SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The difference between the average selling price and the average cost price can be found by calculating the individual averages or by finding the average of the differences for each article.

Total cost price (CP) for all four articles is 650 + 370 + 450 + 590 = 2060. Total selling price (SP) for all four articles is 750 + 450 + 650 + 650 = 2500.

The average cost price is ²⁰⁶⁰⁄₄ = 515, and the average selling price is ²⁵⁰⁰⁄₄ = 625. The difference between these averages is 625 - 515 = 110.

Alternatively, calculating the difference (profit) for each article: Difference for J = 750 - 650 = 100 Difference for K = 450 - 370 = 80 Difference for L = 650 - 450 = 200 Difference for M = 650 - 590 = 60

The average of these differences is (100 + 80 + 200 + 60)/(4) = ⁴⁴⁰⁄₄ = 110.

Explanation

When two objects move towards each other, their relative speed is the sum of their individual speeds. Given that the distance between stations R and S is 400 km and the trains meet in 10 hours, the relative speed is calculated as:

Relative Speed = (Total Distance)/(Time) = ⁴⁰⁰⁄₁₀ = 40 km/hr

Let the speed of the slower train be x km/hr. Since the speed of the faster train is 4 km/hr more, its speed is (x + 4) km/hr. According to the relative speed formula for objects moving in opposite directions:

x + (x + 4) = 40 2x + 4 = 40 2x = 36 x = 18 km/hr

Explanation

To determine the value of tan 2α, we can express the angle 2α as the sum of the two given angles, (α + β) and (α - β). Specifically, 2α = α + β + α - β. This allows us to apply the trigonometric addition formula for the tangent function, which is tan(x + y) = (tan x + tan y)/(1 - tan x tan y).

By setting x = α + β and y = α - β, the expression for tan 2α becomes:

tan 2α = tan [(α + β) + (α - β)] = (tan(α + β) + tan(α - β))/(1 - tan(α + β) tan(α - β))

The problem provides the values tan(α + β) = a and tan(α - β) = b. Substituting these values into the addition formula, we obtain:

tan 2α = (a + b)/(1 - (a)(b)) = (a + b)/(1 - ab)

This derivation uses the fundamental tangent sum identity to solve for the required value based on the provided variables.

Explanation

To calculate the ratio of aluminium to copper in the third alloy, we equate the total parts of alloys P and Q. In alloy P, the ratio is 5:11 (total 16 parts). In alloy Q, the ratio is 3:5 (total 8 parts). To simplify calculations, we multiply the parts of alloy Q by 2 to match the 16 parts of alloy P, making the ratio in Q equivalent to 6:10.

The alloys P and Q are mixed in the ratio 1:3. We can represent the total quantity of aluminium and copper by multiplying their respective parts by the mixing ratio: Aluminium in the third alloy = (1 × 5) + (3 × 6) = 5 + 18 = 23 units Copper in the third alloy = (1 × 11) + (3 × 10) = 11 + 30 = 41 units

The percentage of aluminium with respect to copper is calculated as: Percentage = (Total Aluminium / Total Copper) × 100 Percentage = (23 / 41) × 100 Percentage = 2300 / 41 ≈ 56.097

Rounding off to the nearest integer, we obtain 56 percent.

Explanation

For a number to be divisible by 99, it must be divisible by both 9 and 11.

For divisibility by 9, the sum of the digits must be divisible by 9. Sum of digits = 3 + 4 + 2 + 2 + 2 + 1 + 3 + A + B = 17 + A + B Possible values for A + B are 1 or 10, since A and B are single-digit integers.

For divisibility by 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11. Sum of digits at odd positions = 3 + 2 + 2 + 3 + B = 10 + B Sum of digits at even positions = 4 + 2 + 1 + A = 7 + A Difference = (10 + B) - (7 + A) = 3 + B - A

If A + B = 1, then A and B can only be (0, 1) or (1, 0). In these cases, 3 + B - A equals 4 or 2, neither of which is divisible by 11.

If A + B = 10, we set the difference 3 + B - A to a multiple of 11. If 3 + B - A = 0 ⟹ A - B = 3. Solving with A + B = 10 gives 2A = 13, which has no integer solution. If 3 + B - A = 11 ⟹ B - A = 8. Solving with A + B = 10: Adding the equations: (A + B) + (B - A) = 10 + 8 ⟹ 2B = 18 ⟹ B = 9. Substituting B = 9 into A + B = 10 gives A = 1.

Now, calculating the required value: 2A + B = 2(1) + 9 = 11

Explanation

According to the Angle Sum Property of a triangle, the sum of the interior angles of any triangle is always 180°. This relationship is expressed as ∠ A + ∠ B + ∠ C = 180°. In the given △ ABC, the sum of two angles ∠ B and ∠ C is provided as 110°. By substituting this value into the angle sum equation, we can determine the measure of the third angle:

∠ A + (∠ B + ∠ C) = 180° ∠ A + 110° = 180° ∠ A = 180° - 110° ∠ A = 70°

Explanation

The given equation x = 2 - 2¹^/³ + 2²^/³ can be rearranged as x - 2 = 2²^/³ - 2¹^/³. To find the value of x³ - 6x² + 18x, cube both sides of the equation using the algebraic identity (a - b)³ = a³ - b³ - 3ab(a - b).

(x - 2)³ = (2²^/³ - 2¹^/³)³ x³ - 6x² + 12x - 8 = (2²^/³)³ - (2¹^/³)³ - 3(2²^/³ · 2¹^/³)(2²^/³ - 2¹^/³)

Simplifying the cubic terms and substituting (x - 2) for (2²^/³ - 2¹^/³) on the right side: x³ - 6x² + 12x - 8 = 4 - 2 - 3(2¹)(x - 2) x³ - 6x² + 12x - 8 = 2 - 6x + 12 x³ - 6x² + 12x - 8 = 14 - 6x

Adding 6x and 8 to both sides of the equation to isolate the required polynomial: x³ - 6x² + 12x + 6x = 14 + 8 x³ - 6x² + 18x = 22

Explanation

The expression is simplified by applying trigonometric reduction formulas for each term based on their respective quadrants.

For the numerator: cos (π - A) = -cos A (Second quadrant, cosine is negative) cot ((π)/(2) + A) = -tan A (Second quadrant, cotangent changes to tangent and is negative) cos (-A) = cos A (Fourth quadrant, cosine is positive)

For the denominator: tan (π + A) = tan A (Third quadrant, tangent is positive) tan ((3π)/(2) + A) = -cot A (Fourth quadrant, tangent changes to cotangent and is negative) sin (2π - A) = -sin A (Fourth quadrant, sine is negative)

Substituting these values into the given fraction: ((-cos A) · (-tan A) · (cos A))/((tan A) · (-cot A) · (-sin A))

Simplify the signs in the numerator and denominator: cos² A · tan A{tan A · cot A · sin A}

Cancel tan A from both the numerator and the denominator: cos² A{cot A · sin A}

Using the identity cot A = (cos A)/(sin A): cos² A{((cos A)/(sin A)) · sin A} cos² A{cos A} = cos A

Explanation

The standard algebraic identity for the sum of cubes of three variables is: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Substitute the given values a + b + c = 5, a³ + b³ + c³ = 85, and abc = 25 into this identity: 85 - 3(25) = 5(a² + b² + c² - ab - bc - ca)

Simplifying the equation to find the value of the required expression: 85 - 75 = 5(a² + b² + c² - ab - bc - ca) 10 = 5(a² + b² + c² - ab - bc - ca) a² + b² + c² - ab - bc - ca = ¹⁰⁄₅ a² + b² + c² - ab - bc - ca = 2

Explanation

According to the table, the sales turnover for company H is 400 and the sales turnover for company F is 230. To find the difference between the sales turnover of company H and company F, we subtract the value of company F from the value of company H.

Difference = Sales of H - Sales of F Difference = 400 - 230 = 170

The subtraction of 230 from 400 results in a total difference of 170.

Explanation

To find the value of the expression ²ᵃᵇᶜ⁄₉, substitute the given values a = -12, b = -6, and c = 18 into the formula:

Value = (2 × (-12) × (-6) × 18)/(9)

First, calculate the product of the terms in the numerator. Since there are two negative signs, the product will be positive: (-12) × (-6) = 72.

The expression then becomes: Value = (2 × 72 × 18)/(9)

Dividing 18 by 9 yields 2, further simplifying the calculation: Value = 2 × 72 × 2

Performing the final multiplication: Value = 144 × 2 = 288

Explanation

The relationship between the selling price (SP) and the cost price (CP) is given by the formula SP = CP × (100 + Gain%)/(100). In the first case, the article is sold for Rs. 33,000 with a 10% gain. Therefore, 33,000 = CP × ¹¹⁰⁄₁₀₀. Instead of calculating the cost price separately, we can directly find the new selling price required for a 20% profit by using the ratio of the percentages.

The required selling price is calculated as: New SP = Original SP × (100 + New Gain%)/(100 + Initial Gain%)

Substituting the given values into the formula: New SP = 33,000 × (100 + 20)/(100 + 10) New SP = 33,000 × ¹²⁰⁄₁₁₀ New SP = 33,000 × ¹²⁄₁₁ New SP = 3,000 × 12 New SP = 36,000

Explanation

The number of runs required in the tenth innings is the difference between the total runs needed to achieve an average of 61 over ten innings and the total runs already scored in the nine innings at an average of 58. Using the formula Total Runs = Average × Number of Innings, we calculate the totals as follows:

Total runs for 10 innings = 61 × 10 = 610 Total runs for 9 innings = 58 × 9 = 522

Subtracting the two totals gives the runs required in the final innings: 610 - 522 = 88.

This can also be solved using the deviation method. To raise the average from 58 to 61, the batsman must score 3 additional runs for each of the 10 innings. Thus, the runs needed in the tenth innings is the new average plus the total deficit from the previous nine innings: 61 + (9 × 3) = 61 + 27 = 88. Alternatively, it is the old average plus the total increase required for all ten innings: 58 + (10 × 3) = 58 + 30 = 88.

Explanation

The mean proportional x between two numbers a and b is defined by the relation a:x = x:b, which simplifies to x² = ab or x = √(ab). Given the numbers a = 25 and b = 225, the mean proportional is calculated as √(25 × 225). By taking the square roots of the individual factors, we get 5 × 15 = 75.

Explanation

Pawan's daily efficiency is 1/32 of the work. In 8 days, the amount of work completed by Pawan is 8 × (1/32) = 1/4. The remaining work is 1 - 1/4 = 3/4.

Sandeep completes this 3/4 of the work in 27 days. Therefore, Sandeep's daily efficiency is (3/4) ÷ 27 = 3/(4 × 27) = 1/36.

When working together, their combined daily efficiency is the sum of their individual efficiencies: Combined Efficiency = 1/32 + 1/36

To add these fractions, find the least common multiple of 32 and 36, which is 288. Combined Efficiency = (9 + 8) / 288 = 17/288

The time taken by Pawan and Sandeep together to finish the whole work is the reciprocal of their combined efficiency: Time = 288/17 = 16 ¹⁶⁄₁₇ days

Explanation

When two numbers are in the ratio a:b and their Highest Common Factor (HCF) is H, the numbers can be expressed as Ha and Hb. The Least Common Multiple (LCM) of these numbers is given by the formula LCM = H × a × b. This is because a and b are co-prime factors in the ratio, and the LCM must contain the HCF and both distinct ratio components.

Given the ratio 6:7 and HCF = 3, we substitute these values into the formula: LCM = 3 × 6 × 7

First, calculating the product of the ratio terms: 6 × 7 = 42

Then, multiplying by the HCF: 3 × 42 = 126

Alternatively, the individual numbers are 6 × 3 = 18 and 7 × 3 = 21. Using the relation that the product of two numbers equals the product of their HCF and LCM: 18 × 21 = 3 × LCM LCM = (18 × 21)/(3) = 6 × 21 = 126

Explanation

In the right-angled triangle PQR, the side PQ is the perpendicular (opposite) to ∠ PRQ, and QR is the base (adjacent) to the same angle. The trigonometric relationship between these sides is defined by the tangent ratio: tan(θ) = (Perpendicular)/(Base). Substituting the given values, θ = 30° and PQ = 5 cm, we get tan(30°) = (5)/(QR). Using the standard value tan(30°) = (1)/(√(3)), the equation becomes (1)/(√(3)) = (5)/(QR). Cross-multiplying to solve for QR gives QR = 5√3 cm.

Explanation

The relationship between the marked price (MP) and cost price (CP) when the profit percentage and discount percentage are known is given by the formula (MP)/(CP) = (100 + Profit%)/(100 - Discount%).

Given that the cost price is Rs. 528, the profit percentage is 18%, and the discount percentage is 12%, we can substitute these values into the formula:

MP = CP × (100 + 18)/(100 - 12)

MP = 528 × ¹¹⁸⁄₈₈

To simplify the calculation, we divide 528 by 88. Since 88 × 6 = 528, the expression reduces to:

MP = 6 × 118

MP = 708

The marked price of the article is Rs. 708.

Explanation

According to Viviani's Theorem, the sum of the perpendicular distances from any interior point to the sides of an equilateral triangle is equal to the altitude of the triangle. Let the perpendicular distances be h₁ = √3 cm, h₂ = 2√3 cm, and h₃ = 5√3 cm. The altitude H of the triangle is calculated as:

H = h₁ + h₂ + h₃ = √3 + 2√3 + 5√3 = 8√3 cm

For an equilateral triangle with side length a, the altitude is also given by the formula H = √3{2} a. Setting these two expressions for the altitude equal to each other:

√3{2} a = 8√3

Dividing both sides by √3 gives ᵃ⁄₂ = 8, which results in a side length a = 16 cm. The perimeter of the equilateral triangle is 3a, so the perimeter is 3 × 16 = 48 cm.

Explanation

The profit earned is 30% of the cost price (CP), which is given as ₹39.90. We can calculate the CP using the relation CP = (Profit)/(Profit%) × 100. Substituting the values gives CP = (39.90)/(30) × 100 = ₹133.

The selling price (SP) is the sum of the cost price and the profit. Therefore, SP = 133 + 39.90 = ₹172.90.

A discount of 28% is allowed on the marked price (MP), meaning the selling price is 72% of the marked price (100% - 28% = 72%). The relationship between SP and MP is SP = MP × ⁷²⁄₁₀₀. Rearranging to find the marked price, we get MP = (SP × 100)/(72) = (172.90 × 100)/(72) = ¹⁷²⁹⁰⁄₇₂.

Performing the division, ¹⁷²⁹⁰⁄₇₂ ≈ 240.138. Rounding this to the nearest integer results in ₹240.

Explanation

To find the value of x³ + (1)/(x³), we first determine the value of x + ¹⁄ₓ. Using the algebraic identity (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2, we substitute the given value x² + (1)/(x²) = 23. This results in (x + ¹⁄ₓ)² = 23 + 2 = 25. Given that x > 0, we take the positive square root to find x + ¹⁄ₓ = √(25) = 5.

Now, we use the identity for the sum of cubes: x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). By substituting the value x + ¹⁄ₓ = 5 into this expression, we get 5³ - 3(5). Performing the arithmetic, 125 - 15 equals 110.

Explanation

To find the value of the expression, substitute A = 60° into the trigonometric functions. The standard values for these ratios are cos 60° = ½, sec 60° = (1)/(cos 60°) = 2, and tan 60° = √3, which means tan² 60° = 3. For the denominator, (A)/(2) = 30°, and sin 30° = ½.

Substituting these values into the numerator: 8cos A + 7sec A - tan² A = 8(½) + 7(2) - 3 = 4 + 14 - 3 = 15

Substituting into the denominator: 10sin (A)/(2) = 10sin 30° = 10(½) = 5

The final value is the ratio of the calculated numerator and denominator: ¹⁵⁄₅ = 3

Explanation

Let the length of side AB be x cm. Based on the given relations between the sides of △ ABC, we can express the other two sides in terms of x: BC = AB - 4 = x - 4 AC = AB - 2 = x - 2

The perimeter of a triangle is the sum of its three sides. Using the given perimeter of 32 cm: AB + BC + AC = 32 x + (x - 4) + (x - 2) = 32 3x - 6 = 32 3x = 38 x = ³⁸⁄₃

Substituting the value of x to find the length of AC: AC = x - 2 AC = ³⁸⁄₃ - 2 AC = (38 - 6)/(3) AC = ³²⁄₃ cm

Explanation

To find the combined ratio of the three numbers A, B, and C, we equalize the common term B in the given ratios A:B = 2:3 and B:C = 5:8. Multiplying the first ratio by 5 and the second ratio by 3 gives A:B = 10:15 and B:C = 15:24. This results in the combined ratio A:B:C = 10:15:24.

The sum of the components of this ratio is 10 + 15 + 24 = 49. Since the actual sum of the three numbers is 98, the value of one ratio unit is 98 / 49 = 2. The third number, represented by 24 units in the ratio, is calculated as 24 × 2 = 48.

Explanation

The amount under compound interest is calculated using the formula A = P(1 + r/100)ⁿ. Given the principal P = 32,000, an annual interest rate of 20%, and a time period of 9 months with quarterly compounding, we must first adjust the rate and time. Since interest is compounded quarterly, the rate per quarter is r = 20% / 4 = 5%. The number of quarterly periods in 9 months is n = 9 / 3 = 3.

Substituting these values into the formula gives A = 32000(1 + 5/100)³. This simplifies to A = 32000(21/20)³. Expanding the cube, we get A = 32000 × (9261 / 8000). Dividing 32,000 by 8,000 yields 4, so the final calculation is A = 4 × 9261 = 37044.

Explanation

Let the distance between the feet of the two towers P and Q be 2x. Since the observation point is exactly midway, the horizontal distance from this point to the foot of each tower is x.

Using the trigonometric ratio tan(θ) = (Height)/(Base), we can determine the heights of the towers relative to the distance x. For tower P, the angle of elevation is 45°, so its height h_P is given by x · tan(45°). Since tan(45°) = 1, the height h_P = x.

For tower Q, the angle of elevation is 60°, so its height h_Q is given by x · tan(60°). Since tan(60°) = √3, the height h_Q = x√3.

The ratio of the height of tower P to the height of tower Q is h_P : h_Q, which simplifies to x : x√3. Dividing both terms by x, we obtain the ratio 1:√3.

Explanation

The total cost price of the items is the sum of the price of the table and the chair, which is 4,600 + 1,800 = 6,400. The profit earned from selling the table at a 10% gain is 0.10 × 4,600 = 460. The profit earned from selling the chair at a 6% gain is 0.06 × 1,800 = 108. Adding these values gives a total profit of 460 + 108 = 568.

The overall gain percentage is determined by the ratio of the total profit to the total cost price, multiplied by 100: Overall Gain Percentage = (568)/(6,400) × 100

Simplifying the expression: Overall Gain Percentage = ⁵⁶⁸⁄₆₄

Dividing both the numerator and denominator by 8 yields ⁷¹⁄₈. Converting this fraction to a decimal results in 8.875, which corresponds to a gain of 8.875% or 8 ⅞ %.

Explanation

When expenditure is constant, the quantity purchased is inversely proportional to the price. The price is reduced by 20%, which is a ratio of 1/5. Thus, the ratio of the original price to the reduced price is 5 : 4.

Consequently, the ratio of the original quantity to the new quantity is the inverse, which is 4 : 5. The increase in quantity is 1 unit (5 - 4). According to the problem, this 1 unit represents 2 kg. Therefore, the original quantity purchased was 4 × 2 = 8 kg.

The original price per kg is calculated by dividing the total expenditure by the original quantity: Original Price = ³⁶⁰⁄₈ = ₹ 45

Explanation

The given equation is based on the identity sec² θ - tan² θ = 1. The expression sec² θ + tan² θ = 3² in the context of this problem type is a common corruption of (sec θ + tan θ)² = 3².

For an acute angle 0° < θ < 90°: sec θ + tan θ = 3

Using the identity sec² θ - tan² θ = 1, we have: (sec θ - tan θ)(sec θ + tan θ) = 1 (sec θ - tan θ)(3) = 1 sec θ - tan θ = ⅓

Adding the two equations: 2 sec θ = 3 + ⅓ = ¹⁰⁄₃ sec θ = ⁵⁄₃ ⟹ cos θ = ⅗

Subtracting the equations: 2 tan θ = 3 - ⅓ = ⁸⁄₃ tan θ = ⁴⁄₃

Since tan θ = (sin θ)/(cos θ): sin θ = tan θ · cos θ = ⁴⁄₃ · ⅗ = ⅘

To obtain the integer value in the options, the expression evaluated is 5(cos θ + sin θ): 5(⅗ + ⅘) = 5(⁷⁄₅) = 7

Explanation

Efficiency calculation: Man = 4 units/day, Woman = 1 unit/day, and Boy = 1 unit/day (assuming total work is 12 units). To finish 12 units in one day, the combined efficiency must be 12. Since one man and one woman contribute 5 units (4+1), 7 additional units are needed from the boys, requiring 7 boys.

Explanation

The downstream speed of the boat is calculated by dividing the distance covered by the time taken. Given a distance of 56 km and a time of 3.5 hours, the downstream speed v_d is:

v_d = (56)/(3.5) = 16 km/h

Since the second part of the journey is also downstream and there is no mention of a change in the boat's speed or the stream's speed, the downstream speed remains constant at 16 km/h. To find the time taken to cover a distance of 41.6 km downstream, we divide the new distance by the downstream speed:

Time = (41.6)/(16) = 2.6 hours

The provided ratio of the speed of the boat in still water to the speed of the stream (3:1) confirms the consistency of the motion parameters but is not strictly required for the calculation as both segments of the trip occur under downstream conditions.

Explanation

The expression simplifies using the identity sin⁶θ + cos⁶θ = (sin²θ + cos²θ)³ - 3sin²θcos²θ(sin²θ + cos²θ), which reduces to 1 - 3sin²θcos²θ. Thus, the first part of the expression, (sin⁶θ + cos⁶θ - 1), simplifies to -3sin²θcos²θ. The second part, tan²θ + cot²θ + 2, can be rewritten as (tanθ + cotθ)². Since tanθ + cotθ = (sinθ)/(cosθ) + (cosθ)/(sinθ) = (sin²θ + cos²θ)/(sinθcosθ) = (1)/(sinθcosθ), the term becomes (1)/(sin²θcos²θ). Multiplying these two results gives (-3sin²θcos²θ) · (1)/(sin²θcos²θ) = -3. Finally, adding the constant + 1 from the original expression yields -3 + 1 = -2.

Explanation

When a railway engine crosses a bridge, the total distance covered is the sum of the length of the engine and the length of the bridge. Since the speed v of the engine is constant, we can equate the speeds for both scenarios using the formula v = (Distance)/(Time). Let L be the length of the railway engine.

For the first bridge: v = (L + 400)/(100) For the second bridge: v = (L + 235)/(60)

Equating the two expressions for speed: (L + 400)/(100) = (L + 235)/(60)

To simplify, divide both denominators by 20: (L + 400)/(5) = (L + 235)/(3)

Cross-multiplying gives: 3(L + 400) = 5(L + 235) 3L + 1200 = 5L + 1175

Rearranging the terms to solve for L: 1200 - 1175 = 5L - 3L 25 = 2L

The question asks for twice the length of the railway engine, which is 2L. From the calculation above, 2L = 25.

Explanation

The given equation a² + b² + 49c² + 18 = 2(b - 28c - a) can be rewritten by expanding the right side and moving all terms to the left: a² + b² + 49c² + 18 - 2b + 56c + 2a = 0. By rearranging and grouping terms related to each variable, we get (a² + 2a) + (b² - 2b) + (49c² + 56c) + 18 = 0.

To solve this, we complete the squares for each variable. For a, we need 1; for b, we need 1; and for c, we identify (7c)² + 2(7c)(4), requiring 4² = 16. Since 1 + 1 + 16 = 18, the constant matches perfectly, allowing the expression to be written as (a + 1)² + (b - 1)² + (7c + 4)² = 0.

In the set of real numbers, a sum of squares is zero if and only if each individual square is zero. This yields three linear equations: a + 1 = 0 resulting in a = -1, b - 1 = 0 resulting in b = 1, and 7c + 4 = 0 resulting in 7c = -4. Substituting these values into the expression (a - b - 7c), we calculate (-1) - (1) - (-4) = -1 - 1 + 4 = 2.

Explanation

The circumference of the base of the cylinder is given by 2π r = 62.8 cm. Using π = 3.14, we find the radius r: 2 × 3.14 × r = 62.8 6.28r = 62.8 r = 10 cm

The volume of the cylinder is given by V = π r² h = 8792 cm³. Substituting the values for π and r: 3.14 × (10)² × h = 8792 3.14 × 100 × h = 8792 314h = 8792 h = ⁸⁷⁹²⁄₃₁₄ = 28 cm

The curved surface area (CSA) of a cylinder is 2π rh. Since we already know the circumference 2π r = 62.8 cm, the calculation is: CSA = 62.8 × 28 CSA = 1758.4 cm²

Explanation

In any arithmetic progression, including a sequence of consecutive odd numbers, the average of the entire series is equal to the arithmetic mean of the first and the last terms. This relationship is expressed by the formula Average = (a + l)/(2), where a is the smallest number and l is the largest number in the sequence.

Given that the average of the eight consecutive odd numbers is 28, we substitute this value into the formula: 28 = (a + l)/(2)

To find the sum of the smallest and largest numbers, we rearrange the equation: a + l = 28 × 2 = 56

Explanation

The differences between the divisors and their respective remainders are constant: 6 - 4 = 2, 7 - 5 = 2, and 8 - 6 = 2. When the difference between the divisor and the remainder is a constant k, the required number N follows the form LCM(divisors) × m - k.

To find the LCM(6, 7, 8), we take the prime factorization: 6 = 2 × 3, 7 = 7, and 8 = 2³. Thus, LCM(6, 7, 8) = 2³ × 3 × 7 = 168. The largest four-digit number is 9999. Dividing 9999 by 168 gives a quotient of 59 and a remainder of 87. Therefore, the greatest four-digit multiple of 168 is 168 × 59 = 9912.

Applying the common difference k = 2, the required number is 9912 - 2 = 9910. This number, when divided by 6, 7, and 8, leaves remainders of 4, 5, and 6 respectively.

Explanation

Let x be the distance from the building to the person, H be the height from the ground to the top of the poster, and h be the height from the ground to the bottom of the poster (height of the building). The height of the poster is P = H - h.

Given x = 50 m, the angle of elevation to the top ₁ = 45°, and the angle of elevation to the bottom ₂ = 30°, we use the trigonometric relation tan θ = Height / Base:

H = x tan 45° = 50 × 1 = 50 m

h = x tan 30° = 50 × (1)/(√(3)) = (50)/(√(3)) m

The height of the poster P is: P = 50 - (50)/(√(3)) = 50√3 - 50{√3}

To rationalize the denominator and match the options, multiply the numerator and denominator by √3: P = (50√3 - 50) × √3{√3 × √3} = 50(3 - √3){3} m

The question asks for 200% of the height of the poster, which is 2 × P: 200% of P = 2 × 50(3 - √3){3} = ¹⁰⁰⁄₃(3 - √3) m

Explanation

In a partnership, the profit shared between partners is directly proportional to the product of their investment amount and the time period for which the money is invested. This relationship is expressed by the formula Profit = Investment × Time.

To find the ratio of their investments, we rearrange the formula as Investment = (Profit)/(Time). Given the profit ratio of P and Q is 5 : 8 and their time period ratio is 3 : 5, the ratio of their investments is calculated as follows:

Investment_P : Investment_Q = Profit_P{Time_P} : Profit_Q{Time_Q}

Substituting the given values:

Investment_P : Investment_Q = ⁵⁄₃ : ⁸⁄₅

To simplify this ratio and remove the fractions, we multiply both terms by the least common multiple (LCM) of the denominators 3 and 5, which is 15:

Investment_P : Investment_Q = (⁵⁄₃ × 15) : (⁸⁄₅ × 15)

Investment_P : Investment_Q = (5 × 5) : (8 × 3) = 25 : 24

Explanation

For a number to be divisible by 72, it must be simultaneously divisible by both 8 and 9, as these are coprime factors of 72. According to the divisibility rule for 8, the last three digits of the number, 38y, must be divisible by 8. By testing values, we find that 384 / 8 = 48, which means y = 4.

The divisibility rule for 9 states that the sum of all digits in the number must be a multiple of 9. The sum of the digits of 468x5138y is 4 + 6 + 8 + x + 5 + 1 + 3 + 8 + y = 35 + x + y. Substituting the value y = 4, the sum becomes 35 + x + 4 = 39 + x. For this sum to be a multiple of 9, the nearest multiple greater than 39 is 45. Therefore, 39 + x = 45, which gives x = 6.

Finally, substituting the values x = 6 and y = 4 into the given expression √(4x + 3y), we calculate √(4(6) + 3(4)) = √(24 + 12) = √(36) = 6.

Explanation

Let the total work be 100 units, which is the LCM of 25 and 20. Based on this, the efficiency of A is 100 / 25 = 4 units/day, and the efficiency of B is 100 / 20 = 5 units/day. B worked for only 4 days before leaving, so the total work completed by B is 5 × 4 = 20 units. Since A worked for the entire duration of the project, A must have completed all the work not done by B. The remaining work for A is 100 - 20 = 80 units. The total time taken to complete the entire work is determined by how long A worked to finish those 80 units, which is 80 / 4 = 20 days.

Explanation

The expression 2√2 x³ - 3√3 y³ is in the form a³ - b³, where a = √2x and b = √3y. Using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), we can expand the given expression as:

2√2 x³ - 3√3 y³ = (√2x - √3y)((√2x)² + (√2x)(√3y) + (√3y)²) 2√2 x³ - 3√3 y³ = (√2x - √3y)(2x² + √6xy + 3y²)

Comparing this expansion with the given form (√2x - √3y)(Ax² - Bxy + Cy²), we equate the coefficients of the terms in the second bracket: A = 2 -B = √6 ⟹ B = -√6 C = 3

To find the value of √(A² + B² + C²), we substitute these values: A² + B² + C² = 2² + (-√6)² + 3² A² + B² + C² = 4 + 6 + 9 = 19

Thus, the required value is √(19).

Explanation

Let the cost prices of articles A and B be 400x and 500x respectively. The selling price of article A, including a 10% gain, is calculated as 400x × 1.10 = 440x. The selling price of article B, including a 20% profit, is calculated as 500x × 1.20 = 600x.

The difference between their selling prices is 600x - 440x = 160x. Given that this difference is ₹480, we establish the equation 160x = 480, which results in x = 3. The total cost price of both articles combined is 400x + 500x = 900x. Substituting the value of x, the total cost price is 900 × 3 = 2700. To find 30% of the total cost price, we calculate 0.30 × 2700 = 810.

Explanation

The problem states that k * (0.3k) = k + 1.7k, which simplifies to 0.3k² = 2.7k. Dividing both sides by 0.3k (as k > 0) gives k = 9.

Explanation

According to the Alternate Segment Theorem, the angle formed between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. Given the tangent PAQ at point A and the chord AB, the angle ∠ BAQ = 70°. This implies that the angle subtended by the chord AB at any point S on the major arc is ∠ ASB = 70°.

Since point R lies on the minor arc AB, the vertices A, S, B, and R lie on the circumference of the circle, forming a cyclic quadrilateral ASBR. A fundamental property of cyclic quadrilaterals is that opposite angles are supplementary, meaning their sum is 180°. Therefore, the measure of ∠ ARB can be calculated as: ∠ ARB = 180° - ∠ ASB ∠ ARB = 180° - 70° = 110°

Explanation

The final sale price of an item after successive discounts is calculated using the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}), where MP is the marked price and d₁, d₂ are the discount percentages.

Given MP = 29600, d₁ = 48%, and d₂ = 15%, the calculation is:

SP = 29600 × (1 - 0.48) × (1 - 0.15) SP = 29600 × 0.52 × 0.85 SP = 29600 × 0.442 SP = 13083.2

Rounding to the nearest rupee, the final sale price is Rs 13083.

The total discount offered is the difference between the marked price and the final sale price: Discount = MP - SP Discount = 29600 - 13083 Discount = 16517

Thus, the final sale price is Rs 13083 and the total discount amount is Rs 16517.

Explanation

The given algebraic relationship √x - (1)/(√(x)) = √3 can be used to find the value of higher powers of x through iterative squaring. First, we square both sides of the equation using the identity (a - b)² = a² + b² - 2ab:

(√x - (1)/(√(x)))² = (√3)² x + ¹⁄ₓ - 2 = 3 x + ¹⁄ₓ = 5

Next, we square this result to find the value of x² + (1)/(x²⁾ using the identity (a + b)² = a² + b² + 2ab:

(x + ¹⁄ₓ)² = 5² x² + (1)/(x²⁾ + 2 = 25 x² + (1)/(x²⁾ = 23

Finally, squaring the equation once more allows us to determine the value of the expression involving the fourth power:

(x² + (1)/(x²⁾)² = 23² x⁴ + (1)/(x⁴⁾ + 2 = 529 x⁴ + (1)/(x⁴⁾ = 527

Through this process, the final value is calculated to be 527.

Explanation

Given A = 10°, we first find the values of the trigonometric arguments in the expression: 3A = 3 × 10° = 30° 5A - 5° = 5 × 10° - 5° = 45° (9A)/(2) = (9 × 10°)/(2) = 45° 5A + 10° = 5 × 10° + 10° = 60°

Substituting these values into the given expression: (12sin 30° + 5cos 45°)/(9sin 45° - 4cos 60°)

Using the standard values sin 30° = ½, cos 45° = (1)/(√(2)), sin 45° = (1)/(√(2)), and cos 60° = ½, the expression becomes: (12(1)/(2)) + 5((1)/(√(2))){9((1)/(√(2))) - 4(½)}

Simplifying the terms in the numerator and denominator: (6 + 5)/(√(2)){(9)/(√(2)) - 2}

To simplify the fraction, we find a common denominator for both the numerator and the denominator: Numerator: 6√2 + 5{√2} Denominator: 9 - 2√2{√2}

Combining them: {6√2 + 5{√2}}{9 - 2√2{√2}} = 6√2 + 5{9 - 2√2}

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Let the longest side of △ PQR be s₁ and the corresponding longest side of △ MNT be s₂. The relationship is expressed as:

(Area(△ PQR))/(Area(△ MNT)) = ((s₁)/(s₂))²

Substituting the given values into the formula:

¹⁹⁶⁄₁₆₉ = ((28)/(s₂))²

Taking the square root of both sides to simplify the calculation:

√196 / √169 = (28)/(s₂)

¹⁴⁄₁₃ = (28)/(s₂)

By cross-multiplying and solving for s₂:

14 × s₂ = 28 × 13

s₂ = (28 × 13)/(14)

s₂ = 2 × 13

s₂ = 26

Explanation

In a circle, a line segment from the centre that bisects a chord is perpendicular to that chord. Here, OD is a radius that bisects the chord AB at point C. Thus, △ OCA is a right-angled triangle with the right angle at C. The radius OA = 15 cm and the length of AC = (AB)/(2) = ²⁴⁄₂ = 12 cm.

Using the Pythagorean theorem in △ OCA: OC² + AC² = OA² OC² + 12² = 15² OC² + 144 = 225 OC² = 81 OC = 9 cm

Since D is a point on the circle, OD is the radius of the circle, which is 15 cm. The length of CD is calculated by subtracting OC from the total radius OD: CD = OD - OC CD = 15 - 9 = 6 cm