SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The Triangle Inequality Theorem is a fundamental principle in Euclidean geometry which states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Mathematically, if a, b, and c represent the lengths of the sides of a triangle, then the conditions a + b > c, a + c > b, and b + c > a must all be satisfied. This property is derived from the fact that the shortest distance between any two points is a straight line. If the sum of two side lengths were equal to or less than the third, the three vertices would either lie on a single straight line or the segments would not be long enough to connect, preventing the formation of a closed three-sided polygon.

Explanation

Tara completes 10/15 or 2/3 of the work in 10 days, leaving 1/3 of the work remaining. Since Sara can complete the entire task in 18 days, she will take 1/3 of 18, which is 6 days, to finish the remaining work.

Explanation

Given x + 1/x = 6, we use the identity (x - 1/x)² = (x + 1/x)² - 4 to find (x - 1/x)² = 36 - 4 = 32. Since x > 1, x - 1/x = √32 = 4·√2. The value of x² - 1/x² is (x + 1/x)(x - 1/x) = 6 · 4·√2 = 24·√2.

Explanation

The third proportional c to two numbers a and b is calculated using the formula c = b²/a. Substituting a = 7 and b = 63, we get c = (63 × 63) / 7 = 9 × 63 = 567.

Explanation

The total work is 40 men multiplied by 30 days, which equals 1200 man-days. If 10 men leave, the remaining 30 men will take 1200 divided by 30, resulting in 40 days to complete the work.

Explanation

When the cost price (CP) of M items is equal to the selling price (SP) of N items, the profit percentage is calculated using the standard relation:

Profit % = (M - N)/(N) × 100

In this problem, the cost price of 28 oranges (M = 28) is equal to the selling price of 24 oranges (N = 24). Substituting these values into the formula:

Profit % = (28 - 24)/(24) × 100

Profit % = ⁴⁄₂₄ × 100 = ⅙ × 100

Profit % = ¹⁰⁰⁄₆ = ⁵⁰⁄₃ = 16⅔%

Explanation

The relative speed of the train with respect to the man moving in the same direction is the difference of their speeds: 125 km/h - 17 km/h = 108 km/h. To convert this to meters per second, we multiply by 5/18: 108 * (5/18) = 30 m/s. The time taken for the train to pass the man is the distance (length of the train) divided by the relative speed: 252 m / 30 m/s = 8.4 seconds.

Explanation

Given tan⁴ θ + tan² θ = 1, we can factor it as tan² θ(tan² θ + 1) = 1, which simplifies to tan² θ * sec² θ = 1. Since tan² θ = sin² θ / cos² θ and sec² θ = 1 / cos² θ, we get sin² θ / cos⁴ θ = 1, or sin² θ = cos⁴ θ. Substituting cos⁴ θ = sin² θ into the expression 11(cos⁴ θ + cos² θ) gives 11(sin² θ + cos² θ) = 11(1) = 11.

Explanation

Using the trigonometric identity 2 + 2 cos(x) = 4 cos²(x/2), the innermost root simplifies as follows: √(2 + 2 cos 8θ) = 2 cos 4θ. Repeating the process for the second root gives √(2 + 2 cos 4θ) = 2 cos 2θ, and for the outermost root gives √(2 + 2 cos 2θ) = 2 cos θ.

Explanation

Savings are calculated using the fundamental relation Savings = Income - Expenditure.

For Avinash, the monthly savings are: 50,000 - 18,000 = 32,000

For Radha, the monthly savings are: 60,000 - 24,000 = 36,000

The question requires the ratio of Radha's savings to Avinash's savings. This is expressed as: Radha's Savings : Avinash's Savings = 36,000 : 32,000

To simplify the ratio, first divide both terms by 1,000 to get 36 : 32. Then, divide both terms by their greatest common divisor, which is 4: ³⁶⁄₄ : ³²⁄₄ = 9 : 8

This result represents the simplest integer form of the ratio between their respective savings.

Explanation

When two individuals, A and B, can complete a task in x and y hours respectively, the total time T taken by both of them working together is calculated using the formula T = (x × y)/(x + y). This is derived from the fact that their combined work rate is the sum of their individual rates, ¹⁄ₓ + (1)/(y) = (x + y)/(xy).

Given: Time taken by A, x = 8 hours Time taken by B, y = 12 hours

Substituting these values into the formula: T = (8 × 12)/(8 + 12) T = ⁹⁶⁄₂₀ T = 4.8 hours

Alternatively, using the LCM method, let the total work be the LCM of 8 and 12, which is 24 units. A's efficiency is 24 / 8 = 3 units per hour, and B's efficiency is 24 / 12 = 2 units per hour. Their combined efficiency is 3 + 2 = 5 units per hour. The time taken to complete the job together is 24 / 5 = 4.8 hours.

Explanation

To find the time they take to meet, first convert Rohit's speed to m/s: 5.4 km/h = 5.4 * (5/18) = 1.5 m/s. Since they are walking in opposite directions, their relative speed is the sum of their individual speeds: 2.85 m/s + 1.5 m/s = 4.35 m/s. The time taken to meet is the total distance divided by the relative speed: 600 m / 4.35 m/s = 137.93 seconds, which is approximately 2.3 minutes when rounded to one decimal place.

Explanation

The prime numbers between 32 and 69 are 37, 41, 43, 47, 53, 59, 61, and 67. The sum of these eight numbers is 408, and their average is 408 divided by 8, which equals 51.

Explanation

Since 7 cot P = 24, we have cot P = 24/7. In a right-angled triangle, this means the base is 24 and the perpendicular is 7. Using the Pythagorean theorem, the hypotenuse is √(24² + 7²) = 25, therefore sin P = perpendicular/hypotenuse = 7/25.

Explanation

Ram, Rohan, and Rohit have daily efficiencies of 10, 6, and 5 units respectively (total work 150 units). Rohan and Rohit complete 22 units in 2 days, leaving 128 units which Ram and Rohan finish in 8 days (128 divided by 16). The total time taken to complete the work is 2 plus 8, which equals 10 days.

Explanation

To simplify the expression, the BODMAS rule is applied, which dictates the order of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

The innermost parentheses are evaluated first: 11 × 9 = 99

Next, the expression within the curly braces is solved. Applying the operations within the braces: 121 × 81 ÷ 99 = 121 × ⁸¹⁄₉₉ Simplifying the fraction ⁸¹⁄₉₉ by the common factor of 9 gives ⁹⁄₁₁. The calculation becomes: 121 × ⁹⁄₁₁ = 11 × 9 = 99

Then, simplify the square brackets. Following the rule to process division and multiplication from left to right: 144 ÷ 198 × 99 = ¹⁴⁴⁄₁₉₈ × 99 Since 198 is exactly twice 99, this simplifies to: ¹⁴⁴⁄₂ = 72

Finally, multiply the simplified value inside the bracket by the external coefficient: 2.5 × 72 = 180

Explanation

The total age of individuals is the product of their average age and the number of individuals. For the father and mother, the sum of their ages is 40 × 2 = 80 years. For the entire family consisting of the father, mother, and daughter, the sum of their ages is 30 × 3 = 90 years.

The age of the daughter is the difference between the total age of the three family members and the combined age of the parents: Age of daughter = (Sum of ages of all three) - (Sum of ages of parents) Age of daughter = 90 - 80 = 10 years.

Explanation

The actual profit percentage can be calculated by considering the ratio of the selling price to the actual cost price incurred by the shopkeeper. Let the cost price of a unit weight be 1. For a standard weight of 100 units, the cost price is 100. The shopkeeper sells this at a 15% profit, so the selling price (SP) is 100 × 1.15 = 115.

However, the shopkeeper uses a weight that is 20% less than the standard weight. Thus, the actual weight delivered is 100 × (1 - 0.20) = 80 units. The effective cost price (CP) to the shopkeeper for this transaction is the cost of these 80 units, which is 80.

The actual profit percentage is given by the formula: Profit % = (SP - Actual CP)/(Actual CP) × 100

Substituting the values: Profit % = (115 - 80)/(80) × 100 Profit % = ³⁵⁄₈₀ × 100 Profit % = ³⁵⁰⁄₈ = 43.75%

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if its last three digits are divisible by 8. For the number 148109326, the last three digits are 326.

To find the smallest number to subtract, we divide the last three digits by 8 to find the remainder: 326 ÷ 8 = 40 with a remainder of 6 (8 × 40 = 320; 326 - 320 = 6).

The remainder represents the excess amount that prevents the number from being perfectly divisible by 8. Subtracting this remainder of 6 from the original number will result in a value where the last three digits are 320, which is exactly divisible by 8.

Explanation

The given expression is in the form (a + b)² + (a - b)². Using the standard algebraic identity (a + b)² + (a - b)² = 2(a² + b²), we substitute a = 3x and b = 5 into the formula.

Expanding this, we get: 2((3x)² + 5²) 2(9x² + 25)

Explanation

In a cone, the relationship between slant height l, radius r, and height h is given by the Pythagorean theorem: l² = r² + h². Given that h = r and the slant height l = 5 cm, we substitute these into the equation:

5² = r² + r² 25 = 2r² r² = ²⁵⁄₂

Taking the square root, r = (5)/(√(2)) cm. Since height h is equal to radius r, we also have h = (5)/(√(2)) cm.

The volume V of a cone is calculated using the formula V = ⅓π r² h. Substituting the known values:

V = ⅓ × π × ²⁵⁄₂ × (5)/(√(2)) V = (125π)/(6√(2)) cm³

Explanation

To find the distance between the policeman and the thief after 3 minutes, we use the concept of relative speed. Since both are moving in the same direction, the relative speed is the difference of their individual speeds: Relative speed = 9 km/h - 8 km/h = 1 km/h

To calculate the distance in meters over a period of minutes, we convert the relative speed into meters per minute: 1 km/h = (1000 meters)/(60 minutes) = ⁵⁰⁄₃ m/min

The relative distance covered (the amount by which the gap is closed) in 3 minutes is the product of the relative speed and time: Distance reduced = ⁵⁰⁄₃ m/min × 3 minutes = 50 m

Initially, the thief was 300 m ahead of the policeman. After 3 minutes, the policeman has reduced that gap by 50 m. Therefore, the remaining distance between them is: Remaining distance = 300 m - 50 m = 250 m

Explanation

To find the value of sin t cos t, square both sides of the given equation sin t + cos t = ⅘. Applying the algebraic identity (a + b)² = a² + b² + 2ab, we obtain sin² t + cos² t + 2 sin t cos t = ¹⁶⁄₂₅. Since the fundamental trigonometric identity states that sin² t + cos² t = 1, the equation becomes 1 + 2 sin t cos t = ¹⁶⁄₂₅. Subtracting 1 from both sides gives 2 sin t cos t = ¹⁶⁄₂₅ - 1, which simplifies to 2 sin t cos t = (- 9)/(25). Dividing both sides by 2 results in sin t cos t = (- 9)/(50).

Explanation

To simplify the expression y + 2x - [y - (y - x + y) - (x + y) + y] - 2y, begin by simplifying the terms inside the innermost parentheses. The term (y - x + y) simplifies to (2y - x).

Substitute this back into the square brackets: [y - (2y - x) - (x + y) + y]

Now, expand the signs for all terms within the brackets: [y - 2y + x - x - y + y]

Group the like terms inside the brackets to simplify further: (y - 2y - y + y) + (x - x) = -y + 0 = -y

Now, substitute the simplified value of the brackets back into the main expression: y + 2x - [-y] - 2y

Simplify the signs and combine the remaining terms: y + 2x + y - 2y (y + y - 2y) + 2x = 0 + 2x = 2x

Explanation

The total work to be completed is determined by taking the Least Common Multiple (LCM) of the individual times taken by A, B, and C. The LCM of 16, 24, and 36 is calculated as 144 units.

The efficiency of each individual, representing the amount of work completed per day, is derived by dividing the total work by their respective time taken: Efficiency of A = 144 / 16 = 9 units per day Efficiency of B = 144 / 24 = 6 units per day Efficiency of C = 144 / 36 = 4 units per day

When working together, their combined efficiency is the sum of their individual efficiencies: Combined Efficiency = 9 + 6 + 4 = 19 units per day

The total time required for A, B, and C to complete the job together is the total work divided by their combined efficiency: Time = ¹⁴⁴⁄₁₉ days

To express this as a mixed fraction, we divide 144 by 19, which gives a quotient of 7 and a remainder of 11: 144 = (19 × 7) + 11 Thus, the time taken is 7 ¹¹⁄₁₉ days, which is represented in the options as 7 11 19.

Explanation

The total simple interest earned is the sum of interests from the three individual deposits. Using the formula for simple interest SI = (P × R × T)/(100), where P is the principal, R is the annual rate of interest, and T is the time in years, the equation is:

1620 = (500 × R × 3)/(100) + (650 × R × 5)/(100) + (1250 × R × 7)/(100)

Factoring out the common term (R)/(100):

1620 = (R)/(100) × [ (500 × 3) + (650 × 5) + (1250 × 7) ]

1620 = (R)/(100) × [ 1500 + 3250 + 8750 ]

1620 = (R)/(100) × 13500

1620 = 135 × R

R = ¹⁶²⁰⁄₁₃₅

Dividing 1620 by 135 yields R = 12. Therefore, the rate of interest per annum is 12%.

Explanation

The time taken by the workers to complete the task allows us to determine their individual efficiencies by assuming a total work value. Let the total work be the Least Common Multiple (LCM) of 20, 12, and 5, which is 60 units.

The efficiency of Rakshit is 60 / 20 = 3 units per hour. The efficiency of Ajay is 60 / 12 = 5 units per hour. The combined efficiency of Rakshit, Ajay, and Satish working together is 60 / 5 = 12 units per hour.

The efficiency of Satish can be found by subtracting the individual efficiencies of Rakshit and Ajay from their combined efficiency: Efficiency of Satish = 12 - (3 + 5) = 4 units per hour.

To find the time taken by Satish to complete the work alone, divide the total work by his efficiency: Time = 60 / 4 = 15 hours.

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Since △ ABC ∼ △ DEF, we can apply the following relation:

(Perimeter(△ ABC))/(Perimeter(△ DEF)) = (AB)/(DE)

Given that the perimeter of △ ABC is 40 cm, the perimeter of △ DEF is 12 cm, and the length of side DE is 6 cm, we substitute these values into the formula:

⁴⁰⁄₁₂ = (AB)/(6)

To find AB, we rearrange the equation:

AB = (40 × 6)/(12)

AB = ²⁴⁰⁄₁₂

AB = 20 cm

Explanation

To find the time when Sujata and Anjali meet for the first time on a circular track, we use the concept of relative speed in opposite directions. First, we convert both speeds into meters per second (m/sec). Sujata speed is 2.85 km/h = 2.85 × ⁵⁄₁₈ = (14.25)/(18) = 0.79166... m/sec, which can be written as ¹⁹⁄₂₄ m/sec. Anjali speed is given as 1.5 m/sec, which is ³⁄₂ m/sec.

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. Relative speed vᵣₑₗ = ¹⁹⁄₂₄ + ³⁄₂ = (19 + 36)/(24) = ⁵⁵⁄₂₄ m/sec.

The time taken to meet for the first time is the distance of the circular path divided by the relative speed: Time t = (Distance)/(vᵣₑₗ₎ = (693)/(55){24} = (693 × 24)/(55) seconds.

To convert the time into minutes, we divide the result by 60: Time in minutes = (693 × 24)/(55 × 60) = (693 × 2)/(55 × 5) = (63 × 11 × 2)/(5 × 11 × 5) = ¹²⁶⁄₂₅ = 5.04 minutes.

Explanation

The expression sec²θ + cos²θ = 1 is not a valid trigonometric identity. According to the Pythagorean identities, the relation involving cosine is sin²θ + cos²θ = 1. Since sec θ is the reciprocal of cos θ, the given formula would imply 1/cos²θ + cos²θ = 1. If we let x = cos²θ, the equation becomes 1/x + x = 1, which simplifies to x² - x + 1 = 0. This quadratic equation has no real roots because its discriminant D = (-1)² - 4(1)(1) = -3 is less than zero. Furthermore, since |sec θ| ≥ 1 and |cos θ| ≤ 1, the sum of their squares sec²θ + cos²θ is always greater than or equal to 2 for any real value of θ where sec θ is defined, as the sum of a positive number and its reciprocal x + 1/x is at least 2 for x > 0.

Explanation

The total work is constant and can be expressed by equating the man-days and woman-days for the two scenarios. Let M and W represent the daily efficiency of one man and one woman respectively.

(15M + 25W) × 9.6 = 16W × 27

To simplify, express 9.6 as ⁴⁸⁄₅: (15M + 25W) × ⁴⁸⁄₅ = 432W 15M + 25W = (432 × 5)/(48)W 15M + 25W = 9 × 5W 15M + 25W = 45W 15M = 20W 3M = 4W ⟹ M = ⁴⁄₃W

The total work in terms of women's efficiency is 16W × 27 = 432W units. The daily efficiency of 16 men is: 16M = 16 × ⁴⁄₃W = ⁶⁴⁄₃W units per day.

The number of days required for 16 men to complete the work is: Time = (Total Work)/(Efficiency of 16 men) = (432W)/(64){3W} Time = (432 × 3)/(64) Time = (27 × 3)/(4) = ⁸¹⁄₄ Time = 20.25 days

Explanation

The area of a circle is calculated using the formula A = π r². Given the area is 616 cm², we substitute the values to find the radius: 616 = ²²⁄₇ × r². Solving for r² gives r² = (616 × 7)/(22) = 28 × 7 = 196, which means the radius r = 14 cm.

A perpendicular drawn from the center of a circle to a chord bisects the chord. For the chord XY = 10 cm, the distance from the center to the midpoint of the chord forms a right-angled triangle where the radius is the hypotenuse and half the length of the chord is the base. Let d represent the perpendicular distance. According to the Pythagorean theorem, r² = d² + ((XY)/(2))². Substituting the values 14² = d² + 5² results in 196 = d² + 25. Subtracting 25 from both sides gives d² = 171, therefore d = √(171) cm.

Explanation

The given expression is sin² 25° + sin² 65°. By using the complementary angle identity sin(90° - θ) = cos θ, the term sin 65° can be rewritten as follows: sin 65° = sin(90° - 25°) = cos 25°

Substituting this value back into the original expression, we get: sin² 25° + cos² 25°

According to the fundamental trigonometric identity sin² θ + cos² θ = 1, the sum of the squares of the sine and cosine of the same angle is always equal to 1. Here, with θ = 25°, the expression simplifies directly to 1.

Explanation

Using the algebraic identity (a - b)² = (a + b)² - 4ab, let a = x² and b = (1)/(x²⁾. Substituting the given value x² + (1)/(x²⁾ = 7, the expression becomes:

(x² - (1)/(x²⁾)² = (x² + (1)/(x²⁾)² - 4(x²)((1)/(x²⁾)

(x² - (1)/(x²⁾)² = 7² - 4 = 49 - 4 = 45

Taking the square root of both sides yields x² - (1)/(x²⁾ = ± √(45) = ± 3√5.

To determine the correct sign, consider the condition 0 < x < 1. When a value x is between 0 and 1, its square x² is also between 0 and 1, making x² < 1. Conversely, the reciprocal (1)/(x²⁾ will be greater than 1. Subtracting a larger number from a smaller number results in a negative value. Therefore, x² - (1)/(x²⁾ = - 3√5.

Explanation

The selling price of an article is maximum when the total effective discount offered is the minimum. For any two successive discounts of a% and b%, the effective discount D is given by the formula D = a + b - ᵃᵇ⁄₁₀₀.

In this problem, the sum of the discount percentages in every scheme is 30 (Scheme i: 15+15=30, Scheme ii: 25+5=30, Scheme iii: 20+10=30, and Scheme iv: 30). To find the minimum effective discount among schemes where a+b is constant, we look for the scheme where the product ab is the largest, as this subtracts the most value from the sum.

Calculating the effective discount for Scheme i: D = 15 + 15 - (15 × 15)/(100) = 30 - 2.25 = 27.75%

Calculating the effective discounts for the other schemes: Scheme ii: 25 + 5 - (25 × 5)/(100) = 30 - 1.25 = 28.75% Scheme iii: 20 + 10 - (20 × 10)/(100) = 30 - 2 = 28% Scheme iv: 30 - 0 = 30%

Comparing these values, 27.75% is the minimum effective discount. A lower discount results in a higher final price for the buyer.

Explanation

The area of a triangle is calculated using the formula A = ½ × base × height. When comparing two triangles, the ratio of their areas is equal to the product of the ratios of their bases and heights. This relation is expressed as:

A₁{A₂} = b₁ × h₁{b₂ × h₂}

Given that the ratio of the areas A₁{A₂} is ⁵⁄₃ and the ratio of the heights h₁{h₂} is ⁵⁄₇, we substitute these values into the formula:

⁵⁄₃ = b₁{b₂} × ⁵⁄₇

To isolate the ratio of the bases b₁{b₂}, we multiply both sides by the reciprocal of the height ratio:

b₁{b₂} = ⁵⁄₃ × ⁷⁄₅

By canceling the common factor of 5 in the numerator and denominator, the calculation simplifies to:

b₁{b₂} = ⁷⁄₃

Therefore, the ratio of the bases of the two triangles is 7:3.

Explanation

For any two numbers a and b, the product of the numbers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF). This is expressed by the formula a × b = LCM × HCF. Given that the LCM is 12 and the HCF is 2, the product of the two numbers is 12 × 2 = 24.

The mean proportional of two numbers a and b is defined as the square root of their product, given by √(a × b). Substituting the product calculated above, the mean proportional is √(24). Simplifying this radical expression, we get √(4 × 6), which equals 2√6.

Explanation

The Triangle Inequality Theorem states that for any triangle with side lengths a, b, and c, the length of the third side must satisfy the relationship |a - b| < c < a + b. This means the third side must be strictly greater than the difference between the other two sides and strictly less than the sum of the other two sides.

Given the two side lengths are 30 cm and 42 cm, let a = 30 and b = 42. Substituting these values into the inequality for the third side x: |42 - 30| < x < 42 + 30 12 < x < 72

The values 12 and 72 represent the limits where the triangle would collapse into a straight line (a degenerate triangle). For a valid triangle to exist with a non-zero area, x must be strictly between these two values.

Explanation

The largest 5-digit number is 99,999. To find the largest 5-digit number exactly divisible by 88, we perform long division of 99,999 by 88 to find the remainder.

99,999 = 1136 × 88 + 31

The remainder obtained is 31. By subtracting this remainder from the largest 5-digit number, we get the largest 5-digit number that is a multiple of 88.

99,999 - 31 = 99,968

Verification can also be done using the divisibility rules for 8 and 11, since 88 = 8 × 11. For 99,968, the last three digits (968) are divisible by 8 (968 ÷ 8 = 121). For divisibility by 11, the difference between the sum of digits at odd positions (9 + 9 + 8 = 26) and the sum of digits at even positions (9 + 6 = 15) is 26 - 15 = 11. Since 11 is divisible by 11, the number 99,968 is divisible by 88.

Explanation

To find the value of a⁵ + (1)/(a⁵) when a + ¹⁄ₐ = 7, we use the standard algebraic identity a⁵ + (1)/(a⁵) = (a² + (1)/(a²))(a³ + (1)/(a³)) - (a + ¹⁄ₐ).

First, calculate the square component: a² + (1)/(a²) = (a + ¹⁄ₐ)² - 2 = 7² - 2 = 47

Second, calculate the cube component: a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ) = 7³ - 3(7) = 343 - 21 = 322

Finally, substitute these results into the identity for the fifth power: a⁵ + (1)/(a⁵) = (47 × 322) - 7 a⁵ + (1)/(a⁵) = 15134 - 7 = 15127

The product of 47 and 322 is 15134, and subtracting the original sum of 7 results in 15127.

Explanation

The expression is cos 47° sec 133° + sin 44° csc 136°. We use the reduction formulas for supplementary angles in the second quadrant. For the first term, sec 133° = sec(180° - 47°). Since sec(180° - θ) = -sec θ, we have sec 133° = -sec 47°. Multiplying this by cos 47° yields cos 47° (-sec 47°) = -(cos 47° · (1)/(cos 47°)) = -1.

For the second term, csc 136° = csc(180° - 44°). Since csc(180° - θ) = csc θ, we have csc 136° = csc 44°. Multiplying this by sin 44° yields sin 44° csc 44° = sin 44° · (1)/(sin 44°) = 1. The sum of the two terms is -1 + 1 = 0.

Explanation

To find the time when all three runners meet for the first time, we calculate the time taken for any two distinct pairs to meet and then find their Least Common Multiple (LCM).

The time taken for two runners to meet on a circular track is given by the formula T = L / Vᵣₑₗ, where L is the length of the track and Vᵣₑₗ is their relative speed. When running in the same direction, Vᵣₑₗ is the difference of their speeds; when running in opposite directions, it is the sum of their speeds.

For A and B (same direction), the meeting time tₐ_b = 1200 / (4 - 2) = 1200 / 2 = 600 seconds. For B and C (opposite direction), the meeting time t_b_c = 1200 / (4 + 6) = 1200 / 10 = 120 seconds.

The time when all three meet is the LCM(tₐ_b, t_b_c) = LCM(600, 120) = 600 seconds. To convert this into minutes, we divide by 60: 600 / 60 = 10 minutes.

Explanation

In a simple interest scenario, the interest earned each year remains constant. The difference between the amounts at the end of eleven years and seven years represents the interest accumulated over those four years.

Interest for 4 years = 18,906 - 14,522 = 4,384 Interest for 1 year = (4,384)/(4) = 1,096

The initial sum invested (Principal) is found by subtracting the total interest for seven years from the amount recorded at the end of seven years. Interest for 7 years = 1,096 × 7 = 7,672 Sum Invested = 14,522 - 7,672 = 6,850

Explanation

The area A of an equilateral triangle is given by the formula A = √3{4} s², where s is the side length. Because the area is directly proportional to the square of the side length, the net percentage increase in area when the side increases by x% can be calculated using the successive percentage change formula x + x + (x · x)/(100).

Given that the side increases by 34%, we substitute x = 34 into the formula:

Net increase = 34 + 34 + (34 × 34)/(100) Net increase = 68 + ¹¹⁵⁶⁄₁₀₀ Net increase = 68 + 11.56 = 79.56%

Alternatively, if the original side is 1, the new side is 1.34. The area is proportional to the square of the side, so the new area is proportional to 1.34² = 1.7956. This represents an increase of 0.7956, or 79.56%.

Explanation

The given series 3 + 3² + 3³ + … + 3⁸ is a geometric progression (GP) where the first term a = 3, the common ratio r = 3, and the number of terms n = 8.

The sum of the first n terms of a GP is given by the formula Sₙ = a(rⁿ - 1){r - 1}.

Substituting the values into the formula: S₈ = 3(3⁸ - 1){3 - 1}

First, calculate 3⁸: 3⁴ = 81 3⁸ = 81 × 81 = 6561

Now, substitute this back into the sum formula: S₈ = (3(6561 - 1))/(2) S₈ = (3 × 6560)/(2) S₈ = 3 × 3280 S₈ = 9840

Explanation

Total work can be assumed as the LCM of 30, 40, and 50, which is 600 units.

The efficiency of A is 600 / 30 = 20 units/day. The efficiency of B is 600 / 40 = 15 units/day. The efficiency of C is 600 / 50 = 12 units/day.

Since they work alternatively starting with A, one cycle consists of 3 days (A on day 1, B on day 2, and C on day 3). The total work done in one 3-day cycle is 20 + 15 + 12 = 47 units.

To find the number of complete cycles, divide the total work by the work per cycle: 600 / 47 = 12 cycles with a remainder. Work done in 12 cycles (36 days) is 12 × 47 = 564 units. The remaining work is 600 - 564 = 36 units.

On the 37th day, A works and completes 20 units. Remaining work is 36 - 20 = 16 units. On the 38th day, B works and completes 15 units. Remaining work is 16 - 15 = 1 unit. On the 39th day, it is C's turn. C's efficiency is 12 units/day, so the time taken by C to complete the remaining 1 unit is 1 / 12 day.

Total time taken is 36 + 1 + 1 + ¹⁄₁₂ = 38 ¹⁄₁₂ days.

Explanation

When two objects move in the same direction, their relative speed is the difference between their individual speeds. Here, the relative speed of the constable with respect to the thief is 8 km/h - 6 km/h = 2 km/h. The initial distance between them is 200 m. Since both run for the same duration t until the thief is caught, the distance covered is directly proportional to their speeds (d ∝ v).

The ratio of the speed of the constable to the speed of the thief is 8 : 6, which simplifies to 4 : 3. Consequently, the ratio of the distances covered by them is also 4 : 3. Let the distance covered by the constable be 4x and the distance covered by the thief be 3x. The difference between these distances must equal the initial gap of 200 m. Thus, 4x - 3x = 200 m, which gives x = 200 m. The distance the thief runs before being overtaken is 3x = 3 × 200 = 600 m.

Explanation

The ratio of red, blue, and green pens is given as 7:4:9. To find the number of red pens, we first calculate the total number of parts in the ratio by summing the individual components: 7 + 4 + 9 = 20 parts. Since the total number of pens in the bag is 320, the value of one part is determined by dividing the total pens by the total number of parts: 320 / 20 = 16. The number of red pens is represented by 7 parts of the ratio. Thus, the total number of red pens is 7 × 16 = 112.

Explanation

Total work is calculated as the LCM of the individual time taken by Ravi and Sudha. The LCM of 40 and 60 is 120 units. The efficiency of Ravi is 120 / 40 = 3 units per day, and the efficiency of Sudha is 120 / 60 = 2 units per day.

Since they work on alternate days starting with Sudha, the work done in a single 2-day cycle is the sum of their efficiencies. In the first day Sudha does 2 units and in the second day Ravi does 3 units, totaling 5 units every 2 days. To find the number of cycles required to complete 120 units of work, we divide the total work by the work done per cycle: 120 / 5 = 24 cycles. Because each cycle consists of 2 days, the total time taken to complete the work is 24 × 2 = 48 days.

Explanation

The length of an arc L subtending an angle θ at the centre of a circle with radius r is given by the formula L = (θ)/(360) × 2π r. Given L = 23.1 cm, θ = 18°, and π = ²²⁄₇, we can find the radius r as follows:

23.1 = ¹⁸⁄₃₆₀ × 2 × ²²⁄₇ × r 23.1 = ¹⁄₂₀ × ⁴⁴⁄₇ × r 23.1 = ¹¹⁄₃₅ × r r = (23.1 × 35)/(11) = 2.1 × 35 = 73.5 cm

The area of the circle A is determined by the formula A = π r². Substituting the values of π and r:

A = ²²⁄₇ × (73.5)² A = ²²⁄₇ × 73.5 × 73.5 A = 22 × 10.5 × 73.5 A = 231 × 73.5 A = 16978.50 cm²