SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

To simplify the expression (sin 2A)/(1+cos 2A), we apply the standard trigonometric double angle identities: sin 2A = 2 sin A cos A cos 2A = 2 cos² A - 1

Substituting these values into the given expression: (2 sin A cos A)/(1 + (2 cos² A - 1))

In the denominator, 1 and -1 cancel each other out, leaving: (2 sin A cos A)/(2 cos² A)

By dividing both the numerator and the denominator by 2 cos A, the expression simplifies to: (sin A)/(cos A) = tan A

Explanation

The expression cot A + tan A can be rewritten using sine and cosine as (cos A)/(sin A) + (sin A)/(cos A). Finding a common denominator gives (cos² A + sin² A)/(sin A cos A). By the Pythagorean identity sin² A + cos² A = 1, the denominator simplifies to (1)/(sin A cos A).

Substituting this into the original fraction along with sec A = (1)/(cos A), we have:

(sec A)/(cot A + tan A) = (1)/(cos A){(1)/(sin A cos A)} = (1)/(cos A) × (sin A cos A) = sin A

Squaring this result gives sin² A. From the identity sin² A + cos² A = 1, it follows that sin² A = 1 - cos² A.

Explanation

The expression 1 + tan A tan(A/2) can be simplified using the trigonometric identity tan θ = sin θ / cos θ. By substituting this into the expression, we obtain:

1 + (sin A)/(cos A) · (sin(A/2))/(cos(A/2))

To combine these terms, we use the common denominator cos A cos(A/2):

(cos A cos(A/2) + sin A sin(A/2))/(cos A cos(A/2))

The numerator follows the standard identity cos(x - y) = cos x cos y + sin x sin y. By setting x = A and y = A/2, the numerator simplifies to cos(A - A/2), which is cos(A/2). Substituting this back into the fraction gives:

(cos(A/2))/(cos A cos(A/2))

The term cos(A/2) cancels out in the numerator and denominator, leaving 1 / cos A. According to the reciprocal identity, 1 / cos A is equal to sec A.

Explanation

To find the least value to be added to 1812 so that it becomes divisible by 7, 11, and 14, we first determine the Least Common Multiple (LCM) of the given divisors. Since 14 is a multiple of 7, the LCM of 7 and 14 is 14. Thus, the LCM(7, 11, 14) = LCM(14, 11) = 14 × 11 = 154.

Next, we divide 1812 by 154 to identify the remainder: 1812 = (154 × 11) + 118

The remainder obtained is 118. The least value that must be added to the original number to make it exactly divisible by 154 is the difference between the divisor and the remainder: Value to be added = 154 - 118 = 36

When 36 is added to 1812, the sum is 1848, which is 154 × 12, satisfying the condition of divisibility by 7, 11, and 14.

Explanation

The ratio of the efficiency of A to B is 1.5:1, which simplifies to 3:2. Since efficiency is inversely proportional to the time taken to complete a task, the ratio of the time taken by A to B is 2:3. Let the time taken by A be 2x days and the time taken by B be 3x days. According to the problem, the difference in time is 8 days, so 3x - 2x = 8, which gives x = 8. Therefore, A takes 2 × 8 = 16 days and B takes 3 × 8 = 24 days to complete the work individually.

To find the total work, we take the L.C.M. of 16 and 24, which is 48 units. The efficiency of A is 48 / 16 = 3 units per day, and the efficiency of B is 48 / 24 = 2 units per day. When they work on alternate days starting with A, they complete 3 + 2 = 5 units of work every 2 days. In 9 such cycles (18 days), the work completed is 9 × 5 = 45 units. The remaining work is 48 - 45 = 3 units. On the 19th day, it is A's turn to work. Since A's efficiency is 3 units per day, A will finish the remaining 3 units in exactly 1 day. Thus, the total time taken is 18 + 1 = 19 days.

Explanation

The area of a rhombus is calculated using the formula A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Let d₁ be the length of the larger diagonal. According to the problem, the length of the other diagonal is 70% of the larger one, so d₂ = ⁷⁰⁄₁₀₀ d₁ = ⁷⁄₁₀ d₁.

Substituting the value of d₂ into the area formula gives A = ½ × d₁ × ⁷⁄₁₀ d₁ = ⁷⁄₂₀ d₁². The ratio of the area of the rhombus to the square of the larger diagonal is (A)/(d₁₎_² = 7/20 × d₁²{d₁²} = ⁷⁄₂₀. This results in the ratio 7 : 20.

Explanation

The relationship between the Marked Price (MP) and Cost Price (CP) is established by the formula MP/CP = (100 + P) / (100 - D), where P is the profit percentage and D is the discount percentage. For the first case where the shopkeeper earns a 60% profit with a 20% discount, the ratio is calculated as: MP/CP = (100 + 60) / (100 - 20) = 160 / 80 = 2/1

This ratio shows that the Marked Price is exactly twice the Cost Price (MP = 2 × CP). If the shopkeeper changes the discount to 40%, the new Selling Price (SP) can be found by applying this discount to the Marked Price: SP = MP × (100 - 40) / 100 = (2 × CP) × 0.6 = 1.2 × CP

To find the new profit percentage, compare the new Selling Price to the Cost Price using the formula Profit % = ((SP - CP) / CP) × 100: New Profit % = ((1.2 × CP - CP) / CP) × 100 = (0.2 × CP / CP) × 100 = 20%

Explanation

Given the proportion A/3 = B/2 = C/5, we can assign a constant k such that A = 3k, B = 2k, and C = 5k. Substituting these values into the terms of the required ratio (C + A)² : (A + B)² : (B + C)²:

The first term is (5k + 3k)² = (8k)² = 64k². The second term is (3k + 2k)² = (5k)² = 25k². The third term is (2k + 5k)² = (7k)² = 49k².

The resulting ratio is 64k² : 25k² : 49k². After dividing each term by the common factor k², the ratio simplifies to 64 : 25 : 49.

Explanation

The sum of the ages of the father, mother, and son 5 years ago was 35 × 3 = 105 years. To find their current total age, we add 5 years for each of the three family members: 105 + (5 × 3) = 120 years.

Similarly, the sum of the ages of the father and mother 3 years ago was 46 × 2 = 92 years. Their current total age is found by adding 3 years for each of them: 92 + (3 × 2) = 98 years.

The present age of the son is the difference between the current total age of the family and the current total age of the parents: 120 - 98 = 22 years.

Explanation

When the cost price of A articles is equal to the selling price of B articles, the profit percentage can be calculated using the direct formula: Profit% = (A - B)/(B) × 100 In this scenario, the number of articles for the cost price A is 60, and the number of articles for the selling price B is x. The profit is given as 20%. Substituting these values into the formula: 20 = (60 - x)/(x) × 100 By dividing both sides by 100, the equation simplifies to: 0.2 = (60 - x)/(x) 0.2x = 60 - x Adding x to both sides to group the variables: 1.2x = 60 x = (60)/(1.2) To simplify the division, multiply both the numerator and denominator by 10: x = ⁶⁰⁰⁄₁₂ x = 50

Explanation

The gap between the passing requirement and the candidate's actual score is 55% - 45% = 10% of the total marks. The problem states this gap is equal to 40 marks. By applying the percentage formula where M represents the maximum marks: 10% of M = 40 0.10 × M = 40 M = (40)/(0.10) = 400

Explanation

The first man starts at 11:00 a.m. and runs at 10 km/hr for 2 hours, covering 10 × 2 = 20 km by 1:00 p.m. He then rests until 2:00 p.m., remaining at the 20 km mark. Between 2:00 p.m. and 4:00 p.m., he runs for another 2 hours, covering an additional 10 × 2 = 20 km, reaching a total distance of 40 km from point P. At 4:00 p.m., he begins another 1-hour rest period, staying at 40 km until 5:00 p.m.

The second man starts from point P at 2:00 p.m. and runs non-stop at 15 km/hr. To catch the first man, he must cover the distance the first man has reached. At 4:00 p.m., the second man has covered 15 × 2 = 30 km, which is less than the first man's 40 km. The time t required for the second man to reach the 40 km mark is calculated as t = Distance / Speed = 40 / 15 = 8/3 hours.

Converting 8/3 hours into hours and minutes gives 2 hours and 40 minutes. Adding this duration to the second man's start time of 2:00 p.m. results in 4:40 p.m. Since the first man is resting at the 40 km mark from 4:00 p.m. to 5:00 p.m., he is caught exactly at 4:40 p.m.

Explanation

The difference between Compound Interest (CI) and Simple Interest (SI) for two years (D₂) and three years (D₃) is given by the formulas: D₂ = P × ((R)/(100))² D₃ = P × ((R)/(100))² × (3 + (R)/(100))

By dividing the equation for D₃ by the equation for D₂, the ratio of the differences can be simplified: D₃{D₂} = 3 + (R)/(100)

Substituting the given ratio 23 : 7 into the equation: ²³⁄₇ = 3 + (R)/(100)

Subtracting 3 from both sides to isolate the rate component: (R)/(100) = ²³⁄₇ - 3 (R)/(100) = (23 - 21)/(7) (R)/(100) = ²⁄₇

Multiplying by 100 to find the annual interest rate R: R = ²⁰⁰⁄₇

Explanation

The given equation x²{yz} + y²{zx} + z²{xy} = 3 can be simplified by taking the common denominator xyz: x³ + y³ + z³{xyz} = 3 x³ + y³ + z³ = 3xyz x³ + y³ + z³ - 3xyz = 0

We apply the standard algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Substituting the value from our simplification: (x + y + z)(x² + y² + z² - xy - yz - zx) = 0

For this product to be zero, one of the factors must be zero. In the case where x + y + z = 0, we find the value of the required expression: (x + y + z)³ = 0³ = 0

Explanation

The given equation is x¹^/⁴ + x¹^/⁴ = 2. By combining the terms on the left side, we get 2x¹^/⁴ = 2, which simplifies to x¹^/⁴ = 1. Raising both sides to the power of 4, we find x = 1⁴ = 1.

To find the value of the expression x⁸¹ + (1)/(x⁸¹⁾, we substitute x = 1 into it: 1⁸¹ + (1)/(1⁸¹⁾ = 1 + ¹⁄₁ = 1 + 1 = 2.

Alternatively, if the expression follows the standard algebraic form a + ¹⁄ₐ = 2, the only real solution is a = 1. In this case, setting a = x¹^/⁴ leads to x¹^/⁴ = 1, which again gives x = 1. Substituting this value into any expression of the form xⁿ + (1)/(xⁿ⁾ will always result in 1ⁿ + (1)/(1ⁿ⁾ = 2.

Explanation

Summing the given equations a(a + b + c) = 45, b(a + b + c) = 75, and c(a + b + c) = 105 gives (a + b + c)(a + b + c) = 45 + 75 + 105, which simplifies to (a + b + c)² = 225. Let S = a + b + c, so S² = 225. From the original equations, we find a = 45/S, b = 75/S, and c = 105/S. Squaring these terms gives a² = 2025/S², b² = 5625/S², and c² = 11025/S². Summing these values results in a² + b² + c² = (2025 + 5625 + 11025) / 225, which simplifies to 18675 / 225 = 83.

Explanation

Using the trigonometric identity sin(90° - θ) = cosθ and the definition tanθ = (sinθ)/(cosθ), the given expression sin²(90° - θ) - (sin(90° - θ) sinθ)/(tanθ) can be simplified term by term.

The first term is: sin²(90° - θ) = (cosθ)² = cos²θ

The second term is: (sin(90° - θ) sinθ)/(tanθ) = (cosθ sinθ)/(sinθ){cosθ}

By reciprocating the denominator, the second term becomes: (cosθ sinθ cosθ)/(sinθ) = cos²θ

Combining the simplified terms: cos²θ - cos²θ = 0

Explanation

The problem can be solved by using the trigonometric identity sec² θ - tan² θ = 1. Let the value of the required expression 5 tan θ - 3 sec θ be x. We consider the squared difference of the given and the required expressions:

(5 sec θ - 3 tan θ)² - (5 tan θ - 3 sec θ)²

Expanding both terms:

(25 sec² θ + 9 tan² θ - 30 sec θ tan θ) - (25 tan² θ + 9 sec² θ - 30 sec θ tan θ)

By grouping common coefficients:

25(sec² θ - tan² θ) - 9(sec² θ - tan² θ)

Since sec² θ - tan² θ = 1, the expression simplifies to:

25(1) - 9(1) = 16

Now, substitute the known value of the first expression (5 sec θ - 3 tan θ = 5) into the equation:

5² - x² = 16

25 - x² = 16

x² = 25 - 16 = 9

x = √9 = 3

Explanation

The population figures for the five countries analyzed in this data set are 54 crores, 32 crores, 46 crores, 18 crores, and 24 crores. From these values, it is observed that the most populated country has a population of 54 crores and the least populated country has a population of 18 crores. The difference in population between the most and the least populated country is determined by subtracting the minimum value from the maximum value: 54 - 18 = 36 crores

Explanation

The total GDP of a country is the product of its population and its per capita GDP. According to the data table for this exam shift, Country 5 has a population of 113 million and a per capita GDP of 2800.

Total GDP in million dollars = Population × Per Capita GDP Total GDP in million dollars = 113 × 2800 = 316400

To convert the value from million dollars to crore dollars, we use the relation that 1 crore is equal to 10 million. Therefore, the total GDP in crore dollars is calculated by dividing the total million dollars by 10.

Total GDP in crore dollars = 316400 / 10 = 31640$

Explanation

To find the largest fraction, we can compare their reciprocals. The fraction with the smallest reciprocal value will be the largest.

Reciprocal of 5/113 = 113/5 = 22.6 Reciprocal of 7/120 = 120/7 ≈ 17.14 Reciprocal of 13/145 = 145/13 ≈ 11.15 Reciprocal of 17/160 = 160/17 ≈ 9.41

Since 9.41 is the smallest reciprocal value among the set, the fraction 17/160 is the largest. This is also evident by comparing the approximate decimal values: 17/160 = 0.10625, which is significantly higher than 13/145 ≈ 0.089, 7/120 ≈ 0.058, and 5/113 ≈ 0.044.

Explanation

Nirmit completes 2/3 of the work in 18 days. Therefore, the total time required by Nirmit to complete the entire job (1 unit of work) is calculated by multiplying the days by the reciprocal of the work fraction: 18 × ³⁄₂ = 9 × 3 = 27 days.

Efficiency is inversely proportional to the time taken to complete a task. Since Kashish is twice as efficient as Nirmit, the time taken by Kashish to complete the same job will be half the time taken by Nirmit.

Time taken by Kashish = (Time taken by Nirmit)/(2) = ²⁷⁄₂ days.

Explanation

The rise in the water level of the pool is determined by the total volume of water displaced by the 40 men. The total volume displaced is calculated as the number of men multiplied by the average volume displaced per man:

Total volume displaced V = 40 × 5 = 200 m³

The volume of a rectangular pool is the product of its length, breadth, and height (or rise in water level). Therefore, the rise in water level h is given by the formula:

h = (Total volume displaced)/(Area of the pool base)

The area of the pool base is 30 × 25 = 750 m². Substituting the values into the formula:

h = ²⁰⁰⁄₇₅₀ m = ⁴⁄₁₅ m

To convert the rise from meters to centimeters, multiply by 100:

h (in cm) = ⁴⁄₁₅ × 100 = ⁴⁰⁰⁄₁₅ = ⁸⁰⁄₃ ≈ 26.66 cm

Explanation

The purchase price after applying the two successive discounts on the listed price is calculated as: P = 2375 × (1 - 0.50) × (1 - 0.25) = 2375 × 0.5 × 0.75 = 890.625

The total cost price (CP) is the sum of the purchase price and the repair costs: CP = 890.625 + 165 = 1055.625

The profit percentage is 62.5%, which is equivalent to the fraction ⅝ because 12.5% = ⅛. The selling price (SP) is determined by the formula: SP = CP × (1 + Profit %) = 1055.625 × (1 + ⅝) = 1055.625 × ¹³⁄₈

By calculating the value: SP = (13723.125)/(8) = 1715.390625

Rounding to two decimal places, the selling price of the article is 1715.39.

Explanation

Let the length, breadth, and height be 19x, 11x, and 13x respectively. The difference between length and height is given as 30 cm, which leads to the equation 19x - 13x = 30. Solving this gives 6x = 30, hence x = 5.

The dimensions of the cuboid are: L = 19 × 5 = 95 cm B = 11 × 5 = 55 cm H = 13 × 5 = 65 cm

The volume of a cuboid is calculated by the formula V = L × B × H. Substituting the calculated dimensions: V = 95 × 55 × 65 V = 5225 × 65 = 339625 cm³

Alternatively, using the ratio coefficients directly, the volume is 19 × 11 × 13 × x³ = 2717 × 5³. Calculating this gives 2717 × 125 = 339625 cm³.

Explanation

The age of the coach can be calculated using the relation: Age of new member = Old Average + (New total number of members × Increase in average).

Given: Initial average age = 23 years Initial number of players = 12 New total number of members = 13 (12 players + 1 coach) Increase in average = 2 years

Substituting these values into the formula: Age of Coach = 23 + (13 × 2) Age of Coach = 23 + 26 Age of Coach = 49

Alternatively, this can be solved by comparing the total sum of ages. The sum of the ages of the 12 players is 12 × 23 = 276 years. When the coach is added, the total number of people is 13 and the new average age is 23 + 2 = 25 years. Thus, the new total age is 13 × 25 = 325 years. The coach age is the difference between the new total and the old total: 325 - 276 = 49 years.

Explanation

The effective gain percentage resulting from two successive gains of x% and y% can be calculated using the formula x + y + ˣʸ⁄₁₀₀. For gains of 30% and 20%, the total effective gain is 30 + 20 + (30 × 20)/(100) = 56%.

The final selling price is equal to the cost price plus the total gain, which is 156% of the cost price.

1.56 × CP = 31200

CP = (31200)/(1.56)

CP = 20000

Alternatively, using the multiplier method, the relationship between cost price and selling price is CP × ¹³⁰⁄₁₀₀ × ¹²⁰⁄₁₀₀ = 31200.

CP × 1.3 × 1.2 = 31200

CP × 1.56 = 31200

CP = (31200)/(1.56) = 20000

Explanation

The population of trees increases at a compound rate, which can be calculated using the formula A = P(1 + ʳ⁄₁₀₀)ⁿ. In this formula, A represents the current number of trees (17640), P is the number of trees 2 years ago, r is the annual growth rate (5%), and n is the time period in years (2).

Substituting the given values into the formula: 17640 = P(1 + ⁵⁄₁₀₀)² 17640 = P(1 + ¹⁄₂₀)² 17640 = P(²¹⁄₂₀)² 17640 = P(⁴⁴¹⁄₄₀₀)

To solve for P, we isolate it on one side of the equation: P = (17640 × 400)/(441)

By dividing 17640 by 441, we get 40. P = 40 × 400 P = 16000

Explanation

When two travelers start moving towards each other at the same time and meet at an intermediate point, the ratio of their speeds is the square root of the inverse ratio of the time taken by each to reach their respective destinations after the meeting point. This is expressed by the formula (S₁)/(S₂) = √((T₂)/(T₁)), where S₁ and S₂ are the speeds of the travelers and T₁ and T₂ are the respective times taken after meeting.

In this problem, S₁ (Aman's speed) is 30 km/hr, T₁ (Aman's time to reach Gwalior) is 196 minutes, and T₂ (Kapil's time to reach Delhi) is 225 minutes. Substituting these values into the formula:

(30)/(S₂) = √(²²⁵⁄₁₉₆)

(30)/(S₂) = ¹⁵⁄₁₄

S₂ = (30 × 14)/(15)

S₂ = 2 × 14 = 28 km/hr

Explanation

Simple Interest (SI) is given by the formula SI = (P × R × T)/(100). Since the interest is directly proportional to the principal (P) and time (T), the total interest for 10 years can be split into two 5-year periods.

The interest for the first 5 years on the original principal is half of the total interest for 10 years: SI_fᵢᵣₛₜ__₅ = ³¹³⁰⁄₂ = 1565

For the next 5 years, the principal becomes 5 times the original (5P). Since SI is directly proportional to P, the interest for these 5 years will also become 5 times the interest of the first 5 years: SIₙₑₓₜ__₅ = 5 × 1565 = 7825

The total interest obtained after 10 years is the sum of the interest from both periods: Total Interest = 1565 + 7825 = 9390

Explanation

The given equation can be simplified by dividing each term in the numerator by its respective denominator: ¹¹⁻¹³ˣ⁄ₓ + (11-13y)/(y) + (11-13z)/(z) = 5 (¹¹⁄ₓ - 13) + ((11)/(y) - 13) + ((11)/(z) - 13) = 5

Now, group the terms containing the variables and the constants: ¹¹⁄ₓ + (11)/(y) + (11)/(z) - (13 + 13 + 13) = 5 11(¹⁄ₓ + (1)/(y) + (1)/(z)) - 39 = 5

Add 39 to both sides of the equation: 11(¹⁄ₓ + (1)/(y) + (1)/(z)) = 5 + 39 11(¹⁄ₓ + (1)/(y) + (1)/(z)) = 44

Divide by 11 to find the required value: ¹⁄ₓ + (1)/(y) + (1)/(z) = ⁴⁴⁄₁₁ = 4

Explanation

The given equation is 2x + ⁹⁄ₓ = 9. Multiplying the entire equation by x results in the quadratic equation 2x² + 9 = 9x, which can be rearranged to 2x² - 9x + 9 = 0. Using the factorization method, we split the middle term: 2x² - 6x - 3x + 9 = 0. Factoring by grouping gives 2x(x - 3) - 3(x - 3) = 0, which simplifies to (2x - 3)(x - 3) = 0. This provides two possible values for x: x = 3 or x = ³⁄₂.

To find the minimum value of x² + (1)/(x²), we substitute both roots into the expression. For x = 3, the value is 3² + (1)/(3²) = 9 + ⅑ = ⁸²⁄₉ ≈ 9.11. For x = ³⁄₂, the value is (³⁄₂)² + (1)/((3){2)²} = ⁹⁄₄ + ⁴⁄₉. Finding a common denominator of 36, we get (81 + 16)/(36) = ⁹⁷⁄₃₆ ≈ 2.69. Comparing the two results, ⁹⁷⁄₃₆ is the minimum value.

Explanation

From the given equation (5x - y)/(5x + y) = ³⁄₇, cross-multiplication yields 7(5x - y) = 3(5x + y). Expanding these terms results in 35x - 7y = 15x + 3y. By grouping like terms, we get 20x = 10y, which simplifies to the relationship y = 2x.

The expression to be evaluated is (4x² + y² - 4xy)/(9x² + 16y² + 24xy). Using the algebraic identities for perfect squares, the numerator is (2x - y)² and the denominator is (3x + 4y)². Substituting y = 2x into the numerator, we find (2x - 2x)² = 0² = 0. Since the numerator is zero and the denominator (3x + 8x)² = 121x² is non-zero (given that the original ratio is defined and implies x ≠ 0), the value of the entire expression is 0.

Explanation

The first equation (x + y)² = xy + 1 can be expanded using the identity (x + y)² = x² + 2xy + y². Substituting this into the equation gives x² + 2xy + y² = xy + 1. By rearranging the terms and subtracting xy from both sides, we find that x² + xy + y² = 1.

The second given equation is x³ - y³ = 1. According to the algebraic identity for the difference of cubes, x³ - y³ = (x - y)(x² + xy + y²). By substituting the values derived above into this identity, we get 1 = (x - y)(1). Solving for the required term results in x - y = 1.

Explanation

In a parallelogram with sides a and b and diagonals d₁ and d₂, the sum of the squares of the diagonals is equal to the sum of the squares of all four sides. This relationship is expressed by the formula d₁² + d₂² = 2(a² + b²). Given the sides a = AB = 7 cm and b = BC = 9 cm, and one diagonal d₁ = AC = 8 cm, let the other diagonal be d₂ = BD. Substituting the given values into the formula:

8² + d₂² = 2(7² + 9²) 64 + d₂² = 2(49 + 81) 64 + d₂² = 2(130) 64 + d₂² = 260 d₂² = 260 - 64 d₂² = 196 d₂ = √(196) = 14

The length of the other diagonal is 14 cm.

Explanation

The number of diagonals in a polygon with n sides is determined by the formula (n(n - 3))/(2). This formula arises because from each of the n vertices, a diagonal can be drawn to every other vertex except itself and its two immediate neighbors, resulting in n - 3 possible diagonals per vertex. Since each diagonal connects two vertices, the product n(n - 3) counts every diagonal twice, necessitating the division by 2. For an octagon, n = 8, so the calculation is (8 × (8 - 3))/(2) = (8 × 5)/(2) = 20.

Explanation

Let the center of the quarter circle be the origin O(0, 0) and its radii lie along the positive x-axis and y-axis. Let the side of the square be s = √(5/2) cm.

The two adjacent vertices of the square on the radii are A and B. Let A lie on the x-axis at (d, 0) and B lie on the y-axis at (0, d), where d is the distance from the center. Since A and B are adjacent vertices of the square, the distance AB must equal the side length s. Using the distance formula: d² + d² = s² ⟹ 2d² = s² ⟹ d² = (s²)/(2)

The other two vertices C and D lie on the arc of the quarter circle. Because the square is symmetric with respect to the line y = x, and the side AB has a slope of -1, the sides BC and AD must have a slope of 1. To find the coordinates of C, we move from B(0, d) by a distance s along a line with slope 1. The components of this displacement are (d, d) because the length of the vector (d, d) is √(d² + d²) = √(2d²) = s. Thus: C = (0, d) + (d, d) = (d, 2d) D = (d, 0) + (d, d) = (2d, d)

Since C and D lie on the circumference of the circle with radius R, their coordinates must satisfy x² + y² = R²: R² = d² + (2d)² = 5d²

Substituting d² = (s²)/(2) into the equation for R²: R² = 5 ( (s²)/(2) ) = (5s²)/(2)

Given s = √(5/2), then s² = 5/2. Substituting this value: R² = (5/2) · (5/2) = 25/4 R = √(25/4) = 2.5

Explanation

Using the fundamental trigonometric identity sec² θ = 1 + tan² θ, we can rewrite the given equation sec² θ + tan² θ = 5/3 in terms of tangent alone. By substituting the identity into the equation, we get: 1 + tan² θ + tan² θ = 5/3 1 + 2tan² θ = 5/3 2tan² θ = 5/3 - 1 2tan² θ = 2/3 tan² θ = 1/3

For the primary acute angle, we take the positive square root: tan θ = 1/√3

The value of tan 2θ is then determined using the double angle formula for the tangent function: tan 2θ = (2 tan θ)/(1 - tan² θ)

Substituting the values of tan θ and tan² θ into the formula: tan 2θ = 2(1/√3){1 - 1/3} tan 2θ = 2/√3{2/3} tan 2θ = (2)/(√(3)) × ³⁄₂ tan 2θ = (3)/(√(3)) = √3

Explanation

Let the base of the tower be O and the height of the standing part be OP = h. When the tower breaks at P, its top falls to point Q on the ground. The length of the broken part is PQ, which forms the hypotenuse of the right-angled triangle POQ.

In △ POQ, the angle of elevation at Q is 60°. Using trigonometric ratios: QO = h cot 60° = (h)/(√3) PQ = h csc 60° = (2h)/(√3)

For point R on the opposite side of the tower, the angle of elevation of P is 30°. In △ POR: OR = h cot 30° = h√3

The distance QR is the sum of QO and OR because Q and R are on opposite sides of the tower base: QR = QO + OR = (h)/(√3) + h√3 = 180 Multiplying by √3 gives h + 3h = 180√3, so 4h = 180√3 and h = 45√3 m.

The total height of the tower is the sum of the standing part h and the broken part PQ: Total height = h + (2h)/(√3) = 45√3 + 2(45√3)/√3 Total height = 45√3 + 90 = 45(√3 + 2)

Explanation

Applying the sum-to-product trigonometric identity sin A + sin B = 2 sin (A+B)/(2) cos (A-B)/(2) to the left side of the equation sin 5θ + sin θ = sin 3θ results in 2 sin (5θ + θ)/(2) cos (5θ - θ)/(2) = sin 3θ. This simplifies the expression to 2 sin 3θ cos 2θ = sin 3θ. Dividing both sides by sin 3θ gives the relation 2 cos 2θ = 1, which simplifies to cos 2θ = 1/2. Within the specified range 0 < θ < π/2, the equation cos 2θ = 1/2 is satisfied when 2θ = 60°, leading to θ = 30°. Testing this value in the original equation, sin 30° + sin 150° = 0.5 + 0.5 = 1, while sin 3(30°) = sin 90° = 1, verifying the result.

Explanation

The tax paid by XYZ Limited is calculated based on the profit generated in the specific year and the applicable tax rate. For the year Y7, the profit is the difference between the revenue and the cost:

Profit_Y₇ = Revenue_Y₇ - Cost_Y₇ Profit_Y₇ = 1600 - 1240 = 360 (in Rs 000 crores)

The tax paid is then determined by applying the tax rate of 33% to this profit:

Tax_Y₇ = Tax rate_Y₇ × Profit_Y₇ Tax_Y₇ = 33% × 360 Tax_Y₇ = 0.33 × 360 = 118.8

Therefore, the tax paid by XYZ Limited in year Y7 was Rs 118.8 thousand crores.

Explanation

The Profit After Tax (PAT) for any year is calculated using the formula PAT = (Revenue - Cost) × (1 - Tax Rate).

For year Y2: Profit = 1100 - 850 = 250 PAT = 250 × (1 - 0.22) = 250 × 0.78 = 195

For year Y6: Profit = 1500 - 1200 = 300 PAT = 300 × (1 - 0.30) = 300 × 0.70 = 210

For year Y8: Profit = 1850 - 1400 = 450 PAT = 450 × (1 - 0.15) = 450 × 0.85 = 382.5

Comparing the calculated values, we find that 382.5 > 210 > 195, which establishes the relationship Y8 > Y6 > Y2.

Explanation

The yearly profit is calculated using the formula Profit = Revenue - Cost . Applying this to each year from the provided data:

For Y1: 800 - 600 = 200 For Y2: 1100 - 850 = 250 For Y3: 1200 - 900 = 300 For Y4: 1200 - 950 = 250 For Y5: 1350 - 1050 = 300 For Y6: 1500 - 1200 = 300 For Y7: 1600 - 1240 = 360 For Y8: 1850 - 1400 = 450

The set of profit values obtained for the eight years is {200, 250, 300, 250, 300, 300, 360, 450}. To find the number of distinct values, we identify the unique entries in this set, which are 200, 250, 300, 360, and 450. Counting these unique entries results in a total of 5 distinct profit values.

Explanation

The profit after tax for each year is calculated using the formula PAT = (Revenue - Cost) × (1 - Tax Rate). Using the data provided for each year:

For Y1, PAT is (800 - 600) × (1 - 0.20) = 200 × 0.80 = 160. For Y2, PAT is (1100 - 850) × (1 - 0.22) = 250 × 0.78 = 195. For Y3, PAT is (1200 - 900) × (1 - 0.22) = 300 × 0.78 = 234. For Y4, PAT is (1200 - 950) × (1 - 0.25) = 250 × 0.75 = 187.5. For Y5, PAT is (1350 - 1050) × (1 - 0.30) = 300 × 0.70 = 210. For Y6, PAT is (1500 - 1200) × (1 - 0.30) = 300 × 0.70 = 210. For Y7, PAT is (1600 - 1240) × (1 - 0.33) = 360 × 0.67 = 241.2. For Y8, PAT is (1850 - 1400) × (1 - 0.15) = 450 × 0.85 = 382.5.

Summing these individual annual values gives the total profit after tax: 160 + 195 + 234 + 187.5 + 210 + 210 + 241.2 + 382.5 = 1820.2

Explanation

To determine the largest value among the fractions, we can convert each to its decimal equivalent. The fraction 3/5 is equal to 0.6, while 4/6 simplifies to 2/3, which is approximately 0.667. The fraction 6/12 simplifies to 1/2, which is 0.5, and 1/276 is a very small value, approximately 0.0036. Comparing these decimal values, 0.667 is the greatest, making 4/6 the largest fraction. We can also verify this by comparing 4/6 and 3/5 using cross-multiplication: 4 × 5 = 20 and 3 × 6 = 18. Since 20 is greater than 18, 4/6 is larger than 3/5. Likewise, 4/6 is greater than 6/12 and 1/276.

Explanation

Let the total work be 100%. Based on the information provided: The work done by A, B, and C together is A + B + C = 100%. The work done by A and B together is A + B = 60%. The work done by B and C together is B + C = 70%.

To find the individual work done: Work done by C = (A + B + C) - (A + B) = 100% - 60% = 40%. Work done by A = (A + B + C) - (B + C) = 100% - 70% = 30%. Work done by B = (A + B) - A = 60% - 30% = 30%.

Comparing these values, C completes 40% of the work, while A and B each complete 30%. Since efficiency is defined as the amount of work done in a given time, and all three worked together for the same duration, the person who completed the largest share of the work is the most efficient. C is the most efficient because 40% is the highest individual contribution.

Explanation

When the three spheres are melted and recasted into a single solid sphere, the total volume remains conserved. The volume of a sphere is given by V = ⁴⁄₃π r³.

Sum of volumes of the three small spheres = ⁴⁄₃π (3³ + 4³ + 5³) = ⁴⁄₃π (27 + 64 + 125) = ⁴⁄₃π (216) cm³

Let the radius of the new sphere be R. Then, ⁴⁄₃π R³ = ⁴⁄₃π (216), which gives R³ = 216. Taking the cube root, R = 6 cm.

The surface area of a sphere is given by S = 4π r².

Total surface area of the three small spheres = 4π (3² + 4² + 5²) = 4π (9 + 16 + 25) = 4π (50) = 200π cm²

Surface area of the new large sphere = 4π (6²) = 144π cm²

Decrease in surface area = 200π - 144π = 56π cm²

Percentage decrease in surface area = (56π)/(200π) × 100 = ⁵⁶⁄₂ = 28%

Explanation

The marked price of the sofa set is Rs 4800. After a first discount of 10%, the price becomes: Price after first discount = 4800 × (1 - ¹⁰⁄₁₀₀) = 4800 × 0.9 = 4320

The final selling price after the second discount is Rs 3672. The amount of the second discount is the difference between the price after the first discount and the final selling price: Second discount amount = 4320 - 3672 = 648

The second discount percentage is calculated on the price after the first discount: Second discount % = ⁶⁴⁸⁄₄₃₂₀ × 100 Second discount % = (648)/(43.2) = 15%

Alternatively, using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)): 3672 = 4800 × 0.9 × (1 - (d₂)/(100)) 3672 = 4320 × (1 - (d₂)/(100)) 1 - (d₂)/(100) = ³⁶⁷²⁄₄₃₂₀ = 0.85 (d₂)/(100) = 0.15 d₂ = 15%

Explanation

The amount accrued after 1 year under compound interest is given by the formula A = P(1 + ʳ⁄₁₀₀)ᵗ, where P is the principal, r is the annual rate of interest, and t is the time in years. Since t = 1, the expression simplifies to A = P(1 + ʳ⁄₁₀₀).

Let the initial investments of A, B, and C be 30k, 40k, and 50k respectively, based on the given ratio 3 : 4 : 5. Using the respective interest rates of 20%, 15%, and 10%, the amounts after 1 year are calculated as follows:

For A: A_A = 30k × (1 + ²⁰⁄₁₀₀) = 30k × 1.2 = 36k For B: A_B = 40k × (1 + ¹⁵⁄₁₀₀) = 40k × 1.15 = 46k For C: A_C = 50k × (1 + ¹⁰⁄₁₀₀) = 50k × 1.1 = 55k

The ratio of their amounts after 1 year is 36k : 46k : 55k. By dividing each term by the common factor k, the final ratio is 36 : 46 : 55.

Explanation

The average age of a group is defined as the total sum of ages divided by the number of members. For the initial 120 members with an average age of 60.7 years, the total sum of their ages is 120 × 60.7 = 7284 years. When 30 new members join, the total population becomes 120 + 30 = 150 members, and their new average age is 56.3 years. The total sum of ages for all 150 members is 150 × 56.3 = 8445 years.

To find the total age of the 30 newly joined members, subtract the initial sum from the final sum: 8445 - 7284 = 1161 years. The average age of these new members is then calculated by dividing their total age by their count: 1161 / 30 = 38.7 years.

Alternatively, using the deviation method, the average age of the original 120 members decreased by 60.7 - 56.3 = 4.4 years. The total age "given up" by the original group to reach the new average is 120 × 4.4 = 528 years. This total must be distributed among the 30 new members, meaning their average age is lower than the final average by 528 / 30 = 17.6 years. Thus, the average age of the new members is 56.3 - 17.6 = 38.7 years.