SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Factoring the first quadratic polynomial using the algebraic identity (a - b)² = a² - 2ab + b²: x² - 14x + 49 = (x - 7)²

Similarly, factoring the second quadratic polynomial using the algebraic identity (a + b)² = a² + 2ab + b²: x² + 6x + 9 = (x + 3)²

Multiplying these two factored forms, the given expression is represented as: (x - 7)²(x + 3)²

Taking the square root of this expression gives: √((x - 7)²(x + 3)²) = (x - 7)(x + 3)

Explanation

To find the value of (x² + y²)/(xy), we start by simplifying the given equation:

(x³ + y³)⁽2/3) = x² + y²

By cubing both sides of the equation, we get:

(x³ + y³)² = (x² + y²)³

Expanding both sides using algebraic identities:

x⁶ + 2x³y³ + y⁶ = x⁶ + 3x⁴y² + 3x²y⁴ + y⁶

Subtracting x⁶ and y⁶ from both sides simplifies the equation to:

2x³y³ = 3x⁴y² + 3x²y⁴

Since x ≠ 0 and y ≠ 0, we can divide the entire equation by x²y²:

2xy = 3x² + 3y²

2xy = 3(x² + y²)

Rearranging the terms to find the ratio:

(x² + y²)/(xy) = 2/3

Explanation

The capacity of a hemispherical bowl is given by the volume formula: V = ⅔πr³

Substituting the given radius r = 7 cm and π = 22/7: V = ⅔ × (22/7) × 7³ V = ⅔ × 22 × 7 × 7 V = ⅔ × 1078 V = 2156/3 V = 718 ⅔ cm³

Explanation

The time taken by a leak to empty a full tank, when the inlet pipe takes x hours to fill it and due to the leak it takes y hours, is given by the standard relation:

Time = (x × y) / (y - x)

Here, the inlet pipe takes x = 15 hours to fill the tank, and with the leak, it takes y = 20 hours.

Substituting these values into the formula:

Time = (15 × 20) / (20 - 15) = 300 / 5 = 60 hours

Explanation

The fourth proportional x to three numbers a, b, and c is defined by the proportion a : b = c : x, which gives the formula x = (b × c) / a.

Substituting the given values a = 5, b = 10, and c = 12 into the formula: x = (10 × 12) / 5 x = 120 / 5 x = 24

Explanation

The present value of a loan returned in two equal annual installments at compound interest is given by the relation:

P = I / (1 + r/100) + I / (1 + r/100)²

where P represents the sum borrowed, I is the amount of each installment, and r is the annual rate of interest.

By substituting the given parameters, I = 3025 and r = 10, the relation becomes:

P = 3025 / (1 + 10/100) + 3025 / (1 + 10/100)²

P = 3025 / (11/10) + 3025 / (11/10)²

P = (3025 × 10) / 11 + (3025 × 100) / 121

P = 2750 + 2500

P = 5250

Explanation

Let the total number of votes polled be V. Since there were no invalid votes, the winning candidate received 100% - 35% = 65% of the total votes.

The difference in votes between the winner and the loser is 65% - 35% = 30% of the total votes.

According to the problem, this difference is equal to 1296 votes: 30% of V = 1296 (30 / 100) × V = 1296 V = (1296 × 100) / 30 V = 4320

Explanation

Using the BODMAS rule, the expression is solved in the order of brackets, of, division, multiplication, addition, and subtraction.

First, evaluate the innermost parenthesis containing the "of" operation: 1/4 of 7920 = 1980

Next, evaluate the curly braces: -1980 + 1980 = 0

Now, evaluate the square brackets: -77 + 0 = -77

Next, perform the division: 1980 ÷ 9 = 220

Finally, combine the results of the division and bracket calculations: 220 + [-77] = 220 - 77 = 143

Explanation

Applying the rule of allegation, the overall annual interest rate obtained on the total investment is: (880 ÷ 10,000) × 100 = 8.8%

The interest rates of the two parts are 8% and 10%, while the mean rate is 8.8%. The ratio of the sum lent at 8% to the sum lent at 10% is: (10 - 8.8) : (8.8 - 8) = 1.2 : 0.8 = 3 : 2

The sum lent at 8% is: (3 ÷ (3 + 2)) × 10,000 = 3/5 × 10,000 = 6,000

Explanation

Using the allegation method, we can determine the ratio of Science to Mathematics marks.

Science average is 80% Mathematics average is 90% Combined average is 84%

Applying allegation: Ratio of Science to Mathematics = (90 - 84) : (84 - 80) = 6 : 4 = 3 : 2

Given that the total marks obtained in Science is 54,000, we associate this with the 3 units in our ratio: 3 units = 54,000 1 unit = 18,000

Since the marks obtained in Mathematics correspond to 2 units: 2 units = 2 × 18,000 = 36,000

Explanation

The direct relationship between the Cost Price (CP) and the Marked Price (MP) when given the Discount percentage (D) and Profit percentage (P) is given by:

CP / MP = (100 - D) / (100 + P)

Substitute the given values, D = 20%, P = 12%, and MP = ₹280:

CP / 280 = (100 - 20) / (100 + 12)

CP / 280 = 80 / 112

CP / 280 = 5 / 7

CP = 280 × 5 / 7

CP = 40 × 5 = 200

Explanation

The formula for the total amount to be paid with compound interest is:

A = P × (1 + R/100)ⁿ

Here, the principal amount P is ₹50,000, the annual rate of interest R is 8%, and the time period n is 3 years.

Substituting these values into the formula:

A = 50000 × (1 + 8/100)³

A = 50000 × (1.08)³

A = 50000 × 1.259712

A = 62985.6

Rounding 62985.6 to the nearest rupee gives ₹62,986.

Explanation

To find the speed in kilometres per hour, we use the standard formula:

Speed = Distance / Time

Given: Distance = 200 metres Time = 24 seconds

First, calculate the speed in metres per second:

Speed = 200 / 24 metres per second

To convert the speed from metres per second to kilometres per hour, we multiply by the conversion factor 18/5:

Speed = (200 / 24) × (18 / 5) km/h

Simplifying the expression:

Speed = (200 / 5) × (18 / 24) km/h Speed = 40 × (3 / 4) km/h Speed = 10 × 3 = 30 km/h

Explanation

The fourth proportional x of three numbers a, b, and c is defined by the relation a/b = c/x, which gives the formula:

x = (b × c) / a

Substituting the given values a = 48, b = 72, and c = 108 into the formula:

x = (72 × 108) / 48

Dividing 72 and 48 by their greatest common divisor 24:

x = (3 × 108) / 2

x = 3 × 54

x = 162

Explanation

Let V be the total number of voters in the voting list.

Since 90% of the voters cast their votes and there are no invalid or illegal votes, the total number of votes cast is 90% of V, which represents 100% of the valid votes.

The losing candidate received 30% of the total votes cast. Since this is a two-way election, the winning candidate must have received the remaining 70% of the total votes cast.

The difference in their vote shares is 70% - 30% = 40% of the total votes cast.

We are given that the losing candidate lost by 900 votes, which means the difference between the winner and the loser is 900.

Setting up the relation: 40% of total votes cast = 900 Total votes cast = 900 / 0.40 = 2250

Since the total votes cast represents 90% of the voters on the voting list: 90% of V = 2250 V = 2250 / 0.90 = 2500

Explanation

The discount percentage is calculated by dividing the number of free toys by the total number of toys received by the customer, then multiplying by 100.

Total number of toys received by the customer = 14 purchased toys + 4 free toys = 18 toys

Number of free toys = 4

Discount percentage = (4 / 18) × 100 = (2 / 9) × 100 = 22.22%

Rounding to the nearest integer, the discount percentage is 22%.

Explanation

A number is divisible by 24 if it is divisible by both 3 and 8, as 24 = 3 × 8 and 3 and 8 are co-prime.

To test divisibility of 478656 by 8, we check its last three digits, which are 656. Since 656 ÷ 8 = 82, the number is divisible by 8.

To test divisibility of 478656 by 3, we sum its digits: 4 + 7 + 8 + 6 + 5 + 6 = 36. Since 36 is divisible by 3, the number is divisible by 3.

Since 478656 is divisible by both 3 and 8, it is divisible by 24.

Explanation

Average speed is calculated by dividing the total distance travelled by the total time taken for the journey.

In the first part of the trip, the distance covered is the product of speed and time, which is 50 km/h × 1 h = 50 km.

In the second part of the trip, the distance covered is 60 km/h × 3 h = 180 km.

The total distance for the entire trip is 50 km + 180 km = 230 km.

The total time taken is 1 h + 3 h = 4 h.

Dividing the total distance by the total time gives the average speed: Average Speed = 230 km ÷ 4 h = 57.5 km/h

Explanation

Let the efficiency of a man, a woman, and a boy be M, W, and B respectively.

According to the question, the total work can be represented as: Total Work = 6M × 77 = 2W × 77 = 4B × 77

This gives the relation between their efficiencies: 6M = 2W = 4B

Dividing by the least common multiple of 6, 2, and 4, which is 12: M/2 = W/6 = B/3

Let these ratios equal a constant k, so M = 2k, W = 6k, and B = 3k.

The total work is: 6 × 2k × 77 = 924k

The combined efficiency of 1 man, 1 woman, and 1 boy working together is: 1M + 1W + 1B = 2k + 6k + 3k = 11k

The number of days required by them to complete the work together is: Total Work ÷ Combined Efficiency = 924k ÷ 11k = 84 days

Explanation

Let the total number of items purchased be 100 and the cost price of each item be ₹1. The total cost price is 100 × ₹1 = ₹100.

Since 36% of the items went bad, the number of items successfully sold is 100 - 36 = 64.

To earn an overall profit of 28%, the total selling price of the sold items must be ₹100 × 1.28 = ₹128.

The selling price of each of the 64 sold items is ₹128 ÷ 64 = ₹2.

Since the cost price of each item is ₹1 and the selling price is ₹2, the items were sold at a price ₹1 above the cost price. The percentage above the cost price is (1 / 1) × 100 = 100%.

Explanation

When the three smaller cubes are melted to form a single larger cube, the total volume remains conserved.

The volume of a cube with side s is given by the formula V = s³.

The sum of the volumes of the three individual cubes is: V = 6³ + 8³ + 10³ V = 216 + 512 + 1000 V = 1728 cm³

Let S be the side of the newly formed big cube. Its volume is: S³ = 1728

Taking the cube root on both sides: S = ∛1728 S = 12 cm

Explanation

The volume of a sphere is given by the formula V = 4/3 × π × r³, where r is the radius of the sphere.

Given that the diameter of the sphere is 21 cm, its radius r is 21/2 cm.

Substituting the radius r = 21/2 cm and π = 22/7 into the formula:

V = 4/3 × 22/7 × (21/2) × (21/2) × (21/2)

The terms 3 and 7 in the denominator multiply to 21, which cancels with one 21 in the numerator. The term 4 in the numerator cancels with 2 × 2 in the denominator. The term 22 in the numerator divided by the remaining 2 in the denominator yields 11.

This simplifies the calculation to:

V = 11 × 21 × 21 V = 11 × 441 V = 4851 cm³

Explanation

We are given: cos(x - y) = √3/2

Since x and y are positive acute angles and x ≥ y, the value of (x - y) must be acute or zero. The angle whose cosine is √3/2 in this range is 30°. Therefore, we have: x - y = 30°

We are also given: sin(x + y) = 1

For the range 0° < (x + y) ≤ 90°, the angle whose sine is 1 is 90°. Therefore, we have: x + y = 90°

Adding the two equations: (x - y) + (x + y) = 30° + 90° 2x = 120° x = 60°

Substituting x = 60° into the equation x + y = 90°: 60° + y = 90° y = 30°

Explanation

The formula for the area of a regular hexagon with side length a is:

Area = (3√3 / 2) × a²

A regular hexagon consists of six equilateral triangles of side length a. The area of one such equilateral triangle is (√3 / 4) × a². Multiplying this by six yields the area formula for the entire hexagon.

Substituting the given side length a = 10 cm into the formula:

Area = (3√3 / 2) × 10² Area = (3√3 / 2) × 100 Area = 3√3 × 50 Area = 150√3 cm²

Explanation

The given algebraic expression is a² - 1 - 2x - x².

Grouping the terms involving x and the constant 1 together: a² - 1 - 2x - x² = a² - (x² + 2x + 1)

Using the standard identity (m + n)² = m² + 2mn + n², we can rewrite the terms inside the parentheses: x² + 2x + 1 = (x + 1)²

Substituting this back into the expression gives: a² - (x + 1)²

Now, applying the difference of squares identity, A² - B² = (A + B)(A - B), where A = a and B = x + 1: a² - (x + 1)² = (a + (x + 1))(a - (x + 1)) = (a + 1 + x)(a - 1 - x)

Explanation

The area of a square with side length s is given by the formula: Area = s²

The given area of the square is represented by the quadratic expression 9m² + 12m + 4. This expression can be factored as a perfect square trinomial using the algebraic identity: (a + b)² = a² + 2ab + b²

By rewriting the terms of the expression: 9m² = (3m)² 4 = 2² 12m = 2 × (3m) × 2

Substituting these into the identity yields: 9m² + 12m + 4 = (3m)² + 2 × (3m) × 2 + 2² = (3m + 2)²

Equating this to the area formula: s² = (3m + 2)²

Taking the square root of both sides gives the measure of the side of the square: s = 3m + 2

Explanation

Let the salary of the man be S.

The man spends 10.5% of his salary on daily use items. The remaining salary after this expenditure is: 100% - 10.5% = 89.5% of S

Next, he spends 30% of this remainder on house rent, which means he is left with 70% of the remainder. The final amount left with him is: 70% of 89.5% of S = 12,000

This relation is expressed as: 0.70 × 0.895 × S = 12,000 0.6265 × S = 12,000 S = 12,000 / 0.6265 S ≈ 19,154.03

Rounding to the nearest whole value, the salary is Rs 19,154.

Explanation

Using the algebraic identity for the sum of two cubes:

a³ + b³ = (a + b)(a² - ab + b²)

The given expression p³ + 27 can be written as p³ + 3³. Substituting a = p and b = 3 into the identity yields:

(p + 3)(p² - (p)(3) + 3²) = (p + 3)(p² - 3p + 9)

Explanation

A frequency distribution is a systematic method of organizing and presenting raw data in a tabular or graphical format. It displays the distinct values or intervals of a variable alongside their corresponding frequencies, which represent the number of times each value or interval occurs in the dataset. This allows for an immediate understanding of the pattern, spread, and central tendencies of the data.

Explanation

According to the properties of circles, the lengths of two tangents drawn from an external point to a circle are equal. Since TP and TQ are two tangents drawn from the same external point T to the circle with centre O, their lengths must be equal, meaning TP = TQ. Given that TP = 7 cm, the length of TQ is also 7 cm.

Explanation

For a number to be divisible by 72, it must be divisible by both 8 and 9, as 8 and 9 are coprime factors of 72.

For divisibility by 8, the last three digits of the number, 8n6, must form a three-digit number that is divisible by 8. Testing digit values for n from 1 to 9, the three-digit number 8n6 is divisible by 8 when n = 1 (816), n = 5 (856), or n = 9 (896).

Among these possible values of n, which are 1, 5, and 9, the second largest value is n = 5.

For divisibility by 9, the sum of the digits of the number 5p42978n6 must be divisible by 9. Sum of digits = 5 + p + 4 + 2 + 9 + 7 + 8 + n + 6 Substituting n = 5: Sum of digits = 5 + p + 4 + 2 + 9 + 7 + 8 + 5 + 6 = 46 + p

For 46 + p to be a multiple of 9, the nearest multiple of 9 greater than 46 is 54. 46 + p = 54 p = 8

Using the value of p = 8, we calculate 2p - 1: 2p - 1 = 2 × 8 - 1 = 15

Explanation

The length of the diagonal of a cube with edge length a is given by the formula d = a√3.

Substituting the given diagonal value of 6√3 cm into the formula:

6√3 = a√3

Dividing both sides by √3, we get:

a = 6 cm

Explanation

The volume V of a right circular cone is calculated using the formula V = ⅓πr²h, where r represents the base radius and h represents the height.

Given that the base diameter is 16 cm, the radius r is half of the diameter, which is 8 cm. The height h is 6 cm.

Substituting these values into the formula:

V = ⅓ × π × 8² × 6 V = ⅓ × π × 64 × 6 V = 128π cm³

Explanation

Using the rule of alligation, the ratio of the quantities of the two types of sugar can be determined directly.

Let the quantity of Type 1 sugar be Q₁ and its cost price be CP₁ = 45 per kg. Let the quantity of Type 2 sugar be Q₂ and its cost price be CP₂ = 48 per kg. The mean price of the mixture is M = 46 per kg.

The formula for the ratio of the quantities is: Q₁ / Q₂ = (CP₂ - M) / (M - CP₁)

Substituting the given values into the formula: Q₁ / Q₂ = (48 - 46) / (46 - 45) Q₁ / Q₂ = 2 / 1

Thus, the ratio of the quantity of Type 1 sugar to Type 2 sugar is 2:1.

Explanation

We use the fundamental trigonometric identity: cosec² θ - cot² θ = 1

This can be factored as: (cosec θ + cot θ)(cosec θ - cot θ) = 1

Given that cosec θ + cot θ = 5, we can substitute this into the identity: 5 × (cosec θ - cot θ) = 1 cosec θ - cot θ = 1/5

By adding the two equations: cosec θ + cot θ = 5 cosec θ - cot θ = 1/5

We get: 2 cosec θ = 5 + 1/5 cosec θ = ½ (5 + 1/5)

Explanation

The effective discount for each scheme is calculated to find the most favorable one for the customer.

For Scheme A, the successive discount formula is x + y - xy/100. Effective discount = 15 + 15 - (15 × 15)/100 = 30 - 2.25 = 27.75%

For Scheme B, using the same formula: Effective discount = 12 + 18 - (12 × 18)/100 = 30 - 2.16 = 27.84%

For Scheme C, the customer gets 3 articles free on purchasing 7 articles, which means a total of 10 articles are received. Effective discount = (Free articles / Total articles) × 100 = (3 / 10) × 100 = 30%

Comparing the three schemes, Scheme C offers the maximum discount of 30%, which is higher than the 27.75% from Scheme A and 27.84% from Scheme B. This makes Scheme C the most favorable scheme.

Explanation

The total sum of all 16 numbers is obtained by multiplying their average by the total count: 16 × 23 = 368

The sum of the first 7 numbers is: 7 × 21 = 147

The sum of the last 8 numbers is: 8 × 24 = 192

The combined sum of these 15 numbers (the first 7 and the last 8) is: 147 + 192 = 339

The remaining 8th number is the difference between the total sum of all 16 numbers and the sum of these 15 numbers: 368 - 339 = 29

Explanation

Correct Answer: D

When a solid cuboid is molded into a solid sphere, its volume remains conserved. Therefore, the volume of the cuboid is equal to the volume of the sphere.

Volume of cuboid = length × width × height = 49 × 33 × 24 cm³

Volume of sphere = 4/3 × π × r³

Equating the two volumes: 4/3 × π × r³ = 49 × 33 × 24

Substitute π = 22/7: 4/3 × 22/7 × r³ = 49 × 33 × 24

Solve for r³: r³ = (49 × 33 × 24 × 3 × 7) / (4 × 22)

Simplify the fraction: r³ = (49 × 7) × (33 / 22) × (24 / 4) × 3 r³ = 343 × 1.5 × 6 × 3 r³ = 343 × 27 r³ = 7³ × 3³ r³ = (7 × 3)³ r³ = 21³

Taking the cube root on both sides: r = 21 cm

Explanation

Let the distance Ravina had to travel be R and the distance Suhel had to travel be S. The ratio of their travel distances is R:S = 4:7.

Given that Suhel had to travel a distance of S = 63 km, we can find Ravina's travel distance R using the ratio:

R/S = 4/7

Substituting the value of S:

R/63 = 4/7

R = (4 × 63) / 7

R = 4 × 9

R = 36 km

Explanation

Convert the percentages to fractional forms for quicker calculation: 64% = 64/100 = 16/25 81.25% = 13/16 (since 1/16 = 6.25%, then 13 × 6.25% = 81.25%)

Let the total number of students in the class be T. Number of girls = (16/25) × T Number of girls who chose soccer = (13/16) × (16/25) × T = 143

The factor of 16 in the numerator and denominator cancels out, simplifying the equation to: (13/25) × T = 143 T = (143 × 25) / 13 T = 11 × 25 T = 275

Explanation

The last three years given in the table are 2018, 2019, and 2020.

The profit/loss values for these years are: For 2018: -10 lakh For 2019: -5 lakh For 2020: 35 lakh

To find the total profit or loss over these three years, we sum the respective values: Total Profit/Loss = -10 + (-5) + 35 Total Profit/Loss = -15 + 35 Total Profit/Loss = 20 lakh

Since the resulting value is positive, it represents a net profit of Rs. 20 lakhs.

Explanation

Following the BODMAS rule, brackets and of are solved first, followed by division, multiplication, addition, and subtraction.

First, simplify the term inside the inner parentheses: 16 - 3 = 13

Next, solve the division inside the curly brackets: 650 ÷ 13 = 50

Now, evaluate the of operator: 3/100 of 1000 = (3/100) × 1000 = 30

Substitute these simplified values back into the expression: 450 + 752 - 125 × 30 - 50

Perform the multiplication next: 125 × 30 = 3750

Now, perform the remaining addition and subtraction: 450 + 752 - 3750 - 50 = 1202 - 3750 - 50 = -2548 - 50 = -2598

Explanation

Let the cost price of the earphone be x. The profit percentage is also x%.

Using the standard profit formula: Selling Price = Cost Price × (1 + Profit Percentage / 100)

Substituting the given values: 2000 = x × (1 + x / 100) 2000 = (100x + x²) / 100 200000 = x² + 100x x² + 100x - 200000 = 0

Factorizing the quadratic equation: x² + 500x - 400x - 200000 = 0 x(x + 500) - 400(x + 500) = 0 (x - 400)(x + 500) = 0

Since the cost price must be a positive value, we discard the negative root and obtain x = 400.

Explanation

Under compound interest compounded annually, the amount A after t years is given by the formula: A = P × (1 + r/100)ᵗ

Substituting the given values for 2 years and 3 years: P × (1 + r/100)² = 4624 P × (1 + r/100)³ = 4913

Dividing the second equation by the first equation gives the value of the compounding factor: 1 + r/100 = 4913 / 4624 1 + r/100 = 17 / 16

Substituting this compounding factor back into the equation for 2 years: P × (17/16)² = 4624 P × 289/256 = 4624 P = (4624 × 256) / 289 P = 16 × 256 P = 4096

Explanation

Given: x + 1/x = 10

By multiplying both sides by x, we get: x² + 1 = 10x

The given expression to evaluate is: 7x / (x² + 1 - 8x)

Substitute x² + 1 = 10x into the denominator: 7x / (10x - 8x) = 7x / 2x = 7/2 = 3.5

Explanation

To find the time when the four runners meet at the starting point for the first time, we calculate the Least Common Multiple (LCM) of the individual times they take to complete one round: 400 seconds, 600 seconds, 720 seconds, and 900 seconds.

The prime factorization of each number is: 400 = 2⁴ × 5² 600 = 2³ × 3 × 5² 720 = 2⁴ × 3² × 5 900 = 2² × 3² × 5²

The LCM is obtained by taking the highest power of each prime factor involved: For prime 2, the highest power is 2⁴ For prime 3, the highest power is 3² For prime 5, the highest power is 5²

LCM = 2⁴ × 3² × 5² = 16 × 9 × 25 = 3600 seconds

Explanation

Let V be the total number of enrolled voters.

Since 15% of the voters did not cast their votes, the percentage of voters who polled their votes is 100% - 15% = 85%.

Out of the polled votes, 10% were invalid, meaning 90% of the polled votes were valid. The total number of valid votes is: Valid votes = V × 0.85 × 0.90 = 0.765V

The successful candidate got 55% of the valid votes, which means the losing candidate got 100% - 55% = 45% of the valid votes. The majority by which the successful candidate won is the difference between these two shares: Majority = 55% - 45% = 10% of valid votes

We are given that this majority equals 6120 votes. We can set up the equation: 0.10 × 0.765V = 6120 0.0765V = 6120 V = 6120 ÷ 0.0765 V = 80000

Explanation

Using the algebraic identity (A + B)² = (A - B)² + 4AB, we substitute A = x and B = 2y to get:

(x + 2y)² = (x - 2y)² + 4 × x × 2y (x + 2y)² = (x - 2y)² + 4 × (2xy)

Substituting the given values, x - 2y = 3 and 2xy = 5, into the equation:

(x + 2y)² = 3² + 4 × 5 (x + 2y)² = 9 + 20 (x + 2y)² = 29

Explanation

The given equation can be grouped as: (a² + 1/a²) + (b² + 1/b²) = 4

According to the algebraic property of real numbers, for any non-zero real number x, the sum of its square and the reciprocal of its square is always greater than or equal to 2: x² + 1/x² ≥ 2

This is because (x - 1/x)² ≥ 0, which expands to x² + 1/x² - 2 ≥ 0, meaning x² + 1/x² ≥ 2.

Applying this to both terms in the equation: a² + 1/a² ≥ 2 b² + 1/b² ≥ 2

Adding these two inequalities gives: (a² + 1/a²) + (b² + 1/b²) ≥ 4

For the sum to be exactly equal to 4, both parts must reach their minimum possible value of 2 simultaneously: a² + 1/a² = 2 → a² = 1 b² + 1/b² = 2 → b² = 1

Substituting these values to find the required expression: a⁴ + b⁴ = (a²)² + (b²)² = 1² + 1² = 1 + 1 = 2

Explanation

The number of times the outcome on the die is greater than or equal to 5 is found by summing the individual frequencies of the outcomes 5 and 6.

From the provided table: Frequency of outcome 5 = 175 Frequency of outcome 6 = 190

Calculating the total frequency: 175 + 190 = 365