SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

To find the coefficient of x in the expansion of (x + 11)(3 - 5x), we can expand the product of the two linear terms:

(x + 11)(3 - 5x) = x(3) - x(5x) + 11(3) - 11(5x) = 3x - 5x² + 33 - 55x

Combining the like terms of x gives:

-5x² + (3 - 55)x + 33 = -5x² - 52x + 33

In this simplified polynomial, the term associated with x is -52x, which means the coefficient of x is -52.

Explanation

Time taken by A to complete 4/5 (80%) of the work is 20 days, which means A completes the whole work alone in 20 × 5/4 = 25 days. A's 1-day work rate is 1/25.

The remaining 1/5 (20%) of the work is completed by A and B together in 3 days. Consequently, they can complete the entire work together in 3 × 5 = 15 days, making their combined 1-day work rate 1/15.

B's 1-day work rate is calculated by subtracting A's rate from their combined rate: 1/15 - 1/25 = 5/75 - 3/75 = 2/75

This gives B's time to complete the work as 75/2 = 37.5 days.

Explanation

The prime factorization of the given numbers is: 77 = 7 × 11 275 = 5 × 5 × 11 = 5² × 11

The highest common factor (HCF) of two or more numbers is the product of the lowest powers of their common prime factors. The only prime factor common to both 77 and 275 is 11, which has a power of 1 in both factorization expressions. Thus, the HCF of 77 and 275 is 11.

Explanation

Let the time taken by the Rajdhani train to travel the distance of 648 km be T hours.

Initially, the Express train takes 12 hours more than the Rajdhani train, so the time taken by the Express train is T + 12 hours.

When the speed of the Express train is doubled, the time it takes to cover the same distance is halved. Therefore, the new time taken by the Express train is (T + 12) / 2 hours.

According to the given condition, this new time is 6 hours less than the time taken by the Rajdhani train: (T + 12) / 2 = T - 6

Solving this equation: T + 12 = 2T - 12 T = 24 hours

The Rajdhani train takes 24 hours to travel 648 km.

The speed of the Rajdhani train is: Speed = Distance / Time = 648 / 24 = 27 km/hr

Explanation

By the rule of alligation, the ratio of the quantity of wheat sold at 17% profit to the quantity sold at 27% profit is:

(27 - 21) : (21 - 17) = 6 : 4 = 3 : 2

The total quantity of wheat is 16 quintals. Since 1 quintal equals 100 kg, the total weight of the wheat is:

16 × 100 = 1600 kg

Using the ratio, the quantity of wheat sold at 27% profit is:

1600 × (2 / (3 + 2)) = 1600 × 2/5 = 640 kg

Explanation

According to the divisibility rule of 9, a number is completely divisible by 9 if the sum of its individual digits is divisible by 9.

Let the missing digit in the blank space be x. The sum of the digits of the number 583x437 is:

5 + 8 + 3 + x + 4 + 3 + 7 = 30 + x

For the sum 30 + x to be divisible by 9, it must equal the nearest multiple of 9 greater than or equal to 30, which is 36.

Solving for x:

30 + x = 36 x = 6

Explanation

Let Harishankar's salary be H, Girish's salary be G, and Shirish's salary be S.

According to the given information, Girish's salary is: G = (4/9) × H

Shirish's salary is: S = (5/6) × H

The ratio of Girish's salary to Shirish's salary is: G : S = (4/9)H : (5/6)H

To simplify this ratio, we divide the two fractions: G / S = (4/9) / (5/6) G / S = (4/9) × (6/5) G / S = 24 / 45

By dividing both the numerator and the denominator by their greatest common divisor, which is 3, we get: G / S = 8 / 15

Thus, the ratio is 8:15.

Explanation

To find the values x can take, we solve both inequalities individually.

For the first inequality: 5 - x > 1 + 3x Subtracting 1 and adding x on both sides: 4 > 4x Dividing by 4: x < 1

For the second inequality: 5 + 5x ≥ 2 + 2x Subtracting 2 and 2x on both sides: 3x ≥ -3 Dividing by 3: x ≥ -1

Combining both solution sets, we get the range: -1 ≤ x < 1

Among the options, only 0 lies within this interval.

Explanation

If three terms a, b, and c are in arithmetic progression, the middle term is the arithmetic mean of the other two terms, which gives the relation: 2b = a + c

Substituting the given terms a = 2x - 4, b = 4x + p, and c = 6x - 12 into this formula: 2(4x + p) = (2x - 4) + (6x - 12)

Expanding both sides: 8x + 2p = 8x - 16

Subtracting 8x from both sides: 2p = -16

Dividing by 2: p = -8

Explanation

Since segment PQ is parallel to side BC, triangle APQ is similar to triangle ABC.

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides: Area(APQ) / Area(ABC) = (AP / AB)²

Substituting the given ratio: 16 / 25 = (AP / AB)²

Taking the square root on both sides: AP / AB = 4 / 5

Since point P lies on the side AB, we have AB = AP + PB. Substituting this into the ratio: AP / (AP + PB) = 4 / 5

Cross-multiplying the terms: 5 × AP = 4 × (AP + PB) 5 × AP = 4 × AP + 4 × PB AP = 4 × PB

Dividing both sides by PB: AP / PB = 4 / 1

Explanation

Let the income of the man be 100 units.

Since he spends 59% of his income, his expenditure is 59 units.

His savings are the remaining portion of his income: Savings = Income - Expenditure = 100 - 59 = 41 units

The percentage of his expenditure that he saves is calculated as: Percentage = (Savings / Expenditure) × 100 Percentage = (41 / 59) × 100 = 4100 / 59 ≈ 69.49%

Explanation

Since AB is parallel to CD and the lines AD and BC intersect at E, the alternate interior angles are equal. Specifically, angle BAE is equal to angle CDE, and angle ABE is equal to angle DCE. Additionally, angle AEB and angle DEC are vertically opposite angles and thus equal. Therefore, triangle AEB is similar to triangle DEC.

By the property of similar triangles, the ratio of their corresponding sides is equal: AE / DE = BE / EC

Given that AE = 12 cm, DE = 18 cm, and BC = 12 cm, let the length of EC be x cm. Since BC is the sum of BE and EC, the length of BE can be expressed as 12 - x cm.

Substituting these values into the ratio: 12 / 18 = (12 - x) / x

Simplifying the fraction: 2 / 3 = (12 - x) / x

Cross-multiplying gives: 2x = 3(12 - x) 2x = 36 - 3x 5x = 36 x = 7.2

Explanation

The formula for simple interest is SI = (P × R × T) / 100, where SI is the simple interest, P is the principal sum, R is the annual rate of interest, and T is the time in years.

Rearranging the formula to find the principal sum P: P = (SI × 100) / (R × T)

Substituting the given values: P = (5320 × 100) / (7 × 5) P = 532000 / 35 P = 15200

Explanation

The volume of a cylinder is given by the formula V = π × r² × h, where r is the radius of the base and h is the height of the cylinder.

First, find the radius from the circumference of the base: 2 × π × r = 44 2 × 22/7 × r = 44 r = 7 cm

Next, find the height from the curved surface area (CSA) of the cylinder: CSA = 2 × π × r × h = 660 Since 2 × π × r = 44: 44 × h = 660 h = 15 cm

Finally, calculate the volume of the cylinder: Volume = π × r² × h Volume = 22/7 × 7² × 15 Volume = 22 × 7 × 15 Volume = 2310 cubic cm

Explanation

Let the third number be 100.

Since the first and second numbers are 35% and 50% lesser than the third number respectively: First number = 100 - 35 = 65 Second number = 100 - 50 = 50

To make the second number equal to the first number, it must be increased by 65 - 50 = 15.

The required percentage enhancement is calculated with respect to the second number: Percentage enhancement = (Increase / Second number) × 100 Percentage enhancement = (15 / 50) × 100 = 30%

Explanation

Using the trigonometric identity cos(2π - θ) = cos θ, the given expression can be written as:

cos(11π/6) = cos(2π - π/6)

Applying the identity:

cos(2π - π/6) = cos(π/6)

Since the standard value of cos(π/6) is √3/2, the value of cos(11π/6) is √3/2.

Explanation

The standard trigonometric triple-angle identity for cosine is:

cos3A = 4cos³A - 3cosA

By rearranging this identity to isolate 3cosA, we get:

3cosA = 4cos³A - cos3A

This directly matches the given expression.

Explanation

To simplify the expression cosA / (1 - sinA), multiply both the numerator and the denominator by (1 + sinA):

(cosA × (1 + sinA)) / ((1 - sinA) × (1 + sinA))

= (cosA × (1 + sinA)) / (1 - sin²A)

Using the fundamental trigonometric identity 1 - sin²A = cos²A, the expression becomes:

(cosA × (1 + sinA)) / cos²A

= (1 + sinA) / cosA

= (1 / cosA) + (sinA / cosA)

= secA + tanA

Explanation

The weight to height ratio is calculated by dividing the weight in kilograms by the height in meters.

For Sunita, the ratio is: 60 ÷ 1.5 = 40

The ratios for the others are: Anjali: 55 ÷ 1.2 ≈ 45.83 Vimla: 75 ÷ 1.6 = 46.875 Preeti: 70 ÷ 1.4 = 50

Comparing these values, Sunita has the lowest weight to height ratio of 40.

Explanation

The least percentage profit in the given table is 5% in the year 2011.

The percentage profit of a company is calculated using the formula: Profit % = ((Revenue - Expenditure) / Expenditure) × 100

Given that the expenditure is ₹200 crore: 5 = ((Revenue - 200) / 200) × 100 5 = (Revenue - 200) / 2 10 = Revenue - 200 Revenue = 210 crore

Explanation

The average number of children per house is calculated using the weighted mean formula:

Average = ∑(f × x) / ∑f

Here, x represents the number of children and f represents the number of houses.

First, calculate the total number of children, which is the sum of the products of the number of children and the number of houses: ∑(f × x) = (0 × 3) + (1 × 13) + (2 × 18) + (3 × 3) = 0 + 13 + 36 + 9 = 58

Next, calculate the total number of houses: ∑f = 3 + 13 + 18 + 3 = 37

Divide the total number of children by the total number of houses to find the average: Average = 58 / 37 ≈ 1.57

Explanation

The total number of shares issued by the company is six lakh, which is equal to 6,00,000.

As per the given table, Ejaz holds a 25% share of the company.

The initial number of shares held by Ejaz is calculated as: Initial shares = 25% of 6,00,000 = 0.25 × 6,00,000 = 1,50,000

When Anand sells 15,000 of his shares to Ejaz, the total number of shares held by Ejaz increases by 15,000.

The final number of shares with Ejaz is: Final shares = 1,50,000 + 15,000 = 1,65,000

Explanation

Total interest earned is the sum of the interest gained on each of the two deposits.

The first deposit of Rs 2400 is made on 1st January and earns interest for two half-year periods. Amount at the end of the year = 2400 × (1 + 15/100)² = 2400 × (1.15)² = 2400 × 1.3225 = Rs 3174 Interest from the first deposit = 3174 - 2400 = Rs 774

The second deposit of Rs 2400 is made on 1st July and earns interest for one half-year period. Amount at the end of the year = 2400 × (1 + 15/100) = 2400 × 1.15 = Rs 2760 Interest from the second deposit = 2760 - 2400 = Rs 360

Total interest gained by the customer at the end of the year = 774 + 360 = Rs 1134

Explanation

We are given that x = sec A + tan A.

Expressing sec A and tan A in terms of sin A and cos A: x = 1/cos A + sin A/cos A x = (1 + sin A)/cos A

Multiplying both the numerator and the denominator by (1 - sin A): x = (1 + sin A)(1 - sin A) / [cos A(1 - sin A)]

Using the trigonometric identity 1 - sin² A = cos² A: x = (1 - sin² A) / [cos A(1 - sin A)] x = cos² A / [cos A(1 - sin A)] x = cos A / (1 - sin A)

Explanation

Let the two numbers be a and b.

The mean proportion of two numbers a and b is given by the formula √(ab), and their third proportion is given by the formula b²/a.

Given: √(ab) = 16 b²/a = 1024

Squaring the mean proportion equation: ab = 16² = 256 a = 256/b

Substituting the expression for a into the third proportion equation: b² / (256/b) = 1024 b³ = 1024 × 256 b³ = 2¹⁰ × 2⁸ b³ = 2¹⁸

Taking the cube root on both sides: b = 2⁶ = 64

Substituting the value of b to find a: a = 256 / 64 = 4

The two numbers are 4 and 64.

Explanation

The volume of water transferred from the cone to the cylinder remains constant.

Volume of the cone = (1/3) × π × r₁² × h₁ Volume of the cylinder = π × r₂² × h₂

Equating the two volumes: (1/3) × π × r₁² × h₁ = π × r₂² × h₂ (1/3) × r₁² × h₁ = r₂² × h₂

Given r₁ = 3.5 cm, h₁ = 12 cm, and r₂ = 7 cm: (1/3) × (3.5)² × 12 = 7² × h₂ 4 × 12.25 = 49 × h₂ 49 = 49 × h₂ h₂ = 1 cm

Explanation

The area of an equilateral triangle with side length a is given by the formula:

Area = (√3 / 4) × a²

Substituting the given side length a = 8 cm into the formula:

Area = (√3 / 4) × 8² Area = (√3 / 4) × 64 Area = 16√3 sq cm

Explanation

The slope of the given line 3x - 2y = 6 is m₁ = 3/2. Since the line passing through the origin is perpendicular to it, the product of their slopes must be -1. Therefore, the slope of the perpendicular line is m₂ = -2/3.

The equation of this perpendicular line passing through the origin (0, 0) is: y = (-2/3)x 2x + 3y = 0

To find the coordinates of the intersection point M, we solve the system of equations: 3x - 2y = 6 2x + 3y = 0

From the second equation, we get x = -3y/2. Substituting this into the first equation: 3(-3y/2) - 2y = 6 -9y/2 - 2y = 6 -13y/2 = 6 y = -12/13

Substituting y = -12/13 back into the expression for x: x = -3(-12/13) / 2 = 18/13

This gives the coordinates of M as (18/13, -12/13).

Explanation

Let the weights of Shubha, Govinda, and Reshma be S, G, and R respectively.

The sum of the weights of Shubha, Govinda, and Reshma is: S + G + R = 3 × 65 = 195 kg

The sum of the weights of Shubha and Govinda is: S + G = 2 × 63 = 126 kg

The sum of the weights of Govinda and Reshma is: G + R = 2 × 70 = 140 kg

Adding these two equations gives: S + 2G + R = 126 + 140 = 266 kg

To find the weight of Govinda, subtract the sum of the weights of all three individuals from this combined sum: G = (S + 2G + R) - (S + G + R) G = 266 - 195 = 71 kg

Explanation

The length of an arc of a circle is given by the formula L = (θ/360°) × 2πr, where θ is the central angle, r is the radius of the circle, and π is 22/7.

Substituting θ = 45° and r = 28 cm into the formula:

L = (45°/360°) × 2 × (22/7) × 28

Simplifying the central angle ratio, 45°/360° equals 1/8. The calculation becomes:

L = (1/8) × 2 × 22 × 4

Since 2 × 4 is 8, the 8 in the denominator cancels out with the 8 in the numerator:

L = 22 cm

Explanation

The fundamental relation between two numbers, their HCF, and their LCM is:

Product of two numbers = HCF × LCM

Substituting the given values into the formula: 63 × 77 = 7 × LCM

LCM = (63 × 77) / 7 LCM = 9 × 77 LCM = 693

Explanation

Let the cost price of the electric chimney be C and that of the washing machine be W.

The total cost of both items is: C + W = 57,750

The selling price of the electric chimney at a 34% profit is 1.34 × C. The selling price of the washing machine at a 24% loss is 0.76 × W.

Since the selling prices of both items are equal: 1.34 × C = 0.76 × W 134 × C = 76 × W C / W = 38 / 67

The ratio of the cost price of the electric chimney to the washing machine is 38 : 67. The electric chimney is the cheaper item because it corresponds to fewer parts in the ratio.

The cost price of the electric chimney is: (38 / (38 + 67)) × 57,750 = (38 / 105) × 57,750 = 38 × 550 = 20,900

Explanation

We are given the equation: (7 - 12x) - (3x - 7) = 14

Removing the parentheses by distributing the negative sign: 7 - 12x - 3x + 7 = 14

Combining the like terms on the left side: 14 - 15x = 14

Subtracting 14 from both sides of the equation: -15x = 0

Dividing both sides by -15 yields: x = 0

Explanation

For the quadratic equation ax² + bx + c = 0, the roots can be found by splitting the middle term. In the given equation 6x² - 11x - 35 = 0, the product of the coefficient of x² and the constant term is 6 × (-35) = -210, and their sum is -11. The two factors of -210 that add up to -11 are -21 and 10.

Rewriting the equation by splitting the middle term: 6x² - 21x + 10x - 35 = 0

Grouping and factoring the terms: 3x(2x - 7) + 5(2x - 7) = 0 (3x + 5)(2x - 7) = 0

Setting each factor to zero: 3x + 5 = 0 which gives x = -5/3 2x - 7 = 0 which gives x = 7/2

Explanation

Solving the first inequality: 5x + 2 < 4x + 1 x < -1

Solving the second inequality: 2(x - 4) - 2 ≤ 4x - 1 2x - 8 - 2 ≤ 4x - 1 2x - 10 ≤ 4x - 1 -9 ≤ 2x x ≥ -4.5

Combining both inequalities, the solution set for x is: -4.5 ≤ x < -1

The value -2 lies within this range.

Explanation

Let the marks obtained by the two students be x and y, where x is the higher score.

According to the given conditions: x - y = 10 x = 0.75 × (x + y)

Substituting x = y + 10 into the second equation: y + 10 = 0.75 × (y + 10 + y) y + 10 = 0.75 × (2y + 10) y + 10 = 1.5y + 7.5 0.5y = 2.5 y = 5

Substituting the value of y back into the first equation: x = 5 + 10 = 15

The marks obtained by the two students are 15 and 5.

Explanation

Total collection is the product of the ticket price and the number of visitors.

Let the initial price of the entry ticket be 3x and the new price be 7x. Let the initial number of visitors be 16y and the new number of visitors be 13y.

Initial total collection = 3x × 16y = 48xy New total collection = 7x × 13y = 91xy

Comparing the initial collection to the new collection yields the ratio 48xy:91xy, which simplifies to 48:91. Since 91 is greater than 48, the total collection has increased in the ratio 48:91.

Explanation

Average speed is calculated using the formula:

Average Speed = Distance / Time

Given: Distance = 315 km Time = 2.8 hours

First, calculate the average speed in km/h:

Average Speed = 315 / 2.8 = 112.5 km/h

To convert the speed from km/h to m/s, multiply the value by the conversion factor 5/18:

Average Speed in m/s = 112.5 × 5/18 = 31.25 m/s

Explanation

The n-th term of an arithmetic progression is given by the formula: aₙ = a + (n - 1)d

where a is the first term and d is the common difference.

Using the given values for the 3rd term and the 17th term: a₃ = a + 2d = 17 a₁₇ = a + 16d = -25

Subtracting the equation for the 3rd term from the equation for the 17th term: (a + 16d) - (a + 2d) = -25 - 17 14d = -42 d = -3

Substituting the value of d into the equation for the 3rd term: a + 2(-3) = 17 a - 6 = 17 a = 23

To find which term is -1, we substitute a = 23 and d = -3 into the n-th term formula: aₙ = a + (n - 1)d -1 = 23 + (n - 1)(-3) -24 = (n - 1)(-3) n - 1 = 8 n = 9

Explanation

The relation between the area of a square and its diagonal is given by the formula:

Area = d² / 2

where d is the length of the diagonal.

Substituting the given area of 100 sq cm into the formula:

100 = d² / 2

d² = 200

d = √200 = 10√2 cm

Alternatively, the side of the square s is √100 = 10 cm. The diagonal of a square with side s is s√2, which yields 10√2 cm.

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Substituting the points (4, -8) and (k, 0), and the distance d = 10:

10 = √((k - 4)² + (0 - (-8))²)

100 = (k - 4)² + 64

36 = (k - 4)²

Taking the square root:

k - 4 = ±6

For the positive root: k - 4 = 6 k = 10

For the negative root: k - 4 = -6 k = -2

Explanation

Praful completes ½ of the job in 30 days, which means he can complete the entire job in 60 days. His daily work rate is 1/60 of the job.

Sarabjit completes the remaining ½ of the job in 10 days, which means he can complete the entire job in 20 days. His daily work rate is 1/20 of the job.

Their combined daily work rate is: 1/60 + 1/20 = 1/60 + 3/60 = 4/60 = 1/15

Thus, they can complete the job together in 15 days.

Explanation

Total cost of tickets without any discount is calculated by adding the total cost for adults and children. Cost for 20 adults = 20 × 500 = Rs 10000 Cost for 8 children = 8 × 200 = Rs 1600 Total original cost = 10000 + 1600 = Rs 11600

Since two adults allow one child to enter for free, 20 adults can allow up to 20 / 2 = 10 children to enter for free. Since there are only 8 children in the group, all 8 children will get free entry. Total discount received by the group is the ticket price of the 8 children: Discount = 8 × 200 = Rs 1600

Discount percentage is calculated as: Discount % = (Discount / Total original cost) × 100 Discount % = (1600 / 11600) × 100 = 1600 / 116 = 400 / 29 ≈ 13.79%

Explanation

The average cost of the machines is calculated by dividing the total cost of all machines by the total number of machines purchased.

First, determine the quantity of each type of machine: Machine A = 3 Machine B = 2 Machine C = 8 - (3 + 2) = 3

Next, calculate the total cost by summing the individual costs: Total Cost = (3 × 100000) + (2 × 80000) + (3 × 45000) Total Cost = 300000 + 160000 + 135000 Total Cost = 595000

Now, divide the total cost by the total quantity of machines: Average Cost = 595000 ÷ 8 Average Cost = 74375

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the given points (2, 3) and (-4, 1) into the formula:

m = (1 - 3) / (-4 - 2) = -2 / -6 = 1/3

The equation of a line with slope m passing through a point (x₁, y₁) is given by the point-slope form:

y - y₁ = m(x - x₁)

Substituting the point (2, 3) and the slope m = 1/3:

y - 3 = 1/3(x - 2)

Multiplying both sides by 3 to eliminate the fraction:

3(y - 3) = x - 2

3y - 9 = x - 2

Rearranging the terms to align the variables on one side and the constant on the other:

x - 3y = -7

Explanation

By the Angle Bisector Theorem, the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. Therefore, in ΔABC with the angle bisector AE cutting BC at E, the relation is:

AB / AC = BE / EC

Substituting the given values:

9 / AC = 3.6 / 2.4

Simplifying the ratio of the segments:

3.6 / 2.4 = 3 / 2

Solving for AC:

9 / AC = 3 / 2

AC = (9 × 2) / 3 = 6 cm

Explanation

The effective annual rate of interest, E, is calculated using the formula:

E = (1 + r/n)ⁿ - 1

where r is the nominal annual rate of interest and n is the number of compounding periods per year.

Given that the nominal rate r is 22% (or 0.22) and the interest is compounded half-yearly (n = 2):

E = (1 + 0.22/2)² - 1

E = (1 + 0.11)² - 1

E = (1.11)² - 1

E = 1.2321 - 1

E = 0.2321, which is equal to 23.21%

Explanation

By the Law of Sines in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant:

a / sin(A) = b / sin(B)

Given that the side opposite to the 45° angle has a length of 8√2 cm, we can set a = 8√2 cm, A = 45°, and B = 90°. We need to find the length of the side b opposite to the 90° angle.

Substituting the known values:

(8√2) / sin(45°) = b / sin(90°)

Since sin(45°) = 1/√2 and sin(90°) = 1, this simplifies to:

(8√2) / (1/√2) = b / 1

8√2 × √2 = b

b = 8 × 2 = 16 cm

Explanation

The volume V of a cylinder is calculated using the formula V = π r² h, where r is the radius and h is the height.

Given that the volume V is 2156 cubic cm and the height h is 14 cm, and taking the value of π as 22/7, the equation can be written as:

2156 = (22/7) × r² × 14

Simplifying the equation by dividing 14 by 7:

2156 = 22 × r² × 2

2156 = 44 × r²

Dividing both sides by 44 to find the value of r²:

r² = 2156 ÷ 44

r² = 49

Taking the square root on both sides yields:

r = 7 cm

Explanation

Using the trigonometric identity cot(π + θ) = cot θ, we can rewrite the angle as:

cot(4π/3) = cot(π + π/3)

Since the angle 4π/3 lies in the third quadrant, where the cotangent function is positive, the expression simplifies to:

cot(π/3)

Because cot(π/3) is the reciprocal of tan(π/3), and tan(π/3) = √3, we get:

cot(π/3) = 1/tan(π/3) = 1/√3