SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The final selling price after two successive discounts d₁ and d₂ is determined by the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}). Given the marked price MP = 1000 and the discount rates d₁ = 12 and d₂ = 8, the calculation is as follows:

SP = 1000 × (1 - ¹²⁄₁₀₀) × (1 - ⁸⁄₁₀₀) SP = 1000 × 0.88 × 0.92 SP = 880 × 0.92 SP = 809.6

Rounding 809.6 to the nearest integer gives 810.

Explanation

The profit shared between partners is proportional to their investment ratio after deducting the amount for charity. Let the total profit be P. Since 15% goes to charity, the distributable profit is 85% of P, which is 0.85P.

The distributable profit is shared between A and B in the ratio 7:5. A's share of the profit is given by the formula: A's share = (A's investment ratio)/(Sum of ratios) × Distributable profit

Substituting the given values: 5950 = ⁷⁄₇₊₅ × 0.85P 5950 = ⁷⁄₁₂ × ⁸⁵⁄₁₀₀ × P

To solve for P: P = (5950 × 12 × 100)/(7 × 85) P = (850 × 1200)/(85) P = 10 × 1200 P = 12000

The calculation shows that for A to receive Rs. 5,950 as their share from the 85% distributable profit, the total business profit must be Rs. 12,000.

Explanation

The total cost price (CP) for six bottles is ₹100. To achieve a 20% profit on the transaction, the total selling price (SP) for all six bottles is determined by the formula SPₜₒₜₐₗ = CPₜₒₜₐₗ × (1 + (Profit%)/(100)). Substituting the values, we get SPₜₒₜₐₗ = 100 × ¹²⁰⁄₁₀₀ = 120. To find the selling price for a single bottle, the total selling price is divided by the number of units, resulting in SPᵤₙᵢₜ = ¹²⁰⁄₆ = 20.

Explanation

To find the value of the expression, substitute the standard trigonometric values: tan 45° = 1, cos 30° = √3/2, sec 30° = (2)/(√(3)), and cot 60° = (1)/(√(3)).

Substituting these values into the given expression: ⁴⁄₃(1)² + 3(√3/2)² - 2((2)/(√(3)))² - ¾((1)/(√(3)))²

Evaluating the squares: ⁴⁄₃(1) + 3(¾) - 2(⁴⁄₃) - ¾(⅓) = ⁴⁄₃ + ⁹⁄₄ - ⁸⁄₃ - ¼

Grouping the terms with common denominators for easier calculation: (⁴⁄₃ - ⁸⁄₃) + (⁹⁄₄ - ¼) = -⁴⁄₃ + ⁸⁄₄ = -⁴⁄₃ + 2 = (-4 + 6)/(3) = ⅔

Explanation

Let x be the number of years after which Meenu will be double her daughter's age. Currently, Meenu is 38 years old and her daughter is 8 years old. After x years, Meenu will be 38 + x and her daughter will be 8 + x. We can set up the linear equation based on the condition that Meenu's age will be twice her daughter's age:

38 + x = 2(8 + x)

Expanding the right side: 38 + x = 16 + 2x

Rearranging the terms to solve for x: 38 - 16 = 2x - x 22 = x

After 22 years, Meenu will be 60 years old and her daughter will be 30 years old, satisfying the condition that the mother's age is double the daughter's age.

Explanation

The average speed of an object is defined as the ratio of the total distance traveled to the total time taken. Using the formula:

Average Speed = (Total Distance)/(Total Time)

First, calculate the total distance covered by the train: Total Distance = 700 km + 1012 km = 1712 km

Next, calculate the total time taken: Total Time = 5 hours + 11 hours = 16 hours

The average speed in km/h is: Average Speed (km/h) = ¹⁷¹²⁄₁₆ = 107 km/h

To convert the speed from km/h to m/s, multiply by the conversion factor ⁵⁄₁₈: Average Speed (m/s) = 107 × ⁵⁄₁₈ = ⁵³⁵⁄₁₈ m/s

Converting the improper fraction ⁵³⁵⁄₁₈ into a mixed fraction: 535 ÷ 18 = 29 with a remainder of 13 ⁵³⁵⁄₁₈ = 29 ¹³⁄₁₈ m/s

Explanation

Let the total work be the least common multiple (LCM) of the time taken by R, S, and T individually. The LCM of 4, 6, and 18 is 36. Therefore, let the total work be 36 units.

The efficiency (work done per day) of each person is calculated as: Efficiency of R = 36 / 4 = 9 units/day Efficiency of S = 36 / 6 = 6 units/day Efficiency of T = 36 / 18 = 2 units/day

The combined efficiency of R, S, and T working together is the sum of their individual efficiencies: Combined efficiency = 9 + 6 + 2 = 17 units/day

The total time taken to complete the work together is: Time = (Total Work)/(Combined Efficiency) = ³⁶⁄₁₇

Performing the division: 36 ÷ 17 ≈ 2.1176...

Rounding this result to two decimal places gives 2.12.

Explanation

In right-angled △ ABC, the sides are determined using the Pythagorean theorem AB² + AC² = BC². Given AC = 5 cm and hypotenuse BC = 9 cm, the third side is AB = √(9² - 5²) = √(81 - 25) = √(56) cm. In right-angled △ PQR, with PR = 3 cm and hypotenuse QR = 8 cm, the third side is PQ = √(8² - 3²) = √(64 - 9) = √(55) cm.

Two triangles are congruent if all corresponding sides are equal, and they are similar if their corresponding sides are in the same ratio. For these triangles, the side lengths are {5, √(56), 9} and {3, √(55), 8}. There is no equality between the corresponding sides, and the ratios 5/3, √(56)/√(55), and 9/8 are all different. Since the triangles lack both congruence and similarity, they are not equal in any geometric property, which is expressed as △ ABC ≠ △ PQR.

Explanation

First, simplify the expressions for p and q by rationalizing the denominators. For p, multiplying both the numerator and denominator by the conjugate √2 + 1 gives: p = (√2 + 1)²{(√2 - 1)(√2 + 1)} = 2 + 1 + 2√2{2 - 1} = 3 + 2√2 Similarly, for q, multiplying by the conjugate √2 - 1 gives: q = (√2 - 1)²{(√2 + 1)(√2 - 1)} = 2 + 1 - 2√2{2 - 1} = 3 - 2√2 From these values, calculate the sum and product: p + q = (3 + 2√2) + (3 - 2√2) = 6 pq = (3 + 2√2)(3 - 2√2) = 3² - (2√2)² = 9 - 8 = 1 The required expression is (p²)/(q) + (q²)/(p) = (p³ + q³)/(pq). Since pq = 1, the expression simplifies to p³ + q³. Using the algebraic identity p³ + q³ = (p + q)³ - 3pq(p + q), substitute the calculated values: p³ + q³ = 6³ - 3(1)(6) = 216 - 18 = 198

Explanation

The angle 120° lies in the second quadrant of the unit circle, where the cosine function is negative. Using the supplementary angle identity cos(180° - θ) = -cos θ, we can rewrite the expression by setting θ = 60°. This gives cos 120° = cos(180° - 60°) = -cos 60°. Since the standard trigonometric value of cos 60° is 0.5, the value of cos 120° is -0.5. Alternatively, applying the identity cos(90° + θ) = -sin θ with θ = 30° results in cos(90° + 30°) = -sin 30°, which also equals -0.5.

Explanation

The volume of a cone is expressed as V_cₒₙₑ = ⅓ π r² h₁ and the volume of a cylinder is expressed as V_c_yₗᵢₙ_dₑᵣ = π r² h₂. Given that the radii are equal and the volumes are equal, we equate the two formulas:

⅓ π r² h₁ = π r² h₂

Since π r² is common to both sides and non-zero, it can be cancelled:

⅓ h₁ = h₂

To find the ratio of their heights h₁ : h₂, we rearrange the equation:

h₁{h₂} = ³⁄₁

This calculation shows that the height of the cone must be three times the height of the cylinder to maintain equal volumes when their radii are identical.

Explanation

To find the average quantity of food item C used during the four months, we sum the quantities consumed in March, April, May, and June and divide the total by the number of months.

Average = (Total Quantity of C)/(Number of Months)

Based on the table, the quantities for food item C are 280 kg in March, 295 kg in April, 280 kg in May, and 315 kg in June.

Total Quantity = 280 + 295 + 280 + 315 = 1170 kg

Average = ¹¹⁷⁰⁄₄ = 292.5 kg

Explanation

The fourth proportional x to three numbers a, b, and c is determined by the ratio a:b = c:x, which leads to the formula x = (b × c)/(a). Given the numbers 144, 192, and 216, we substitute them into the formula:

x = (192 × 216)/(144)

The ratio of the first two terms 144:192 can be simplified by dividing both by 48, resulting in 3:4. Since the same ratio must apply to the next pair, we have 216:x = 3:4. Dividing 216 by 3 gives 72, and multiplying this unit value by 4 gives:

x = 72 × 4 = 288

Explanation

Applying the BODMAS rule, first solve the expression within the parentheses. The mixed fractions are converted as 2³⁄₁₀ = ²³⁄₁₀ and 2⅗ = ²⁶⁄₁₀, giving a sum of ²³⁺²⁶⁄₁₀ = ⁴⁹⁄₁₀. Next, solve the of operation: ³⁄₇ of ⁴⁹⁄₁₀ = ³⁄₇ × ⁴⁹⁄₁₀ = (3 × 7)/(10) = ²¹⁄₁₀.

The expression now simplifies to ⁷⁄₁₀ ÷ ²¹⁄₁₀ + ⅕ ÷ ⁷⁄₅ - ²⁄₇. Proceed with the division operations by multiplying by the reciprocals: ⁷⁄₁₀ × ¹⁰⁄₂₁ = ⅓ and ⅕ × ⁵⁄₇ = ⅐. Substituting these back into the expression yields ⅓ + ⅐ - ²⁄₇. Combining the terms with the denominator 7 results in ⅓ - ⅐. Finally, using the common denominator of 21, the calculation is (7 - 3)/(21) = ⁴⁄₂₁.

Explanation

To determine the value of tan 15°, we use the identity tan (A - B) = (tan A - tan B)/(1 + tan A tan B) by substituting the standard angles A = 45° and B = 30°. Given that tan 45° = 1 and tan 30° = (1)/(√(3)), the formula becomes:

tan 15° = tan (45° - 30°) = (1 - 1)/(√(3)){1 + (1)((1)/(√(3)))}

Multiplying both the numerator and the denominator by √3 to eliminate the complex fraction gives:

tan 15° = √3 - 1{√3 + 1}

To rationalize the expression, multiply the numerator and the denominator by the conjugate of the denominator, which is √3 - 1:

tan 15° = (√3 - 1)²{(√3 + 1)(√3 - 1)}

Applying the square of a binomial (a - b)² = a² + b² - 2ab in the numerator and the difference of squares (a + b)(a - b) = a² - b² in the denominator:

tan 15° = 3 + 1 - 2√3{3 - 1} = 4 - 2√3{2}

Factoring out a 2 from the numerator and simplifying the fraction results in:

tan 15° = 2 - √3

Explanation

To find the total time required, we first determine the individual typing rates of Suresh and Anil. Suresh's rate is 40 / 5 = 8 pages per hour. Anil's rate is 32 / 6 = 16/3 pages per hour. Since they are working simultaneously on two different computers, their combined rate of work is the sum of their individual rates.

Combined rate = 8 + ¹⁶⁄₃ = (24 + 16)/(3) = ⁴⁰⁄₃ pages per hour.

The time needed to type a 110-page assignment is calculated using the formula Time = (Total Work)/(Combined Rate).

Time = 110 ÷ ⁴⁰⁄₃ = 110 × ³⁄₄₀ = ³³⁄₄ = 8.25 hours.

Converting 0.25 hours into minutes, we get 0.25 × 60 = 15 minutes. Therefore, the total time required is 8 hours and 15 minutes.

Explanation

Since the policeman and the thief start running at the same time and the chase ends when the policeman catches the thief, the time of travel for both is constant. When time is constant, the distance covered is directly proportional to the speed.

Ratio of speeds of Policeman and Thief = Vₚ : Vₜ = 32 : 25

Ratio of distances covered by Policeman and Thief = Dₚ : Dₜ = 32 : 25

The difference in the distance covered by the policeman and the thief must be equal to the initial gap between them. Difference in distance units = 32 - 25 = 7 units

Given that the initial distance gap is 210 m: 7 units = 210 m 1 unit = ²¹⁰⁄₇ = 30 m

The distance run by the thief before being overtaken corresponds to 25 units: Dₜ = 25 × 30 = 750 m

Explanation

The total distance traveled is the sum of the distances covered in each segment of the journey, calculated using the formula Distance = Speed × Time. For the cycling segment, D₁ = 8.5 × 4 = 34 km. For the auto segment, D₂ = 20 × 1.5 = 30 km. For the walking segment, D₃ = 4 × y = 4y km. The total distance is 34 + 30 + 4y = 64 + 4y.

The total time for the journey is the sum of the individual durations: T = 4 + 1.5 + y = 5.5 + y. Average speed is defined as the total distance divided by the total time. Given the average speed is 10 km/h, we use the relation:

10 = (64 + 4y)/(5.5 + y)

Multiplying both sides by (5.5 + y) yields: 10(5.5 + y) = 64 + 4y 55 + 10y = 64 + 4y

Rearranging the equation to solve for y: 10y - 4y = 64 - 55 6y = 9 y = ⁹⁄₆ = 1.5

Explanation

The area of a sector of a circle is calculated using the formula A = (θ)/(360°) × π r², where θ is the central angle and r is the radius. Given the area A = 1848 m², the central angle θ = 270°, and π = ²²⁄₇, we can substitute these values into the formula:

1848 = ²⁷⁰⁄₃₆₀ × ²²⁄₇ × r²

First, simplify the fraction representing the portion of the circle: ²⁷⁰⁄₃₆₀ = ¾

Now, substitute this back into the equation: 1848 = ¾ × ²²⁄₇ × r² 1848 = ⁶⁶⁄₂₈ × r²

To solve for r², rearrange the equation: r² = (1848 × 28)/(66)

Dividing 1848 by 66 gives 28. Therefore: r² = 28 × 28 r = √(28²) r = 28 m

Explanation

Let the total work be 60 units. Since 1 boy can finish the work in 20 days, the efficiency of a boy is E_B = 60 / 20 = 3 units/day. Since 2 women can finish the work in 30 days, the efficiency of 2 women is 60 / 30 = 2 units/day, making the efficiency of 1 woman E_W = 1 unit/day.

The combined efficiency of a woman and a boy is E_W + E_B = 1 + 3 = 4 units/day. The time taken by them to finish the work is 60 / 4 = 15 days. According to the problem, a man and a woman finish the work in half this time, which is 15 / 2 = 7.5 days.

The combined efficiency of a man and a woman is therefore 60 / 7.5 = 8 units/day. By subtracting the woman's efficiency, the efficiency of a man is E_M = 8 - 1 = 7 units/day. The total efficiency of 4 men is 4 × 7 = 28 units/day. The time taken by 4 men to finish the work is 60 / 28 = 15 / 7, which is approximately 2.14 days.

Explanation

A number is divisible by 18 if it is simultaneously divisible by both 2 and 9. For the 11-digit number 7823326867X to be divisible by 2, the last digit X must be an even number. For the number to be divisible by 9, the sum of its digits must be a multiple of 9.

The sum of the digits is 7 + 8 + 2 + 3 + 3 + 2 + 6 + 8 + 6 + 7 + X = 52 + X. To make this sum divisible by 9, we look for the smallest multiple of 9 greater than or equal to 52, which is 54 (9 × 6 = 54). Setting the equation 52 + X = 54 gives X = 2. Since 2 is an even number, it satisfies the condition for divisibility by 2 as well, making the entire number divisible by 18.

Explanation

The work-equivalence formula relates the number of persons M, days D, hours per day H, and work done W as (M₁ · D₁ · H₁)/(W₁) = (M₂ · D₂ · H₂)/(W₂).

For the first scenario, the parameters are M₁ = N + 18, D₁ = 20, H₁ = 7.5, and W₁ = 48. For the second scenario, the remaining work is 100 - 48 = 52%, and the parameters are M₂ = N + 12, D₂ = 30, H₂ = 6.5, and W₂ = 52.

Substituting these values into the formula: ((N + 18) · 20 · 7.5)/(48) = ((N + 12) · 30 · 6.5)/(52)

Simplifying the constants: ((N + 18) · 150)/(48) = ((N + 12) · 195)/(52)

Reducing the fractions by their greatest common divisors (6 for the first fraction and 13 for the second): (N + 18) · ²⁵⁄₈ = (N + 12) · ¹⁵⁄₄

Multiplying both sides by 8 to eliminate the denominators: 25(N + 18) = 2 · 15(N + 12) 25(N + 18) = 30(N + 12)

Dividing both sides by 5: 5(N + 18) = 6(N + 12) 5N + 90 = 6N + 72 90 - 72 = 6N - 5N N = 18

Explanation

The time taken to complete a task is directly proportional to the amount of work done. If Mohit completes ⅔ of the work in 24 days, the time required to complete the entire work (1 unit) is 24 × ³⁄₂ = 36 days. To find the time required for ⅑ of the work, we multiply the total time by the required fraction: 36 × ⅑ = 4 days. This can also be solved using the direct proportion (W₁)/(T₁) = (W₂)/(T₂), where (2/3)/(24) = (1/9)/(T₂). Solving for T₂ gives T₂ = (1/9 × 24)/(2/3) = ²⁴⁄₉ × ³⁄₂ = ⁸⁄₃ × ³⁄₂ = 4 days.

Explanation

To find the value of x⁶ - (1)/(x⁶), we utilize the algebraic identity x⁶ - (1)/(x⁶) = (x³ + (1)/(x³))(x³ - (1)/(x³)).

First, calculate x³ + (1)/(x³) from the given x + ¹⁄ₓ = 2√2 using the shortcut formula k³ - 3k: x³ + (1)/(x³) = (2√2)³ - 3(2√2) = 16√2 - 6√2 = 10√2

Next, find x - ¹⁄ₓ using the identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4: (x - ¹⁄ₓ)² = (2√2)² - 4 = 8 - 4 = 4 Given x > 1, then x - ¹⁄ₓ = √4 = 2.

Now, calculate x³ - (1)/(x³) from x - ¹⁄ₓ = 2 using the formula m³ + 3m: x³ - (1)/(x³) = (2)³ + 3(2) = 8 + 6 = 14

Finally, multiply the two results to find the value of the expression: x⁶ - (1)/(x⁶) = (10√2) × 14 = 140√2

Explanation

To find the value of the given expression, we first determine the value of cot x from the provided equation. Starting with:

(sin x - cos x)/(sin x + cos x) = ⅖

By cross-multiplying, we get: 5(sin x - cos x) = 2(sin x + cos x) 5sin x - 5cos x = 2sin x + 2cos x 3sin x = 7cos x

Dividing both sides by 3sin x, we obtain the value of cot x: (cos x)/(sin x) = ³⁄₇ ⟹ cot x = ³⁄₇

Squaring both sides gives cot² x = ⁹⁄₄₉. Now, we substitute this into the required expression:

(1 + cot² x)/(1 - cot² x) = (1 + 9)/(49){1 - ⁹⁄₄₉}

Simplifying the numerator and denominator: (49 + 9)/(49){(49 - 9)/(49)} = ⁵⁸⁄₄₀

Reducing the fraction: ⁵⁸⁄₄₀ = ²⁹⁄₂₀ = 1.45

Explanation

The given expression follows the algebraic form (a² + b² - ab)/(a³ + b³) where a = 432 and b = 247. According to the algebraic identity for the sum of two cubes, the denominator can be factored as a³ + b³ = (a + b)(a² - ab + b²). Substituting this identity into the expression gives:

(a² - ab + b²)/((a + b)(a² - ab + b²))

The term (a² - ab + b²) appears in both the numerator and the denominator, allowing them to be cancelled out. This simplifies the entire expression to:

(1)/(a + b)

By substituting the numerical values of a and b back into the simplified formula, we get:

(1)/(432 + 247) = ¹⁄₆₇₉

Explanation

The speed v is defined by the formula v = ᵈ⁄ₜ, where d is distance and t is time. On the first day, the relationship is established as v = ˣ⁄ₜ. On the second day, the speed vₙₑ_w is calculated using the new distance 2.5x and the new time 0.75t:

vₙₑ_w = (2.5x)/(0.75t)

By substituting the initial relation v = ˣ⁄ₜ into the equation, the expression becomes:

vₙₑ_w = (2.5)/(0.75) v

Simplifying the numerical coefficient by multiplying both numerator and denominator by 100 results in:

vₙₑ_w = ²⁵⁰⁄₇₅ v = ¹⁰⁄₃ v

Explanation

The beating distance in a race is defined as the distance the slower runner still has to travel at the exact moment the winner crosses the finish line. To find this, we first determine the speed of the slower runner, Mohan.

Mohan's speed v = (Total Distance)/(Time taken by Mohan) = ¹¹⁷⁰⁄₉₀ = 13 m/s.

Raman reaches the final point in 65 seconds. At this specific point in time, Mohan is still on the track. The time remaining for Mohan to finish the race is the difference between their finishing times: Time Difference = 90 - 65 = 25 seconds.

The distance by which Raman beats Mohan is the distance Mohan covers during this remaining time interval: Beating Distance = Speed of Mohan × Time Difference Beating Distance = 13 × 25 = 325 m.

Explanation

Let the total work be the Least Common Multiple (LCM) of the times taken by A and B, which is 6 units. The efficiency of A is 6 / 3 = 2 units per hour, and the efficiency of B is 6 / 6 = 1 unit per hour. Working on alternate hours, they complete 2 + 1 = 3 units of work every 2 hours. To complete the total work of 6 units, the number of 2-hour cycles required is 6 / 3 = 2. Consequently, the total time taken is 2 × 2 = 4 hours.

Explanation

According to the Intersecting Secants Theorem (Power of a Point), when two secants are drawn to a circle from an external point P, the product of the length of the entire secant segment and its external part is equal for both secants. The relationship is given by the formula: PA · PB = PC · PD

Given the first secant segment PA = 200 mm and the chord length AB = 120 mm, the length of the external segment PB is: PB = PA - AB = 200 - 120 = 80 mm

For the second secant, the length from the external point to the further intersection point is given as PC = 160 mm. Let PD represent the length of the external segment of this secant. Substituting the known values into the theorem: 200 · 80 = 160 · PD 16000 = 160 · PD PD = 100 mm

The length of the chord CD is the difference between the total length of the secant segment PC and its external part PD: CD = PC - PD = 160 - 100 = 60 mm

Explanation

Using the algebraic identity (a + b)² + (a - b)² = 2(a² + b²), we substitute a = sin θ and b = cos θ. The expression simplifies to 2(sin² θ + cos² θ). According to the fundamental trigonometric identity, sin² θ + cos² θ = 1 for any value of θ. Substituting this identity into the simplified expression gives 2(1) = 2. Alternatively, expanding both squares results in (sin² θ + 2 sin θ cos θ + cos² θ) + (sin² θ - 2 sin θ cos θ + cos² θ). The cross-product terms 2 sin θ cos θ and -2 sin θ cos θ cancel each other out, leaving 2 sin² θ + 2 cos² θ, which simplifies to 2(sin² θ + cos² θ) = 2.

Explanation

To calculate the overall profit percentage, we first determine the actual cost price of the quantity sold. The grocer buys sugar at ₹50 per 1000 g, so the cost price of 1 g is ₹0.05. Since he gives 900 g instead of 1000 g, his actual cost price (CP) for the quantity sold is 900 × 0.05 = ₹45.

The grocer sells this 900 g quantity for ₹48. Therefore, the selling price (SP) is ₹48. The profit earned is SP - CP = 48 - 45 = ₹3. The profit percentage is calculated on the actual cost price using the formula:

Profit % = (Profit)/(CP) × 100

Substituting the values, we get:

Profit % = ³⁄₄₅ × 100 = ¹⁄₁₅ × 100 = 6.666...%

Rounding to one decimal place, the approximate profit percentage is 6.7%.

Explanation

A number is divisible by 36 if it satisfies the divisibility rules for both 4 and 9, as 4 and 9 are coprime factors of 36. To be divisible by 4, the last two digits of the number must form a value divisible by 4. In 55512, the last two digits are 12, which is divisible by 4 (12 = 4 × 3). To be divisible by 9, the sum of the digits of the number must be a multiple of 9. For 55512, the sum of the digits is 5 + 5 + 5 + 1 + 2 = 18. Since 18 is divisible by 9 (18 = 9 × 2), the number meets both criteria. Therefore, 55512 is divisible by 36.

Explanation

The length of an arc of a sector is given by the formula l = (θ)/(360°) × π d, where θ is the central angle and d is the diameter of the circle.

Substituting the given values, where θ = 108°, d = 7 cm, and π = ²²⁄₇:

l = ¹⁰⁸⁄₃₆₀ × ²²⁄₇ × 7

l = ¹⁰⁸⁄₃₆₀ × 22

Simplifying the fraction ¹⁰⁸⁄₃₆₀ by dividing both the numerator and the denominator by their greatest common divisor, 36:

(108 ÷ 36)/(360 ÷ 36) = ³⁄₁₀

Now, multiply this by 22:

l = ³⁄₁₀ × 22 = ⁶⁶⁄₁₀ = 6.6 cm

Explanation

The simple interest is calculated using the formula SI = (P × R × T)/(100), where P is the principal amount, R is the rate of interest per annum, and T is the time period in years. Given P = 6,000, R = 8%, and T = 21 months, we first convert the time into years by dividing by 12, which gives T = ²¹⁄₁₂ years.

Substituting these values into the formula: SI = (6000 × 8 × 21)/(100 × 12)

Simplifying the calculation: SI = 60 × 8 × ⁷⁄₄ SI = 60 × 2 × 7 SI = 120 × 7 = 840

Explanation

The direct variation relationship is expressed by the equation A = k(B + 18), where k is the constant of proportionality. Using the initial values provided, 108 = k(36 + 18), which simplifies to 108 = 54k. Dividing both sides by 54 yields k = 2.

To find the value of A when B = 68, substitute the constant k and the new value of B back into the variation formula:

A = 2(68 + 18) A = 2(86) A = 172

Explanation

The volume V of a conical tent is given by the formula V = ⅓ π r² h. Substituting the given radius r = 7 m and vertical height h = 24 m, and taking π = ²²⁄₇, the calculation is V = ⅓ × ²²⁄₇ × 7² × 24. Simplifying this, the 7 in the denominator cancels one 7 from the r² term, and 24 divided by 3 gives 8. This leads to V = 22 × 7 × 8, which calculates to 1232 m³. Since each bag of rice occupies 2 m³, the number of full bags that can fit into the tent is 1232 / 2 = 616.

Explanation

Average speed is the ratio of total distance to total time. First, find the time taken for the first segment using the formula t₁ = (d₁)/(s₁) = ²⁴⁄₄₈ = 0.5 hours. Next, calculate the distance covered in the second segment using d₂ = s₂ × t₂ = 80 × 1.5 = 120 km. The total distance for the entire journey is 24 + 120 = 144 km, and the total time taken is 0.5 + 1.5 = 2 hours. Substituting these values into the average speed formula, we get ¹⁴⁴⁄₂ = 72 km/h.

Explanation

The given expression is (cos A)/(1 - sin A) + (cos A)/(1 + sin A) = 4. Factoring out cos A and simplifying the terms inside the parentheses by taking a common denominator, we get: cos A ( (1 + sin A + 1 - sin A)/((1 - sin A)(1 + sin A)) ) = 4 Using the identity (1 - sin A)(1 + sin A) = 1 - sin² A = cos² A, the equation simplifies to: cos A ( (2)/(cos² A) ) = 4 (2)/(cos A) = 4 cos A = ²⁄₄ = ½ In the interval 0° < A < 90°, the cosine function takes the value ½ when A = 60°.

Explanation

The relationship between the Marked Price (MP) and the Cost Price (CP) can be directly determined using the ratio formula: (MP)/(CP) = (100 + Gain%)/(100 - Discount%)

Given the Cost Price is ₹9,000, the gain percentage is 10, and the discount percentage is 10, we substitute these values into the formula: (MP)/(9000) = (100 + 10)/(100 - 10) (MP)/(9000) = ¹¹⁰⁄₉₀

Simplifying the fraction on the right side: (MP)/(9000) = ¹¹⁄₉

To find the Marked Price, multiply the Cost Price by this ratio: MP = 9000 × ¹¹⁄₉ MP = 1000 × 11 MP = 11,000

Alternatively, the Selling Price (SP) is calculated by adding the 10% gain to the cost price: 9000 + 10% of 9000 = 9900. Since this SP represents the value after a 10% discount on the MP, 90% of MP = 9900. Solving for MP gives 9900 / 0.9 = 11,000.

Explanation

According to the divisibility rule for 3, a number is divisible by 3 if and only if the sum of its digits is divisible by 3. For the number 34783, the sum of the digits is 3 + 4 + 7 + 8 + 3 = 25. Since 25 is not divisible by 3 (25 = 3 × 8 + 1), the number 34783 is not divisible by 3.

Explanation

When two circles touch each other externally at a point P, the angle subtended by the segment of the direct common tangent AB at the point of contact P is always 90°. This can be shown by drawing a common internal tangent at P that intersects AB at a point M. Since the lengths of tangents from an external point to a circle are equal, MA = MP and MB = MP. This implies MA = MB = MP, meaning M is the midpoint of AB and the circumcenter of △ APB. Since the median MP is half the length of the side AB, △ APB must be a right-angled triangle with ∠ APB = 90°.

In △ APB, the sum of all interior angles is 180°. Using the given value ∠ PAB = 65° and the derived property ∠ APB = 90°, we calculate the third angle as follows: ∠ ABP = 180° - (∠ APB + ∠ PAB) ∠ ABP = 180° - (90° + 65°) ∠ ABP = 180° - 155° = 25°

Explanation

The number of distinct meeting points on a circular track for two persons moving in opposite directions is determined by the simplified ratio of their speeds. Let the speeds be v₁ and v₂. We simplify the ratio v₁ : v₂ to x : y such that x and y are co-prime integers.

Given v₁ = 20 m/s and v₂ = 30 m/s, the ratio is 20 : 30 = 2 : 3. Here, x = 2 and y = 3. For motion in opposite directions, the number of distinct meeting points is calculated as x + y. Substituting the simplified values, we get 2 + 3 = 5.

This can also be understood through relative speed and time. The relative speed of the two persons is v₁ + v₂ = 20 + 30 = 50 m/s. The time taken for each meeting is the track length divided by the relative speed: 100 / 50 = 2 seconds. The time required for the two persons to return to the starting point simultaneously is the least common multiple of their lap times. The lap times are 100 / 20 = 5 seconds and 100 / 30 = 10/3 seconds. The LCM of 5 and 10/3 is 10 seconds. Within this cycle of 10 seconds, they meet 10 / 2 = 5 times. Because they return to the starting point at the end of the 10-second cycle, these 5 meeting points are the only distinct locations where they cross before the pattern repeats.

Explanation

The algebraic identity for the given expression is a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). To find the value of (ab + bc + ca), we use the square of a trinomial identity:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values a + b + c = 12 and a² + b² + c² = 50:

12² = 50 + 2(ab + bc + ca) 144 = 50 + 2(ab + bc + ca) 144 - 50 = 2(ab + bc + ca) 94 = 2(ab + bc + ca) ab + bc + ca = 47

Now, substitute these values into the cubic identity:

a³ + b³ + c³ - 3abc = 12 × (50 - 47) a³ + b³ + c³ - 3abc = 12 × 3 a³ + b³ + c³ - 3abc = 36

Explanation

The Marked Price (MP) of the saree is calculated by applying the 16% profit markup to the Cost Price (CP) of ₹500. Using the formula MP = CP × (1 + (Profit%)/(100)), we get MP = 500 × (1 + ¹⁶⁄₁₀₀) = 500 × 1.16 = 580.

The Selling Price (SP) is obtained after applying a discount of x% on the MP. The relationship is SP = MP × (1 - ˣ⁄₁₀₀). Substituting the known values, we have 493 = 580 × (1 - ˣ⁄₁₀₀). Dividing both sides by 580 gives 1 - ˣ⁄₁₀₀ = ⁴⁹³⁄₅₈₀ = 0.85. Solving for x results in ˣ⁄₁₀₀ = 1 - 0.85 = 0.15, which means x = 15.

Explanation

The total weight of any group is determined by the formula Total Weight = Number of Students × Average Weight.

Initially, the total weight of the 32 students is 32 × 18.5 = 592 kg.

After 4 students leave the group, 28 students remain (32 - 4 = 28). The new total weight for these 28 students is 28 × 15.5 = 434 kg.

The combined weight of the 4 students who left is the difference between the initial total weight and the remaining total weight: 592 - 434 = 158 kg.

To find the average weight of the 4 students who left, divide their total weight by the number of students: ¹⁵⁸⁄₄ = 39.5 kg.

Explanation

The expression tan² θ + tan⁴ θ can be simplified by factoring out tan² θ, resulting in tan² θ (1 + tan² θ). Using the fundamental trigonometric identity 1 + tan² θ = sec² θ, we can substitute sec² θ for the term in the parentheses and (sec² θ - 1) for the initial tan² θ. The expression becomes:

(sec² θ - 1) sec² θ = sec⁴ θ - sec² θ

Given that sec θ = ⁴⁄₃, we calculate the squared and fourth-power values: sec² θ = (⁴⁄₃)² = ¹⁶⁄₉ sec⁴ θ = (¹⁶⁄₉)² = ²⁵⁶⁄₈₁

Substituting these values into the derived formula: ²⁵⁶⁄₈₁ - ¹⁶⁄₉ = ²⁵⁶⁄₈₁ - ¹⁴⁴⁄₈₁ = ¹¹²⁄₈₁

Explanation

The expression is simplified by applying the BODMAS rule to solve the operations within the parentheses and brackets first.

The first group in the numerator is (12 + 5 - 48/16 + 71). Evaluating the division gives 12 + 5 - 3 + 71, which sums to 85.

The second group is (72/36 + 6 × 7)/11. Evaluating the inner terms gives (2 + 42)/11, which simplifies to 44/11 = 4.

The third group is [(51 + 4 - 13) + (13 - 12 × 7)]. The first set of inner parentheses yields 55 - 13 = 42. The second set yields 13 - 84 = -71. Adding these together gives 42 + (-71) = -29.

Now, combining these parts into the numerator structure: 85 + (4 × -29). Performing the multiplication first gives 85 - 116, which equals -31.

Placing this numerator over the given denominator results in (-31)/232.

Explanation

To find the value of cot 15° using the identity cot (A - B) = (cot A cot B + 1)/(cot B - cot A), we substitute A = 45° and B = 30°, as 45° - 30° = 15°. Utilizing the standard trigonometric values cot 45° = 1 and cot 30° = √3, the formula yields:

cot 15° = cot (45° - 30°) = (cot 45° cot 30° + 1)/(cot 30° - cot 45°)

cot 15° = (1)(√3) + 1{√3 - 1} = √3 + 1{√3 - 1}

To simplify the expression, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate √3 + 1:

cot 15° = (√3 + 1)(√3 + 1){(√3 - 1)(√3 + 1)}

Applying the algebraic identities (a + b)² = a² + 2ab + b² and (a - b)(a + b) = a² - b²:

cot 15° = 3 + 2√3 + 1{3 - 1} = 4 + 2√3{2}

Dividing the terms in the numerator by the denominator results in the final value:

cot 15° = 2 + √3

Explanation

To find a relation independent of α, we use the trigonometric identity sec² α - tan² α = 1.

From the first given equation, P cos α = 3, we can write cos α = (3)/(P). Since sec α = (1)/(cos α), it follows that sec α = (P)/(3).

From the second given equation, 4 tan α = Q, we can write tan α = (Q)/(4).

Substituting these expressions into the trigonometric identity:

((P)/(3))² - ((Q)/(4))² = 1

Squaring the terms yields:

(P²)/(9) - (Q²)/(16) = 1

This equation represents the relationship between P and Q that is independent of α.