form a linear pair.

⇒ ∠OBD + ∠DBC = 180°

⇒ ∠OBD = 180° − 120° = 60°

⇒ In ΔOBD, OB and OD are both radii of the same circle, so OB = OD.

⇒ Angles opposite to equal sides are equal, hence ∠ODB = ∠OBD = 60°.

⇒ Using the sum of angles in a triangle, ∠BOD = 180° − (60° + 60°) = 60°.

⇒ Because CD is a tangent at D, the radius OD is perpendicular to it. Therefore, ∠ODC = 90°.

⇒ In right-angled ΔODC, ∠DOC = 60°.

⇒ tan(∠DOC) = CD ÷ OD

⇒ tan(60°) = 7√3 ÷ R

⇒ √3 = 7√3 ÷ R

⇒ R = 7 cm

⇒ The diameter of the circle = 2 × R = 2 × 7 = 14 cm.

∴ The correct answer is 14 cm.

Additional Information

Tangent-Radius Theorem

A tangent to a circle always forms a right angle (90°) with the radius drawn to the exact point of tangency. This geometric property is universally useful for creating right-angled triangles to solve for missing lengths.

Isosceles Triangle Property

When two sides of a triangle are equal (such as the radii of a circle drawn from the center), the angles strictly opposite to those sides must also be equal. This frequently helps deduce that a triangle is equilateral.

Exterior Angle Theorem

The exterior angle of a triangle equals the sum of its two interior opposite angles. In ΔOBD, ∠DBC acts as an exterior angle, ensuring that ∠DBC = ∠BOD + ∠ODB (120° = 60° + 60°).

Shortcut Trick

Use the direct algebraic identity: (a + b)² + (a − b)² = 2(a² + b²)

Here, the given values are x = 4 + √6 and y = 4 − √6.

Substituting a = 4 and b = √6 into the identity directly:

⇒ x² + y² = 2 × (4² + (√6)²)

⇒ x² + y² = 2 × (16 + 6) = 2 × 22 = 44

∴ The correct answer is 44.

Alternate Method

Given:

x = 4 + √6 and y = 4 − √6

Formula Used:

x² + y² = (x + y)² − 2xy

Calculations:

⇒ Sum of x and y (x + y) = (4 + √6) + (4 − √6) = 8

⇒ Product of x and y (xy) = (4 + √6) × (4 − √6)

⇒ Using (a + b)(a − b) = a² − b²

⇒ xy = 4² − (√6)² = 16 − 6 = 10

⇒ Now substituting these values into the formula:

⇒ x² + y² = (8)² − 2 × 10

⇒ x² + y² = 64 − 20 = 44

∴ The correct answer is 44.

Additional Information

Difference of Squares

The formula a² − b² = (a + b)(a − b) is very useful for multiplying conjugate expressions directly without expanding.

Conjugate Surds

Expressions of the form a + √b and a − √b are known as conjugate surds. Their sum and product always result in rational numbers.

Subtraction of Perfect Squares

(a + b)² − (a − b)² = 4ab. This identity helps directly compute the difference between squares of conjugates.

Sum of Cubes Identity

x³ + y³ = (x + y)(x² − xy + y²) or equivalently (x + y)³ − 3xy(x + y).

Shortcut Trick

Convert to the same angle using complementary property: cos 63° = sin(90° − 27°) = sin 27°.

Simplify expression: cosec 27° − sin 27° = (1 − sin² 27°) / sin 27° = cos² 27° / sin 27°.

From given cos 27° = p / q, we find sin 27° = √(1 − p²/q²) = √(q² − p²) / q.

Substitute these values directly: (p / q)² ÷ [√(q² − p²) / q] = p² / [q√(q² − p²)].

∴ The correct answer is p² / (q√(q² − p²)).

Alternate Method

Given:

cos 27° = p / q

Concept Used:

In a right-angled triangle, cos θ = Base / Hypotenuse

By Pythagoras Theorem: Perpendicular = √(Hypotenuse² − Base²)

cos(90° − θ) = sin θ

cosec θ = Hypotenuse / Perpendicular

Calculations:

⇒ Base = p, Hypotenuse = q

⇒ Perpendicular = √(q² − p²)

⇒ We need to find the value of: cosec 27° − cos 63°

⇒ Using complementary angle relation, cos 63° = cos(90° − 27°) = sin 27°

⇒ sin 27° = Perpendicular / Hypotenuse = √(q² − p²) / q

⇒ cosec 27° = Hypotenuse / Perpendicular = q / √(q² − p²)

⇒ Expression = [q / √(q² − p²)] − [√(q² − p²) / q]

⇒ Expression = [q² − (q² − p²)] / [q√(q² − p²)]

⇒ Expression = [q² − q² + p²] / [q√(q² − p²)]

⇒ Expression = p² / [q√(q² − p²)]

∴ The correct answer is p² / (q√(q² − p²)).

Additional Information

Trigonometric Identities

Fundamental identities include sin² θ + cos² θ = 1 and cosec θ = 1 ÷ sin θ.

Complementary Angles

The trigonometric ratios of complementary angles are related as sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Pythagoras Theorem

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

Shortcut Trick

The average of the first n even numbers can be directly determined using the formula: (n + 1).

Here, the number of even numbers (n) = 92.

⇒ Average = 92 + 1 = 93

∴ The correct answer is 93.

Alternate Method

Given:

Total number of terms (n) = 92

The series represents the first 92 even numbers: 2, 4, 6, 8, ..., 184

Formula Used:

For any uniformly spaced series (Arithmetic Progression), Average = (First Term + Last Term) ÷ 2

Average of First 92 Even Numbers

1^st Term 2

4

Average 93

182

92^nd Term 184

... ...

Avg = (2 + 184) ÷ 2 = 93

Calculations:

⇒ First term of the series (a) = 2

⇒ Last term of the series (l) = 92 × 2 = 184

⇒ Average = (First Term + Last Term) ÷ 2

⇒ Average = (2 + 184) ÷ 2

⇒ Average = 186 ÷ 2 = 93

∴ The correct answer is 93.

Additional Information

Average of first n Odd Numbers

The average of the first n odd numbers is always equal to n.

Average of first n Natural Numbers

The average of the first n natural numbers is evaluated as (n + 1) ÷ 2.

Average of Uniformly Spaced Numbers

For any consecutive series or Arithmetic Progression, the average is exactly midway between the limits: Average = (First term + Last term) ÷ 2.

Shortcut Trick

Apply the Brahmagupta−Fibonacci algebraic identity: (ax − by)² + (bx + ay)² = (a² + b²)(x² + y²)

For the given expression (5z − 7y)² + (7z + 5y)², substitute a = 5, b = 7, x = z, and y = y.

Sum = (5² + 7²)(z² + y²) = (25 + 49)(z² + y²) = 74(z² + y²)

Expression becomes: 74z² + 74y² − 49z²

Result = 25z² + 74y²

∴ The correct answer is 25z² + 74y².

Alternate Method

Given:

(5z − 7y)² + (7z + 5y)² − 49z²

Formula Used:

(a − b)² = a² − 2ab + b²

(a + b)² = a² + 2ab + b²

Calculations:

⇒ Expanding the first term: (5z − 7y)² = (5z)² − 2 × (5z) × (7y) + (7y)² = 25z² − 70zy + 49y²

⇒ Expanding the second term: (7z + 5y)² = (7z)² + 2 × (7z) × (5y) + (5y)² = 49z² + 70zy + 25y²

⇒ Adding both expansions: (25z² − 70zy + 49y²) + (49z² + 70zy + 25y²)

⇒ Grouping similar terms: (25z² + 49z²) + (−70zy + 70zy) + (49y² + 25y²)

⇒ The middle terms cancel out: 74z² + 0 + 74y²

⇒ Now, subtract the remaining 49z² from the original expression: 74z² + 74y² − 49z²

⇒ Final simplification: (74 − 49)z² + 74y² = 25z² + 74y²

∴ The correct answer is 25z² + 74y².

Additional Information

Brahmagupta−Fibonacci Identity

This identity expresses the product of two sums of squares as a sum of squares: (a² + b²)(x² + y²) = (ax − by)² + (bx + ay)². It is incredibly useful for instantly expanding and grouping complex crossed algebraic pairs without calculating individual terms.

Standard Algebraic Identities

Other essential identities for quick simplification include: (a + b)² + (a − b)² = 2(a² + b²) and (a + b)² − (a − b)² = 4ab.

BODMAS Rule in Algebra

Always follow the order of operations when simplifying polynomials: evaluate brackets and exponent powers first, then handle the addition or subtraction of grouped coefficients.

Shortcut Trick

In a right-angled triangle, if the hypotenuse is 29 and the base is 21, the third side directly forms the standard Pythagorean triplet (20, 21, 29).

Thus, Perpendicular = 20.

cos θ = 21/29 and sin θ = 20/29

cos² θ − sin² θ = (21/29)² − (20/29)² = (441 − 400) / 841 = 41/841

∴ The correct answer is 41/841.

Alternate Method

Given:

In right-angled triangle PQR, right-angled at R:

Hypotenuse (PQ) = 29 cm

Base (QR) = 21 cm

∠Q = θ

Formula Used:

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

cos θ = Base / Hypotenuse

sin θ = Perpendicular / Hypotenuse

Calculation:

⇒ PQ² = PR² + QR²

⇒ 29² = PR² + 21²

⇒ 841 = PR² + 441

⇒ PR² = 841 − 441 = 400

⇒ PR = √400 = 20 cm

⇒ cos θ = QR / PQ = 21/29

⇒ sin θ = PR / PQ = 20/29

Now, substitute the values into the required expression:

⇒ cos² θ − sin² θ = (21/29)² − (20/29)²

⇒ cos² θ − sin² θ = (441/841) − (400/841)

⇒ cos² θ − sin² θ = (441 − 400) / 841 = 41/841

∴ The correct answer is 41/841.

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy the equation a² + b² = c². Commonly asked triplets in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (20, 21, 29).

Trigonometric Identities

The three fundamental identities are: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Double Angle Formula

The term evaluated in the question, cos² θ − sin² θ, is mathematically equivalent to the double angle formula cos(2θ).

Given:

$\frac{a^5 \times b^3 \times c^7}{a^4 \times b^8 \times c^9}$

Formula used:

x^m / xⁿ = x^m-n

Calculation:

⇒ a^5-4 × b^3-8 × c^7-9

⇒ a¹ × b^-5 × c^-2

∴ The correct answer is option (2).

Shortcut Trick

For a number to be divisible by 24, it must be divisible by both 3 and 8.

Since 8 is an even number, any number divisible by it must also be even, eliminating 38211 and 38121.

Check the last 3 digits of the remaining numbers for divisibility by 8.

For 32118: 118 ÷ 8 = 14.75 → Not divisible.

For 31128: 128 ÷ 8 = 16 → Completely divisible.

∴ The correct answer is 31128.

Alternate Method

Given:

Numbers to check: 38211, 38121, 32118, 31128

Concept Used:

Divisibility rule of 24: A number is divisible by 24 if it is divisible by its co-prime factors, 3 and 8.

Calculations:

⇒ Step 1: Check divisibility by 3 (sum of the digits must be a multiple of 3).

⇒ Sum of digits for all options = 3 + 8 + 2 + 1 + 1 = 15.

⇒ Since 15 is divisible by 3, all given numbers are divisible by 3.

⇒ Step 2: Check divisibility by 8 (last three digits must form a number divisible by 8).

⇒ 38211 → The last digit is odd, so it is not divisible by 8.

⇒ 38121 → The last digit is odd, so it is not divisible by 8.

⇒ 32118 → The last three digits are 118. 118 ÷ 8 = 14.75 → Not divisible.

⇒ 31128 → The last three digits are 128. 128 ÷ 8 = 16 → Divisible.

⇒ Thus, 31128 is the only number divisible by both 3 and 8.

∴ The correct answer is 31128.

Additional Information

Divisibility by Composite Numbers

To check divisibility by a composite number, test the divisibility of its co-prime factors. For example, since 24 = 3 × 8 (where 3 and 8 are co-prime), a number divisible by 24 must be divisible by both 3 and 8.

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its digits is a multiple of 3.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is a multiple of 8.

Shortcut Trick

Speed of P in m/s = 12 × (5/18) = 10/3 m/s.

Time taken by P to finish the race = 400 ÷ (10/3) = 120 seconds.

Time taken by Q = 120 + 13 = 133 seconds.

Distance run by Q = 400 − 20 = 380 m.

Speed of Q = 380 ÷ 133 = 20/7 m/s ⇒ (20/7) × (18/5) = 72/7 = 10.29 km/hr.

∴ The correct answer is 10.29 km/hr.

Alternate Method

Given:

Total distance of race = 400 m

Speed of P = 12 km/hr

Start distance given to Q = 20 m

Time by which P beats Q = 13 seconds

Formula Used:

Time = Distance ÷ Speed

Speed conversion: 1 km/hr = 5/18 m/s

Calculations:

⇒ Speed of P = 12 × (5/18) = 10/3 m/s

⇒ Time taken by P to complete the race = 400 ÷ (10/3) = 120 seconds

⇒ P gives Q a start of 20 m, so distance covered by Q = 400 − 20 = 380 m

⇒ P beats Q by 13 seconds, so Time taken by Q = Time taken by P + 13

⇒ Time taken by Q = 120 + 13 = 133 seconds

⇒ Speed of Q in m/s = Distance ÷ Time = 380 ÷ 133 = 20/7 m/s

⇒ Speed of Q in km/hr = (20/7) × (18/5) = 72/7 km/hr

⇒ Speed of Q ≈ 10.2857 km/hr

⇒ Rounding up to two decimal places, Speed of Q = 10.29 km/hr

∴ The correct answer is 10.29 km/hr.

Additional Information

Start Distance in Races

If A gives B a start of x meters in a race of L meters, it means A has to run the complete L meters while B only has to run (L − x) meters.

Start Time in Races

If A beats B by t seconds, it implies B takes t seconds more than A to complete his designated track distance.

Speed Conversions

To convert km/hr into m/s, multiply the value by 5/18. To convert m/s into km/hr, multiply the value by 18/5.

Shortcut Trick

Present ratio of ages = 4 : 3

Ratio of ages after 4 years = 6 : 5

Notice that the increase in ratio parts is the same for both friends: 6 − 4 = 2 parts, and 5 − 3 = 2 parts.

⇒ 2 parts = 4 years ⇒ 1 part = 2 years.

Current sum of their ages = (4 + 3) parts = 7 × 2 = 14 years.

Sum of their ages after 22 years from now = 14 + (2 × 22) = 14 + 44 = 58 years.

∴ The correct answer is 58 years.

Alternate Method

Given:

Ratio of present ages of the two friends = 4 : 3

Ratio of their ages after 4 years = 6 : 5

Calculations:

Let the present ages of the two friends be 4x and 3x years respectively.

⇒ After 4 years, their ages will be (4x + 4) and (3x + 4) years.

⇒ According to the given condition: (4x + 4) ÷ (3x + 4) = 6 ÷ 5

⇒ 5 × (4x + 4) = 6 × (3x + 4)

⇒ 20x + 20 = 18x + 24

⇒ 20x − 18x = 24 − 20

⇒ 2x = 4

⇒ x = 2

⇒ The present ages of the two friends are 4 × 2 = 8 years and 3 × 2 = 6 years.

⇒ Present sum of their ages = 8 + 6 = 14 years.

⇒ After 22 years, the age of the first friend = 8 + 22 = 30 years.

⇒ After 22 years, the age of the second friend = 6 + 22 = 28 years.

⇒ Sum of their ages after 22 years = 30 + 28 = 58 years.

∴ The correct answer is 58 years.

Additional Information

Ratio Difference Trick for Age Problems

If the difference between the respective ratio parts of the ages for multiple individuals is equal, you can directly equate the difference in ratio parts to the change in years. If the difference is unequal, you must first cross-multiply the difference of alternate ratios to equalize them before applying the shortcut.

Future and Past Ages Sum Rule

If the present sum of the ages of n people is S, their sum after t years will be S + n × t, and their sum t years ago was S − n × t. Every individual adds or subtracts t years simultaneously.

Cross-Multiplication Method

For proportions formatted as a/b = c/d, you can determine the unknown variables by multiplying diagonally across the equal sign resulting in: a × d = b × c. This fundamental property transforms ratio equations into manageable linear equations.

Shortcut Trick

Using the 5-12-13 triplet for sin A = 5/13, we get P = 5, H = 13, and B = 12.

Thus, cos A = 12/13 and tan A = 5/12.

Numerator = 12/13 − 2×(5/12) = (144 − 130) / 156 = 14/156.

Denominator = 5/13 + 3×(5/12) = (60 + 195) / 156 = 255/156.

Value = Numerator ÷ Denominator = 14/255.

∴ The correct answer is 14/255.

Alternate Method

Given:

sin A = 5/13

Formula Used:

In a right-angled triangle:

sin A = Perpendicular ÷ Hypotenuse

cos A = Base ÷ Hypotenuse

tan A = Perpendicular ÷ Base

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

Calculations:

⇒ Let Perpendicular = 5 and Hypotenuse = 13

⇒ Base = √(13² − 5²) = √(169 − 25) = √144 = 12

⇒ cos A = 12/13

⇒ tan A = 5/12

⇒ Numerator = cos A − 2×tan A = 12/13 − 2×(5/12) = 12/13 − 10/12

⇒ Numerator = (144 − 130) / 156 = 14/156

⇒ Denominator = sin A + 3×tan A = 5/13 + 3×(5/12) = 5/13 + 15/12

⇒ Denominator = (60 + 195) / 156 = 255/156

⇒ Required Value = (14/156) ÷ (255/156) = 14/255

∴ The correct answer is 14/255.

Additional Information

Pythagorean Triplets

These are sets of three integers that strictly satisfy Pythagoras Theorem. Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Trigonometric Identities

The most fundamental identity is sin² A + cos² A = 1, from which values of cos A can be directly determined without drawing a triangle.

Tangent Relation

The tangent of an angle can always be derived from sine and cosine using the relation tan A = sin A ÷ cos A.

Shortcut Trick

Ratio of Loss to Gain = 1 : 2

Total difference in parts = 1 + 2 = 3 parts

Difference in Selling Prices = ₹1070 − ₹680 = ₹390

3 parts = ₹390 ⇒ 1 part (Loss) = ₹130

Cost Price = SP₁ + Loss = 680 + 130 = ₹810

∴ The correct answer is ₹810.

Alternate Method

Given:

Selling Price 1 (SP₁) = ₹680

Selling Price 2 (SP₂) = ₹1070

Gain at SP₂ is double the loss at SP₁.

Formula Used:

Loss = Cost Price (CP) − Selling Price (SP)

Gain = Selling Price (SP) − Cost Price (CP)

Calculations:

⇒ Let the Cost Price of the mobile phone be CP.

⇒ Loss when sold at ₹680 = CP − 680

⇒ Gain when sold at ₹1070 = 1070 − CP

⇒ According to the given condition: Gain = 2 × Loss

⇒ 1070 − CP = 2 × (CP − 680)

⇒ 1070 − CP = 2CP − 1360

⇒ 3CP = 1070 + 1360

⇒ 3CP = 2430

⇒ CP = 2430 ÷ 3 = ₹810

∴ The correct answer is ₹810.

Additional Information

Cost Price (CP) & Selling Price (SP)

Cost Price is the amount paid to purchase an article. Selling Price is the amount at which the article is sold to the customer.

Profit & Loss Basics

Profit occurs when the Selling Price is greater than the Cost Price (Profit = SP − CP). Loss occurs when the Cost Price is greater than the Selling Price (Loss = CP − SP).

Marked Price & Discount

Discount is a reduction given on the Marked Price (MP) of an article before selling. It is calculated as: Selling Price = MP − Discount.

Shortcut Trick

Radius (r) = 56 ÷ 2 = 28 cm.

Slant height (l) = CSA ÷ (πr) = 3080 ÷ (22/7 × 28) = 35 cm.

Using Pythagorean triplet (3, 4, 5) × 7 → (21, 28, 35), height (h) = √(35² − 28²) = 21 cm.

Volume = 1/3 × πr²h = 1/3 × 22/7 × 28² × 21 = 17248 cm³.

∴ The correct answer is 17248 cm³.

Alternate Method

Given:

Diameter of the base = 56 cm

Curved Surface Area (CSA) = 3080 cm²

Formula Used:

Radius (r) = Diameter ÷ 2

CSA = πrl (where l is the slant height)

l² = r² + h²

Volume = (1/3)πr²h

Calculations:

⇒ Radius (r) = 56 ÷ 2 = 28 cm

⇒ πrl = 3080

⇒ (22/7) × 28 × l = 3080

⇒ 88 × l = 3080

⇒ l = 3080 ÷ 88 = 35 cm

⇒ h = √(l² − r²)

⇒ h = √(35² − 28²)

⇒ h = √(1225 − 784) = √441 = 21 cm

⇒ Volume = (1/3) × (22/7) × 28 × 28 × 21

⇒ Volume = 22 × 4 × 28 × 7

⇒ Volume = 17248 cm³

∴ The correct answer is 17248 cm³.

Additional Information

Total Surface Area of a Cone

The total surface area includes the curved surface and the circular base: TSA = πr(r + l).

Volume of a Cylinder

A cylinder with the same base radius and height as a cone has exactly three times its volume. Volume = πr²h.

Frustum of a Cone

When a cone is cut by a plane parallel to its base, the lower part is a frustum. Volume = (1/3)πh(R² + r² + Rr).

Shortcut Trick

From the addition theorem: P(A) = P(A ∪ B) + P(A ∩ B) − P(B)

⇒ P(A) = 3/4 + 1/4 − 1/2 = 1/2

Now, calculate P(A) × P(B) = 1/2 × 1/2 = 1/4

Since P(A ∩ B) = P(A) × P(B) = 1/4, the events A and B are independent.

∴ The correct answer is Independent.

Alternate Method

Given:

P(A ∪ B) = 3/4

P(A ∩ B) = 1/4

P(B) = 1/2

Formula Used:

Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Independence Rule: Events are independent if P(A ∩ B) = P(A) × P(B)

Calculation:

⇒ 3/4 = P(A) + 1/2 − 1/4

⇒ 3/4 = P(A) + 1/4

⇒ P(A) = 3/4 − 1/4

⇒ P(A) = 2/4 = 1/2

Now check the condition for independent events:

⇒ P(A) × P(B) = 1/2 × 1/2

⇒ P(A) × P(B) = 1/4

⇒ We are given P(A ∩ B) = 1/4.

⇒ Therefore, P(A ∩ B) = P(A) × P(B)

This perfectly satisfies the condition for independent events.

∴ The correct answer is Independent.

Additional Information

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. The key mathematical condition is: P(A ∩ B) = P(A) × P(B).

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time, meaning they share no common outcomes. The condition is: P(A ∩ B) = 0.

Addition Rule of Probability

For any two overlapping events A and B, the probability of either event occurring is given by the formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Shortcut Trick Cumulative Frequencies: 9, 25, 49, 64, 68. Total frequency N = 68 ⇒ N ÷ 2 = 34. The first cumulative frequency > 34 is 49, hence the median class is 40-60. Median = l + [(N/2 − CF) ÷ f] × h Median = 40 + [(34 − 25) ÷ 24] × 20 = 40 + (9 ÷ 24) × 20 = 47.5 ∴ The correct answer is 47.5. Alternate Method Given: Class Intervals: 0-20, 20-40, 40-60, 60-80, 80-100 Frequencies: 9, 16, 24, 15, 4 Formula Used: Median = l + [(N/2 − CF) ÷ f] × h Where: l = lower limit of median class N = total frequency CF = cumulative frequency of the preceding class f = frequency of the median class h = class size Calculations: ⇒ First, let us construct the cumulative frequency (CF) distribution: ⇒ 0-20: 9 ⇒ 20-40: 9 + 16 = 25 ⇒ 40-60: 25 + 24 = 49 ⇒ 60-80: 49 + 15 = 64 ⇒ 80-100: 64 + 4 = 68 ⇒ Total frequency (N) = 68 ⇒ N ÷ 2 = 68 ÷ 2 = 34 ⇒ The cumulative frequency just greater than 34 is 49, corresponding to the class 40-60. ⇒ Median class = 40-60 ⇒ Here, l = 40, CF = 25, f = 24, h = 20 ⇒ Median = 40 + [(34 − 25) ÷ 24] × 20 ⇒ Median = 40 + (9 ÷ 24) × 20 ⇒ Median = 40 + (3 ÷ 8) × 20 ⇒ Median = 40 + (60 ÷ 8) ⇒ Median = 40 + 7.5 = 47.5 ∴ The correct answer is 47.5. Additional Information Mode of Grouped Data Mode = l + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h, where f1 is the modal class frequency, f0 is the preceding class frequency, and f2 is the succeeding class frequency. Mean of Grouped Data Mean = Σ(fixi) ÷ Σfi, where xi represents the class mark or midpoint of each interval, calculated as (lower limit + upper limit) ÷ 2. Empirical Relationship Mode = 3 × Median − 2 × Mean. This formula is highly useful to find one of the measures of central tendency when the other two are known for a moderately skewed distribution.

Shortcut Trick

Present population = Initial Population × (Multiplying Factor)² = 9487500 × 0.92 × 0.92

Instead of full calculation, apply the Digital Sum method to find the correct option.

Digital sum of 9487500 = 9 + 4 + 8 + 7 + 5 = 33 → 6 and 0.92 = 9 + 2 = 11 → 2

Resulting Digital Sum = 6 × 2 × 2 = 24 → 6

Only the value 8030220 has a digital sum of 6 (8 + 0 + 3 + 0 + 2 + 2 + 0 = 15 → 6).

∴ The correct answer is 8030220.

Alternate Method

Given:

Population 2 years ago (P) = 9487500

Rate of decrease (R) = 8% per year

Time (n) = 2 years

Formula Used:

A = P × (1 − R / 100)ⁿ

Calculations:

⇒ Present Population (A) = 9487500 × (1 − 8 / 100)²

⇒ A = 9487500 × (92 / 100)²

⇒ A = 9487500 × (23 / 25) × (23 / 25)

⇒ A = 9487500 × (529 / 625)

⇒ A = 15180 × 529

⇒ A = 8030220

∴ The correct answer is 8030220.

Additional Information

Successive Percentage Decrease

If a value decreases by x% and y% successively, the net percentage change is given by: Net Decrease = −x − y + (x × y) / 100 %. For a repeated 8% decrease, the overall effective reduction is −8 − 8 + 0.64 = −15.36%.

Digital Sum Concept

The digital sum is found by repeatedly adding the digits of a number until a single digit remains. In multiplication, the Digital Sum of the product equals the product of the Digital Sums of the individual multipliers. It is a highly reliable shortcut for verifying complex decimal and fraction multiplications.

Population Growth & Decay Formula

When a population uniformly changes at R% per year for n years, the new population is directly calculated using the compound interest formula approach: A = P × (1 ± R / 100)ⁿ.

Shortcut Trick

Equating the fractions: (3/10) × Adi = (3/5) × Dia

Ratio of Adi to Dia = (3/5) : (3/10) = 2 : 1

Total parts = 2 + 1 = 3 parts, which corresponds to ₹1440.

Dia's share = 1 part = 1440 ÷ 3 = ₹480.

∴ The correct answer is ₹480.

Alternate Method

Given:

Total amount of Adi and Dia = ₹1440

(3/10) of Adi's amount = (3/5) of Dia's amount

Adi (2 parts) Dia (1 part)

Total Amount = ₹1440

₹960 ₹480

Calculations:

⇒ Let Adi's amount be A and Dia's amount be D.

⇒ (3/10) × A = (3/5) × D

⇒ A / D = (3/5) × (10/3)

⇒ A / D = 30 / 15 = 2 / 1

⇒ This implies A = 2D

⇒ We know that A + D = 1440

⇒ 2D + D = 1440

⇒ 3D = 1440

⇒ D = 1440 ÷ 3 = 480

∴ The correct answer is ₹480.

Additional Information

Ratio & Proportion

When a given quantity is divided into a ratio a : b, the first part is (a / (a + b)) × Total and the second part is (b / (a + b)) × Total.

Equality of Fractions

If fractions multiplied by quantities are equal (e.g., x × A = y × B), their inverse ratio yields the ratio of the quantities (A : B = y : x). This is highly useful for directly converting fractional relations into ratios.

Shortcut Trick

To find the LCM of exponential numbers, directly pick the highest power for each distinct base across all given terms.

Bases present in the given expressions: 2, 9, 13, and 19.

Highest power for 2 is 3 → 2³

Highest power for 9 is 3 → 9³

Highest power for 13 is 2 and for 19 is 2 → 13² × 19²

∴ The correct answer is 2³ × 9³ × 13² × 19².

Alternate Method

Given:

First term = 2³ × 9² × 13

Second term = 2² × 13² × 19

Third term = 9³ × 13² × 19²

Formula Used:

The Least Common Multiple (LCM) of expressions is the product of all distinct bases raised to their highest respective powers.

Calculation:

⇒ Let us list the powers of all distinct bases for each given term:

⇒ For base 2: The powers are 3, 2, and 0. The highest power is 3.

⇒ For base 9: The powers are 2, 0, and 3. The highest power is 3.

⇒ For base 13: The powers are 1, 2, and 2. The highest power is 2.

⇒ For base 19: The powers are 0, 1, and 2. The highest power is 2.

⇒ Taking the product of these highest powers, we get the required LCM.

⇒ LCM = 2³ × 9³ × 13² × 19²

∴ The correct answer is 2³ × 9³ × 13² × 19².

Additional Information

Highest Common Factor (HCF)

To find the HCF of numbers in exponential form, select the lowest power of each common prime factor across all numbers.

Product of Two Numbers

For any two numbers A and B: A × B = LCM(A, B) × HCF(A, B). Note that this rule applies only to exactly two numbers, not three or more.

Co-prime Numbers

Two numbers are co-prime if their HCF is 1. The LCM of two co-prime numbers is simply their product.

Shortcut Trick

Let the assumed average price be ₹51 per kg.

Net deviation of 9 kg maize = 9 × (68 − 51) = +153

Distribute this net deviation over the total mixture (8 kg + 9 kg = 17 kg).

Average increase = 153 ÷ 17 = +9

Actual average = 51 + 9 = ₹60 per kg.

∴ The correct answer is ₹60.

Alternate Method

Given:

Quantity of first type (Q₁) = 8 kg, Cost (P₁) = ₹51 per kg

Quantity of second type (Q₂) = 9 kg, Cost (P₂) = ₹68 per kg

Formula Used:

Average price = (Q₁ × P₁ + Q₂ × P₂) ÷ (Q₁ + Q₂)

Calculations:

⇒ Total cost of 8 kg maize = 8 × 51 = ₹408

⇒ Total cost of 9 kg maize = 9 × 68 = ₹612

⇒ Total cost of the mixture = 408 + 612 = ₹1020

⇒ Total quantity of the mixture = 8 + 9 = 17 kg

⇒ Average price = 1020 ÷ 17 = ₹60 per kg

∴ The correct answer is ₹60.

Additional Information

Rule of Alligation

It helps to find the exact ratio in which two ingredients at given prices must be mixed to produce a mixture of a desired average price. Ratio = (Dearer Price − Average Price) : (Average Price − Cheaper Price).

Weighted Average

When different quantities have varying numeric values, the combined average is given by: A = (w₁x₁ + w₂x₂ + ...) ÷ (w₁ + w₂ + ...).

Replacement Method

If V is the total volume of pure liquid, and x volume is repeatedly replaced with water n times, the final volume of pure liquid is: Final = Initial × (1 − x ÷ V)ⁿ.

Shortcut Trick

Total time from 13 Feb to 27 Apr = 15 (Feb) + 31 (Mar) + 27 (Apr) = 73 days.

Time in years = 73 ÷ 365 = 1/5 year.

SI = (P × R × T) ÷ 100

⇒ SI = (2000 × 6.5 × 1) ÷ (100 × 5)

⇒ SI = 20 × 1.3 = ₹26

∴ The correct answer is ₹26.

Alternate Method

Given:

Principal (P) = ₹2000

Rate of Interest (R) = 6.5% p.a.

Time Period = From 13 February 2023 to 27 April 2023

Formula Used:

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

Calculations:

⇒ The year 2023 is an ordinary year, so February has 28 days.

⇒ Days left in February = 28 − 13 = 15 days

⇒ Days in March = 31 days

⇒ Days in April = 27 days

⇒ Total Days (T) = 15 + 31 + 27 = 73 days

⇒ Converting time into years, T = 73 ÷ 365 = 1/5 years

⇒ SI = (2000 × 6.5 × 1/5) ÷ 100

⇒ SI = (2000 × 6.5) ÷ 500

⇒ SI = 4 × 6.5

⇒ SI = 26

∴ The correct answer is ₹26.

Additional Information

Total Amount Formula

The total amount at the end of the term is the sum of the initial Principal and the Simple Interest earned: Amount = P + SI.

Day Calculation Rule

When calculating interest for a specific period of days, typically either the depositing date or the withdrawal date is counted, but not both.

Rule of 72

It is a quick shortcut to estimate the number of years required to double your invested money at a given annual fixed rate. (Years ≈ 72 ÷ Rate).

Shortcut Trick

Arrange the given data in ascending order: 11, 12, 13, 13, 14, 15, 16, 17, 18, 19.

Since the number of terms n = 10 (even), the median is the simple average of the two middle terms.

Median = Average of the 5^th and 6^th terms.

Median = (14 + 15) ÷ 2 = 14.5.

∴ The correct answer is 14.5 years.

Alternate Method

Given:

The ages of 10 students (in years): 12, 13, 16, 18, 14, 19, 13, 17, 15, and 11.

Formula Used:

For an even number of observations n, Median = [(n/2)^th term + (n/2 + 1)^th term] ÷ 2

11

12

13

13

14

15

16

17

18

19

Median Position

Average = 14.5

Calculation:

⇒ Total number of observations, n = 10 (even number).

⇒ First, sort the given ages in ascending order:

⇒ 11, 12, 13, 13, 14, 15, 16, 17, 18, 19

⇒ Median = [(10/2)^th term + (10/2 + 1)^th term] ÷ 2

⇒ Median = [5^th term + 6^th term] ÷ 2

⇒ Here, the 5^th term is 14, and the 6^th term is 15.

⇒ Median = (14 + 15) ÷ 2

⇒ Median = 29 ÷ 2 = 14.5

∴ The correct answer is 14.5 years.

Additional Information

Mean

The mean is the simple average of a given set of numbers. It is calculated by Mean = Sum of all observations ÷ Total number of observations.

Median

The median is the exact middle value when a data set is ordered from least to greatest. If n is odd, Median = ((n+1)÷2)^th term. If n is even, the Median is the average of the two central terms.

Mode

The mode is the value that appears most frequently in a data set. A set of data may have one mode, multiple modes, or no mode at all.

Shortcut Trick

For a 'Buy X Get Y free' scheme, the effective discount percentage is given by:

Discount % = [Free Items ÷ Total Items] × 100

Here, Paid Items (X) = 5 and Free Items (Y) = 1

⇒ Discount % = [1 ÷ (5 + 1)] × 100

⇒ Discount % = (1 ÷ 6) × 100 = 16.67%

∴ The correct answer is 16.67%.

Alternate Method

Given:

Offer: Buy 5 water bottles, get 1 for free.

Formula Used:

Discount % = (Total Discount ÷ Total Marked Price) × 100

Calculations:

Let the Marked Price (MP) of 1 water bottle be 10 units.

⇒ Total MP of 6 bottles (5 paid + 1 free) = 6 × 10 = 60 units

Since the customer pays for only 5 bottles, the Selling Price (SP) becomes:

⇒ Total SP of 6 bottles = 5 × 10 = 50 units

⇒ Discount = MP − SP = 60 − 50 = 10 units

⇒ Discount % = (10 ÷ 60) × 100

⇒ Discount % = (1 ÷ 6) × 100 = 16.666... ≈ 16.67%

∴ The correct answer is 16.67%.

Additional Information

Successive Discounts

If two successive discounts of a% and b% are given, the equivalent single discount is [a + b − (a × b) ÷ 100]%.

Marked Price vs Cost Price

The ratio of Marked Price (MP) to Cost Price (CP) when profit and discount are given is MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %).

Discount Percentage

Discount is always calculated on the Marked Price (MP) unless specified otherwise. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

Ratio of present ages of Sweeta and Suman = 4 : 5

Ratio of their ages 6 years ago = 2 : 3

Difference in ratio parts = (4 − 2) = 2 parts for Sweeta, and (5 − 3) = 2 parts for Suman.

Since the difference in parts is equal, 2 parts difference corresponds to 6 years.

⇒ 1 part = 3 years ⇒ Sweeta's present age = 4 × 3 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given:

Ratio of present ages of Sweeta and Suman = 4 : 5

Ratio of their ages 6 years ago = 2 : 3

Calculations:

Let the present ages of Sweeta and Suman be 4x and 5x years respectively.

6 years ago, Sweeta's age = (4x − 6) years

6 years ago, Suman's age = (5x − 6) years

According to the question,

⇒ (4x − 6) / (5x − 6) = 2 / 3

By cross-multiplying,

⇒ 3 × (4x − 6) = 2 × (5x − 6)

⇒ 12x − 18 = 10x − 12

⇒ 12x − 10x = 18 − 12

⇒ 2x = 6

⇒ x = 3

Sweeta's present age = 4x

⇒ 4 × 3 = 12 years

∴ The correct answer is 12 years.

Additional Information

Equalising Ratio Differences

If the difference between numerator parts and denominator parts isn't equal, multiply the ratios by the difference of their own parts to balance them before equating to the given actual values.

Cross-Multiplication Method

When a ratio a : b changes to c : d by adding or subtracting x to both, the common variable value is found quickly by equating differences: |ad − bc| = x|c − d|.

Past and Future Ages

If present age is n, then t years ago age was (n − t) and t years hence age will be (n + t). The difference in ages of any two people always remains strictly constant over any period of time.

Shortcut Trick

We can use the multiplying factor method for successive percentage changes.

First year multiplying factor = 100% + 6% = 1.06

Second year multiplying factor = 100% + 20% = 1.20

Initial Sale × 1.06 × 1.20 = 254400

Initial Sale = 254400 ÷ 1.272 = 200000

∴ The correct answer is ₹200000.

Alternate Method

Given:

Sale increase in the 1^st year (r₁) = 6%

Sale increase in the 2^nd year (r₂) = 20%

Present sale = ₹254400

Formula Used:

Present Value = Past Value × (1 + r₁/100) × (1 + r₂/100)

Calculation:

Let the sale two years ago be x.

⇒ x × (1 + 6/100) × (1 + 20/100) = 254400

⇒ x × (106/100) × (120/100) = 254400

⇒ x × (53/50) × (6/5) = 254400

⇒ x × (318/250) = 254400

⇒ x = (254400 × 250) ÷ 318

⇒ x = 800 × 250

⇒ x = 200000

∴ The correct answer is ₹200000.

Additional Information

Successive Percentage Change

When a value undergoes successive percentage changes of a% and b%, the net percentage change is given by: Net Change = a + b + (a × b) ÷ 100.

Multiplying Factor Method

An increase of r% is equivalent to multiplying the base value by (1 + r/100). For successive increases, we multiply the base value sequentially by the respective factors to reach the final value quickly.

Reverse Percentage Calculation

To find an original value from a final value after multiple percentage changes, divide the final value by the overall multiplying factor instead of improperly subtracting percentages.

Shortcut Trick

Area enclosed by three equal circles touching each other = r²(√3 − π/2) where r = 21 ÷ 2 = 10.5 cm.

Enclosed Area = (10.5)² × (1.732 − 11/7)

Enclosed Area = (110.25 × 1.732) − (110.25 × 11 ÷ 7)

Enclosed Area = 190.953 − 173.25 = 17.703 ≈ 17.70 cm²

∴ The correct answer is 17.70 cm².

Alternate Method

Given:

Diameter of each ring = 21 cm

Radius (r) = 21 ÷ 2 = 10.5 cm

Concept Used:

When three identical circles touch each other externally, their centers form an equilateral triangle with side length equal to 2r.

Area of enclosed space = Area of Equilateral Triangle − Area of 3 Sectors enclosed inside the triangle

Calculations:

⇒ Side of the equilateral triangle (a) = r + r = 10.5 + 10.5 = 21 cm

⇒ Area of equilateral triangle = (√3 ÷ 4) × a²

⇒ Area of equilateral triangle = (1.732 ÷ 4) × 21²

⇒ Area of equilateral triangle = 0.433 × 441 = 190.953 cm²

⇒ The equilateral triangle has three interior angles of 60°, so the three sectors combined form a semi-circle (180°).

⇒ Area of the 3 sectors = (1 ÷ 2) × π × r²

⇒ Area of the 3 sectors = (1 ÷ 2) × (22 ÷ 7) × (10.5)²

⇒ Area of the 3 sectors = 11 × 1.5 × 10.5 = 173.25 cm²

⇒ Area of enclosed space = 190.953 − 173.25 = 17.703 cm²

⇒ Rounding off to nearest 2 decimal places, we get 17.70 cm².

∴ The correct answer is 17.70 cm².

Additional Information

Area of Enclosed Space Between Three Circles

When three equal circles of radius r touch each other externally, the area of the enclosed space is mathematically given as r²(√3 − π/2).

Perimeter of the Enclosed Space

The boundary of the enclosed space is formed by three inward arcs of 60° each. Its total perimeter is equal to the circumference of a semi-circle, which is πr.

Radius of Inscribed Circle

The radius of the largest possible circle that can be inscribed exactly in the empty space between three touching identical circles is r((2 ÷ √3) − 1).

Shortcut Trick

The angle subtended by an arc at the major arc is half of the angle subtended at the centre.

⇒ Interior opposite angle = 134° ÷ 2 = 67°.

For any cyclic quadrilateral, the exterior angle is directly equal to the interior opposite angle.

⇒ ∠CBP = Interior opposite angle = 67°.

∴ The correct answer is 67°.

Alternate Method

Given:

∠AOC = 134° (Angle subtended by arc ABC at the centre)

AB is produced to P

Formula Used:

Angle subtended at circumference = Angle at centre ÷ 2

Sum of opposite angles in a cyclic quadrilateral = 180°

Angles on a straight line add up to 180°

134°

67°

113°

67°

O A B C D P

Calculations:

⇒ Let D be a point on the major arc AC. Points A, B, C, and D form a cyclic quadrilateral.

⇒ The angle subtended by arc AC at the centre is ∠AOC = 134°.

⇒ The angle subtended by the same arc at the major arc is ∠ADC.

⇒ ∠ADC = ∠AOC ÷ 2

⇒ ∠ADC = 134° ÷ 2 = 67°

⇒ Since ABCD is a cyclic quadrilateral, the sum of its opposite angles is 180°.

⇒ ∠ABC + ∠ADC = 180°

⇒ ∠ABC + 67° = 180°

⇒ ∠ABC = 180° − 67° = 113°

⇒ Line segment ABP is a straight line, so angles ∠ABC and ∠CBP form a linear pair.

⇒ ∠ABC + ∠CBP = 180°

⇒ 113° + ∠CBP = 180°

⇒ ∠CBP = 180° − 113° = 67°

∴ The correct answer is 67°.

Additional Information

Angle at Centre Theorem

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

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral. The sum of either pair of opposite angles is exactly 180°.

Exterior Angle Property

If one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle. In this question directly, ∠CBP = ∠ADC = 67°.

Shortcut Trick

Sum of length and breadth = Perimeter ÷ 2 = 120 ÷ 2 = 60 m.

Value of 1 ratio part = 60 ÷ (3 + 2) = 60 ÷ 5 = 12.

Length = 3 × 12 = 36 m, Breadth = 2 × 12 = 24 m.

Area = 36 × 24 = 864 m².

∴ The correct answer is 864 m².

Alternate Method

Given:

Ratio of length (L) and breadth (B) = 3 : 2

Perimeter of the park = 120 m

Formula Used:

Perimeter of a rectangle = 2 × (L + B)

Area of a rectangle = L × B

Length (L) = 3x Breadth (B) = 2x Area = L × B Perimeter = 2(L + B) = 120 m

Calculation:

Let the length and breadth of the rectangular park be 3x and 2x respectively.

⇒ 2 × (3x + 2x) = 120

⇒ 2 × 5x = 120

⇒ 10x = 120

⇒ x = 12

⇒ Length = 3x = 3 × 12 = 36 m

⇒ Breadth = 2x = 2 × 12 = 24 m

⇒ Area = 36 × 24 = 864 m²

∴ The correct answer is 864 m².

Additional Information

Properties of a Rectangle

Opposite sides are equal and parallel. Every interior angle is 90°. Length of diagonal = √(L² + B²).

Square

A special type of rectangle where all four sides are equal. Perimeter = 4a. Area = a², where 'a' is the side length.

Parallelogram

A quadrilateral with opposite sides parallel. Area = Base × Height. Perimeter = 2(a + b), where 'a' and 'b' are adjacent sides.

The pattern followed here is:

Logic: All symbol moving clockwise direction.

Hence, the correct answer is "Option 2".

Question: Shourya delivered parcels to five locations A, B, C, D and E while prioritising delivery of heavier parcels first. Which location was delivered in the end?

Statements:

(I) The parcel for location C was heavier than all other parcels.

Statement (I) tells us that C was the heaviest, meaning it was delivered first. However, it provides no information about the relative weights of A, B, D, and E. 

Therefore, we cannot determine which location was delivered last.

(II) The parcel for location D was lighter than the parcels assigned to locations B and A.

Statement (II) tells us that D was lighter than B and A. But it doesn't give us any information about the weights of C and E, or the relative weights of B and A.

Therefore, we cannot determine which location was delivered last.

Analysis of Statements (I) and (II) together:

Even when we combine the information from both statements, we still cannot determine which location was delivered last. We know C was delivered first, and D was lighter than B and A, but we don't know the relative weights of B, A, and E, or how they compare to D.

Thus, the data provided in both statements (I) and (II) are not sufficient to answer the question.

Hence, the correct answer is "Option 4".

Given: "Like seasoned gardeners, Gardener G practices crop rotation to maintain soil fertility, leading to thriving and diverse plant growth."

Inferences:

1. The thriving and diverse plant growth in Gardener G's garden is likely influenced by the practice of crop rotation.

The statement explicitly links Gardener G's crop rotation practice to "thriving and diverse plant growth." The phrase "leading to" indicates a causal relationship. Therefore, it is logical to infer that the crop rotation is a likely influence on the healthy plants. This inference directly aligns with the information provided.

2. Gardener G is consulted by other gardeners in the area for best gardening practices.

The statement mentions "seasoned gardeners" and that Gardener G is "like" them. This implies experience and skill. However, it does not explicitly state or imply that other gardeners seek Gardener G's advice. While possible, it is not a direct or necessary inference from the given statement.

Inference 1 is directly supported by the statement's causal link between crop rotation and plant growth. Inference 2 is a potential interpretation but is not explicitly derived from the statement.

Hence, the correct answer is "Option 1."

The logic followed here is:

Option 1) 286 / 11 = 26 (Whole number)

Option 2) 104 / 4 = 26 (Whole number)

Option 3) 494 / 19 = 26 (Whole number)

Option 4)  521 / 20 = 26.05 (Not a whole number)

Thus, the pair 521 - 20 does not follow the same pattern as the other pairs.

Hence, the correct answer is "Option 4".

The given code language is:

Thus, code for 'B' is 3.

Hence, the correct answer is "Option 3".

The figure which complete the given figure is:

Hence, the correct answer is "Option 1".

The correct option for embedded is:

Hence, the correct answer is "Option 2".

Given:

98 ÷ 7 + 288 × 2 − 144 × 3 − 152 ÷ 2 + 120 = ?

Conditions:

1. '+' and '-' are interchanged.
2. '×' and '÷' are interchanged.

Step 1: Replace the operators according to the given conditions.

The equation becomes: 98 × 7 − 288 ÷ 2 + 144 ÷ 3 + 152 × 2 − 120 = ?

Applying BODMAS rule.

→ 98 × 7 − 288 ÷ 2 + 144 ÷ 3 + 152 × 2 − 120

→ 98 × 7 − 144 + 48 + 152 × 2 − 120

→ 686 − 144 + 48 + 304 − 120

→ 542 + 48 + 304 − 120

→ 894 - 120 = 774

Therefore, 774 is the correct answer.

Hence, the correct answer is "Option 1".

Given: 98, 50, 26, 14, ?, 5

The logic followed here is:

Logic: Dividing the previous number by 2 and adding 1.

Step 1: 98 ÷  2 + 1 = 49 + 1 = 50

Step 2: 50 ÷ 2 + 1 = 25 + 1 = 26

Step 3: 26 ÷ 2 + 1 = 13 + 1 = 14

Step 4: 14 ÷ 2 + 1 = 7 + 1 = 8

Step 5: 8 ÷ 2 + 1 = 4 + 1 = 5

Therefore, the missing number is 8.

Hence, the correct answer is 'Option 4"..

Given Letters : R, F, I, G, E, D

Only one meaningful six-letter word can be formed using the letters R, F, I, G, E, D.

Here, the only meaningful six-letter word is "FRIDGE".

Hence, the correct answer is "Option 1"..

The logic followed here is:

1. Set goals -This is the initial step where you identify and define your financial objectives.
2. Assess -Evaluate your current financial situation to understand where you stand in relation to your goals.
3. Budget - Create a budget to plan your income and expenses, ensuring that you allocate resources towards achieving your goals.
4. Invest - Invest your money according to your budget and goals to grow your wealth.
5. Review -Periodically review your financial plan to track progress and make necessary adjustments.

So, the order would be: '1, 5, 2, 4, 3'.

Hence, the correct answer is "Option 1".

Given: 

Question: Five houses of five different colours (blue, green, yellow, red and violet), with equal space between every two adjacent houses, are constructed in a row facing north. Which coloured house is exactly in the middle of the row?

Statements:

(I) Green and yellow coloured houses are constructed at the extreme ends of the row.

Statement I tells us that green and yellow are at the extreme ends. However, it doesn't give us the positions of the other colors (red, violet, and blue).

Therefore, we cannot determine which color is in the middle.

(II) Violet and blue coloured houses are immediate neighbouring houses of green and yellow coloured houses, respectively.

Statement II tells us that violet is next to green and blue is next to yellow. But it doesn't tell us the exact order or which end they are on.

Therefore, we cannot determine the middle house.

Combining both statements, we can deduce the arrangement of the houses:

From statement I, we know either Green -  _  -  _  -  _  -  Yellow

or Yellow  -  _  -  _  -  _  -  Green.

From statement II, we know that Violet is next to Green and Blue is next to Yellow. So the arrangement must be:

Green - Violet -  _  - Blue - Yellow

or Yellow - Blue -  _  - Violet - Green.

Since we have all 5 colors, the remaining color, Red, has to be in the middle.

Thus, Data in Statements I and II together is sufficient to answer the question

Hence, the correct answer is "Option 4".

Given: U – V + W ÷ Y x Z’

U – V means U is brother of V

V + W means V is mother of W

W ÷ Y means W is father of Y

Y x Z means Y is wife of Z

Thus, 'U' is Wife's Father's Mother's Brother of 'Z'.

Hence, the correct answer is "Option 1".

Given:

Cause: Heavy rains yesterday in City X have caused flooding in many areas including the railway station and airport.

Effects:

I. Railways has announced a delay in the departure of three trains scheduled to start from City X this morning.

The cause states that the railway station was flooded. Flooding can disrupt railway operations, leading to delays. Therefore, it is logical to assume that the railway delay is a direct effect of the heavy rains and flooding. Effect I is a possible consequence of the given cause.

II. A number of flights were unable to land in the airport of City X last evening.

The cause also states that the airport was flooded. Flooding can make it dangerous or impossible for planes to land. Therefore, it is logical to assume that flights being unable to land is a direct effect of the heavy rains and flooding. Effect II is also a possible consequence of the given cause.

III. Finance ministry has extended the date for filing tax returns by a month.

The cause is related to weather and transportation disruptions. Extending the date for filing tax returns is a financial decision made by the Finance Ministry. There is no direct or logical link between heavy rains causing flooding and the Finance Ministry extending tax return deadlines. Effect III is not a probable consequence of the given cause.

Effects I and II are direct and logical consequences of the flooding caused by heavy rains. Effect III is unrelated to the given cause.

Hence, the answer correct is "Option 3".

The logic followed here is:

Position of Paras from the top = 5^th

Position of Paras from the bottom = 24^th

So,

Total number of students = Position of Paras from the top + Position of Paras from the bottom - 1

= 5 + 24 - 1

= 29 - 1 = 28.

Hence, the correct answer is "Option 4".

The logic followed here is:

: ORW :: RUZ : %

Option 1) # = UYD % = KNS

Here, not follow same logic.

Option 2) # = VYD % = LNS

Here, not follow same logic.

Option 3) # = VYD % = KNS

Here, follow the same logic.

Option 4) # = VYD % = KNK

Here, not follow same logic.

Hence, the correct answer is "Option 3".

Given: U – V x W ÷ Y + Z

U – V means U is brother of V

V x W means V is wife of W

W ÷ Y means W is father of Y

Y + Z means Y is mother of Z

Thus, U is Mother's mother's brother of Z.

Hence, correct answer is "Option 3".

The correct mirror image is:

Hence, the correct answer is "Option 4".

Given: Seven people A, B, C, D, E, F and G scored different marks (out of 100) in an exam.

G scored the lowest marks. B scored second highest marks, which were 80. A scored 72.

So the order, so far, is _ > 80 (B) > 72 (A) > _ > _ >  _ > G.

E scored more than F but less than C. Marks scored by C were a multiple of 11 and less than 70.

The sum of marks scored by C and F was 120. C scored a multiple of 11 and less than 70. The multiples of 11 less than 70 are 11, 22, 33, 44, 55, and 66.

E scored less than C, and the sum of C and F is 120.

Let's test the possible values of C. If C = 66, then F = 120 - 66 = 54. This satisfies the condition that E is between F and C.

C scored 66 and F scored 54.

The current order is  _ > 80 (B) > 72 (A) > 66 (C) > E > 54 (F) > G.

D is the only remaining person.

So the final order is  D > 80 (B) > 72 (A) > 66 (C) > E > 54 (F) > G.

Here, F scored 54.

Hence, the correct answer is "Option 3".

Given: 3571869

Arrange in ascending order: 1 3 5 6 7 8 9

The digits that remain unchanged in their positions are:

Original : 3 5 7 1 8 6 9

Ascending order: 1 3 5 6 7 8 9

Thus, the positions of one digits remain unchanged.

Hence, the correct answer is "Option 2".

The logic followed here is:

₹¥$£  - Currency symbols

ABCD  - Alphabetical letters

× + − ÷  - Mathematical operators not Text.

1234  - Numerical digits

Hence, the correct answer is "Option 3".$

The pattern followed here is:

Logic: 1. New element added in middle upper side figure.

2. Same symbol is increase by one in middle left and middle right.  

3. Same symbol is increase by one in bottom left corner and shifted to bottom right corner.

Hence, the correct answer is "Option 3".

The least possible venn diagram is:

Conclusions:

(I) Some toys are dolls → False ( there is no direct relation between toys and dolls. so, it can be a possibility)

(II) No cycle is a robot → False ( there is no direct relation between cycle and robot. so, it can be a possibility)

Thus, Neither conclusion (I) nor (II) is true.

Hence, the correct answer is "Option 4".

Given:

Statement: Many people have complained to the Finance Ministry regarding a delay in receiving refunds of their Income Tax returns.

Course of Action I:

The Finance Ministry must institute an enquiry into the complaints.

Follows logically.

An enquiry is a reasonable first step to investigate the issue.

Course of Action II:

The Finance Ministry must compensate taxpayers by paying interest.

Does not necessarily follow.

Compensation can only be decided after confirming the cause and responsibility for the delay.

Thus, only Course of Action I logically follows.

Hence, the correct answer is "Option 2".

The correct mirror image is:

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: 1^st number × 9 + 6 = 2^nd number.

17 is related to 159

→ 17 × 9 + 6

→ 153 + 6

→ 159. ( 2^nd number)

And

11 is related to 105

→ 11 × 9 + 6

→ 99 + 6

→ 105. ( 2^nd number)

Similarly,

13 is related to 

→ 13 × 9 + 6

→ 117 + 6

→ 123. 

Hence, the correct answer is "Option 3".

The logic followed here is:

Option A) 

1 element is repeat one.

2^nd element is repeat thrice.

Option B)

1 element is repeat one.

2^nd element is repeat thrice.

Option C)

1 element is repeat one.

2^nd element is repeat thrice.

Option D)

1 element is repeat one.

2^nd element is repeat thrice.

Option E)

1 element is repeat twice.

2^nd element is repeat twice.

Not follow the same logic.

Hence, the correct answer is "Option 4".

The code language is:

Thus, code for music is 'jo'.

Hence, the correct answer is "Option 1".

The logic followed here is:

3. Seed 
4. Seedling  
5. Plant  
6. Pollination 
7. Fruit 

So, the order would be: 3, 4, 1, 2, 5

Hence, the correct answer is "Option 1".

Given: M, N, O, X, Y and Z are sitting around a circular table facing the centre.

1. O is an immediate neighbour of Y.

2. Z sits second to the right of O.

3. X is the immediate neighbour of both M and N.

4. M is not an immediate neighbour of O.

Thus, 'O' is sitting to the immediate left of Y.

Hence, the correct answer is "Option 2"

The correct answer is Option (4).

Key Points

• Indirect Speech: He said that he would need a copy of the documents.
• Direct to Indirect Speech:
• Pronoun Change:
• "I" is changed to "he" to match the subject of the reporting verb.
• Phrasing:
• "shall" changed to "would" according to the subject.
• Sentence Structure:
• The phrase "a copy of the documents" remains unchanged as it is the object of the need.
• Punctuation and Format:
• No quotation marks are used in indirect speech.
• The reporting clause "He said..." starts the sentence.

Therefore, the correct answer is Option (4).

Correct Sentence: "He said he would need a copy of the documents."

Additional Information

• Option 1: "He said that he would need a copy of the documents."
• Changes "shall" to "would," which may not accurately convey the original future intent in some contexts.
• Option 2: "He asked if I would need a copy of the documents."
• Changes the statement to a question, which is incorrect in this context.
• Option 3: "He mentioned that he needed a copy of the documents."
• Changes the future tense "shall" to the past tense "needed," which inaccurately alters the original meaning.

The correct answer is Option 4.

Key Points

• The paragraph starts by mentioning backward areas in Andhra Pradesh.
• Sentence A follows since it identifies one of such backward areas:
• "Rayalaseema is one of them."
• Sentence B is next because it explains the result of degradation in Rayalaseema:
• "Massive degradation of natural resources and extensive changes in land use have resulted in poor crop productivity in Rayalaseema."
• Sentence D follows as it elaborates on the technical details of soil degradation:
• "Soil Organic Carbon (SOC), which is crucial for maintaining soil quality, has decreased by 84% in the region. Poor soil fertility and soil degradation, like in Rayalaseema region, are the major causes of food shortage."
• Sentence C is the last before the concluding sentence as it introduces the efforts of the National Soil Survey and Land use planning organisation:
• "The National Soil Survey and Land use planning organisation is doing its job by giving suggestions to the farmers of Rayalaseema."
• The concluding sentence (6) ties back to the suggestions provided.
• The correct sequence is ABDC.

Coherent paragraph after rearrangement:

"There are some backward areas in Andhra Pradesh. Rayalaseema is one of them. Massive degradation of natural resources and extensive changes in land use have resulted in poor crop productivity in Rayalaseema. Soil Organic Carbon (SOC), which is crucial for maintaining soil quality, has decreased by 84% in the region. Poor soil fertility and soil degradation, like in Rayalaseema region, are the major causes of food shortage. The National Soil Survey and Land use planning organisation is doing its job by giving suggestions to the farmers of Rayalaseema. The advice is about which crops are to be cultivated and which not."

The correct answer is: Option 1 i.e. democracy.

Key Points

• The word "democracy" means a system of government by the whole population, typically through elected representatives. (लोकतंत्र)
• Example: India is the world's largest democracy, where citizens elect their leaders through voting.
• "Government by the people" refers to a form of government in which the people have the authority to deliberate and decide legislation (direct democracy) or to choose governing officials to do so (representative democracy). (जनता द्वारा सरकार)
• Example: The United States is a government by the people, where citizens have the power to elect their leaders.
• Hence, we can infer that "government by the people" is best replaced by "democracy".

Therefore, the correct answer is: Option 1 i.e. democracy.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Oligarchy (गुटतंत्र): A small group of people having control of a country, organization, or institution.
• Example: The country was run by an oligarchy of wealthy families.
• Theocracy (धर्मतंत्र): A system of government in which priests rule in the name of God or a god.
• Example: Iran is an example of a theocracy where religious leaders have significant influence over the government.
• Autocracy (स्वेच्छाचारिता): A system of government by one person with absolute power.
• Example: North Korea is often cited as an example of an autocracy due to the centralization of power in one leader.

The correct answer is: Option 1.

Key Points

• The word "Thrifty" means using money and other resources carefully and not wastefully. (मितव्ययी)
• Example: Being thrifty, she always looks for the best deals and discounts.
• "Extravagant" refers to spending much more than is necessary or wise, often in a wasteful manner. (अपव्ययी)
• Example: His extravagant lifestyle quickly depleted his savings.
• Hence, we can infer that the opposite of 'thrifty' is 'extravagant'.

Therefore, the correct answer is: Option 1.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Thankful (आभारी): Feeling or expressing gratitude.
• Example: She was thankful for all the support she received from her friends.
• Intolerant (असहिष्णु): Not willing to accept views, beliefs, or behaviors that differ from one's own.
• Example: He was intolerant of any criticism towards him.
• Indifferent (उदासीन): Having no particular interest or sympathy; unconcerned.
• Example: She was indifferent to the outcome of the election.

The correct answer is: Option 2.

Key Points

• The word "sitting on the fence" means to be undecided or not to take sides in a dispute. (निर्णय न कर पाना या किसी पक्ष में न जाना)
• Example: Many voters are still sitting on the fence, unable to decide who to vote for.
• "over the fence" refers to something that is beyond a boundary or limit, but it is not an idiom used for indecision. (सीमा के पार)
• Example: The ball went over the fence into the neighbor's yard.
• "by the fence" means near the fence, and is not an idiom related to indecision. (बाड़ के पास)
• Example: She stood by the fence watching the sunset.
• "breaking a leg" is an idiom used to wish someone good luck, especially before a performance. (शुभकामनाएँ देना)
• Example: Break a leg! I know you'll do great in your play tonight.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: over the fence (सीमा के पार): This phrase refers to something beyond a boundary, but it does not relate to indecision.
• Example: The ball went over the fence into the neighbor's yard.
• Option 3: by the fence (बाड़ के पास): This phrase means near the fence, and is not related to indecision.
• Example: She stood by the fence watching the sunset.
• Option 4: breaking a leg (शुभकामनाएँ देना): This idiom is used to wish someone good luck, typically before a performance.
• Example: Break a leg! I know you'll do great in your play tonight.

The correct answer is 'finest'.

Key Points

• The superlative form "finest" is used to denote the highest degree of quality among all potential options.
• In this context, "finest" appropriately describes the highest level of tailored protection provided by prescription safety glasses for individuals with vision correction needs.
• Options 1, 2, and 4 are incorrect as they do not convey the intended superlative degree that "finest" does.

Therefore, the correct answer is Option 3.

Complete Sentence: "Prescription safety glasses provide the finest tailored protection for individuals with vision correction needs."

Additional Information

• Option 1: "finer" is incorrect because "finer" is a comparative form used when comparing two items, while the context here implies the highest degree, requiring the superlative form.
• Option 2: "more finely" is incorrect because "more finely" is an adverbial phrase that does not fit the grammatical structure or context of the sentence.
• Option 4: "fine" is incorrect because "fine" is a positive degree adjective and does not convey the superlative degree needed in this context.

The correct answer is: Option 4.

Key Points

• The word "Indispensable" means something that is absolutely necessary or essential. (अनिवार्य)
• Example: Good health is indispensable for a happy life.
• The word "Indispensible" is incorrectly spelt. The correct spelling is "Indispensable". (अनिवार्य)
• Example: Good leadership is indispensable for the success of any organization.

Therefore, the correct answer is: Option 4.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Indespensable (अनिवार्य): This is an incorrect spelling of "Indispensable".
• Example: N/A
• Indespensible (अनिवार्य): This is an incorrect spelling of "Indispensable".
• Example: N/A
• Indispenscibe (अनिवार्य): This is an incorrect spelling of "Indispensable".
• Example: N/A

The correct answer is: Option 2.

Key Points

• The word "Narcissist" means someone who has an excessive interest in or admiration of themselves. (आत्ममुग्ध व्यक्ति)
• Example: John is such a narcissist; he spends hours looking at himself in the mirror.
• "Narcissist" refers to a person who loves and admires themselves the most, often to the point of ignoring others. (आत्ममुग्ध व्यक्ति)
• Example: Her narcissistic behavior made it difficult for her to form genuine friendships.
• Hence, we can infer that the one-word substitute for "Someone who loves and admires himself the most" is 'Narcissist'.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Recluse (एकांतवासी): A person who lives a solitary life and tends to avoid other people.
• Example: The old man lived as a recluse in a small cabin in the woods.
• Altruist (परोपकारी): A person who has an unselfish concern for the welfare of others.
• Example: She is known as an altruist because of her endless charitable work.
• Omniscient (सर्वज्ञ): Knowing everything.
• Example: The novel is narrated by an omniscient narrator who knows everything about the characters' thoughts and feelings.

The correct answer is Option 1.

Key Points

• The sentence begins with stating an outstanding feature of the 21^st century.
• Segment Q follows as it starts explaining the improvement:
• "the improvement of the income and"
• Segment R is next as it continues the thought:
• "living conditions of the"
• Segment S follows as it connects the working man's living conditions with a result:
• "working man leading to"
• Segment P is last before the concluding segment as it explains the result:
• "greater security and education which"
• The concluding segment (6) ties back to what the greater security and education have meant.
• The correct sequence is Q, R, S, P.

Coherent sentence after rearrangement:

"An outstanding feature of the 21^st century has been the improvement of the income and living conditions of the working man leading to greater security and education which has meant greater political power."

The correct answer is '2' i.e. The hikers trekked for hours through the dense forest before reaching the mountain peak. 

Key Points

• This sentence describes a sequence of actions that the hikers undertook.
• The order of the clauses logically follows the chronology of the actions.
• The subject of the sentence "the hikers" should be followed by the main verb "trekked".
• The phrase "for hours" indicates the duration of the action and follows directly after the verb 'trekked'.
• The prepositional phrase "through the dense forest" shows where the action took place and modifies the verb.
• Finally, the infinitive phrase "before reaching the mountain peak" shows the concluding action after the hike.

Additional Information

• The placement of adverbial phrases (like "for hours" and "through the dense forest") is important for clarity and flow.
• The structure used in the correct sentence typically follows the pattern: Subject + Main Verb + Additional information describing the action (such as adverbials of time, place, or manner).
• In English, the placement of time-related information generally comes after the verb or at the end of the sentence.
• Understanding the role and typical placement of these sentence components can help construct clear, coherent sentences.

The correct answer is: Option 1.

Key Points

• The word "Pedestrian" means someone who walks on foot. (पैदल यात्री)
• Example: The pedestrian crossed the street at the crosswalk.
• "Pedestrian" refers to someone who travels by walking, usually on roads or pavements. (पैदल यात्री)
• Example: Pedestrians should always use the sidewalk for safety.

Therefore, the correct answer is: Option 1.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Auditor (लेखापरीक्षक): A person who conducts an audit.
• Example: The auditor found several discrepancies in the financial report.
• Decanter (शराब की बोतल): A vessel used to hold the decantation of a liquid.
• Example: The waiter poured the wine from the decanter into the glasses.
• Stalker (पीछा करने वाला): A person who harasses or persecutes someone with unwanted and obsessive attention.
• Example: The celebrity had to get a restraining order against the stalker.

The correct answer is: Option 2.

Key Points

• The word "Practitionaire" is not correctly spelled in English. The correct spelling is "Practitioner" which means someone who is actively engaged in an art, discipline, or profession, especially medicine. (व्यवसायिक)
• Example: She is a well-respected practitioner in the field of cardiology.
• "Practitioner" is the correct word and it means someone who practices a learned profession, such as in law or medicine. (व्यवसायिक)
• Example: The practitioner attended to the patient with utmost care and professionalism.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Apparent (स्पष्ट): Clearly visible or understood; obvious.
• Example: Her disappointment was apparent to everyone in the room.
• Recommend (सिफारिश करना): To suggest that someone or something would be good or suitable for a particular job or purpose.
• Example: I highly recommend this book for its insightful content.
• Memento (स्मृति चिन्ह): An object kept as a reminder or souvenir of a person or event.
• Example: She kept the ticket stub as a memento of their first concert together.

The correct answer is 'a colour'.

Key Points

• The denotation of a word is its direct, explicit meaning or definition, which is objective and not influenced by feelings or experiences.
• In this context, the word "blue" is used to describe the colour of the blueberries in the basket.
• Options 1, 2, and 4 refer to connotations of the word "blue", which are the emotional or cultural associations with the word, but they do not represent its denotation.

Therefore, the correct answer is Option 3.

Complete Sentence: "The denotation of the word 'blue' is a colour."

Additional Information

• Option 1: "gloominess" is incorrect because it reflects a connotation rather than the direct meaning.
• Option 2: "sadness" is incorrect; it also represents a connotation rather than the direct meaning.
• Option 4: "negativity" is incorrect because it is another connotation and does not denote the colour blue.

The correct answer is Option 1.

Key Points

• Indirect Speech: The soldier told his fellow soldiers that he had buried their brave comrade there.
• Direct to Indirect Speech:
• Pronoun Change:
• "I" is changed to "he" to match the subject of the reporting verb.
• "our" is changed to "their" to align with the perspective of the fellow soldiers.
• Tense Change:
• "have buried" is changed to "had buried" to signify that the action is being reported in the past.
• Word Change:
• "here" is changed to "there" to properly reflect the change in location when reporting.
• Punctuation and Format:
• No quotation marks are used in indirect speech.
• The reporting clause "The soldier told his fellow soldiers that..." starts the sentence.

Therefore, the correct answer is Option 1.

Correct Sentence: "The soldier told his fellow soldiers that he had buried their brave comrade there."

Additional Information

• Option 2: "The soldier told us that I had buried their brave comrade here."
• Incorrectly changes the audience to "us" instead of "his fellow soldiers."
• Does not change "I" to "he," and "here" remains "here" instead of "there," which is not correct in reported speech.
• Option 3: "The soldier told his fellow soldiers that I had buried our brave comrade there."
• Does not change "I" to "he," and "our" remains "our" instead of "their," which is not correct in reported speech.
• Option 4: "The soldier told us that he had buried our brave comrade there."
• Incorrectly changes the audience to "us" instead of "his fellow soldiers."
• "our" remains "our" instead of "their," which is not correct in reported speech.

The correct answer is: Option 3.

Key Points

• The word "ameliorate" means to make something better or improve a situation. (सुधारना)
• Example: The government took several measures to ameliorate the living conditions of the poor.
• "Exacerbate" refers to making a bad situation worse. (बिगाड़ना)
• Example: The heavy rains exacerbated the flooding problem in the city.
• Hence, we can infer that the opposite of 'ameliorate' is 'exacerbate'.