SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

By the Triangle Inequality Theorem, the length of the third side c of a triangle with known sides a and b must satisfy the inequality: |a - b| < c < a + b

Substituting the given side lengths, a = 14 cm and b = 9 cm: |14 - 9| < c < 14 + 9 5 < c < 23

The length of the third side must be strictly greater than 5 cm and strictly less than 23 cm. The value 17 cm satisfies this inequality.

Explanation

Net percentage change = a - b - ab/100

Where a is the percentage markup above the cost price (10%) and b is the percentage discount on the marked price (10%).

Substituting the values into the formula: Net percentage change = 10 - 10 - (10 × 10)/100 = -1%

The negative sign indicates a net decrease in the value compared to the cost price, which represents a loss of 1%. The actual marked price of ₹550 is redundant for calculating the percentage.

Explanation

In a right-angled triangle, tan θ = opposite/adjacent = 3/4.

By the Pythagorean theorem, hypotenuse = √(opposite² + adjacent²) = √(3² + 4²) = √25 = 5.

This gives sin θ = opposite/hypotenuse = 3/5.

Substituting sin θ into the given expression:

(1 + sin θ) / (1 - sin θ) = (1 + 3/5) / (1 - 3/5) = (8/5) / (2/5) = 8/2 = 4.

Explanation

To find the present value (PV) of a future amount of ₹9,360 after 2 years at a 20% annual compound interest rate, we use the formula PV = FV / (1 + r)ⁿ. Substituting the values, we get PV = 9,360 / (1.20)² = 9,360 / 1.44 = ₹6,500.

Explanation

The net percentage change can be calculated using the successive percentage formula: Net Change = x + y + (xy/100). Substituting x = 18 (markup) and y = -24 (discount), we get 18 - 24 + (18 * -24)/100 = -6 - 4.32 = -10.32%, which represents a loss of 10.32%.

Explanation

By the rule of alligation, the ratio of the quantities of two components in a mixture is determined by the formula:

Ratio of Water to Milk = (Cost Price of Milk - Mean Price) / (Mean Price - Cost Price of Water)

Assuming the cost of water is ₹0 per litre and the mixture is sold at its cost price of ₹30 per litre:

Cost Price of Milk = ₹40 Cost Price of Water = ₹0 Mean Price of Mixture = ₹30

Substituting these values into the formula:

Ratio of Water to Milk = (40 - 30) / (30 - 0) = 10 / 30 = 1 / 3

Thus, the ratio of water and milk in the mixture is 1 : 3.

Explanation

In the equilateral triangle ABC, all sides are equal and each angle measures 60°, so AB = BC = AC and ∠ACB = 60°. Since side BC is produced to point D such that BC = CD, and we know BC = AC, it follows that AC = CD.

The points B, C, and D lie on a straight line, which means ∠ACB and ∠ACD form a linear pair. Therefore, ∠ACD = 180° - ∠ACB = 180° - 60° = 120°.

In triangle ACD, because AC = CD, it is an isosceles triangle where the angles opposite to the equal sides are equal, meaning ∠CAD = ∠CDA. Since the sum of angles in triangle ACD is 180°, we have ∠CAD + ∠CDA + ∠ACD = 180°. Substituting the known values gives 2 × ∠CAD + 120° = 180°, which simplifies to 2 × ∠CAD = 60°, and thus ∠CAD = 30°.

Explanation

Use the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values, a + b + c = 13 and ab + bc + ca = 45, into the identity: 13² = a² + b² + c² + 2(45)

169 = a² + b² + c² + 90

Subtract 90 from both sides to find the value of a² + b² + c²: a² + b² + c² = 169 - 90 = 79

Explanation

Let the cost price of one item be CP, the marked price be MP, and the selling price be SP.

From the first condition, the marked price of 85 items is equal to the cost price of 153 items: 85 × MP = 153 × CP MP / CP = 153 / 85 MP / CP = 9 / 5

From the second condition, the selling price of 104 items is equal to the marked price of 65 items: 104 × SP = 65 × MP SP / MP = 65 / 104 SP / MP = 5 / 8

To find the relationship between the selling price and the cost price, multiply the two ratios: SP / CP = (SP / MP) × (MP / CP) SP / CP = (5 / 8) × (9 / 5) SP / CP = 9 / 8

This means if the cost price is 8 units, the selling price is 9 units.

The profit percentage is calculated as: Profit % = (SP - CP) / CP × 100% Profit % = (9 - 8) / 8 × 100% Profit % = 1/8 × 100% = 12.5%

Explanation

The formula for Simple Interest is: SI = (P × R × T) / 100

Where: P is the principal sum = 450 R is the rate of interest per annum = 4.5% SI is the simple interest accumulated = 81 T is the time period in years

Rearranging the formula to solve for time T: T = (SI × 100) / (P × R)

Substituting the values into the equation: T = (81 × 100) / (450 × 4.5) T = 8100 / 2025 T = 4 years

Explanation

To find the ratio of Ram's salary to Ghanshyam's salary, the compound ratio of the two given ratios is calculated.

Let R represent Ram's salary, S represent Shyam's salary, and G represent Ghanshyam's salary.

The given ratios are: R/S = 3/4 S/G = 5/6

Multiplying these two ratios yields the ratio of Ram's salary to Ghanshyam's salary: R/G = (R/S) × (S/G) R/G = (3/4) × (5/6) R/G = 15/24

Simplifying the fraction by dividing the numerator and denominator by 3: R/G = 5/8

Thus, the ratio of Ram's salary to Ghanshyam's salary is 5 : 8.

Explanation

The volume of a cuboidal box is calculated using the formula: Volume = length × breadth × height

Substituting the given values into this formula: 1875 = 25 × 15 × height 1875 = 375 × height height = 1875 / 375 height = 5 cm

Explanation

According to the Intersecting Chords Theorem, when two chords AB and CD of a circle intersect internally at a point P, the products of the segments of the chords are equal:

PA × PB = PC × PD

The length of chord AB is the sum of its segments PA and PB:

AB = PA + PB

Given AB = 16 cm and PB = 6 cm, the segment PA is:

PA = 16 - 6 = 10 cm

Substituting the values of PA, PB, and PD into the intersecting chords formula:

10 × 6 = PC × 12

60 = PC × 12

PC = 60 ÷ 12 = 5 cm

Explanation

To find the average marks of the five students, use the average formula:

Average = (Sum of all marks) ÷ (Total number of students)

First, calculate the sum of the marks obtained by Ajay, Pronoy, Mahendra, Jharna, and Dipti: Sum = 85 + 72 + 70 + 84 + 94 = 405

Since there are 5 students in total, divide the sum by 5: Average = 405 ÷ 5 = 81

Explanation

The cost price of the article is CP = ₹80 and the marked price is MP = ₹125.

A discount of 40% is allowed on the marked price.

Discount = 40% of 125 = (40 / 100) × 125 = ₹50

The selling price is: SP = MP - Discount = 125 - 50 = ₹75

Since the selling price is less than the cost price, there is a loss: Loss = CP - SP = 80 - 75 = ₹5

The loss percentage is: Loss % = (Loss / CP) × 100% = (5 / 80) × 100% = (1 / 16) × 100% = 6.25%

Explanation

The collinearity of the points representing the heights of Subhas' eyes, the top of the tree, and the top of the building allows the use of the slope formula. Let Subhas' position be the origin x = 0.

The coordinates representing the tops of the three objects are: Subhas' eyes: (0, h) Top of the tree: (7.5, 3.15) Top of the building: (7.5 + 45, 11.25) = (52.5, 11.25)

Since these three points lie on a straight line, the slope of the line segment between the tree and the building must equal the slope of the line segment between Subhas' eyes and the tree:

(11.25 - 3.15) / (52.5 - 7.5) = (3.15 - h) / (7.5 - 0)

8.1 / 45 = (3.15 - h) / 7.5

0.18 = (3.15 - h) / 7.5

3.15 - h = 0.18 × 7.5

3.15 - h = 1.35

h = 3.15 - 1.35

h = 1.8

Explanation

Let the radius of the circle be R. A perpendicular drawn from the centre of a circle to a chord bisects the chord.

For the smaller chord of length 10 cm, the bisected length is 10/2 = 5 cm. Since its distance from the centre is 12 cm, we can use the Pythagorean theorem in the right-angled triangle formed by the radius, the perpendicular distance, and half the chord: R² = 12² + 5² R² = 144 + 25 = 169 R = 13 cm

For the larger chord of length 24 cm, the bisected length is 24/2 = 12 cm. Let its distance from the centre be d. Using the Pythagorean theorem: R² = d² + 12² 169 = d² + 144 d² = 25 d = 5 cm

Since the chords lie on opposite sides of the centre, the total distance between them is the sum of their perpendicular distances from the centre: Distance = 12 + 5 = 17 cm.

Explanation

By the fundamental trigonometric identity, sin²θ + cos²θ = 1. Since θ is an acute angle, sin θ must be positive.

Therefore, we have:

sin θ = √(1 - cos²θ)

Substituting the given value of cos θ = 39/89:

sin θ = √(1 - (39/89)²) sin θ = √((89² - 39²) / 89²)

Using the difference of squares formula, a² - b² = (a - b)(a + b):

89² - 39² = (89 - 39)(89 + 39) = 50 × 128 = 6400

Thus:

sin θ = √(6400 / 89²) = 80/89

Explanation

The discount rate is calculated using the formula:

Discount Percentage = ((Marked Price - Selling Price) ÷ Marked Price) × 100

Given: Marked Price = ₹1,350 Selling Price = ₹1,188

Discount = 1,350 - 1,188 = ₹162

Discount Percentage = (162 ÷ 1,350) × 100 = 12%

Explanation

We use the algebraic identity: (a + 1/a)³ = a³ + 1/a³ + 3(a + 1/a)

Rearranging this formula gives: a³ + 1/a³ = (a + 1/a)³ - 3(a + 1/a)

By substituting the given value a + 1/a = 5 into the equation, we get: a³ + 1/a³ = 5³ - 3(5) a³ + 1/a³ = 125 - 15 a³ + 1/a³ = 110

Explanation

From the given relation: tan θ × tan 2θ = 1

This can be written as: tan 2θ = 1 / tan θ tan 2θ = cot θ tan 2θ = tan(90° - θ)

By comparing the angles: 2θ = 90° - θ 3θ = 90° θ = 30°

Substituting θ = 30° into the expression cot 5θ: cot 5θ = cot(5 × 30°) cot 5θ = cot 150° cot 150° = cot(180° - 30°) cot 150° = -cot 30° cot 150° = -√3

Explanation

To find the overall profit percentage, we can use the multiplying factor method.

The profit factor for selling at an 8% profit is: 1 + 8/100 = 1.08

The factor due to using 25% less weight, which means selling 75% of the actual quantity for the same price, is: 1 / (1 - 25/100) = 1 / 0.75 = 4/3

The overall multiplier for the transaction is the product of these two factors: Overall multiplier = 1.08 × 4/3 = 4.32 / 3 = 1.44

To convert this multiplier back to a percentage profit: Profit percentage = (1.44 - 1) × 100 = 44%

Explanation

First, calculate the total number of days from 22 May 1993 to 2 August 1993, with both days included: Number of days in May = 10 days (from 22 May to 31 May) Number of days in June = 30 days Number of days in July = 31 days Number of days in August = 2 days (1 August and 2 August) Total days = 10 + 30 + 31 + 2 = 73 days

Since 1993 is a non-leap year, the total number of days in the year is 365. Time T in years = 73/365 = 1/5 year

Using the simple interest formula: SI = (P × R × T) / 100

Substitute the values P = 12,300, R = 12, and T = 1/5 into the formula: SI = (12300 × 12 × 1/5) / 100 SI = 123 × 12 / 5 SI = 1476 / 5 SI = 295.2

Rounding off 295.2 to the nearest tens place gives 300.

Explanation

In a circle, the perpendicular drawn from the centre to a chord bisects the chord. Let O be the centre of the circle, and PQ be the chord of length 3 cm. The perpendicular from O meets PQ at a point M, which bisects PQ.

Therefore, PM = PQ/2 = 3/2 = 1.5 cm.

The radius of the circle OP is 1.7 cm. In the right-angled triangle OMP, we apply the Pythagorean theorem:

OP² = OM² + PM²

1.7² = OM² + 1.5²

2.89 = OM² + 2.25

OM² = 2.89 - 2.25

OM² = 0.64

OM = √0.64 = 0.8 cm

The distance of the line segment PQ from the centre of the circle is 0.8 cm.

Explanation

When two people run in opposite directions on a circular track of length L, the time taken to meet is given by the formula: Time = Track length / Relative speed, where relative speed is the sum of their individual speeds.

For the first meeting, the initial speeds are 10 m/s and 40 m/s. The relative speed is 10 + 40 = 50 m/s. The time taken for the first meeting is 400 / 50 = 8 seconds.

After the first meeting, A's speed doubles to 20 m/s and B's speed halves to 20 m/s. The relative speed for this stage is 20 + 20 = 40 m/s. The time taken between the first and second meeting is 400 / 40 = 10 seconds.

After the second meeting, A's speed doubles to 40 m/s and B's speed halves to 10 m/s. The relative speed for this stage is 40 + 10 = 50 m/s. The time taken between the second and third meeting is 400 / 50 = 8 seconds.

The total time elapsed from the start until they meet for the third time is the sum of these individual time intervals: 8 + 10 + 8 = 26 seconds.

Explanation

The net annual growth rate of the population is the difference between the annual birth rate and the annual death rate.

Net growth rate (r) = 6% - 3% = 3% per annum

The population after n years is given by the compound interest formula: P_present = P × (1 + r/100)ⁿ

Substituting the initial population P = 5,00,000, rate r = 3%, and time n = 2 years: P_present = 5,00,000 × (1 + 3/100)² P_present = 5,00,000 × (1.03)² P_present = 5,00,000 × 1.0609 P_present = 5,30,450

Explanation

Convert the mixed fractions into improper fractions: 2 ½ = 5/2 1 ⅞ = 15/8 9 ⅜ = 75/8 11 ⅔ = 35/3

Evaluate the expression following the BODMAS rule, which prioritizes the "of" operator before division.

First, solve the "of" operation inside the second bracket: 11 ⅔ of ⅛ = 35/3 × 1/8 = 35/24

Next, simplify the first bracket: 2 ½ ÷ 1 ⅞ = 5/2 ÷ 15/8 = 5/2 × 8/15 = 4/3

Next, simplify the second bracket: 9 ⅜ ÷ (35/24) = 75/8 ÷ 35/24 = 75/8 × 24/35 = 45/7

Finally, divide the two simplified bracket terms: 4/3 ÷ 45/7 = 4/3 × 7/45 = 28/135

Explanation

The prime numbers lying between 69 and 92 are 71, 73, 79, 83, and 89.

To find their average, use the formula:

Average = Sum of values ÷ Number of values

Sum of the prime numbers = 71 + 73 + 79 + 83 + 89 = 395

Number of prime numbers = 5

Average = 395 ÷ 5 = 79

Explanation

By the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of its length. Since P and Q are the midpoints of AB and AC respectively, PQ = ½ BC, which means BC = 2 PQ.

The point R lies on PQ such that PR:RQ = 1:3. Given that PR = 6 cm, the length of RQ is calculated as: RQ = 3 × PR = 3 × 6 = 18 cm

The total length of PQ is: PQ = PR + RQ = 6 + 18 = 24 cm

Substituting the value of PQ into the midpoint relation gives: BC = 2 × PQ = 2 × 24 = 48 cm

Explanation

The volume of a sphere is given by the formula:

V = 4/3 π r³

The diameter of the sphere is 14 cm, so the radius r is:

r = 14 / 2 = 7 cm

Substituting the values into the formula with π = 22/7:

V = 4/3 × 22/7 × 7³ V = 4/3 × 22/7 × 343 V = 4/3 × 22 × 49 V = 4312/3 cm³

Converting this improper fraction into a mixed fraction:

4312/3 = 1437 ⅓ cm³

Explanation

Correct Answer: D

Given that x = 7 + 3√5, we first calculate x² using the identity (a + b)² = a² + b² + 2ab:

x² = (7 + 3√5)² x² = 7² + (3√5)² + 2 × 7 × 3√5 x² = 49 + 45 + 42√5 x² = 94 + 42√5

Next, we find 1/x by rationalizing the denominator:

1/x = 1 / (7 + 3√5) Multiplying the numerator and denominator by (7 - 3√5): 1/x = (7 - 3√5) / (7² - (3√5)²) 1/x = (7 - 3√5) / (49 - 45) 1/x = (7 - 3√5) / 4

Now, we square 1/x to find 1/x²:

1/x² = ((7 - 3√5) / 4)² 1/x² = (7² + (3√5)² - 2 × 7 × 3√5) / 16 1/x² = (49 + 45 - 42√5) / 16 1/x² = (94 - 42√5) / 16 1/x² = (47 - 21√5) / 8

To find the value of x² + 1/x², we add the two expressions:

x² + 1/x² = (94 + 42√5) + (47 - 21√5) / 8

Expressing the first term with a common denominator of 8:

x² = (8 × (94 + 42√5)) / 8 x² = (752 + 336√5) / 8

Adding the two fractions:

x² + 1/x² = (752 + 336√5 + 47 - 21√5) / 8 x² + 1/x² = (799 + 315√5) / 8

Explanation

Total percentage of salary spent on various expenses is: 12% + 8% + 20% = 40%

The remaining balance percentage is: 100% - 40% = 60%

Given that this remaining balance equals ₹36,000, we can set up the relation for the total salary (S): 60% of S = 36,000 0.60 × S = 36,000 S = 36,000 / 0.60 = 60,000

Explanation

Let the total work be represented by the least common multiple of the individual days, which is LCM(9, 18, 30) = 90 units.

The daily work rates or efficiencies are calculated as: Rate of A, B, and C combined = 90 ÷ 9 = 10 units/day Rate of A alone = 90 ÷ 18 = 5 units/day Rate of B alone = 90 ÷ 30 = 3 units/day

The daily rate of C alone is: Rate of C = Combined rate - Rate of A - Rate of B Rate of C = 10 - 5 - 3 = 2 units/day

The number of days C alone takes to complete the total work of 90 units is: Time = Total work ÷ Rate of C = 90 ÷ 2 = 45 days.

Explanation

Using the basic trigonometric identity, sin²A + cos²A = 1, we can find the value of cos A: cos A = √(1 - sin²A) = √(1 - (3/5)²) = √(1 - 9/25) = √(16/25) = 4/5

Using the relation tan A = sin A / cos A, we get: tan A = (3/5) / (4/5) = 3/4

Now, substitute the values of cos A and tan A into the given expression: cos A + tan A - 1 = 4/5 + 3/4 - 1

To simplify, find the common denominator of 5 and 4, which is 20: cos A + tan A - 1 = 16/20 + 15/20 - 20/20 = (16 + 15 - 20) / 20 = 11/20

Explanation

The given expression is the standard trigonometric triple-angle identity: sin(3A) = 3 sin A - 4 sin³A

From the given equation: 4 sin²A - 3 = 0 sin²A = 3/4

Since A lies in the first quadrant (0 ≤ A ≤ 90°), sin A must be positive: sin A = √3/2 This corresponds to A = 60°.

Substituting A = 60° into the identity: sin(3 × 60°) = sin(180°) = 0

Alternatively, substituting the value of sin A directly into the expression: 3(√3/2) - 4(√3/2)³ = 3√3/2 - 4(3√3/8) = 3√3/2 - 3√3/2 = 0

Explanation

To find the school with the highest percentage of income from donations, the donation amount is divided by the total income for each school and multiplied by 100.

For School S: Donation income = 300 lakh rupees Total income = 300 + 200 + 125 + 120 + 200 = 945 lakh rupees Percentage of income from donations = (300 / 945) × 100 ≈ 31.75%

This percentage is the highest among all the schools, as the respective calculation for the other schools yields: School P: (140 / 792) × 100 ≈ 17.68% School Q: (250 / 954) × 100 ≈ 26.21% School R: (200 / 960) × 100 ≈ 20.83% School T: (220 / 930) × 100 ≈ 23.66%

Explanation

Let the speed of the stream be s km/h.

The speed of the boat upstream is 20 - s km/h, and the speed downstream is 20 + s km/h.

The total time taken for the round trip of 22.5 km each way is 3 hours. This gives the relation: 22.5 / (20 - s) + 22.5 / (20 + s) = 3

Factoring out 22.5 and simplifying the terms inside the parentheses: 22.5 × ( (20 + s + 20 - s) / (20² - s²) ) = 3 22.5 × ( 40 / (400 - s²) ) = 3 900 / (400 - s²) = 3 300 = 400 - s² s² = 100 s = 10

Explanation

The Cost Price (CP) of an item is given by the formula CP = SP ÷ (1 ± r/100), where SP is the selling price and r is the percentage gain or loss.

For the first item, which is sold at a 25% loss: CP₁ = 42,600 ÷ (1 - 0.25) = 42,600 ÷ 0.75 = ₹56,800

For the second item, which is sold at a 20% gain: CP₂ = 42,600 ÷ (1 + 0.20) = 42,600 ÷ 1.2 = ₹35,500

Total Cost Price (Total CP) = 56,800 + 35,500 = ₹92,300 Total Selling Price (Total SP) = 2 × 42,600 = ₹85,200

Since Total CP is greater than Total SP, the transaction results in an overall loss.

Overall Loss = Total CP - Total SP = 92,300 - 85,200 = ₹7,100

Overall Loss Percentage = (Overall Loss ÷ Total CP) × 100 Overall Loss Percentage = (7,100 ÷ 92,300) × 100 ≈ 7.69%

Rounding to one decimal place, the overall loss is 7.7%.

Explanation

The amount A received from simple interest is calculated using the formula A = P × (1 + RT/100), where P is the principal, R is the rate of interest per annum, and T is the time in years.

For the lower rate of interest investment: P = ₹4,500, R = 5%, and T = 4 years. A₁ = 4500 × (1 + (5 × 4)/100) = 4500 × 1.20 = ₹5,400.

For the higher rate of interest investment: P = ₹5,500, R = 8%, and T = 4 years. A₂ = 5500 × (1 + (8 × 4)/100) = 5500 × 1.32 = ₹7,260.

The percentage of the first amount with respect to the second amount is: Percentage = (A₁ / A₂) × 100 Percentage = (5400 / 7260) × 100 ≈ 74.38%

Rounding to the nearest integer gives 74%.

Explanation

In a circle, the tangent at any point is perpendicular to the radius through the point of contact. Therefore, the radius (r), the tangent segment (t), and the line segment joining the centre of the circle to the external point X (y) form a right-angled triangle where y is the hypotenuse.

Using Pythagoras' theorem: y² = r² + t²

Given that the radius r = 20 cm and the tangent length t = 21 cm, substituting these values into the equation yields: y² = 20² + 21² y² = 400 + 441 y² = 841 y = √841 y = 29

Explanation

Using the algebraic identity (x - 1/x)² = x² + 1/x² - 2, we substitute the given value to get (x - 1/x)² = 29 - 2 = 27. Taking the square root on both sides gives x - 1/x = ±√27, which simplifies to ±3√3.

Explanation

From the given table, the columns 1, 2, and 3 represent the companies, while the rows A, B, and C represent the products. The maximum production value in the entire table is 40. This value is located at the intersection of row C (Product C) and column 3 (Company 3), representing a production of 40 units.

Explanation

The volume of a solid sphere is calculated using the formula V = 4/3 π r³, where r is the radius of the sphere.

Given that the diameter of the sphere is 42 cm, the radius r is half of the diameter: r = 42 ÷ 2 = 21 cm

Substituting the values of the radius and π = 22/7 into the volume formula: V = 4/3 × 22/7 × 21³ V = 4/3 × 22/7 × 21 × 21 × 21

Since 3 × 7 = 21, the denominator cancels out one of the factors of 21 in the numerator: V = 4 × 22 × 21 × 21 V = 88 × 441 V = 38808 cm³

Explanation

The difference between compound interest and simple interest for a period of 3 years is given by the standard formula:

Difference = P × (r/100)² × (3 + r/100)

where P is the principal sum and r is the annual rate of interest.

Substituting the given values, Difference = 310 and r = 10:

310 = P × (10/100)² × (3 + 10/100)

310 = P × (1/10)² × (3 + 0.1)

310 = P × (1/100) × 3.1

310 = P × (31/1000)

P = 310 × (1000/31)

P = 10 × 1000

P = 10000

The sum of money is £10,000.

Explanation

The length of the direct common tangent of two circles with radii r₁ and r₂ and distance between their centres d is given by the formula:

Length of direct common tangent = √(d² - (r₁ - r₂)²)

Given that: r₁ = 8 cm r₂ = 5 cm d = 11 cm

Substitute these values into the formula:

Length = √(11² - (8 - 5)²) Length = √(121 - 3²) Length = √(121 - 9) Length = √112 Length = √(16 × 7) Length = 4√7 cm

Explanation

Using the algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Let a = 80, b = 70, and c = 50.

The given expression can be written in terms of a, b, and c as: (a³ + b³ + c³ - 3abc) / (a² + b² + c² - ab - bc - ca)

By substituting the identity into the numerator, the terms (a² + b² + c² - ab - bc - ca) in the numerator and denominator cancel out, simplifying the expression to: a + b + c

Substituting the values of a, b, and c: 80 + 70 + 50 = 200

Explanation

In a circle, any angle subtended by a diameter at the circumference is a right angle. Since EF is the diameter of the circle, the angle ∠EAF subtended by EF at point A is 90°.

In triangle AEF, the sum of all interior angles is 180°. Since ∠EAF is 90°, the sum of the remaining two angles, ∠AEF and ∠AFE, must be 90°.

Given that the two chords AE and AF are equal in length, triangle AEF is an isosceles triangle. In an isosceles triangle, the angles opposite to the equal sides are also equal, which means ∠AEF = ∠AFE.

Dividing the remaining 90° equally between these two equal angles gives ∠AEF = 90° ÷ 2 = 45°.

Explanation

To find the population in the year 2021, we use the formula for successive percentage increases:

P_final = P_initial × (1 + R₁/100) × (1 + R₂/100)

Given: P_initial = 30,000 R₁ = 15% R₂ = 25%

Substituting the values:

P₂021 = 30,000 × (1 + 15/100) × (1 + 25/100) P₂021 = 30,000 × 1.15 × 1.25 P₂021 = 34,500 × 1.25 P₂021 = 43,125

Explanation

The mean proportion between two numbers a and b is given by the geometric mean formula: Mean Proportion = √(a × b)

Substituting the given values, a = 0.049 and b = 0.9: Mean Proportion = √(0.049 × 0.9) Mean Proportion = √0.0441

To simplify the square root, express the decimal value as a fraction: 0.0441 = 441 / 10000

Taking the square root of both the numerator and the denominator: √(441 / 10000) = 21 / 100 = 0.21

Explanation

The relation between Marked Price (MP), Cost Price (CP), Profit percentage, and Discount percentage is given by the formula:

MP/CP = (100 + Profit%) / (100 - Discount%)

For the first case, profit is 45% and discount is 45%:

MP/CP = (100 + 45) / (100 - 45) = 145 / 55 = 29/11

For the second case, the discount is 50%. Let the new profit percentage be P:

MP/CP = (100 + P) / (100 - 50)

Substituting the value of MP/CP:

29/11 = (100 + P) / 50

1450/11 = 100 + P

P = 1450/11 - 100

P = 350/11 = 31 9/11 %