SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

To solve the equation sin 3A = cos (A - 26°), we use the trigonometric identity relating sine and cosine functions: sin θ = cos (90° - θ). By applying this identity to the left side of the equation, we can express the entire equation in terms of the cosine function:

cos (90° - 3A) = cos (A - 26°)

Since both sides are expressed as cosine functions and 3A is given as an acute angle, we can equate the angles directly:

90° - 3A = A - 26°

Next, we rearrange the equation to isolate terms containing A on one side and constant values on the other:

90° + 26° = A + 3A

116° = 4A

Solving for A:

A = (116°)/(4)

A = 29°

Checking the condition provided in the question, if A = 29°, then 3A = 3 × 29° = 87°. Since 87° is less than 90°, the condition that 3A is an acute angle is satisfied.

Explanation

In an isosceles triangle, the angles opposite to the equal sides are equal. Since XY = XZ, the angles opposite to these sides, ∠ Z and ∠ Y respectively, must be equal. Given that ∠ Y = 80°, it follows that ∠ Z = 80°. According to the angle sum property of a triangle, the sum of the interior angles is always 180°. Therefore, ∠ X + ∠ Y + ∠ Z = 180°. Substituting the known values into the equation gives ∠ X + 80° + 80° = 180°, which simplifies to ∠ X + 160° = 180°. Subtracting 160° from both sides yields ∠ X = 20°.

Explanation

A system of linear equations in the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂ results in a unique solution if the ratio of the coefficients of x is not equal to the ratio of the coefficients of y, satisfying the condition (a₁)/(a₂) ≠ (b₁)/(b₂). For the given system, the first equation is 3x + 2y = 7 and the second is 2x + 3y = 7. Identifying the coefficients, we have a₁ = 3, b₁ = 2, a₂ = 2, and b₂ = 3. Comparing the ratios of these coefficients, we find (a₁)/(a₂) = ³⁄₂ and (b₁)/(b₂) = ⅔. Since ³⁄₂ ≠ ⅔, the two lines represented by these equations are not parallel and intersect at exactly one point in the Cartesian plane. This intersection point represents the single set of values for x and y that satisfies both equations simultaneously.

Explanation

The profit of a company is determined by the formula Profit = Expenditure × (Profit%)/(100), derived from the given income relation. For Company A in the year 2005, the expenditure is ₹40 crore and the profit percentage is 60%. Calculating the profit gives 40 × ⁶⁰⁄₁₀₀ = 24 crore ₹. For Company B in the same year, the expenditure is ₹35 crore and the profit percentage is 48%. Calculating the profit gives 35 × ⁴⁸⁄₁₀₀ = 16.8 crore ₹. The difference between the profits of Company A and Company B in 2005 is 24 - 16.8 = 7.2 crore ₹.

Explanation

To determine the difference between the sales of the two specified categories, we extract the relevant values from the provided data table. Delhi recorded sales of 252 units in the month of June. Chennai recorded sales of 247 units in the month of July.

The difference is calculated by subtracting the smaller value from the larger value as follows:

Difference = Sales of Delhi (June) - Sales of Chennai (July) Difference = 252 - 247 = 5

The numerical gap between the units sold in Delhi during June and the units sold in Chennai during July is exactly 5 units.

Explanation

To find the required ratio, first calculate the total number of people who take tea more than 3 times a day across all four zones: Total (More than 3 times) = 1732 + 1842 + 6045 + 2741 = 12360

Next, calculate the total number of people who do not take tea at all (the Never category) across all four zones: Total (Never) = 317 + 127 + 223 + 293 = 960

The ratio of people who take tea more than 3 times a day to those who do not take tea at all is: Ratio = 12360 : 960

Dividing both terms by 10 gives: 1236 : 96

Further simplifying by dividing both terms by 12: 1236 ÷ 12 = 103 96 ÷ 12 = 8

The simplified ratio is 103:8.

Explanation

The cost per advertisement is calculated using the formula Cost per Advertisement = (Total Money Spent)/(Total Number of Advertisements). The total money spent on one advertisement from each of the three categories for a given year is the sum of these individual costs.

For the year 2018: Sum₂₀₁₈ = (3,50,000)/(10) + (3,60,000)/(12) + (2,10,000)/(15) Sum₂₀₁₈ = 35,000 + 30,000 + 14,000 = 79,000

For the year 2021: Sum₂₀₂₁ = (7,60,000)/(19) + (4,80,000)/(24) + (4,40,000)/(22) Sum₂₀₂₁ = 40,000 + 20,000 + 20,000 = 80,000

The required ratio of the total money spent in 2021 to that in 2018 is: Ratio = 80,000 : 79,000 = 80 : 79

Explanation

The radius R of a circle whose area is equal to the sum of the areas of two circles with radii r₁ and r₂ is determined by the relationship R = √(r₁² + r₂²), derived from π R² = π r₁² + π r₂². Substituting the given values r₁ = 8 cm and r₂ = 15 cm into the formula yields R = √(8² + 15²) = √(64 + 225) = √(289) = 17 cm. The perimeter P of a circle is calculated using the formula P = 2π R. By substituting the calculated radius R = 17 cm and using T as the notation for the constant π, the perimeter is P = 2 × T × 17 = 34T cm.

Explanation

Let the current age of Akash be A and the current age of Ghanshyam be G. The problem states that G = 3A.

Ten years ago, their ages were G - 10 and A - 10 respectively. According to the second condition, G - 10 = 4(A - 10).

Substituting the first equation into the second: 3A - 10 = 4(A - 10) 3A - 10 = 4A - 40 4A - 3A = 40 - 10 A = 30

The current age of Akash is 30 years, so the current age of Ghanshyam is G = 3 × 30 = 90 years. The difference between their current ages is 90 - 30 = 60 years.

Explanation

Given tan A + cot A = 2, we square both sides of the equation to obtain (tan A + cot A)² = 2². Using the identity (x + y)² = x² + y² + 2xy, the expression expands to tan² A + cot² A + 2 tan A cot A = 4. Since cot A is the reciprocal of tan A, their product tan A cot A is equal to 1. Substituting this value, the equation becomes tan² A + cot² A + 2(1) = 4. Subtracting 2 from both sides results in tan² A + cot² A = 4 - 2 = 2. Alternatively, the equation tan A + (1)/(tan A) = 2 is a standard form x + ¹⁄ₓ = 2, which is satisfied only when x = 1. Thus, tan A = 1, and substituting this into the required expression gives 1² + 1² = 2.

Explanation

Expanding the expression using the algebraic identity (a + b)² = a² + b² + 2ab, we get: (sin A + csc A)² = sin² A + csc² A + 2 sin A csc A (cos A + sec A)² = cos² A + sec² A + 2 cos A sec A

Since csc A = (1)/(sin A) and sec A = (1)/(cos A), the terms 2 sin A csc A and 2 cos A sec A simplify to 2(1) = 2. Adding the two expanded expressions results in: sin² A + cos² A + csc² A + sec² A + 2 + 2

Using the fundamental Pythagorean identity sin² A + cos² A = 1, the expression becomes: 1 + csc² A + sec² A + 4 = 5 + csc² A + sec² A

Now, substitute the trigonometric identities csc² A = 1 + cot² A and sec² A = 1 + tan² A: 5 + (1 + cot² A) + (1 + tan² A) = 5 + 1 + 1 + tan² A + cot² A = 7 + tan² A + cot² A

Explanation

The given expression follows the algebraic form (a³ - b³)/(a² + ab + b²), where a = 453 and b = 383. According to the algebraic identity for the difference of two cubes, a³ - b³ = (a - b)(a² + ab + b²). Substituting this into the expression, we get:

((a - b)(a² + ab + b²))/(a² + ab + b²)

The common term (a² + ab + b²) in the numerator and denominator cancels out, leaving the result as a - b. Substituting the original values:

453 - 383 = 70

Explanation

The selling price SP of an article after successive discounts of d₁% and d₂% is calculated using the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}), where MP is the marked price. Given MP = 1600, d₁ = 17, and SP = 747, the equation is set up as follows:

747 = 1600 × (1 - ¹⁷⁄₁₀₀) × (1 - ˣ⁄₁₀₀) 747 = 1600 × 0.83 × (1 - ˣ⁄₁₀₀) 747 = 1328 × (1 - ˣ⁄₁₀₀)

To isolate the term containing x, divide the selling price by the price after the first discount:

1 - ˣ⁄₁₀₀ = ⁷⁴⁷⁄₁₃₂₈ 1 - ˣ⁄₁₀₀ = 0.5625

Solving for x:

ˣ⁄₁₀₀ = 1 - 0.5625 ˣ⁄₁₀₀ = 0.4375 x = 43.75

Explanation

When two trains moving in opposite directions cross each other, the total distance to be covered is the sum of their lengths, and the relative speed is the sum of their individual speeds.

Total distance D = L₁ + L₂ = 132 + 108 = 240 m.

Relative speed Vᵣₑₗ = S₁ + S₂ = 32 + 40 = 72 km/h.

To convert the relative speed into meters per second (m/s), multiply by ⁵⁄₁₈: Vᵣₑₗ = 72 × ⁵⁄₁₈ = 4 × 5 = 20 m/s.

The time taken to cross each other is given by the formula T = (D)/(Vᵣₑₗ₎. T = ²⁴⁰⁄₂₀ = 12 s.

Explanation

When a number N is divided by a divisor D₁ leaving a remainder R₁, and the same number is divided by another divisor D₂ which is a factor of D₁, the new remainder R₂ is simply the remainder obtained by dividing R₁ by D₂.

In this case, the number is divided by 55 with a remainder of 28. Since 11 is a factor of 55 (11 × 5 = 55), the remainder when the same number is divided by 11 is the remainder of 28 ÷ 11.

Performing the division: 28 = (11 × 2) + 6

The remainder is 6.

Explanation

Let the total monthly income of Vipul be I. According to the problem, he spends 30% on rent, leaving a remainder of 70% of I.

From this remainder, he spends 60% on groceries. Therefore, the expenditure on groceries is 0.60 × 0.70I = 0.42I. The remaining amount after the grocery expense is 40% of the first remainder, which is 0.40 × 0.70I = 0.28I.

Next, he spends 40% of this second remainder on others, which means he saves the remaining 60% of it. His savings can be expressed as: Savings = 0.60 × 0.28I = 0.168I

Given that his savings are £ 7056, we set up the equation: 0.168I = 7056 I = (7056)/(0.168) = 42000

The amount spent on groceries is 42% of the total income: Grocery Expenditure = 0.42 × 42000 Grocery Expenditure = 42 × 420 = 17640

Explanation

The total cost of a mixture is the sum of the costs of its individual components. Using the weighted average formula Q₁ P₁ + Q₂ P₂ = (Q₁ + Q₂) Pₘᵢₓₜᵤᵣₑ, where Q represents quantity and P represents price:

10 × 26 + 6 × x = (10 + 6) × 35

260 + 6x = 16 × 35

260 + 6x = 560

6x = 560 - 260

6x = 300

x = ³⁰⁰⁄₆ = 50

Explanation

The volume V of a cube with side length a is given by the formula V = a³. To find the side length when the volume is 13,824 cm³, we calculate the cube root of the volume:

a = ³√(13824)

To solve this efficiently, we can use the method of estimation and unit digits. The number 13,824 ends in the digit 4. In the sequence of cubes of single digits, only 4³ = 64 ends in 4, so the unit digit of the cube root must be 4. Next, we look at the thousands part of the number, which is 13. We know that 2³ = 8 and 3³ = 27. Since 13 lies between 8 and 27, the tens digit of the cube root must be 2.

Combining these, we get a = 24. We can verify this by calculating 24 × 24 × 24. Since 24² = 576, then 576 × 24 = 13824.

Explanation

Let the time taken by pipe B to fill the tank be 5x hours. Since pipe A takes ⅘ of the time required by pipe B, the time taken by pipe A is 4x hours.

The combined rate of pipes A and B filling the tank is ¹⁄₄ₓ + ¹⁄₅ₓ = ⁹⁄₂₀ₓ. Therefore, the time taken by A and B together, T_A_B, is ²⁰ˣ⁄₉ hours.

The problem states that when outlet pipe C is also opened, the time taken, T_A_B_C, is ⅘ more than T_A_B. This means T_A_B_C = T_A_B + ⅘T_A_B = ⁹⁄₅T_A_B. Substituting the value of T_A_B, we get T_A_B_C = ⁹⁄₅ × ²⁰ˣ⁄₉ = 4x hours.

It is given that T_A_B_C = 40 hours. Equating the two expressions for T_A_B_C, we have 4x = 40, which gives x = 10. Substituting x back into the individual times, pipe A takes 40 hours and pipe B takes 50 hours.

The net rate when all three pipes are open is given by (1)/(A) + (1)/(B) - (1)/(C) = (1)/(T_A_B_C₎. Substituting the known values: ¹⁄₄₀ + ¹⁄₅₀ - (1)/(C) = ¹⁄₄₀. By subtracting ¹⁄₄₀ from both sides, we get ¹⁄₅₀ - (1)/(C) = 0, which leads to C = 50 hours.

Explanation

The mean proportional x between two numbers p and q is defined by the relation x = √(pq), which can be rewritten as pq = x². Given that the mean proportional is 12, the product of the two numbers must satisfy pq = 12² = 144. For the values p = 24 and q = 6, the product is 24 × 6 = 144. Since this product matches the square of the mean proportional, these values satisfy the mathematical requirement.

Explanation

In a configuration of two concentric circles, let R = 26 cm be the radius of the larger circle and r = 10 cm be the radius of the smaller circle. When a chord of the larger circle is tangent to the smaller circle, the radius of the smaller circle drawn to the point of tangency is perpendicular to the chord and bisects it.

Let L be the length of the chord. The radius of the larger circle, the radius of the smaller circle, and half of the chord length form a right-angled triangle. According to the Pythagorean theorem, R² = r² + (L/2)². Substituting the given values:

26² = 10² + (L/2)² 676 = 100 + (L/2)² (L/2)² = 576 L/2 = √(576) = 24 cm

The total length of the chord is L = 2 × 24 = 48 cm.

Explanation

The ratio of the cost prices of the sofa and the table can be determined using the method of alligation. Let the profit percentage of the sofa be Pₛ = 10%, the profit percentage of the table be Pₜ = 15.25%, and the overall profit percentage be Pₘ = 13%.

The ratio of the cost price of the sofa (CPₛ) to the cost price of the table (CPₜ) is given by: CPₛ : CPₜ = (Pₜ - Pₘ) : (Pₘ - Pₛ) CPₛ : CPₜ = (15.25 - 13) : (13 - 10) CPₛ : CPₜ = 2.25 : 3

To simplify the ratio, multiply both sides by 100: 225 : 300 = 3 : 4

The total cost price is represented by 3 + 4 = 7 units. Given the total cost price is 28,000: 7 units = 28,000 1 unit = 4,000

The difference between the cost price of the table and the sofa is the difference in their ratio units: Difference = |4 - 3| = 1 unit Difference = 1 × 4,000 = 4,000

Explanation

From the given equation 3 tan A = 4, we have tan A = ⁴⁄₃. In a right-angled triangle, tan A is the ratio of the perpendicular side to the base. Taking the perpendicular P = 4 and the base B = 3, the hypotenuse H is calculated using the Pythagorean theorem:

H = √(P² + B²) = √(4² + 3²) = √(16 + 9) = √(25) = 5

The trigonometric ratios for sin A and cos A are: sin A = (P)/(H) = ⅘ cos A = (B)/(H) = ⅗

Substituting these values into the required expression: 4 sin A + 3 cos A = 4 ( ⅘ ) + 3 ( ⅗ ) = ¹⁶⁄₅ + ⁹⁄₅ = ²⁵⁄₅ = 5

Explanation

The cash price of the cell phone is Rs. 600. Since a down payment of Rs. 300 is made, the remaining principal amount to be financed is Rs. 600 - Rs. 300 = Rs. 300. The buyer pays Rs. 360 after two months to clear the debt, meaning the interest paid is Rs. 360 - Rs. 300 = Rs. 60. The time period for this interest is 2 months, which is equivalent to 2/12 = 1/6 years. Using the simple interest formula I = (P × R × T)/(100), where I is interest, P is principal, R is the annual rate, and T is time in years, we have 60 = (300 × R × (1/6))/(100). Simplifying the equation gives 60 = 3 × R × ⅙, which reduces to 60 = (R)/(2). Thus, R = 120. The rate of interest charged under the scheme is 120% per annum.

Explanation

The degree measure of an arc of a circle is the angle subtended by it at the center. According to the inscribed angle theorem, 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. In this case, the minor arc PQ subtends an angle ∠ POQ = 80° at the center O.

Both points R and S lie on the major arc of PQ. Therefore, the angles ∠ PRQ and ∠ PSQ are inscribed angles subtended by the same minor arc PQ in the same segment. Applying the theorem:

∠ PRQ = ½ ∠ POQ = ½ × 80° = 40° ∠ PSQ = ½ ∠ POQ = ½ × 80° = 40°

Since angles in the same segment of a circle are equal, ∠ PRQ and ∠ PSQ must be identical. Each angle is half the central angle, resulting in a pair of 40° and 40°.

Explanation

The simple interest SI is calculated using the formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time period. Since both loans have the same rate (8%) and the same duration (5 years), the total interest received is the sum of the interests from both principals.

Let P₁ = 3000 be the sum lent to Raju and P₂ be the sum lent to Ravi. The total simple interest is given by: SIₜₒₜₐₗ = ((P₁ + P₂) × R × T)/(100)

Substituting the given values into the formula: 1600 = ((3000 + P₂) × 8 × 5)/(100) 1600 = ((3000 + P₂) × 40)/(100) 1600 = (3000 + P₂) × 0.4

To find the combined principal, divide the total interest by 0.4: 3000 + P₂ = (1600)/(0.4) 3000 + P₂ = 4000

Subtracting the principal lent to Raju from the total principal gives the sum lent to Ravi: P₂ = 4000 - 3000 P₂ = 1000

Explanation

Two straight lines are said to be coinciding when they occupy the exact same position in space and represent the same set of points. In coordinate geometry, for a system of two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the lines coincide if the condition (a₁)/(a₂) = (b₁)/(b₂) = (c₁)/(c₂) is satisfied. This proportionality indicates that the two equations are essentially multiples of each other and define the same line. Since an intersection point is defined as any point that lies on both lines simultaneously, and every point on a coinciding line is also a point on the other, there are infinitely many points of intersection. A straight line by definition consists of an infinite number of points, so two lines that overlap perfectly share all those points.

Explanation

The expression is simplified by applying the trigonometric half-angle identities 1 - cosθ = 2sin²(θ/2), 1 + cosθ = 2cos²(θ/2), and sinθ = 2sin(θ/2)cos(θ/2). Substituting these into the rearranged expression:

((1 - cosθ) + sinθ)/((1 + cosθ) + sinθ) = 2sin²(θ/2) + 2sin(θ/2)cos(θ/2){2cos²(θ/2) + 2sin(θ/2)cos(θ/2)}

Factoring the numerator and denominator to isolate common terms:

(2sin(θ/2) [sin(θ/2) + cos(θ/2)])/(2cos(θ/2) [sin(θ/2) + cos(θ/2)])

Canceling the common factor [sin(θ/2) + cos(θ/2)] and the numerical coefficient 2 reduces the expression to:

(sin(θ/2))/(cos(θ/2)) = tan(θ/2)

Explanation

Let the ages of A and B one year ago be x and 2x respectively. Consequently, their present ages are x + 1 and 2x + 1. After 9 years from now, their ages will be (x + 1) + 9 = x + 10 and (2x + 1) + 9 = 2x + 10.

According to the given condition, the ratio of the ages of B and A after 9 years will be 4 : 3. This can be expressed as:

(2x + 10)/(x + 10) = ⁴⁄₃

By cross-multiplying, we get: 3(2x + 10) = 4(x + 10) 6x + 30 = 4x + 40 2x = 10 x = 5

Substituting the value of x back into the expressions for their present ages, we find that A is 5 + 1 = 6 years old and B is 2(5) + 1 = 11 years old. The sum of their present ages is 6 + 11 = 17 years.

Explanation

The gain is given as ⅙ of the selling price (SP). To calculate the gain percentage, the profit must be expressed as a percentage of the cost price (CP). Let the selling price be 6 units. According to the problem, the gain is 1 unit (1/6 of 6).

The cost price is determined by the formula CP = SP - Gain. Substituting the unit values, we get CP = 6 - 1 = 5 units.

The gain percentage is calculated using the formula: Gain Percentage = (Gain)/(CP) × 100

Substituting the values into the formula: Gain Percentage = ⅕ × 100 = 20%

The specific selling price of ₹25404 is provided in the question but is not required for the calculation since the gain percentage is derived directly from the fractional relationship between the gain and the selling price.

Explanation

The given trigonometric equation is 10sin²θ + 6cos²θ = 7. To solve for tanθ, we can rewrite the equation in terms of tanθ by dividing every term by cos²θ.

(10sin²θ)/(cos²θ) + (6cos²θ)/(cos²θ) = (7)/(cos²θ)

10tan²θ + 6 = 7sec²θ

Using the fundamental identity sec²θ = 1 + tan²θ, we substitute:

10tan²θ + 6 = 7(1 + tan²θ)

10tan²θ + 6 = 7 + 7tan²θ

Rearranging the terms to isolate tan²θ on one side:

10tan²θ - 7tan²θ = 7 - 6

3tan²θ = 1

tan²θ = ⅓

Taking the square root, and considering that θ is in the first quadrant (0° < θ < 90°) where the tangent function is positive:

tanθ = (1)/(√(3))

Explanation

The dimensions of the cuboid follow the ratio L:B:H = 3:2:1. Since the breadth B is given as 12 cm, we can determine the scaling factor x by setting 2x = 12, which yields x = 6.

Using this scaling factor, the length L = 3x = 18 cm and the height H = 1x = 6 cm. The volume V of a cuboid is given by the formula V = L × B × H. Substituting the calculated dimensions results in V = 18 × 12 × 6 = 1296 cm³.

Explanation

The sum of the interior angles of a triangle is always 180°. Given that the angles are in the ratio 1:2:3, let the angles be 1x, 2x, and 3x. According to the angle sum property, 1x + 2x + 3x = 180°, which simplifies to 6x = 180°. Solving for x gives x = 30°. The smallest angle is 1x = 30° and the largest angle is 3x = 90°. The difference between the largest and smallest angle is 3x - 1x = 2x. Substituting the value of x, the difference is 2 × 30° = 60°.

Explanation

Given A = 15°, we first determine the values of the angles used in the expression: 3A = 3 × 15° = 45° 2A = 2 × 15° = 30° 4A = 4 × 15° = 60°

Substituting these into the given expression, we use the standard trigonometric values: cos 45° = (1)/(√(2)) sin 30° = ½ sin 60° = √3{2}

Calculating the numerator: 11√2 cos 45° + 10 sin 30° = 11√2 × (1)/(√(2)) + 10 × ½ = 11 + 5 = 16

Calculating the denominator: 7√3 sin 60° = 7√3 × √3{2} = 7 × ³⁄₂ = ²¹⁄₂

The value of the full expression is: (16)/(21){2} = 16 × ²⁄₂₁ = ³²⁄₂₁

Explanation

Let the two numbers be 2x and 5x. The difference between the parts of the ratio is 5 - 2 = 3 units. Given that the absolute difference between the numbers is 210, we calculate the value of a single unit:

1 unit = ²¹⁰⁄₃ = 70

The smaller number corresponds to 2 units in the ratio. Substituting the value of one unit: Smaller number = 2 × 70 = 140

Explanation

The gain percentage can be calculated directly using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x is the markup percentage and y is the discount percentage. Since a discount reduces the price, y is taken as a negative value.

Substituting x = 25 and y = -18:

Net % = 25 - 18 + (25 × (-18))/(100)

Net % = 7 - ⁴⁵⁰⁄₁₀₀

Net % = 7 - 4.5 = 2.5%

Alternatively, assuming the cost price is 100, the marked price is 125 (25% above cost). A discount of 18% on the marked price means the selling price is 82% of 125.

Selling Price = 125 × ⁸²⁄₁₀₀ = ⁵⁄₄ × 82 = ⁴¹⁰⁄₄ = 102.5

The gain percentage is the difference between the selling price and the cost price: 102.5 - 100 = 2.5%.

Explanation

The rate of work for each pipe is determined by the reciprocal of the time taken to fill or empty the cistern. Pipe A fills at a rate of 1/4 cistern per hour, Pipe B fills at 1/5 cistern per hour, and Pipe C empties at 1/3 cistern per hour. When all three pipes are opened together, the net filling rate per hour is:

Net Rate = ¼ + ⅕ - ⅓

To solve this, find the common denominator for 4, 5, and 3, which is 60:

Net Rate = ¹⁵⁄₆₀ + ¹²⁄₆₀ - ²⁰⁄₆₀ = (15 + 12 - 20)/(60) = ⁷⁄₆₀ cistern per hour

The time required to fill the cistern in hours is the reciprocal of the net rate:

Time (hours) = ⁶⁰⁄₇

To convert this time into minutes, multiply the value by 60:

Time (minutes) = ⁶⁰⁄₇ × 60 = ³⁶⁰⁰⁄₇ ≈ 514.285

Rounding to the nearest minute, the total time taken is 514 minutes.

Explanation

The algebraic identity states that if x + y + z = 0, then x³ + y³ + z³ = 3xyz, which can also be written as x³ + y³ + z³ - 3xyz = 0.

Given the condition a + b = c, we can rearrange it to a + b - c = 0. By letting x = a, y = b, and z = -c, we satisfy the condition x + y + z = 0. Substituting these values into the cubic identity:

a³ + b³ + (-c)³ - 3(a)(b)(-c) = 0

Simplifying the terms, we get:

a³ + b³ - c³ + 3abc = 0

Explanation

The initial savings of the person are determined by the formula S = I - E, where I is income and E is expenditure. Given I = 16,500 and E = 12,000, the initial savings are 16,500 - 12,000 = 4,500.

The change in savings is given by Δ S = Δ I - Δ E. The increase in income is 24% of 16,500, which is 0.24 × 16,500 = 3,960. The expenditure decreases by 10%, meaning the change in expenditure is -10% of 12,000, which is -1,200. Substituting these values into the change formula, we get Δ S = 3,960 - (-1,200) = 3,960 + 1,200 = 5,160.

The percentage increase in savings is calculated as (Δ S)/(S) × 100. Plugging in the values results in (5,160)/(4,500) × 100 = (516)/(4.5) ≈ 114.67%. Rounding to the nearest integer, the percentage increase is 115%.

Explanation

The Midpoint Theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half its length. Applying this to Δ ABC where D, E, and F are the midpoints of AB, BC, and AC respectively, the side lengths of the medial triangle Δ DEF are determined as DE = ½ AC, EF = ½ AB, and DF = ½ BC.

Congruent triangles must have identical dimensions and equal corresponding side lengths. Since each side of Δ ABC is exactly twice the length of the corresponding side in Δ DEF, the triangles maintain a 2:1 proportionality. This relationship defines them as similar triangles (Δ ABC ∼ Δ EFD) rather than congruent ones. The area of Δ ABC is 2² = 4 times the area of Δ DEF, further confirming that they are not equal in size.

Explanation

According to the divisibility rule for 3, a number is divisible by 3 if the sum of its digits is a multiple of 3. For the number 436P5, the sum of the digits is 4 + 3 + 6 + P + 5 = 18 + P. To make the total sum divisible by 3, the value of P must be such that (18 + P) is a multiple of 3. Since 18 itself is divisible by 3, P must also be a digit that is a multiple of 3. The possible single-digit values for P are 0, 3, 6, and 9. Comparing these values, the largest possible value for P is 9.

Explanation

The relationship is expressed as the proportion (A)/(C) = (B)/(D). To solve for D, the formula is rearranged as D = (B × C)/(A). Substituting the given values A = 25, B = 40, and C = 30 into the equation:

D = (40 × 30)/(25)

D = ¹²⁰⁰⁄₂₅ = 48

Alternatively, the ratio A:C simplifies as 25:30 = 5:6. Since the problem states A:C = B:D, the ratio 40:D must also equal 5:6. Multiplying both terms of the simplified ratio 5:6 by 8 to match the value of B yields 40:48.

Explanation

According to the Internal Angle Bisector Theorem, the ratio of the segments BD and DC is equal to the ratio of the sides AB and AC. Therefore, BD/DC = AB/AC = 6/12 = 1/2. Dividing the length of BC (9 cm) in the ratio 1:2 gives BD = (1/3) * 9 = 3 cm.

Explanation

The sum of the first n natural numbers is calculated using the arithmetic progression formula Sₙ = ⁿ⁽ⁿ⁺¹⁾⁄₂. For the first 12 natural numbers, we substitute n = 12 into the formula to find the total sum:

S₁₂ = (12 × (12 + 1))/(2) = (12 × 13)/(2) = 6 × 13 = 78

The number 6 is a composite number because it has more than two factors, specifically 1, 2, 3, and 6. By applying the rules of divisibility, we see that 78 is divisible by 6 because it is an even number (divisible by 2) and the sum of its digits (7 + 8 = 15) is divisible by 3. Dividing 78 by 6 yields exactly 13 with no remainder (78 ÷ 6 = 13). Therefore, 6 is the composite number that divides the sum of the first 12 natural numbers.

Explanation

Let O be the common center of the concentric circles with radii R for the outer circle and r for the inner circle. Let p be the perpendicular distance from the center O to the line cutting the circles. A theorem in geometry states that a perpendicular from the center of a circle to a chord bisects the chord.

For the inner circle, the chord length is 6 cm, so the length of the segment from the foot of the perpendicular to the circle's edge is 6/2 = 3 cm. Applying the Pythagorean theorem to the right triangle formed by the radius, the perpendicular distance, and the half-chord: r² = p² + 3² = p² + 9

For the outer circle, the chord length is 18 cm, so the length of the segment from the foot of the perpendicular to the circle's edge is 18/2 = 9 cm. Applying the Pythagorean theorem to the corresponding right triangle: R² = p² + 9² = p² + 81

To find the difference in the squares of the radii, subtract the first equation from the second: R² - r² = (p² + 81) - (p² + 9) R² - r² = 81 - 9 = 72

Explanation

The average score of the entire class is calculated using the weighted average formula. In a class of 30 students, 12 students scored an average of 62, while the remaining 30 - 12 = 18 students scored an average of 78. To simplify the arithmetic, the ratio of the number of students in these two groups can be expressed as 12 : 18, which reduces to 2 : 3.

The class average score A is determined by: A = ((2 × 62) + (3 × 78))/(2 + 3) A = (124 + 234)/(5) A = ³⁵⁸⁄₅ = 71.6

Alternatively, using the total sum method, the aggregate marks of the 30 students are calculated as: Total Marks = (12 × 62) + (18 × 78) = 744 + 1404 = 2148 Dividing this total by the number of students results in 2148 / 30 = 71.6.

Explanation

The policeman and the thief are moving in the same direction, so their relative speed is the difference between their individual speeds.

Relative speed vᵣₑₗ = 82 km/h - 64 km/h = 18 km/h

To convert this speed into meters per second, multiply by ⁵⁄₁₈: vᵣₑₗ = 18 × ⁵⁄₁₈ = 5 m/s

The time t required for the policeman to catch the thief is the time taken to cover the initial gap of 1300 m at this relative speed: t = (Distance)/(Relative Speed) t = ¹³⁰⁰⁄₅ = 260 seconds

Explanation

The length of a direct common tangent to two circles with radii R and r and distance between centers D is given by √(D² - (R - r)²). When two circles touch each other externally, the distance between their centers D is equal to the sum of their radii R + r. Substituting D = R + r into the formula, the length of the direct common tangent simplifies to 2√(Rr).

Given R = 18 cm and r = 12 cm, the length of the tangent is: L = 2√(18 × 12) L = 2√(216) L = 2√(36 × 6) L = 2 × 6 √6 L = 12√6 cm

Explanation

The problem can be solved efficiently using the rule of allegation or the concept of deviation. First, calculate the average interest rate for the entire sum. The total interest of ₹352 on a principal of ₹4,000 for one year gives an average annual rate R: R = (352 × 100)/(4000 × 1) = 8.8%

Applying the rule of allegation between the two interest rates of 8% and 10% to find the ratio of the principal amounts: Ratio of sum at 8% to sum at 10% = (10 - 8.8) : (8.8 - 8) = 1.2 : 0.8 = 3 : 2

The total sum of ₹4,000 is divided into 3 + 2 = 5 parts. Each part is equal to 4000 / 5 = 800. Since the sum lent at 10% corresponds to 2 parts in the ratio: Sum at 10% = 2 × 800 = 1600

Alternatively, use the deviation method. If the entire ₹4,000 were lent at the lower rate of 8%, the annual interest would be 4000 × 0.08 = 320. However, the actual interest received is ₹352. The difference of 352 - 320 = 32 arises because a portion of the money was lent at 10% instead of 8%. The difference in interest rates is 10% - 8% = 2%. If x is the sum lent at 10%, then: 2% of x = 32 0.02x = 32 x = (32)/(0.02) = 1600

Explanation

To determine if a circle can pass through the points X, Y, and Z, we must check their relative positions. Given the distances XY = 11 cm, YZ = 13 cm, and XZ = 24 cm, we observe that XY + YZ = 11 + 13 = 24 cm. Since the sum of the lengths of two segments XY and YZ is exactly equal to the length of the third segment XZ, the points X, Y, and Z are collinear, meaning they lie on a single straight line.

A fundamental theorem in geometry states that one and only one circle passes through any three non-collinear points. If three points are collinear, the perpendicular bisectors of the segments connecting them (such as the bisector of XY and the bisector of YZ) are parallel to each other. Because these perpendicular bisectors never intersect, there is no point that is equidistant from X, Y, and Z to serve as the center of a circle. Consequently, it is impossible for a circle to pass through three points that lie on a straight line.