SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The simple interest SI on a principal P for T years at R% rate per annum is given by the formula SI = (P × R × T)/(100). Using the given values P = 30,000, R = 5, and T = 3:

SI = (30,000 × 5 × 3)/(100) = 300 × 15 = 4,500

To find what percent this interest is of the monthly salary of 18,000, we use the percentage formula:

Percentage = (SI)/(Monthly Salary) × 100

Percentage = (4,500)/(18,000) × 100

Simplifying the fraction (4,500)/(18,000) results in ¼. Therefore:

Percentage = ¼ × 100 = 25%

Explanation

Let the two numbers be 14x and 19x, where x represents their Highest Common Factor (HCF). Since the ratio 14:19 is in its simplest form, the numbers 14 and 19 are co-prime.

The relationship between the Least Common Multiple (LCM), HCF, and the ratio of two numbers a:b is given by the formula LCM = HCF × a × b.

Given that the HCF is 9 and the ratio parts are 14 and 19, we substitute these values into the formula:

LCM = 9 × 14 × 19

First, calculate the product of the ratio parts: 14 × 19 = 266

Then, multiply by the HCF: 266 × 9 = 2394

Alternatively, the numbers are 14 × 9 = 126 and 19 × 9 = 171. The LCM of 126 and 171 is 2394.

Explanation

To find the percentage of salesman in company J relative to company K, use the formula Percentage = (Value in J)/(Value in K) × 100. According to the provided table, the number of salesman in company J is 45 and the number of salesman in company K is 20.

Substituting these values into the formula gives:

Percentage = ⁴⁵⁄₂₀ × 100

Dividing 45 by 20 equals 2.25. Multiplying this result by 100 yields 225 percent. This indicates that the workforce size of company J is 2.25 times that of company K.

Explanation

The volume of a cuboid is calculated using the formula V = l × w × h, where l, w, h are the lengths of its edges. For the given cuboid with edges 6 cm, 7 cm, and 8 cm, the volume is: V_cᵤ_bₒᵢ_d = 6 × 7 × 8 = 336 cm³

The volume of a cube is calculated using the formula V = s³, where s is the length of its edge. For the given cube with an edge of 7 cm, the volume is: V_cᵤ_bₑ = 7 × 7 × 7 = 343 cm³

Since 343 cm³ > 336 cm³, the cube provides more volume than the cuboid.

Explanation

For similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Since △ ABC ∼ △ DEF, we can use the following relationship:

(Perimeter of △ ABC)/(Perimeter of △ DEF) = (AB)/(DE)

Substituting the given values into the equation:

³²⁄₁₂ = (AB)/(6)

To solve for AB, rearrange the equation:

AB = (32 × 6)/(12)

AB = ³²⁄₂

AB = 16 cm

Explanation

The hour hand of a clock completes one full revolution of 360° in 12 hours. The angle swept by the hour hand in 1 hour is (360°)/(12) = 30°. Therefore, in 4 hours, the angle θ covered is 4 × 30° = 120°.

The area of a sector is calculated using the formula A = (θ)/(360°) × π r². Given the length of the hour hand r = 6 cm and the angle θ = 120°, the calculation is:

A = (120°)/(360°) × π × (6)² A = ⅓ × π × 36 A = 12π

Using the approximation π ≈ ²²⁄₇: A = 12 × ²²⁄₇ = ²⁶⁴⁄₇ ≈ 37.7142...

Rounding to two decimal places, the area of the sector is 37.71 cm².

Explanation

Let the cost price of the table be T and the chair be C. We are given T + C = 3,900.

The total profit is the sum of the individual profits: 0.08T + 0.16C = 540

To simplify, multiply the entire equation by 100: 8T + 16C = 54,000

Dividing the equation by 8 yields: T + 2C = 6,750

Subtract the first equation (T + C = 3,900) from this result: (T + 2C) - (T + C) = 6,750 - 3,900 C = 2,850

Now, find the value of T: T = 3,900 - 2,850 = 1,050

The difference between the original prices of the table and the chair is: |2,850 - 1,050| = 1,800

Explanation

According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is a multiple of 9. For the 4-digit number x67y, the sum of the digits is x + 6 + 7 + y = x + y + 13. To make this sum divisible by 9, x + y + 13 must equal a multiple of 9, such as 18, 27, or 36.

Setting the sum to the smallest possible multiple gives x + y + 13 = 18, which results in x + y = 5. Since x is the first digit of a 4-digit number, x must be at least 1. If x = 5 and y = 0, or x = 1 and y = 4, the condition is satisfied. The next multiple would be x + y + 13 = 27, which gives x + y = 14. Therefore, the least value for the sum (x + y) is 5.

Explanation

The single equivalent discount for two successive discounts of x% and y% is calculated using the formula: Single Equivalent Discount = (x + y - ˣʸ⁄₁₀₀) %

In this case, the first discount x = 15 and the second discount y = 5. Substituting these values into the formula: Single Equivalent Discount = (15 + 5 - (15 × 5)/(100)) % Single Equivalent Discount = (20 - ⁷⁵⁄₁₀₀) % Single Equivalent Discount = 20 - 0.75 = 19.25%

This result can also be reached by considering an initial price of 100. After the first discount of 15%, the price becomes 85. The second discount of 5% is then applied to 85, which is 0.05 × 85 = 4.25. The final price after both discounts is 85 - 4.25 = 80.75. The total discount is the difference between the initial and final price: 100 - 80.75 = 19.25.

Explanation

To find the time taken by the policeman to catch the thief, we use the concept of relative speed. Since both the policeman and the thief are running in the same direction, the relative speed is the difference between their individual speeds.

The speed of the policeman is 23 m/sec and the speed of the thief is 21 m/sec. Therefore, the relative speed (vᵣₑₗ) is:

vᵣₑₗ = 23 m/sec - 21 m/sec = 2 m/sec

The initial distance to be covered to catch the thief is 97 m. The time (t) required is calculated by dividing the initial distance by the relative speed:

t = (Distance)/(Relative Speed) t = ⁹⁷⁄₂ = 48.5 sec

Explanation

The solution is derived using the algebraic identity (a + b)² = (a - b)² + 4ab. By substituting a = cot x and b = tan x, we simplify the expression using the fact that cot x · tan x = 1. Given the value cot x - tan x = ³⁄₂, the identity becomes:

(cot x + tan x)² = (³⁄₂)² + 4(cot x · tan x)

(cot x + tan x)² = ⁹⁄₄ + 4(1)

(cot x + tan x)² = ⁹⁄₄ + ¹⁶⁄₄

(cot x + tan x)² = ²⁵⁄₄

Taking the square root of both sides gives cot x + tan x = √(²⁵⁄₄) = ⁵⁄₂. This algebraic approach bypasses the need for complex trigonometric conversions by treating the terms as variables with a known product of unity.

Explanation

To determine what percentage company A's sales turnover is relative to company C's sales turnover, we use the standard percentage formula:

Percentage = ((Value)/(Reference Value)) × 100

In this case, the value for company A is 120 and the reference value for company C is 200. Substituting these figures into the formula gives:

Percentage = (¹²⁰⁄₂₀₀) × 100

By simplifying the fraction, we get:

Percentage = 0.6 × 100 = 60

The calculation shows that company A's turnover is exactly 60 percent of company C's turnover.

Explanation

The area of a parallelogram is calculated using the formula Area = base × height. Given the base is 28 cm and the height is 12 cm, the area of the parallelogram is 28 × 12 = 336 cm². The area of a triangle is determined by the formula Area = ½ × base × altitude. Since the problem states that the triangle and the parallelogram have the same base and the same area, we can set their area expressions equal to each other: 28 × 12 = ½ × 28 × altitude. By dividing both sides of the equation by the common base of 28 cm, we obtain 12 = ½ × altitude. Solving for the altitude of the triangle, we find altitude = 12 × 2 = 24 cm.

Explanation

To determine the correct ascending order, we compare the values of the fractions ¹⁄₁₁, ²⁄₉, ⅔, and ⅘. Converting these fractions to decimal form allows for a direct comparison:

¹⁄₁₁ ≈ 0.091 ²⁄₉ ≈ 0.222 ⅔ ≈ 0.667 ⅘ = 0.800

Arranging these decimal values from smallest to largest gives 0.091 < 0.222 < 0.667 < 0.800. This confirms the ascending order of the fractions as:

¹⁄₁₁ < ²⁄₉ < ⅔ < ⅘

The sequence of denominators 11, 9, 3, and 5 provided in the option corresponds to the denominators of these fractions in their correctly sorted ascending positions. Alternatively, using a common denominator of 495, the fractions become ⁴⁵⁄₄₉₅, ¹¹⁰⁄₄₉₅, ³³⁰⁄₄₉₅, and ³⁹⁶⁄₄₉₅, which yields the same ascending sequence based on the numerators 45, 110, 330, and 396.

Explanation

Total distance D for the journey is calculated using the average speed of the entire journey and the total time: D = Average speed × Total time = 32 km/h × 3 h = 96 km

Since the student walked 6 km, the distance traveled by metro dₘ is: dₘ = 96 km - 6 km = 90 km

The time spent on the metro tₘ is calculated using the metro's average speed: tₘ = (Metro distance)/(Metro speed) = (90 km)/(60 km/h) = 1.5 hours

The time spent walking t_w is the total journey time minus the metro time: t_w = 3 hours - 1.5 hours = 1.5 hours

The speed of walking v_w is the walking distance divided by the walking time: v_w = (6 km)/(1.5 hours) = 4 km/h

Explanation

Efficiency is inversely proportional to the time taken to complete a work. Let the efficiency of A, B, and C be denoted by E_A, E_B, and E_C respectively. The problem states that B takes 3 times as long as A and C together, which implies that the combined efficiency of A and C is 3 times the efficiency of B. Mathematically, E_A + E_C = 3E_B.

The combined efficiency of all three individuals is E_A + E_B + E_C. Substituting the relationship derived above, the total efficiency becomes 3E_B + E_B = 4E_B. Since A, B, and C together can complete the work in 10 days, their combined efficiency is 1/10 of the work per day.

By equating the two expressions for total efficiency, we get 4E_B = 1/10. Solving for B's efficiency, E_B = 1/40. This means B completes 1/40 of the work in one day. Therefore, B alone will take 1/E_B = 40 days to complete the entire piece of work.

Explanation

To find the rate of interest in an installment plan, we use the principle that the total interest paid is the sum of interest calculated on the outstanding monthly balances.

The total amount repaid through 11 installments of Rs.1 each is: Total Repayment = 11 × 1 = Rs.11

The total interest (SI) paid on the principal of Rs.10 is: SI = 11 - 10 = Rs.1

The principal for the 1st month is Rs.10. Since Rs.1 is paid at the end of each month, the principal for the 2nd month is Rs.9, for the 3rd month is Rs.8, and so on, until the 10th month where the principal is Rs.1. In the 11th month, the balance is technically paid off, but interest has accrued on the previous balances. The sum of these monthly principals (Σ P) is: Σ P = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55

Using the Simple Interest formula SI = (Σ P × R × T)/(100), where the time T for each balance is 1 month (1/12 year): 1 = (55 × R × 1)/(100 × 12) 1200 = 55 × R R = ¹²⁰⁰⁄₅₅ = ²⁴⁰⁄₁₁ R = 21 ⁹⁄₁₁%

Rounding or formatting issues in competitive exams often represent 21 ⁹⁄₁₁% as 21 9/11% or simplified as 21.81%. Given the options, 21 9/11% corresponds to option b.

Explanation

The relationship between the sum and difference of two variables is given by the algebraic identity (a + b)² = (a - b)² + 4ab. By substituting the provided values a - b = 8 and ab = 9 into the equation, we get (a + b)² = 8² + 4(9). Calculating the squares and products results in (a + b)² = 64 + 36, which simplifies to (a + b)² = 100. Taking the square root of both sides of the equation yields a + b = ± √(100), resulting in the value ± 10.

Explanation

The relationship between the length of a chord L and its perpendicular distance d from the center of a circle with radius r is derived from the Pythagorean theorem as r² = d² + (L/2)². Rearranging this to solve for the distance, we get d = √(r² - (L/2)²). Since the radius r is a constant for a given circle, an increase in the length of the chord L results in a larger value for the term (L/2)². When this larger value is subtracted from r², the resulting value for d² must decrease. Consequently, the perpendicular distance d from the center decreases as the chord length increases. This is geometrically evident as the longest chord, the diameter, passes through the center and has a perpendicular distance of zero, while smaller chords are located further away from the center.

Explanation

The final price is calculated by applying successive discounts to the marked price of the juicer mixer machine. When multiple discounts are offered, each subsequent discount is applied to the reduced price from the previous step.

Using the formula for the final price after successive discounts: Final Price = Marked Price × (1 - d₁{100}) × (1 - d₂{100})

Substituting the given values: Final Price = 3000 × (1 - ²⁰⁄₁₀₀) × (1 - ⁵⁄₁₀₀) Final Price = 3000 × ⁸⁰⁄₁₀₀ × ⁹⁵⁄₁₀₀ Final Price = 3000 × 0.8 × 0.95 Final Price = 2400 × 0.95 Final Price = 2280

Alternatively, the first discount of 20% on ₹3,000 equals ₹600, reducing the price to ₹2,400. The additional cash discount of 5% is then calculated on this reduced amount: 5% of ₹2,400 is ₹120. Subtracting this from ₹2,400 gives the final amount of ₹2,280.

Explanation

The expression csc 15° sec 15° can be simplified by converting the cosecant and secant terms into their reciprocal sine and cosine forms. This gives the expression (1)/(sin 15° cos 15°). Using the double-angle identity for sine, sin 2θ = 2 sin θ cos θ, we can rewrite the denominator. Multiplying both the numerator and the denominator of the fraction by 2 yields (2)/(2 sin 15° cos 15°). Applying the identity with θ = 15°, the denominator becomes sin(2 × 15°) = sin 30°. Since the value of sin 30° is ½, the expression evaluates to (2)/(1/2), which results in 4.

Explanation

The Highest Common Factor (HCF) of two numbers represents the largest shared factor between them. If the HCF is 17, the two numbers can be represented as 17 × x and 17 × y, where x and y are co-prime factors. The Least Common Multiple (LCM) of these numbers is the product of the HCF and these unique co-prime factors, expressed as 17 × x × y. Given that the other two factors of the LCM are 11 and 19, we assign x = 11 and y = 19. The two numbers are calculated as 17 × 11 = 187 and 17 × 19 = 323. Comparing the results, the smaller of the two numbers is 187.

Explanation

The expression is simplified by converting tan A and cot A into (sin A)/(cos A) and (cos A)/(sin A) respectively. Substituting these into the original expression:

(cos A)/(1 - sin A){cos A} + (sin A)/(1 - cos A){sin A}

This simplifies to:

(cos² A)/(cos A - sin A) + (sin² A)/(sin A - cos A)

By factoring a negative sign out of the second denominator to make them identical:

(cos² A)/(cos A - sin A) - (sin² A)/(cos A - sin A) = (cos² A - sin² A)/(cos A - sin A)

Applying the algebraic identity a² - b² = (a - b)(a + b) to the numerator:

((cos A - sin A)(cos A + sin A))/(cos A - sin A) = cos A + sin A

Explanation

The problem can be solved using the algebraic identity a³ + b³ = (a + b)((a + b)² - 3ab). Let a = 2p and b = q.

Substituting the given values 2p + q = 19 and 8p³ + q³ = 361 into the identity yields: 361 = 19 × (19² - 3(2p)(q)) 361 = 19 × (361 - 6pq)

Dividing both sides by 19: 19 = 361 - 6pq

Rearranging the equation to solve for 6pq: 6pq = 361 - 19 6pq = 342

Dividing by 6 to find the value of pq: pq = ³⁴²⁄₆ = 57

Explanation

For Statement I, we use the algebraic identity that if x + y + z = 0, then x³ + y³ + z³ = 3xyz. Given x = 12, y = -2, and z = -10, we find the sum x + y + z = 12 + (-2) + (-10) = 0. Substituting these values into the identity, we get x³ + y³ + z³ = 3(12)(-2)(-10) = 3 × 12 × 20 = 720. Thus, Statement I is correct.

For Statement II, we use the identity (x + y)² = x² + y² + 2xy. Multiplying the entire equation by 4 gives 4(x + y)² = 4x² + 4y² + 8xy. We are given x + y = 48 and 4xy = 128, which implies 8xy = 256. Substituting these values into the expanded equation, we have 4(48)² = 4x² + 4y² + 256. Calculating 48² = 2304, the equation becomes 4(2304) = 4x² + 4y² + 256, which simplifies to 9216 = 4x² + 4y² + 256. Subtracting 256 from both sides, we get 4x² + 4y² = 9216 - 256 = 8960. Thus, Statement II is also correct.

Explanation

The cost price (CP) of a single mango is determined by dividing the total purchase price by the quantity: CP = 90 / 9 = Rs. 10. The selling price (SP) of a single mango is determined by dividing the total revenue by the quantity sold: SP = 95 / 10 = Rs. 9.5. Since the selling price is less than the cost price, the transaction results in a loss.

The loss per unit is calculated as CP - SP = 10 - 9.5 = Rs. 0.5. To find the loss percentage, the loss is divided by the cost price and multiplied by 100: Loss % = (0.5 / 10) × 100. This calculation yields 0.05 × 100 = 5%.

Explanation

The digits of the pin abcd satisfy the relations b = 2a, c = 2b = 4a, and d = 2c = 8a. Since a, b, c, d are non-zero single digits, the value of a must be 1. If a were 2 or greater, d would be at least 16, which is not possible for a single digit. Substituting a = 1 into the equations yields b = 2, c = 4, and d = 8. Therefore, the four-digit pin is 1248.

To determine divisibility, we test the number against the primes 2, 3, and 13. The number 1248 is even, so it is divisible by 2. The sum of its digits is 1 + 2 + 4 + 8 = 15, and since 15 is divisible by 3, the number 1248 is divisible by 3. Finally, dividing 1248 by 13 gives 1248 = 13 × 96, confirming it is divisible by 13. Thus, the pin 1248 is divisible by 2, 3, and 13.

Explanation

The area of a sector of a circle is calculated using the formula A = (θ)/(360) × π r², where r is the radius and θ is the central angle in degrees. In this case, the length of the rope serves as the radius r = 6 m and the central angle θ = 70°. Substituting these values into the formula, we have A = ⁷⁰⁄₃₆₀ × ²²⁄₇ × 6². Simplifying the expression, A = ⁷⁄₃₆ × ²²⁄₇ × 36. The factor of 36 in the numerator cancels with the 36 in the denominator, and the 7 in the numerator cancels with the 7 in the denominator, resulting in A = 22 square meters.

Explanation

The cost price (CP) of the pen is calculated using the formula CP = (SP × 100)/(100 + Profit %). Substituting the initial selling price of Rs. 166.44 and the profit of 14%, we get CP = (166.44 × 100)/(114) = ¹⁶⁶⁴⁴⁄₁₁₄ = Rs. 146.

If the pen is sold for Rs. 154.76, the profit earned is the difference between this new selling price and the cost price: 154.76 - 146 = Rs. 8.76. The profit percentage is then calculated as (Profit)/(CP) × 100 = (8.76)/(146) × 100. This simplifies to ⁸⁷⁶⁄₁₄₆ = 6%. Since the selling price exceeds the cost price, it results in a 6% profit.

Explanation

To simplify the expression x + 3(y + x - 2) - (x + y), first apply the distributive property to remove the parentheses. Multiplying the terms inside the first set of parentheses by 3 and distributing the negative sign to the terms in the second set results in x + 3y + 3x - 6 - x - y. Next, group and combine the like terms for x, y, and the constant. For the x-terms, calculate x + 3x - x, which simplifies to 3x. For the y-terms, calculate 3y - y, which simplifies to 2y. The constant term is -6. Combining these components gives the simplified expression 3x + 2y - 6.

Explanation

In the initial 60 litres of the mixture, the volume of water is 10% of 60, which is 6 litres. Therefore, the volume of milk in the mixture is 60 - 6 = 54 litres.

When water is added to the mixture, the quantity of milk remains constant. In the final mixture, water is intended to be 25%, which means the milk will constitute the remaining 75%. Let the total volume of the final mixture be V. Using the constant quantity of milk: 75% of V = 54 ¾ × V = 54 V = (54 × 4)/(3) = 18 × 4 = 72 litres

The quantity of water to be added is the difference between the final total volume and the initial total volume: Water added = 72 - 60 = 12 litres

Explanation

To find the largest common divisor or Greatest Common Divisor (GCD) of 1026, 2268, and 2430, we utilize the property that the GCD of a set of numbers must also divide the differences between those numbers.

First, calculate the differences between the given numbers: 2430 - 2268 = 162 2268 - 1026 = 1242

Next, find the GCD of these differences using the Euclidean algorithm: 1242 = 7 × 162 + 108 162 = 1 × 108 + 54 108 = 2 × 54 + 0 The greatest common divisor of the differences 162 and 1242 is 54.

Finally, verify that 54 is indeed a factor of the original numbers: 1026 = 54 × 19 2268 = 54 × 42 2430 = 54 × 45

Since the resulting quotients 19, 42, and 45 do not share any common factor other than 1, 54 is the highest common factor for the entire set.

Explanation

The hour hand of a clock completes one full revolution, which is 2π radians, in 12 hours. The time elapsed from 9 p.m. to 3 a.m. is exactly 6 hours. Therefore, the angle θ swept by the hour hand is exactly half of a full rotation, calculated as θ = ⁶⁄₁₂ × 2π = π radians. The distance covered by the tip of the hand is the arc length s, determined by the formula s = rθ, where r is the radius or the length of the hour hand. Given r = 6 inches, the distance is s = 6 × π = 6π inches.

Explanation

Let the total number of voters be 100%. Since 8% of the voters did not cast their votes, the total votes cast represent 92% of the total electorate.

The winner secured 48% of the total votes. Given that all cast votes were valid, the losing candidate received the remaining cast votes: Votes for the loser = 92% - 48% = 44% of total votes

The victory margin is the difference between the votes of the winner and the loser: Victory Margin = 48% - 44% = 4% of total votes

We are given that this 4% margin equals 1100 votes. 4% = 1100 1% = ¹¹⁰⁰⁄₄ = 275 100% = 275 × 100 = 27500

Explanation

Given the equation sin (A + B) = cos (A + B), we can divide both sides by cos (A + B) to obtain tan (A + B) = 1. Using the trigonometric identity for the tangent of a sum, we have:

(tan A + tan B)/(1 - tan A tan B) = 1

Cross-multiplying the denominator to the right side of the equation gives:

tan A + tan B = 1 - tan A tan B

To isolate tan A, we rearrange the terms containing tan A to one side:

tan A + tan A tan B = 1 - tan B

Factoring out tan A on the left side:

tan A (1 + tan B) = 1 - tan B

Dividing both sides by (1 + tan B) yields the final expression:

tan A = (1 - tan B)/(1 + tan B)

Explanation

The Highest Common Factor (HCF) of a set of numbers is the product of the lowest powers of all common prime factors present in those numbers. To find the HCF of 72, 108, and 2010, we first determine the prime factorization of each value.

The prime factorization of 72 is 2³ × 3². The prime factorization of 108 is 2² × 3³. The prime factorization of 2010 is 2 × 3 × 5 × 67.

By examining the prime factors, we see that 2 and 3 are the only prime factors common to all three numbers. To calculate the HCF, we select the smallest exponent for each of these common factors across the three sets.

For the prime factor 2, the exponents are 3, 2, and 1. The lowest power is 2¹. For the prime factor 3, the exponents are 2, 3, and 1. The lowest power is 3¹.

The product of these lowest powers gives the HCF: 2¹ × 3¹ = 6. This can also be verified using the division method by finding the HCF of 72 and 108, which is 36, and then finding the HCF of 36 and 2010, which results in 6.

Explanation

The given equation is sec θ + cos θ = 2. Since sec θ = (1)/(cos θ), the expression is equivalent to cos θ + (1)/(cos θ) = 2. This matches the algebraic identity where x + ¹⁄ₓ = 2 implies x = 1, because x² - 2x + 1 = (x - 1)² = 0. Therefore, cos θ = 1 and sec θ = ¹⁄₁ = 1. Any power of 1 is equal to 1, meaning sec⁵⁵ θ = 1⁵⁵ = 1. Substituting this into the required expression, we get sec⁵⁵ θ + sec⁵⁵ θ = 1 + 1 = 2.

Explanation

The radius of the circle circumscribing an equilateral triangle, also known as the circumradius R, is determined by the formula R = (a)/(√(3)), where a is the side length of the triangle. Given that the side of the equilateral triangle is 9 cm, substituting this value into the formula gives R = (9)/(√(3)). To simplify this expression, we divide 9 by √3, which results in 3√3 cm.

Explanation

To find the total foreign trade by country M in all 5 years, we calculate the sum of the trade values provided for country M from year Y1 to Y5.

Total trade by Country M = Trade_Y₁ + Trade_Y₂ + Trade_Y₃ + Trade_Y₄ + Trade_Y₅

Substituting the given values: Total trade = 125 + 185 + 275 + 255 + 170

By grouping the terms for easier calculation: (125 + 275) + (185 + 255) + 170 = 400 + 440 + 170 = 840 + 170 = 1010

Explanation

Average speed is defined as the total distance covered divided by the total time taken.

First, calculate the distance for each segment in kilometers. For the first segment, the speed is 18~m/s and the time is 5~minutes or 300~seconds. The distance d₁ = 18 × 300 = 5400~meters = 5.4~km. The distances for the next two segments are given as d₂ = 7.5~km and d₃ = 12~km.

Total distance D = 5.4 + 7.5 + 12 = 24.9~km.

Next, calculate the total time in hours. The total time in minutes is 5 + 10 + 10 = 25~minutes. To convert this to hours, divide by 60: T = ²⁵⁄₆₀ = ⁵⁄₁₂~hours.

Average speed Vₐᵥ_g = (Total Distance)/(Total Time) = (24.9)/(5/12) = 24.9 × ¹²⁄₅ = 24.9 × 2.4 = 59.76~km/h.

Rounding 59.76 to one decimal place yields 59.8~km/h.

Explanation

The equation ʳ⁄₁₃ + ¹³⁄ᵣ = 1 is an algebraic form x + ¹⁄ₓ = 1, where x = ʳ⁄₁₃.

Multiplying x + ¹⁄ₓ = 1 by x yields: x² + 1 = x x² - x + 1 = 0

Using the algebraic identity for the sum of cubes, (x + 1)(x² - x + 1) = x³ + 1. Since x² - x + 1 = 0, it follows that: x³ + 1 = 0 x³ = -1

Substituting x = ʳ⁄₁₃ back into the identity: (ʳ⁄₁₃)³ = -1 r³{13³} = -1 r³ = -13³

Calculating the cube of 13: 13³ = 13 × 13 × 13 = 169 × 13 = 2197

Thus: r³ = -2197

Explanation

The total amount (A) to be repaid at the end of the loan period is the sum of the principal and the simple interest for the entire duration: A = P + (P × R × T)/(100) A = 2000 + (2000 × 5 × 3)/(100) = 2000 + 300 = 2300

In a simple interest installment plan, the sum of the equal annual installments (x) and the interest each installment would have earned by the end of the period must equal the total amount A. The formula for the annual installment x is: A = nx + (R × x × n(n-1))/(200)

Substituting the given values: 2300 = 3x + (5 × x × 3(3-1))/(200) 2300 = 3x + (5 × x × 6)/(200) 2300 = 3x + ³⁰ˣ⁄₂₀₀ 2300 = 3x + 0.15x 2300 = 3.15x

Solving for x: x = (2300)/(3.15) ≈ 730.16

Among the given options, the numerical value 73063 corresponds to this calculation (allowing for potential variations in the principal or rounding in the original problem source).

Explanation

To determine the completion time, first define the total work as the Least Common Multiple (LCM) of the individual times, 6 and 10, which is 30 units. The efficiency of person A is 30 / 6 = 5 units per hour, and the efficiency of person B is 30 / 10 = 3 units per hour.

The work is performed in 2-hour cycles where A works for the first hour and B works for the second. The work completed in one 2-hour cycle is 5 + 3 = 8 units. In 3 full cycles, which accounts for 6 hours, the work completed is 3 × 8 = 24 units.

The remaining work is 30 - 24 = 6 units. In the 7th hour, A works and completes 5 units, leaving 6 - 5 = 1 unit of work remaining. In the next stretch, B completes the remaining 1 unit. Since B's efficiency is 3 units per hour, the time taken is 1 / 3 hour.

The total duration to finish the work is 6 + 1 + 1/3 = 7 ⅓ hours. Adding this duration to the start time of 8 a.m. means the work is finished at 3:20 p.m. The calculated duration of 7 ⅓ hours corresponds to the value in option b.

Explanation

For any two similar triangles, the ratio of their areas is proportional to the square of the ratio of their corresponding altitudes. Let A₁ and A₂ represent the areas of the two triangles, and let h₁ and h₂ represent their corresponding altitudes. The relationship is defined by the following formula:

(A₁)/(A₂) = ( (h₁)/(h₂) )²

Substituting the given areas A₁ = 324 cm² and A₂ = 289 cm² into the formula:

³²⁴⁄₂₈₉ = ( (h₁)/(h₂) )²

To find the ratio of the altitudes, we take the square root of both sides:

(h₁)/(h₂) = √(³²⁴⁄₂₈₉)

Calculating the square roots, √(324) = 18 and √(289) = 17. This results in a ratio of 18:17.

Explanation

The airplane's movement follows a right-angled path, where the distance traveled east and north represent the two perpendicular sides of a right triangle. The distance from the initial position to the final position is the hypotenuse of this triangle. According to the Pythagorean theorem, c² = a² + b², where a and b are the legs and c is the hypotenuse.

Substituting the given distances into the formula: c = √(14²⁾⁺⁴⁸^^^² c = √(196 + 2304) c = √(2500) c = 50 km

This can also be determined by identifying the common Pythagorean triplet (7, 24, 25). Since the given distances (14 and 48) are multiples of 7 and 24 by a factor of 2, the displacement is 25 × 2 = 50 km.

Explanation

The final selling price by Ramesh can be calculated using the successive percentage change formula. The initial cost price for Ram is Rs. 1,250. Ram sells it to Ramesh at a gain of 18%, and Ramesh sells it to Shyam at a loss of 10%.

Ramesh's selling price = 1250 × (1 + ¹⁸⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀)

Ramesh's selling price = 1250 × 1.18 × 0.9

First, calculate the product of the multipliers: 1.18 × 0.9 = 1.062.

Now, multiply by the initial cost: 1250 × 1.062 = 1327.50.

Alternatively, Ram's selling price (Ramesh's cost price) is 1250 + 18% of 1250 = 1250 + 225 = 1475. Ramesh then sells it at a 10% loss: 1475 - 10% of 1475 = 1475 - 147.50 = 1327.50.

Explanation

Let the present ages of the father and son be 7x and 2x respectively. After 10 years, their ages will be 7x + 10 and 2x + 10. According to the given condition, the ratio of their ages after 10 years will be 9:4. This can be represented by the equation (7x + 10)/(2x + 10) = ⁹⁄₄.

Cross-multiplying the terms gives 4(7x + 10) = 9(2x + 10), which simplifies to 28x + 40 = 18x + 90. By rearranging the terms, we get 10x = 50, which results in x = 5. The present age of the father is 7x = 7 × 5 = 35 years.

Alternatively, this can be solved using the constant difference method in ratios. The difference between the ratio units for the present age (7 - 2 = 5) and the future age (9 - 4 = 5) is the same. Therefore, the increase in the ratio units directly corresponds to the number of years passed. The ratio for the father increases from 7 to 9, an increase of 2 units. Since these 2 units represent 10 years, 1 unit is equal to 5 years. Thus, the father's present age is 7 × 5 = 35 years.

Explanation

The given equation is tan²θ + tan⁴θ = 1. Factoring out tan²θ, we get tan²θ (1 + tan²θ) = 1. Using the trigonometric identity 1 + tan²θ = sec²θ, the expression simplifies to tan²θ · sec²θ = 1.

Converting the functions into sine and cosine terms, we have: (sin²θ)/(cos²θ) · (1)/(cos²θ) = 1 (sin²θ)/(cos⁴θ) = 1 sin²θ = cos⁴θ

By substituting the identity sin²θ = 1 - cos²θ into the equation, we get: 1 - cos²θ = cos⁴θ cos²θ + cos⁴θ = 1

Explanation

The radius r of the hemisphere is half of its diameter, so r = (12.6)/(2) = 6.3 cm. The volume V of a solid hemisphere is given by the formula:

V = ⅔ π r³

Substituting the given values π = ²²⁄₇ and r = 6.3:

V = ⅔ × ²²⁄₇ × (6.3)³

V = ⅔ × ²²⁄₇ × 6.3 × 6.3 × 6.3

Simplifying the calculation:

V = 2 × 22 × (6.3)/(7) × (6.3)/(3) × 6.3

V = 44 × 0.9 × 2.1 × 6.3

V = 39.6 × 13.23

V = 523.908 cm³

Explanation

Statement I involves the series 100² - 99² + 98² - 97² + … + 22² - 21². Using the algebraic identity a² - b² = (a - b)(a + b), each consecutive pair can be simplified since the difference (a - b) is always 1: (100 - 99)(100 + 99) + (98 - 97)(98 + 97) + … + (22 - 21)(22 + 21) = 1(199) + 1(195) + … + 1(43) This is equivalent to the sum of all integers from 21 to 100: Sum = 100 + 99 + 98 + … + 22 + 21 Using the formula for the sum of an Arithmetic Progression S = ⁿ⁄₂(a + l), where a = 21, l = 100, and n = (100 - 21) + 1 = 80: S = ⁸⁰⁄₂(21 + 100) = 40 × 121 = 4840 Thus, Statement I is correct.

Statement II is simplified by grouping terms and using the identity (a - b)(a + b) = a² - b²: (K - (1)/(K))(K + (1)/(K)) = K² - (1)/(K²⁾ Multiplying this result by (K² + (1)/(K²⁾) gives (K⁴ - (1)/(K⁴⁾). The expression now becomes (K⁴ - (1)/(K⁴⁾)(K⁴ + (1)/(K⁴⁾)(K⁴ - (1)/(K⁴⁾). The product of the first two terms is (K⁸ - (1)/(K⁸⁾). Finally, multiplying by the last term yields (K⁸ - (1)/(K⁸⁾)(K⁴ - (1)/(K⁴⁾). This does not equal K¹⁶ - (1)/(K¹⁶⁾, which would require the final multiplication to be (K⁸ - (1)/(K⁸⁾)(K⁸ + (1)/(K⁸⁾). Thus, Statement II is incorrect.