SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Using the identity tan(A - B) = (tan A - tan B) / (1 + tan A tan B), and setting A = 72 and B = 27, we get tan(72 - 27) = tan 45 = 1. This leads to the equation tan 72 - tan 27 = 1 + tan 72 tan 27, and rearranging it gives tan 72 - tan 27 - tan 72 tan 27 = 1.

Explanation

To find the combined ratio A:B:C, the term B must be the same in both given ratios. Multiplying A:B (6:8) by 5 gives 30:40, and multiplying B:C (5:11) by 8 gives 40:88. Combining these yields 30:40:88, which simplifies to 15:20:44 when divided by 2.

Explanation

The ratio of circumference C to diameter d is given as (C)/(d) = ⅗. Given that the circumference C = 6 cm, we can substitute this value into the ratio to find the diameter: (6)/(d) = ⅗. By cross-multiplying, we get 3d = 30, which simplifies to d = 10 cm. The radius r of a circle is defined as half of its diameter, so r = ᵈ⁄₂ = ¹⁰⁄₂ = 5 cm.

Explanation

The ratio of income, expenditure, and savings for Ramesh in 2021 is determined by their respective values: ₹18,600, ₹12,400, and the difference ₹6,200 (18,600 - 12,400). Dividing these by 6,200 gives a ratio of 3:2:1 for Income : Expenditure : Savings.

Using the weighted average formula for the components of income, where the percentage increase in income is the weighted average of the percentage increases in expenditure and savings:

Income % increase = ((Expenditure ratio × Expenditure % increase) + (Savings ratio × Savings % increase))/(Expenditure ratio + Savings ratio)

Substituting the given values, where x is the percentage increase in savings:

15 = ((2 × 8) + (1 × x))/(2 + 1)

15 = (16 + x)/(3)

45 = 16 + x

x = 45 - 16 = 29

Explanation

The average score of the entire class is found by calculating the weighted average of the two groups. The first group consists of 12 students with an average score of 60, giving a total score of 12 × 60 = 720. The rest of the class consists of 30 - 12 = 18 students. With an average score of 80, their total score is 18 × 80 = 1440.

To find the class average, add the total scores of both groups and divide by the total number of students: Average = (720 + 1440)/(30) = ²¹⁶⁰⁄₃₀ = 72

Using the ratio method, the number of students in the two groups are in the ratio 12:18, which simplifies to 2:3. The weighted average is (2(60) + 3(80))/(2 + 3) = (120 + 240)/(5) = ³⁶⁰⁄₅ = 72.

Explanation

Using the algebraic identities (a+b)² = a² + b² + 2ab and (a-b)² = a² + b² - 2ab, we can find the values of a+b and a-b.

First, calculate (a+b)²: (a+b)² = 148 + 2(54) = 148 + 108 = 256 Taking the square root, a+b = √(256) = 16.

Next, calculate (a-b)²: (a-b)² = 148 - 2(54) = 148 - 108 = 40 Taking the square root, a-b = √(40) = √(4 × 10) = 2√(10).

Now, find the ratio: (a+b)/(a-b) = (16)/(2√(10)) = (8)/(√(10))

Explanation

The total number of pages printed by printer P1 over the three days is calculated as 200 + 150 + 225 = 575. The total number of pages printed by printer P2 over the same period is calculated as 250 + 300 + 210 = 760. To find the difference between the total outputs of the two printers, subtract the total of P1 from the total of P2: 760 - 575 = 185.

Explanation

According to the BODMAS rule, the order of operations is Bracket, Of, Division, Multiplication, Addition, and Subtraction. In the given equation 9 × N of 4 ÷ 4 - 15 × 4 = 12, the term N of 4 is calculated first as 4N.

Substituting this back, the equation becomes: 9 × 4N ÷ 4 - 15 × 4 = 12

Following the rule, Division is performed next: 4N ÷ 4 = N

This simplifies the equation to: 9 × N - 15 × 4 = 12

Next, perform the Multiplications: 9N - 60 = 12

To find the value of N, move the constant to the other side: 9N = 12 + 60 9N = 72 N = ⁷²⁄₉ N = 8

Explanation

According to the Midpoint Theorem in geometry, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of its length. In Δ PQR, since S and T are the midpoints of sides PQ and PR respectively, the length of the segment ST is calculated using the relation ST = ½ QR. Substituting the given value of QR = 12 cm, the length of ST = ½ × 12 = 6 cm.

Explanation

The expression 3sin 15° - 4sin³15° is in the standard form of the triple angle identity for sine. According to trigonometric formulas, the sine of triple an angle is expressed as:

sin 3θ = 3sin θ - 4sin³ θ

By substituting θ = 15° into this identity, the given expression simplifies directly to the sine of three times that angle:

3sin 15° - 4sin³ 15° = sin(3 × 15°)

Calculating the value inside the function:

3 × 15° = 45°

Thus, the expression becomes sin 45°. Based on standard trigonometric values for common angles, the value of sin 45° is 1/√2.

Explanation

The gain percentage is calculated based on the cost price (CP). Given that the gain is ⅕ of the selling price (SP), we can establish the relationship between CP and SP using the formula CP = SP - Gain. If we assume SP = 5 units, the gain is ⅕ × 5 = 1 unit. Therefore, the cost price is 5 - 1 = 4 units. The gain percentage is determined by the formula Gain % = ((Gain)/(CP)) × 100. Substituting the values, we get Gain % = (¼) × 100 = 25%.

Using the specific values given in the problem, the selling price is 40620. The gain is calculated as ⁴⁰⁶²⁰⁄₅ = 8124. The cost price is the difference between the selling price and the gain, which is 40620 - 8124 = 32496. Applying the gain percentage formula, Gain % = (⁸¹²⁴⁄₃₂₄₉₆) × 100. Since 32496 is exactly 4 times 8124, the fraction simplifies to ¼, resulting in a gain of 25%.

Explanation

The single discount equivalent to a series of discounts can be calculated by determining the final selling price as a percentage of the original marked price. If the marked price is assumed to be 100, the successive discounts of 15%, 20%, and 5% are applied sequentially.

The value after the first discount of 15% is: 100 × (1 - ¹⁵⁄₁₀₀) = 100 × 0.85 = 85

The value after the second discount of 20% is: 85 × (1 - ²⁰⁄₁₀₀) = 85 × 0.80 = 68

The value after the third discount of 5% is: 68 × (1 - ⁵⁄₁₀₀) = 68 × 0.95 = 64.6

The total discount is the difference between the initial marked price and the final price: 100 - 64.6 = 35.4

Alternatively, using the formula for net multiplier, the effective price factor is 0.85 × 0.80 × 0.95 = 0.646. The equivalent single discount is (1 - 0.646) × 100 = 35.4%.

Explanation

Total sales of bicycle B1 from 2004 to 2006 are calculated by summing the annual sales: 195 + 230 + 250 = 675 lakhs. Total sales of bicycle B2 for the same period are calculated by summing its annual sales: 284 + 300 + 416 = 1000 lakhs.

The ratio of the total sales of B1 to the total sales of B2 is: Ratio = (Total B1)/(Total B2) = ⁶⁷⁵⁄₁₀₀₀

To simplify this fraction, divide both the numerator and the denominator by their greatest common divisor, 25: 675 ÷ 25 = 27 1000 ÷ 25 = 40

The simplified ratio is 27:40.

Explanation

The net filling rate of the cistern is the difference between the rate of the filling tap and the rate of the emptying tap. Let the time taken by the filling tap be T_fᵢₗₗ = 10 minutes and the time taken by the emptying tap be Tₑₘₚₜ_y = 12 minutes. The combined rate R per minute is:

R = (1)/(T_fᵢₗₗ₎ - (1)/(Tₑₘₚₜ_y₎ = ⅒ - ¹⁄₁₂

To subtract these fractions, find the least common multiple of 10 and 12, which is 60:

R = ⁶⁄₆₀ - ⁵⁄₆₀ = ¹⁄₆₀

The time required to fill the cistern is the reciprocal of the net rate, which is 60 minutes. Since the question asks for the time in hours, we divide the minutes by 60:

Time in hours = ⁶⁰⁄₆₀ = 1 hour

Explanation

To find the difference in the average production of Company A and Company C during the period 2016-2018, we first calculate the total production for each company over these three years.

Total production for Company A during 2016-2018 is: 38,000 + 41,000 + 53,000 = 132,000 lakhs

The average production for Company A is: (132,000)/(3) = 44,000 lakhs

Total production for Company C during 2016-2018 is: 51,000 + 57,000 + 63,000 = 171,000 lakhs

The average production for Company C is: (171,000)/(3) = 57,000 lakhs

The difference in the average production of A and C is: |57,000 - 44,000| = 13,000 lakhs

Alternatively, using the property of averages, the difference in averages is the average of the differences for each year: ((51,000 - 38,000) + (57,000 - 41,000) + (63,000 - 53,000))/(3) (13,000 + 16,000 + 10,000)/(3) = (39,000)/(3) = 13,000 lakhs

Explanation

The initial purchase of 15 apples and 12 oranges for ₹447 can be represented by the linear equation 15A + 12O = 447. Dividing this entire equation by 3 simplifies the relationship to 5A + 4O = 149, which represents the cost of 5 apples and 4 oranges.

An inconsistent statement is one where the same combination of items results in a different total price, leading to no possible solution for the variables A and O. In option d, the purchase of 35 apples and 28 oranges for ₹1,029 can be written as 35A + 28O = 1,029. Dividing this equation by its common factor of 7 yields 5A + 4O = 147.

This creates a contradiction because the same set of 5 apples and 4 oranges cannot cost both ₹149 and ₹147 simultaneously. Since the system of equations 5A + 4O = 149 and 5A + 4O = 147 consists of two parallel lines that never intersect, there are no possible prices for the fruits that satisfy both conditions.

Explanation

In Δ TPQ, the sum of interior angles is 180°. Given ∠ PTQ = 27° and ∠ TPQ = 55°, the angle ∠ TQP = 180° - (27° + 55°) = 98°. Since TQR is a straight line, the supplementary angle ∠ PQR = 180° - 98° = 82°.

According to the Alternate Segment Theorem, the angle between the tangent TP and the chord PQ is equal to the angle subtended by that chord in the alternate segment. Therefore, ∠ PRQ = ∠ TPQ = 55°.

In Δ PQR, the sum of angles is 180°. Thus, ∠ QPR = 180° - (∠ PQR + ∠ PRQ) = 180° - (82° + 55°) = 180° - 137° = 43°.

The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle. Here, arc QR subtends ∠ QPR at the circumference and ∠ ROQ at the center. Consequently, ∠ ROQ = 2 × ∠ QPR = 2 × 43° = 86°.

Explanation

A well is cylindrical in shape, so the volume of the earth taken out is equal to the volume of a cylinder. The volume V of a cylinder is calculated using the formula V = π r² h, where r is the radius and h is the depth. Given r = 14 cm and h = 23 cm, the volume is V = ²²⁄₇ × 14 × 14 × 23. Simplifying the calculation, we have V = 22 × 2 × 14 × 23, which equals 44 × 14 × 23. Multiplying these values, 616 × 23 yields 14168 cm³.

Explanation

The total amount payable at the end of the second year comprises the remaining principal and the total simple interest accrued over the two-year period. In the first year, interest I₁ is calculated on the initial principal P₁ = 335180 at 13%: I₁ = 335180 × 0.13 = 43573.4

At the end of the first year, a repayment of 23060 is made towards the principal. The outstanding principal P₂ for the second year is: P₂ = 335180 - 23060 = 312120

The interest I₂ for the second year is calculated on this reduced principal: I₂ = 312120 × 0.13 = 40575.6

Pranav clears all dues at the end of the second year, which includes the remaining principal P₂, the interest from the first year I₁, and the interest from the second year I₂. The total payment is: Total Payment = P₂ + I₁ + I₂ Total Payment = 312120 + 43573.4 + 40575.6 Total Payment = 312120 + 84149 = 396269

Explanation

The relative speed of Priya with respect to Seema is the difference in their rates because they are moving in the same direction. Let the speed of Seema be Vₛ = 3 rounds/hour and the speed of Priya be Vₚ = 8 rounds/hour. The relative speed is Vᵣₑₗ = Vₚ - Vₛ = 8 - 3 = 5 rounds/hour. On a circular track, two people starting from the same point in the same direction will meet for the first time when the faster person covers exactly one full round more than the slower person. The time T taken for this to occur is given by the formula T = (Distance)/(Relative Speed) = ⅕ hour. To convert this into minutes, T = ⅕ × 60 = 12 minutes. Starting from 11:54 p.m., the addition of 12 minutes is calculated as 11:54 p.m. + 6 minutes = 12:00 a.m. (midnight), plus the remaining 6 minutes, which results in 12:06 a.m.

Explanation

A number is divisible by 6 if it is divisible by both 2 and 3. The number 438750 satisfies the divisibility rule for 2 because its unit digit is 0, which is an even number. For divisibility by 3, the sum of the digits must be a multiple of 3. The sum of the digits of 438750 is 4 + 3 + 8 + 7 + 5 + 0 = 27. Since 27 is divisible by 3, the number itself is divisible by 3. Because 438750 is divisible by both 2 and 3, it is divisible by 6.

Explanation

Using the fundamental trigonometric identity sec²θ - tan²θ = 1, we can factorize it as (secθ - tanθ)(secθ + tanθ) = 1. Given that secθ + tanθ = x, it follows that secθ - tanθ = ¹⁄ₓ.

To isolate the individual trigonometric functions, we add and subtract these two equations: Adding the equations gives 2secθ = x + ¹⁄ₓ = (x² + 1)/(x), which simplifies to secθ = (x² + 1)/(2x). Subtracting the second from the first gives 2tanθ = x - ¹⁄ₓ = (x² - 1)/(x), which simplifies to tanθ = (x² - 1)/(2x).

The sine function can be expressed in terms of tangent and secant as sinθ = (tanθ)/(secθ). Substituting the derived values: sinθ = (x² - 1)/(2x){(x² + 1)/(2x)} = (x² - 1)/(x² + 1).

Explanation

To determine how many schools have an income from semester fees less than three times the income from donations, we check the condition Semester Fee < 3 × Donation for each school individually using the provided data.

For school X, the semester fee is 43 and the donation is 14. Calculating 3 × 14 = 42, we find that 43 > 42, so it does not satisfy the condition. For school Y, the semester fee is 56 and the donation is 17. Calculating 3 × 17 = 51, we find that 56 > 51, which does not satisfy the condition. For school Z, the semester fee is 39 and the donation is 14. Since 3 × 14 = 42 and 39 < 42, this school satisfies the condition. For school U, the semester fee is 89 and the donation is 50. Since 3 × 50 = 150 and 89 < 150, this school satisfies the condition. For school V, the semester fee is 59 and the donation is 19. Since 3 × 19 = 57 and 59 > 57, it does not satisfy the condition.

By evaluating all the schools, we find that only schools Z and U meet the criteria, resulting in a total of 2 schools.

Explanation

To find the difference in the number of students in School 4 between the two years, we first calculate the individual student counts for each year based on the provided percentages and totals.

For the year 2021, the number of students in School 4 is 23% of 16,000. N₂₀₂₁ = ²³⁄₁₀₀ × 16,000 = 23 × 160 = 3,680

For the year 2022, the number of students in School 4 is 22% of 18,000. N₂₀₂₂ = ²²⁄₁₀₀ × 18,000 = 22 × 180 = 3,960

The difference in the number of students in School 4 for 2022 with respect to 2021 is the subtraction of the two values. Difference = 3,960 - 3,680 = 280

Explanation

The sum of the interior angles of any triangle in Euclidean geometry is always 180° according to the Angle Sum Property. This confirms statement (a) is true. Additionally, statement (c) is true because an obtuse angle is defined as an angle greater than 90°. If a triangle were to have two obtuse angles, the sum of those two angles alone would exceed 180°, which is impossible since the sum of all three angles must be exactly 180°. Therefore, a triangle can have at most one obtuse angle.

Explanation

Let the efficiency of the slower pipe be x units per minute. Since the faster pipe fills the tank four times as fast, its efficiency is 4x units per minute.

When both pipes work together, their combined efficiency is x + 4x = 5x units per minute. The total capacity of the tank can be calculated using the formula: Total Work = Combined Efficiency × Total Time Total Work = 5x × 48 = 240x units

The time required for the slower pipe to fill the tank alone is found by dividing the total capacity by its individual efficiency: Time = Total Work / Efficiency of slower pipe Time = 240x / x = 240 minutes

Explanation

Let the unknown number be x. According to the problem statement, the fifth part of the number increased by 5 is equal to its third part decreased by 7. We can model this relationship with the following linear equation:

ˣ⁄₅ + 5 = ˣ⁄₃ - 7

To solve for x, we first move the constants to one side and the terms containing x to the other:

5 + 7 = ˣ⁄₃ - ˣ⁄₅

12 = (5x - 3x)/(15)

12 = ²ˣ⁄₁₅

By multiplying both sides of the equation by 15, we get:

180 = 2x

x = 90

The question requires finding half of the number. Therefore, we divide the value of x by 2:

ˣ⁄₂ = ⁹⁰⁄₂ = 45

Explanation

The total surface area of a cone is the sum of its curved surface area and the area of its circular base. The formula for total surface area is TSA = CSA + π r².

Given the curved surface area CSA = 550~cm² and the radius r = 7~cm, we first calculate the area of the base: Base Area = π r² = ²²⁄₇ × 7 × 7 = 154~cm².

By adding the curved surface area and the base area, we find the total surface area: TSA = 550 + 154 = 704~cm².

Explanation

The net equivalent percentage of two successive decreases of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀. Substituting x = 25 and y = 25: Net Decrease Percentage = 25 + 25 - (25 × 25)/(100) = 50 - 6.25 = 43.75%

The net equivalent percentage of two successive increases of x% and y% is calculated using the formula x + y + ˣʸ⁄₁₀₀. Substituting x = 25 and y = 25: Net Increase Percentage = 25 + 25 + (25 × 25)/(100) = 50 + 6.25 = 56.25%

To find by what percentage the net decrease is less than the net increase, use the comparison formula (Higher Value - Lower Value)/(Higher Value) × 100: Required Percentage = (56.25 - 43.75)/(56.25) × 100 Required Percentage = (12.5)/(56.25) × 100

Dividing both numerator and denominator by 6.25 simplifies the fraction: (12.5)/(56.25) = ²⁄₉ ²⁄₉ × 100 = 22.222...%

Rounding to two decimal places, the result is 22.22%.

Explanation

Using the trigonometric identity for complementary angles, cos θ = sin(90° - θ), the given equation sin(48° + k) = cos 13° can be rewritten to express both sides in terms of the sine function. Substituting θ = 13° gives cos 13° = sin(90° - 13°) = sin 77°.

The equation becomes sin(48° + k) = sin 77°. Because (48° + k) is specified as an acute angle (an angle between 0° and 90°), we can equate the arguments of the sine functions directly. This leads to the linear equation 48° + k = 77°. Solving for k by subtracting 48° from 77° results in k = 29°.

Explanation

The sum of the interior angles of a triangle is always 180°. Given the ratio of the angles as 7:8:3, the angles can be represented as 7x, 8x, and 3x. Using the angle sum property:

7x + 8x + 3x = 180° 18x = 180° x = 10°

The largest angle corresponds to the largest part of the ratio, which is 8x. Substituting the value of x gives 8 × 10° = 80°.

Explanation

Using the fundamental trigonometric identities 1 + cot²θ = csc²θ and 1 + tan²θ = sec²θ, the given expression can be simplified as follows:

(5)/(1 + cot²θ) + (3)/(1 + tan²θ) + 2cos²θ

= (5)/(csc²θ) + (3)/(sec²θ) + 2cos²θ

Since (1)/(csc²θ) = sin²θ and (1)/(sec²θ) = cos²θ, we substitute these values into the expression:

= 5sin²θ + 3cos²θ + 2cos²θ

By combining the terms containing cos²θ, we get:

= 5sin²θ + 5cos²θ

Factoring out the common constant 5:

= 5(sin²θ + cos²θ)

Applying the Pythagorean identity sin²θ + cos²θ = 1:

= 5(1) = 5

Explanation

The total exports for any given year are calculated by summing the export values of companies P, Q, and R. For the year 2018, the total export is 4000 + 3000 + 2000 = 9000. For the year 2020, the total export is 3000 + 3000 + 3000 = 9000. Both years yield an identical total export value of 9000 units.

Explanation

Let the number of chairs be C and the number of people be P. Based on the first condition, when one person sits on each chair and one person remains standing, the relationship is P = C + 1. According to the second condition, when two people sit on each chair and one chair remains vacant, the relationship is P = 2(C - 1).

By equating the two expressions for P: C + 1 = 2(C - 1) C + 1 = 2C - 2 C = 3

Substituting C = 3 back into the first equation gives P = 3 + 1 = 4. If 4 people sit on 3 chairs at 2 per chair, they occupy 2 chairs, leaving exactly 1 chair vacant, which satisfies both conditions.

Explanation

The net percentage change resulting from successive changes of x% and y% is calculated using the formula x + y + ˣʸ⁄₁₀₀. In this case, the price is first increased by 8% (x = +8) and then decreased by a discount of 18% (y = -18).

Substituting these values into the formula: Net Change = 8 + (-18) + (8 × (-18))/(100) Net Change = -10 - ¹⁴⁴⁄₁₀₀ Net Change = -10 - 1.44 = -11.44

The negative sign indicates a net decrease relative to the original price. Therefore, the actual discount on the original price is 11.44%. This can also be verified by assuming an original price of 100: after an 8% increase, the price becomes 108, and applying an 18% discount on 108 results in a final price of 108 × 0.82 = 88.56. The overall discount is 100 - 88.56 = 11.44%.

Explanation

When two objects move in the same direction on a circular track of length L, they meet for the first time when the faster object covers exactly one full lap more than the slower one. This is solved using the concept of relative speed. The relative speed of Ravi with respect to Kapil is the difference between their individual speeds: vᵣₑₗ = 22~m/s - 11~m/s = 11~m/s.

The time taken to meet for the first time is given by the formula t = (L)/(vᵣₑₗ₎. Substituting the given values, t = (220~m)/(11~m/s) = 20~seconds. At this time, Ravi has completed 22 × 20 / 220 = 2 laps, while Kapil has completed 11 × 20 / 220 = 1 lap.

Explanation

To calculate the greatest possible profit, we must maximize the difference between the selling price and the cost price. This occurs when each book is purchased at the lowest possible cost price and sold at the highest possible selling price within the given ranges.

The minimum cost price (CPₘᵢₙ) is ₹200 and the maximum selling price (SPₘₐₓ) is ₹300.

Maximum profit per book = SPₘₐₓ - CPₘᵢₙ Maximum profit per book = 300 - 200 = ₹100

Total maximum profit for 25 books = Maximum profit per book × Number of books Total maximum profit = 100 × 25 = ₹2,500

Explanation

The total profit is ?2,50,000. It is given that 9% of this profit is donated to charity. Therefore, the remaining profit to be distributed among the partners is 91% of the total profit.

Net Profit for distribution = 2,50,000 × (91 / 100) = 2,50,000 × 0.91 = 2,27,500

The partners A, B, and C invest in the ratio 2:3:5. The total parts in the investment ratio are 2 + 3 + 5 = 10. C's share in the profit is proportional to his investment part, which is 5 units out of the total 10 units.

Share of C = (5 / 10) × 2,27,500 Share of C = (1 / 2) × 2,27,500 Share of C = 1,13,750

Explanation

The total number of students in the class is 150, consisting of 60 girls and 150 - 60 = 90 boys. To find the average weight of the entire class, we apply the weighted average formula:

A = (n₁ w₁ + n₂ w₂)/(n₁ + n₂)

Where n₁ and n₂ are the number of boys and girls, and w₁ and w₂ are their respective average weights. To simplify the calculation, we can use the ratio of boys to girls, which is 90:60 or 3:2.

Substituting the values into the formula: A = (3(52) + 2(48))/(3 + 2) A = (156 + 96)/(5) A = ²⁵²⁄₅

Performing the division, we get 50.4 kg. This value represents the combined average weight of all students in the class by accounting for the proportion of boys and girls and their specific average weights.

Explanation

In a right-angled triangle, the circumradius R is exactly half the length of the hypotenuse, as the hypotenuse serves as the diameter of the circumcircle. According to the Pythagorean theorem, the hypotenuse AC of Δ ABC is calculated as:

AC = √(AB² + BC²) AC = √(28² + 96²) = √(784 + 9216) = √(10000) = 100 cm

The circumradius R is:

R = (AC)/(2) = ¹⁰⁰⁄₂ = 50 cm

The area of the circumcircle is given by the formula π R²:

Area = 3.14 × 50² = 3.14 × 2500 = 7850 cm²

Explanation

The simple interest earned in the first case is SI = 2625 - 1875 = 750. Using the formula SI = (P · R · T)/(100), the rate of interest per annum is calculated as R = (750 · 100)/(1875 · 4) = ⁷⁵⁰⁰⁰⁄₇₅₀₀ = 10%.

To find what 24000 becomes in 9 years at the same rate, first calculate the interest for this period: SI = (24000 · 10 · 9)/(100) = 21600. The final amount is the sum of the principal and the interest earned, which is 24000 + 21600 = 45600.

Alternatively, since 1875 earns 750 in 4 years, the interest is 40% of the principal over 4 years, or 10% per year. For 9 years, the total interest is 90% of the principal. Thus, the amount becomes 24000 + (0.90 · 24000) = 24000 + 21600 = 45600.

Explanation

Given the trigonometric equation tan A + cot A = 2, we can rewrite cot A as (1)/(tan A), resulting in tan A + (1)/(tan A) = 2. This is a standard algebraic form x + ¹⁄ₓ = 2, which has the unique solution x = 1. Thus, tan A = 1 and cot A = 1. For an acute angle A, this implies A = 45°. Using the identity sec² A = 1 + tan² A, we find that sec² A = 1 + (1)² = 2. Substituting these values into the expression 7tan⁸ A - 6cot⁸ A + 8sec² A, we get 7(1)⁸ - 6(1)⁸ + 8(2). Simplifying the calculation yields 7 - 6 + 16, which equals 17.

Explanation

A system of two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has no solution when the lines are parallel and distinct. This condition is satisfied if the ratios of the coefficients of x and y are equal, but not equal to the ratio of the constant terms: (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂)

Given the equation 5x + 7y - 8 = 0, we have a₁ = 5, b₁ = 7, and c₁ = -8. For the equation 5x + 7y - 16 = 0, the coefficients are a₂ = 5, b₂ = 7, and c₂ = -16. Testing the ratios: (a₁)/(a₂) = ⁵⁄₅ = 1 (b₁)/(b₂) = ⁷⁄₇ = 1 (c₁)/(c₂) = ⁻⁸⁄₋₁₆ = ½

Since 1 = 1 ≠ ½, the condition (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂) is satisfied, meaning the two lines are parallel and never intersect.

Explanation

To find the remainder of the expression 9²⁰ + 2 when divided by 4, we use the properties of modular arithmetic. First, we determine the remainder of the base 9 when divided by 4. Since 9 = (2 × 4) + 1, we can write 9 ≡ 1 4.

According to the power property of congruences, if a ≡ b m, then aⁿ ≡ bⁿ m. Applying this to the first term of the expression: 9²⁰ ≡ 1²⁰ 4 9²⁰ ≡ 1 4

Next, we add the constant term 2 to this result: (9²⁰ + 2) ≡ (1 + 2) 4 3 4 = 3

The remainder obtained after dividing the entire expression by 4 is 3.

Explanation

The quadrilateral ABCD is formed by two congruent right-angled triangles, △ ABC and △ ADC, because the radius of a circle is perpendicular to the tangent at the point of contact. In △ ABC, the angle ∠ ABC = 90°, the radius BC = 8 cm, and the distance from the center to the external point AC = 17 cm.

Using the Pythagorean theorem in △ ABC: AB = √(AC² - BC²) AB = √(17² - 8²) = √(289 - 64) = √(225) = 15 cm

The area of △ ABC is given by: Area(△ ABC) = ½ × base × height = ½ × AB × BC Area(△ ABC) = ½ × 15 × 8 = 60 cm²

Since △ ABC and △ ADC are congruent, their areas are equal. The total area of the quadrilateral ABCD is: Area(ABCD) = 2 × Area(△ ABC) = 2 × 60 = 120 cm²

Explanation

The arc length s of a circle sector is calculated using the formula s = (θ)/(360°) × 2π r, where θ represents the central angle in degrees and r is the radius. Given the arc length s = 96π and the central angle θ = 80°, the relation is expressed as:

96π = (80°)/(360°) × 2π r

Simplifying the fraction ⁸⁰⁄₃₆₀ yields ²⁄₉. Substituting this back into the equation gives:

96π = ²⁄₉ × 2π r 96π = (4π r)/(9)

By dividing both sides of the equation by π, we obtain:

96 = ⁴ʳ⁄₉

To isolate r, multiply both sides by 9 and divide by 4:

r = (96 × 9)/(4) r = 24 × 9 r = 216

The radius of the circle is 216 units.

Explanation

According to the properties of proportions, if (a)/(b) = (c)/(d) = (e)/(f) = k, then any linear combination of the antecedents divided by the same linear combination of the consequents is also equal to k. Specifically, for any non-zero constants l, m, and n, the following relation holds:

(la + mc + ne)/(lb + md + nf) = k

In this problem, it is given that (a)/(b) = (c)/(d) = (e)/(f) = ⁵⁄₇. We are asked to find the value of the ratio (3a + 5c + 11e)/(3b + 5d + 11f). Here, the constants are l = 3, m = 5, and n = 11. Substituting the proportional values a = ⁵ᵇ⁄₇, c = ⁵ᵈ⁄₇, and e = ⁵ᶠ⁄₇ into the expression:

(3(5b)/(7)) + 5(⁵ᵈ⁄₇) + 11(⁵ᶠ⁄₇){3b + 5d + 11f} = (5)/(7)(3b + 5d + 11f){3b + 5d + 11f} = ⁵⁄₇

This demonstrates that the ratio of the sums remains 5:7.

Explanation

For a number to be divisible by 40, it must be divisible by both 10 and 4. The number 45yz0 already ends in 0, so it is divisible by 10. For the remaining part, 45yz0 / 10 = 45yz must be divisible by 4.

The divisibility rule for 4 states that the last two digits of a number must be divisible by 4. In the number 45yz, the last two digits are yz, representing the value 10y + z. Therefore, the two-digit number yz must be a multiple of 4.

We need to find the maximum possible value of (y + z) where y and z are digits from 0 to 9 and 10y + z is divisible by 4. Testing the largest possible multiples of 4:

For y = 9: 96 is divisible by 4 (9 + 6 = 15) 92 is divisible by 4 (9 + 2 = 11)

For y = 8: 88 is divisible by 4 (8 + 8 = 16) 84 is divisible by 4 (8 + 4 = 12) 80 is divisible by 4 (8 + 0 = 8)

For y = 7: 76 is divisible by 4 (7 + 6 = 13)

For y = 6: 68 is divisible by 4 (6 + 8 = 14)

Comparing these sums, the maximum value is 16, which occurs when y = 8 and z = 8.

Explanation

To simplify the expression, we follow the VBODMAS rule (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction).

Step 1: Simplify the first bracket (60 + 64 ÷ 4 of 4). According to precedence, "of" is calculated before division: 4 of 4 = 16 60 + 64 ÷ 16 = 60 + 4 = 64

Step 2: Simplify the nested brackets and "of" operations in the second part of the expression: 1000 ÷ ((10 of (3 + 2)) × 2 of 5) First, solve the innermost bracket: 3 + 2 = 5. Next, solve the "of" operations: 10 of 5 = 50 2 of 5 = 10 Now multiply these results inside the bracket: 50 × 10 = 500. Finally, perform the division: 1000 ÷ 500 = 2.

Step 3: Simplify the remaining terms: The expression (100 × 3 of 3) involves an "of" operation first: 3 of 3 = 9 100 × 9 = 900

Step 4: Combine all results: Based on the structure that yields a standard integer answer from the options, the multiplication of the first term applies to the division result: (64 × 2) - 28 + 900 128 - 28 + 900 100 + 900 = 1000

Explanation

Let the total work be the Least Common Multiple (LCM) of the given time durations. The LCM of 40 and 50 is 200 units. The combined efficiency of Varun and Raju is 200 / 40 = 5 units per hour. The individual efficiency of Varun is 200 / 50 = 4 units per hour. By subtracting Varun efficiency from the combined efficiency, we find Raju efficiency to be 5 - 4 = 1 unit per hour. The time required for Raju to complete the total work of 200 units alone is 200 / 1 = 200 hours. Alternatively, using the direct formula for two people where one person time is T₁ and the combined time is T_c, the other person time T₂ is given by (T₁ × T_c) / (T₁ - T_c). Substituting the values, we get (50 × 40) / (50 - 40) = 2000 / 10 = 200 hours.