SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The final concentration of a mixture is determined by the weighted average of the concentrations of its components. Using the formula for weighted averages:

C_fᵢₙₐₗ = ((w_A × C_A) + (w_B × C_B) + (w_C × C_C))/(w_A + w_B + w_C)

Given the concentrations C_A = 20%, C_B = 30%, C_C = 40%, and the mixing ratio 3 : 5 : a, we substitute these values into the equation where the final concentration is 30%:

30 = ((3 × 20) + (5 × 30) + (a × 40))/(3 + 5 + a)

30 = (60 + 150 + 40a)/(8 + a)

30(8 + a) = 210 + 40a

240 + 30a = 210 + 40a

240 - 210 = 40a - 30a

30 = 10a

a = 3

Alternatively, since the concentration of acid B is exactly 30%, it does not affect the final average if the weighted average of acids A and C is also 30%. For A (20%) and C (40%) to average out to 30%, the weights must balance the equal deviations (10% below and 10% above the target). Thus, the weight of A must equal the weight of C, which means 3 = a.

Explanation

The given expression follows the algebraic identity (a³ - b³)/(a² + ab + b²) = a - b. By observing the terms, let a = 0.58 and b = 0.1. The numerator is 0.58 × 0.58 × 0.58 - 0.001, which is equivalent to a³ - b³ since 0.1³ = 0.001. The denominator is 0.58 × 0.58 + 0.058 + 0.01, which corresponds to a² + ab + b² because 0.58 × 0.1 = 0.058 and 0.1² = 0.01. Applying the identity, the entire fraction simplifies to a - b. Therefore, the calculation is 0.58 - 0.1, which equals 0.48.

Explanation

To find the total gain percentage for successive increases, use the effective percentage formula: Total Gain % = a + b + ᵃᵇ⁄₁₀₀

In this case, both the first increase (a) and the second increase (b) are 50%. Substituting these values into the formula: Total Gain % = 50 + 50 + (50 × 50)/(100) Total Gain % = 100 + ²⁵⁰⁰⁄₁₀₀ Total Gain % = 100 + 25 = 125%

Alternatively, if the initial salary is assumed to be 100, the first 50% increase makes it 150. The second 50% increase is calculated on this new amount: 50% of 150 is 75. Thus, the final salary is 150 + 75 = 225. The total gain is 225 - 100 = 125, which represents a 125% increase from the original salary.

Explanation

The given expression is secθ(1 + cot²θ)(csc²θ - cot²θ){csc³θ}. Using the fundamental trigonometric identities 1 + cot²θ = csc²θ and csc²θ - cot²θ = 1, we can substitute these values directly into the expression.

The numerator becomes secθ · csc²θ · 1. Dividing this by the denominator csc³θ, the expression simplifies to: secθ · csc²θ{csc³θ} = (secθ)/(cscθ)

By converting the terms into sine and cosine, we get: (1/cosθ)/(1/sinθ) = (sinθ)/(cosθ) = tanθ

Given that cotθ = 4/3, and knowing that tanθ = (1)/(cotθ), we calculate: tanθ = (1)/(4/3) = 3/4

Explanation

The expression given is 77⁷⁷ + 77. To find the remainder when divided by 78, we use modular arithmetic properties. Since 77 is one less than 78, we can express it as 77 ≡ -1 78.

Substituting this into the expression: 77⁷⁷ + 77 ≡ (-1)⁷⁷ + (-1) 78

Because the exponent 77 is an odd number, (-1)⁷⁷ simplifies to -1. The congruence then becomes: -1 + (-1) 78 = -2 78

To find the standard positive remainder, we add the divisor to the negative result: 78 - 2 = 76

Therefore, the remainder when 77⁷⁷ + 77 is divided by 78 is 76.

Explanation

The single equivalent discount D for two successive discounts d₁ and d₂ is given by the formula D = (d₁ + d₂ - (d₁ × d₂)/(100))%.

Substituting the given values d₁ = 40 and d₂ = 30: D = (40 + 30 - (40 × 30)/(100))% D = (70 - ¹²⁰⁰⁄₁₀₀)% D = (70 - 12)% D = 58%

Alternatively, if the initial marked price is assumed to be 100, the first discount of 40% reduces the price to 60. The second discount of 30% is then applied to the remaining 60, which equals 18 (0.30 × 60). The final price becomes 60 - 18 = 42. The total discount amount is the difference between the initial price and the final price, which is 100 - 42 = 58. Expressed as a percentage of the initial price, this is 58%.

Explanation

In a circle, the perpendicular dropped from the center to a chord bisects that chord. This construction forms a right-angled triangle where the hypotenuse is the radius of the circle (r), one leg is the distance of the chord from the center (d), and the other leg is half the length of the chord (L/2).

Applying the Pythagorean theorem: r² = d² + (L/2)²

Substitute the given values, where r = 5√(13) cm and d = 10 cm: (5√(13))² = 10² + (L/2)² 25 × 13 = 100 + (L/2)² 325 = 100 + (L/2)²

Subtracting 100 from both sides gives: 225 = (L/2)²

Taking the square root of both sides: 15 = L/2

To find the full length of the chord (L), multiply by 2: L = 15 × 2 = 30 cm.

Explanation

The annual installment x required to discharge a debt A due in n years at simple interest rate R percent per annum is given by the formula A = nx + (Rx × n(n-1))/(200). This formula accounts for the fact that each installment, except the last one, earns interest from the time it is paid until the end of the n-year period.

Substituting the given values A = 26,160, n = 4 years, and R = 6% into the formula:

26160 = 4x + (6 × x × 4 × (4 - 1))/(200)

26160 = 4x + (6 × x × 12)/(200)

26160 = 4x + ⁷²ˣ⁄₂₀₀

26160 = 4x + 0.36x

26160 = 4.36x

x = (26160)/(4.36)

x = ²⁶¹⁶⁰⁰⁰⁄₄₃₆

Dividing 2,616,000 by 436 gives 6,000. Thus, an annual installment of ₹6,000 will discharge the debt of ₹26,160 in 4 years at 6% simple interest per annum.

Explanation

To simplify the expression 21 - [6 + 7 - {3.22 - (1.1 × 0.2)}], apply the BODMAS rule, which prioritizes operations inside brackets in the order of circular (), curly {}, and then square [].

First, calculate the product inside the circular brackets: 1.1 × 0.2 = 0.22

Next, solve the expression within the curly braces using the result from the previous step: {3.22 - 0.22} = 3

Then, evaluate the terms within the square brackets: [6 + 7 - 3] = 13 - 3 = 10

Finally, perform the subtraction outside the brackets to find the final value: 21 - 10 = 11

Explanation

According to the order of operations (BODMAS), we first perform the division operations in the given equation 45 ÷ 3 + 4 × A - 48 ÷ 24 + 4 = 1.

Substituting the results of the divisions: 15 + 4 × A - 2 + 4 = 1

Next, we group the constant terms and simplify the linear equation: 15 - 2 + 4 + 4A = 1 17 + 4A = 1

Now, isolate the term containing A: 4A = 1 - 17 4A = -16

Solving for A: A = ⁻¹⁶⁄₄ A = -4

Explanation

The length of the common external tangent Lₑₓₜ between two circles with radii r₁ and r₂ and distance d between their centers is given by the formula Lₑₓₜ² = d² - (r₁ - r₂)². The length of the common internal tangent Lᵢₙₜ is given by the formula Lᵢₙₜ² = d² - (r₁ + r₂)².

To find the product of the radii, we subtract the equation for the internal tangent from the equation for the external tangent to eliminate the distance d: Lₑₓₜ² - Lᵢₙₜ² = [d² - (r₁ - r₂)²] - [d² - (r₁ + r₂)²] Lₑₓₜ² - Lᵢₙₜ² = (r₁ + r₂)² - (r₁ - r₂)²

Using the algebraic identity (a + b)² - (a - b)² = 4ab, the expression simplifies to: Lₑₓₜ² - Lᵢₙₜ² = 4r₁r₂

Substituting the given values Lₑₓₜ = 9 and Lᵢₙₜ = 5: 9² - 5² = 4r₁r₂ 81 - 25 = 4r₁r₂ 56 = 4r₁r₂

Dividing by 4 to solve for the product of the radii: r₁r₂ = 14

Explanation

Since Δ PQR ≅ Δ MON, the corresponding parts of congruent triangles (CPCT) are equal. This implies that corresponding angles and corresponding sides must match in the order they are named.

Comparing corresponding angles, we have ∠ PQR = ∠ MON. Substituting the given values: 82° = (5y - 8)° 5y = 82 + 8 5y = 90 y = 18

Comparing corresponding sides, we have QR = ON. Substituting the given values and the calculated value of y: 24 = 3x + y 24 = 3x + 18 3x = 24 - 18 3x = 6 x = 2

The value of (x + y) is: 2 + 18 = 20

Explanation

For a number to be divisible by 5, the units digit # must be either 0 or 5. However, since the number must also be divisible by 8, it must be an even number, which restricts the units digit # to 0. The divisibility rule for 8 states that the number formed by the last three digits must be divisible by 8. In the 7-digit number 62684#, the last three digits are 40. We substitute the value # = 0 and test the digit = 4 from the given options. The resulting three-digit number is 440. Since 440 = 8 × 55, it satisfies the condition for divisibility by 8. Thus, the digits for * and # are 4 and 0 respectively.

Explanation

The area of a square is given by the formula A = s², where s is the length of the side. To find the side length from the expression 16x² + 40x + 25, we recognize it as a perfect square trinomial of the form a² + 2ab + b² = (a + b)². Identifying the terms, we see that 16x² = (4x)² and 25 = 5², while the middle term 40x matches 2(4x)(5). Therefore, the area can be written as (4x + 5)², and the side length s is 4x + 5 units.

The perimeter of a square is defined as P = 4s. Substituting the side length into this formula gives P = 4(4x + 5), which simplifies to 16x + 20. Factoring out a 2 from this expression results in 2(8x + 10).

Explanation

The average of the squares of the first n natural numbers is derived from the sum formula S = ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆. By dividing the sum by the number of terms n, the formula for the average becomes ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆.

For the first 45 natural numbers, substituting n = 45 into the formula: Average = ((45 + 1)(2 × 45 + 1))/(6) Average = (46 × 91)/(6)

Simplifying the fraction by dividing 46 and 6 by 2: Average = (23 × 91)/(3) Average = ²⁰⁹³⁄₃ Average = 697.666...

Rounding to two decimal places, the result is 697.67.

Explanation

When a rectangular paper is rolled along its breadth to form a cylinder, the breadth of the paper becomes the circumference of the cylinder's base, and the length of the paper becomes the height of the cylinder. Given the length is 50 cm and the breadth is 22 cm, we have the height h = 50 cm and the circumference C = 22 cm.

Using the formula for circumference C = 2 π r, where r is the radius: 2 × ²²⁄₇ × r = 22 r = (22 × 7)/(2 × 22) = ⁷⁄₂ = 3.5 cm

The volume V of a cylinder is given by the formula V = π r² h. Substituting the values: V = ²²⁄₇ × (⁷⁄₂)² × 50 V = ²²⁄₇ × ⁴⁹⁄₄ × 50 V = 11 × 7 × 25 V = 77 × 25 = 1,925 cm³

Explanation

To find the value of (x - y)/(x + y), we use the algebraic identities for the square of a sum and the square of a difference. First, calculate (x + y)² and (x - y)² using the given values x² + y² = 280 and xy = 120:

(x + y)² = x² + y² + 2xy = 280 + 2(120) = 280 + 240 = 520 (x - y)² = x² + y² - 2xy = 280 - 2(120) = 280 - 240 = 40

Next, find the ratio of the squares: (x - y)²{(x + y)²} = ⁴⁰⁄₅₂₀ = ⁴⁄₅₂ = ¹⁄₁₃

Taking the square root of both sides to find the required ratio: (x - y)/(x + y) = √(¹⁄₁₃) = (1)/(√(13))

Explanation

The speed of the boat in downstream is the sum of the speed of the boat in still water and the speed of the stream, while the speed in upstream is the difference between the two.

Downstream speed v_d = 18 + 3 = 21 km/h Upstream speed vᵤ = 18 - 3 = 15 km/h

The total time taken for the round trip is the sum of the time taken to travel downstream and the time taken to travel upstream over the same distance of 105 km.

Total Time T = (Distance)/(v_d) + (Distance)/(vᵤ) T = ¹⁰⁵⁄₂₁ + ¹⁰⁵⁄₁₅ T = 5 + 7 = 12 hours

Explanation

Let the principal be P, the rate of interest be R, and the time be T years. Using the simple interest formula SI = (P × R × T)/(100), we are given that SI = (P)/(5) and R = 5T. Substituting these values into the formula:

(P)/(5) = (P × 5T × T)/(100)

⅕ = (5T²)/(100)

T² = ¹⁰⁰⁄₂₅ = 4

T = 2 years

The original rate of interest is R = 5 × 2 = 10% per annum. The new rate of interest is Rₙₑ_w = 10% + 3% = 13% per annum.

The simple interest on ₹5,600 for 6 years at the new rate is:

SI = (5600 × 13 × 6)/(100)

SI = 56 × 78

SI = 4368

Explanation

The final selling price after multiple successive discounts is calculated using the formula SP = MP × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100). Here, the marked price MP is ₹1,000 and the successive discounts are 20%, 30%, and 10%.

Substituting these values into the formula: SP = 1000 × (1 - 20/100) × (1 - 30/100) × (1 - 10/100) SP = 1000 × (80/100) × (70/100) × (90/100) SP = 1000 × 0.8 × 0.7 × 0.9 SP = 1000 × 0.504 SP = 504

The product of the three discount factors results in a final price multiplier of 0.504, which when applied to the marked price of 1,000, yields 504.

Explanation

Let the total capacity of the tank be the Least Common Multiple (LCM) of the given times 10, 15, and 12, which is 60 units. Based on this capacity, the efficiency of pipe P is 60 / 10 = 6 units per hour, and the efficiency of pipe Q is 60 / 15 = 4 units per hour. Since pipe R is an outlet pipe, its efficiency is 60 / 12 = -5 units per hour.

Between 6 a.m. and 9 a.m., only pipes P and Q are operational. In these 3 hours, the amount of water filled is (6 + 4) × 3 = 30 units. The remaining capacity to be filled is 60 - 30 = 30 units. From 9 a.m. onwards, all three pipes P, Q, and R are open, resulting in a net filling rate of 6 + 4 - 5 = 5 units per hour. The time required to fill the remaining 30 units is 30 / 5 = 6 hours. Starting from 9 a.m., adding 6 hours brings the time to 3 p.m.

Explanation

For a system of two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution, the lines must be parallel and distinct. This occurs when the ratios of the coefficients of x and y are equal, but not equal to the ratio of the constant terms, as defined by the condition (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂).

Given the equations 2x + 7ky - 8 = 0 and x + y - 9 = 0, we identify the coefficients as a₁ = 2, b₁ = 7k, c₁ = -8 and a₂ = 1, b₂ = 1, c₂ = -9. Applying the condition for no solution, we set the ratio of the coefficients of x equal to the ratio of the coefficients of y:

²⁄₁ = ⁷ᵏ⁄₁

Solving this equation for k gives 2 = 7k, which simplifies to k = ²⁄₇. We also verify the third part of the condition: (a₁)/(a₂) = 2 and (c₁)/(c₂) = ⁻⁸⁄₋₉ = ⁸⁄₉. Since 2 ≠ ⁸⁄₉, the lines are parallel and not coincident, confirming that there is no solution when k = ²⁄₇.

Explanation

The length of the transverse common tangent (L) between two circles is calculated using the formula L = √(d² - (r₁ + r₂)²), where d represents the distance between the centers of the circles and r₁ and r₂ represent their radii.

Given values: r₁ = 5 cm r₂ = 10 cm d = 17 cm

First, calculate the sum of the radii: r₁ + r₂ = 5 + 10 = 15 cm

Now, substitute the values into the formula: L = √(17² - 15²) L = √(289 - 225) L = √(64) L = 8 cm

The calculation follows the Pythagorean triplet (8, 15, 17), where 17 is the hypotenuse (distance between centers) and 15 is one leg (sum of radii), making the length of the tangent the other leg, which is 8 cm.

Explanation

Let the total number of enrolled voters be x. According to the problem, 85% of these voters cast their votes, and 2% of those cast votes were invalid, meaning 98% were valid. Sunita received 78% of these valid votes, which equals 64,974.

The relationship can be expressed by the following equation: x × ⁸⁵⁄₁₀₀ × ⁹⁸⁄₁₀₀ × ⁷⁸⁄₁₀₀ = 64974

To find x, we rearrange the equation: x = (64974 × 100 × 100 × 100)/(85 × 98 × 78)

Simplifying the calculation: 64974 ÷ 78 = 833 x = (833 × 10⁶)/(85 × 98)

Further simplifying by dividing 833 and 98 by their common factor 49 (since 833 = 49 × 17 and 98 = 49 × 2): x = (17 × 10⁶)/(85 × 2)

Since 85 = 17 × 5: x = (1 × 10⁶)/(5 × 2) x = (1,000,000)/(10) x = 1,00,000

Explanation

Using the trigonometric identity for complementary angles sin(90° - θ) = cos θ and cos(90° - θ) = sin θ, we can simplify each fraction in the expression.

In the first term, since 58° + 32° = 90°, then sin 58° = sin(90° - 32°) = cos 32°. Substituting this into the first fraction gives: (3 cos 32°)/(5 cos 32°) = ⅗

In the second term, since 62° + 28° = 90°, then cos 62° = cos(90° - 28°) = sin 28°. Substituting this into the second fraction gives: (2 sin 28°)/(5 sin 28°) = ⅖

Summing these two values results in: ⅗ + ⅖ = ⁵⁄₅ = 1

Explanation

Given 7tanθ = 3, so tanθ = 3{7}.

Dividing both the numerator and denominator of the expression by cosθ:

(5sinθ - cosθ)/(5sinθ + 2cosθ) = (5tanθ - 1)/(5tanθ + 2)

Substituting tanθ = 3{7}:

= (5 × 3)/(7) - 1{5 × 3{7} + 2} = (15)/(7) - {7{7}}{15{7} + 14{7}} = (8)/(7){29{7}} = ⁸⁄₂₉

Explanation

The given equation 7cos²θ + 5sin²θ = 6 can be simplified using the identity sin²θ + cos²θ = 1: 5(sin²θ + cos²θ) + 2cos²θ = 6 5(1) + 2cos²θ = 6 2cos²θ = 1 ⟹ cos²θ = 1/2

Since 0° < θ < 90°, cosθ = 1/√2, which gives θ = 45°. At this value, sec 45° = √2 and tan 45° = 1.

Now, simplify the required expression using the identity sec²θ - tan²θ = 1: √(secθ + tanθ)√(secθ - tanθ) = √(((secθ + tanθ)(secθ + tanθ))/((secθ - tanθ)(secθ + tanθ))) = √((secθ + tanθ)²(sec²θ - tan²θ)) = secθ + tanθ

Substituting the values for θ = 45° into the complete expression: 1 + (sec 45° + tan 45°) = 1 + √2 + 1 = √2 + 2

Explanation

The concentration of milk in a mixture is the ratio of the volume of milk to the total volume of the mixture.

For the first man, the total volume of the mixture is 10 + 6 = 16 litres. The proportion of milk in this mixture is ¹⁰⁄₁₆, which simplifies to ⅝.

For the second man, the total volume of the mixture is 8 + 4 = 12 litres. The proportion of milk in this mixture is ⁸⁄₁₂, which simplifies to ⅔.

To find the ratio of milk in the two mixtures, we compare these two proportions: Ratio = ⅝ : ⅔

By cross-multiplying to find a common denominator, the ratio becomes (5 × 3) : (2 × 8), which results in 15 : 16.

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) when a discount is offered and a profit is earned is derived from the fact that the Selling Price (SP) is common to both. Specifically, SP = MP × (100 - D%)/(100) and SP = CP × (100 + P%)/(100). Equating these gives the standard ratio:

(CP)/(MP) = (100 - Discount%)/(100 + Profit%)

Given a discount of 20% and a gain of 10%, substitute these values into the formula:

(CP)/(MP) = (100 - 20)/(100 + 10)

(CP)/(MP) = ⁸⁰⁄₁₁₀

Dividing both terms by 10 simplifies the fraction to:

(CP)/(MP) = ⁸⁄₁₁

Explanation

The area A of a sector of a circle is given by A = ½rl and the perimeter P is given by P = l + 2r, where r is the radius and l is the arc length. Substituting l = 63 - 2r into the area formula:

209 = ½r(63 - 2r) 418 = 63r - 2r² 2r² - 63r + 418 = 0

Using the quadratic formula r = -b ± √(b² - 4ac){2a}: r = 63 ± √(63² - 4 · 2 · 418){4} r = 63 ± √(3969 - 3344){4} = 63 ± √(625){4} r = (63 ± 25)/(4)

The values for r are 22 and 9.5. Since the radius is a non-negative integer, we take r = 22 cm.

The arc length l is: l = 63 - 2(22) = 19 cm

The central angle θ in degrees is calculated using the relation l = (θ)/(360) · 2π r: 19 = (θ)/(360) · 2 · ²²⁄₇ · 22 θ = (19 · 360 · 7)/(2 · 22 · 22) = ⁴⁷⁸⁸⁰⁄₉₆₈ ≈ 49.4628°

Rounding to one decimal place, the central angle is 49.5°.

Explanation

To find the smallest number to be added to 999 so that it becomes exactly divisible by 99, we first calculate the remainder when 999 is divided by 99.

Dividing 999 by 99: 999 = (99 × 10) + 9

The remainder r is 9. For a number to be exactly divisible by a divisor d after adding a value x, the required value is calculated as x = d - r. Substituting the given values: x = 99 - 9 = 90

By adding 90 to 999, we get 1089. Verifying the divisibility: 1089 ÷ 99 = 11

Since the division is exact, 90 is the smallest number that must be added.

Explanation

To find the value of x³ - (1)/(x³), we first determine the value of x - ¹⁄ₓ using the given sum x + ¹⁄ₓ = 2√(10). We apply the algebraic identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4. Substituting the given value results in: (x - ¹⁄ₓ)² = (2√(10))² - 4 = 40 - 4 = 36

Since x > 1, the difference x - ¹⁄ₓ must be positive, which gives x - ¹⁄ₓ = √(36) = 6.

Next, we use the identity for the difference of cubes, x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ). Substituting the value 6 into this formula yields: x³ - (1)/(x³) = 6³ + 3(6) x³ - (1)/(x³) = 216 + 18 = 234

Explanation

The number of valid votes received by candidate Q is determined by applying the percentages of participation and validity to the total electorate. Candidate P received 78 percent of the valid votes, which implies candidate Q received the remaining 22 percent of the valid votes.

The percentage of the electorate that cast a vote is 100 percent minus 10 percent, which equals 90 percent. The percentage of polled votes that were valid is 100 percent minus 15 percent, which equals 85 percent.

The formula for the number of valid votes for Q (V_Q) is: V_Q = Total × (% Polled) × (% Valid) × (% Q's share)

Substituting the given values: V_Q = 75,60,000 × ⁹⁰⁄₁₀₀ × ⁸⁵⁄₁₀₀ × ²²⁄₁₀₀ V_Q = 75,60,000 × 0.90 × 0.85 × 0.22

Performing the calculation: 75,60,000 × 0.90 = 68,04,000 (Polled votes) 68,04,000 × 0.85 = 57,83,400 (Valid votes) 57,83,400 × 0.22 = 12,72,348 (Valid votes for Q)

Explanation

The Intersecting Chords Theorem states that when two chords of a circle intersect at a point inside the circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. For chords MN and PQ intersecting at O, the relation is MO × ON = PO × OQ.

Given the lengths MO = 9 cm, ON = 5 cm, and OQ = 6 cm, we substitute these into the formula:

9 × 5 = OP × 6 45 = OP × 6 OP = ⁴⁵⁄₆ OP = 7.5 cm

Explanation

In a triangle ABC right-angled at C, the interior angle C is 90°. According to the angle sum property of a triangle, the sum of all interior angles is 180°, which means A + B + C = 180°. Substituting C = 90°, we find that A + B = 90°.

The expression to be evaluated is sin A cos B cot (A + B). By substituting the value of the sum of angles A and B, we have:

sin A cos B cot (90°)

The trigonometric value of cot 90° is 0. Given that tan A = √3, we know A = 60° and therefore B = 30°. The terms sin 60° and cos 30° are both equal to √3/2. Substituting these values into the expression:

(√3/2) · (√3/2) · 0 = 0

Since any finite product involving zero as a factor is zero, the value of the expression is 0.

Explanation

To find the least number of square tiles required to pave a rectangular courtyard, the side of each square tile must be the largest possible. This is determined by finding the Highest Common Factor (HCF) of the length and breadth of the courtyard.

First, convert the dimensions to centimeters: Length L = 4 m 95 cm = 495 cm Breadth B = 16 m 65 cm = 1665 cm

Next, find the HCF of 495 and 1665. The prime factorization of 495 is 3² × 5 × 11. The prime factorization of 1665 is 3² × 5 × 37. The HCF(495, 1665) = 3² × 5 = 45 cm. Thus, the side of the largest square tile is 45 cm.

The total number of tiles required is the area of the courtyard divided by the area of one square tile: Number of tiles = (Area of courtyard)/(Area of one tile) = (495 × 1665)/(45 × 45)

Simplifying the calculation: ⁴⁹⁵⁄₄₅ = 11 ¹⁶⁶⁵⁄₄₅ = 37 Number of tiles = 11 × 37 = 407

Explanation

The area of a circular track is defined as the difference between the area of the outer circle and the area of the inner circle. Given the outer radius R = 28 m and the inner radius r = 25 m, the area A is given by the formula:

A = π R² - π r² = π(R² - r²)

Applying the difference of squares identity a² - b² = (a + b)(a - b), we get:

A = π(28 + 25)(28 - 25) A = π(53)(3) = 159π m²

The cost of levelling is provided as ₹28 per m². The total cost is calculated by multiplying the area of the track by the unit rate:

Total Cost = Area × Rate Total Cost = 159π × 28

Using the standard approximation π = ²²⁄₇:

Total Cost = 159 × ²²⁄₇ × 28 Total Cost = 159 × 22 × 4 Total Cost = 159 × 88 Total Cost = 13992

Explanation

Let the efficiency of the slower pipe be x. Since the faster pipe is one and a half times as fast, its efficiency is 1.5x, or ³⁄₂x. The combined efficiency of both pipes working together is the sum of their individual efficiencies: Efficiencyₜₒₜₐₗ = x + ³⁄₂x = ⁵⁄₂x

The total capacity of the tank can be expressed as the product of the combined efficiency and the time taken to fill it together: Total Work = ⁵⁄₂x × 15 = ⁷⁵⁄₂x

The time taken by the faster pipe alone is calculated by dividing the total work by the efficiency of the faster pipe: Time_fₐₛₜₑᵣ = (Total Work)/(Efficiency_fₐₛₜₑᵣ₎ = (75)/(2)x{³⁄₂x} Time_fₐₛₜₑᵣ = ⁷⁵⁄₂ × ⅔ = ⁷⁵⁄₃ = 25 minutes

Explanation

The expression is evaluated using the trigonometric identities for complementary angles, specifically csc(90° - θ) = sec θ and sin(90° - θ) = cos θ.

The first term is (sec 35°)/(csc 55°). Since 55° = 90° - 35°, we can rewrite the denominator as csc(90° - 35°), which equals sec 35°. This simplifies the fraction to (sec 35°)/(sec 35°) = 1.

The second term is (cos 25°)/(sin 65°). Since 65° = 90° - 25°, we can rewrite the denominator as sin(90° - 25°), which equals cos 25°. This simplifies the fraction to (cos 25°)/(cos 25°) = 1.

Substituting these values back into the given expression, we get 1 - 1 = 0.

Explanation

The perimeter of an equilateral triangle is given by the formula P = 3a, where a represents the length of one side. Given that the perimeter is 36 units, the side length is calculated as a = ³⁶⁄₃ = 12 units. The area of an equilateral triangle is determined by the formula Area = √3{4} a². By substituting the side length into this formula, we get Area = √3{4} × 12² = √3{4} × 144. Simplifying this expression results in 36√3 square units.

Explanation

The given equations are asin³ X + bcos³ X = sin Xcos X and asin X = bcos X. We can rewrite the first equation by factoring the powers:

(asin X)sin² X + (bcos X)cos² X = sin Xcos X

Substituting the relation asin X = bcos X into the first term:

(bcos X)sin² X + (bcos X)cos² X = sin Xcos X

Factor out bcos X:

bcos X(sin² X + cos² X) = sin Xcos X

Using the fundamental trigonometric identity sin² X + cos² X = 1:

bcos X = sin Xcos X

Since X ≠ 90°, cos X ≠ 0, we can divide both sides by cos X:

b = sin X

Now, substitute b = sin X back into the second given relation asin X = bcos X:

asin X = (sin X)cos X

Since X ≠ 0°, sin X ≠ 0, we can divide both sides by sin X:

a = cos X

To find the value of a² + b²:

a² + b² = (cos X)² + (sin X)² a² + b² = cos² X + sin² X = 1

Explanation

The total gain percentage of a dishonest merchant can be calculated using the product of the profit factors from the price markup and the weight manipulation. The merchant makes a 10% profit on the selling price and uses a weight that is 12% less than the true weight.

Let the cost price of 100 units of weight be 100. Due to the 10% profit, the selling price for 100 units of weight is 100 × (1 + 0.10) = 110 . However, the merchant only provides 88 units of weight (12% less than 100). Thus, the merchant's actual cost for the quantity sold is 88.

The total gain percentage is calculated as: Total Gain % = (Selling Price - Actual Cost)/(Actual Cost) × 100 Total Gain % = (110 - 88)/(88) × 100 Total Gain % = ²²⁄₈₈ × 100 Total Gain % = ¼ × 100 = 25%

Explanation

Let v_V be the speed of Vishu and T_A be the time taken by Atul to complete the 900 m race.

In the first scenario, Atul gives Vishu a 20-second head-start and beats him by 135 m. This means that while Atul runs 900 m in time T_A, Vishu runs 900 - 135 = 765 m in T_A + 20 seconds.

In the second scenario, Atul gives Vishu a start of 189 m and beats him by 8 seconds. This means while Atul runs 900 m in time T_A, Vishu runs 900 - 189 = 711 m in T_A + 8 seconds.

Since Vishu's speed v_V is constant, we can find it by calculating the ratio of the difference in distance to the difference in time between the two scenarios: v_V = (765 - 711) / ((T_A + 20) - (T_A + 8)) v_V = 54 / 12 = 4.5 m/s

The time taken by Vishu to complete the full 900 m race is: T_V = 900 / 4.5 = 200 seconds

Converting 200 seconds into minutes and seconds: 200 / 60 = 3 minutes and 20 seconds

Explanation

The standard division algorithm is expressed as Dividend = (Divisor × Quotient) + Remainder. According to the problem, the remainder is 32. Since the divisor is 2 times the remainder, we calculate the divisor as 2 × 32 = 64. The problem also states that the divisor is 4 times the quotient, which means the quotient is 64 / 4 = 16. Substituting these values into the division formula, the dividend is (64 × 16) + 32. Performing the multiplication, 64 × 16 equals 1024. Adding the remainder results in 1024 + 32 = 1056.

Explanation

The time T is determined by the condition that the simple interest on a sum is three-fifths of that sum at an interest rate of 10% per annum. Using the standard simple interest formula I = (P · R · T)/(100), where I is interest, P is principal, R is rate, and T is time, we substitute the given values: (P · 10 · T)/(100) = ⅗ P

By cancelling P from both sides and simplifying the expression, we find the duration in years: (T)/(10) = ⅗ T = 6 years

The objective is to find the interest accrued on a principal of ₹16,000 at a simple interest rate of 14% per annum for this period of 6 years. Applying the simple interest formula again: Interest = (16,000 · 14 · 6)/(100) Interest = 160 · 84 Interest = 13,440

Explanation

The weighted average formula for two groups is given by A = n₁A₁ + n₂A₂{n₁ + n₂}, where n represents the number of students and A represents their average score. Given that the average of the first group of 12 students is 62 and the average of the remaining 18 students is also 62, the calculation becomes A = (12 × 62 + 18 × 62)/(30). Factoring out the common term, we get A = (62(12 + 18))/(30) = (62 × 30)/(30) = 62. Mathematically, if all subgroups of a population share the same mean, the mean of the entire population is necessarily equal to that same value.

Explanation

In a rhombus ABCD, the diagonals AC and BD are perpendicular bisectors of each other. The locus of all points P in the plane such that PA = PC is the perpendicular bisector of the segment AC. Since the diagonal BD is the perpendicular bisector of diagonal AC, any point O satisfying OA = OC must lie on the line containing the diagonal BD.

Given that O is an interior point of the rhombus and lies on the line BD, it must lie on the segment BD between the vertices B and D. Consequently, the points D, O, and B are collinear, and the rays OD and OB extend in opposite directions from the point O. The degree measure of the straight angle ∠ DOB is therefore 180°.

To find five-ninth of the degree measure of ∠ DOB: Value = ⁵⁄₉ × 180° Value = 5 × 20° Value = 100°

Explanation

The salary growth follows an arithmetic progression where the annual increment represents the common difference d. The salary after n years of service is given by the formula Sₙ = S₀ + n × d, where S₀ is the starting salary and d is the fixed annual increment.

Using the provided data: S₆ = S₀ + 6d = 28,400 S₁₅ = S₀ + 15d = 52,997

The difference in salary between the fifteenth year and the sixth year represents the total increments earned over those nine years: (S₀ + 15d) - (S₀ + 6d) = 52,997 - 28,400 9d = 24,597

To find the annual increment d, divide the total difference by the number of years: d = (24,597)/(9) d = 2,733

Explanation

Let the shares of P, Q, R, and S be represented by 4x, 9x, 2x, and 3x respectively. The problem states that Q receives ₹560 more than R, which corresponds to the difference in their respective ratio parts: 9x - 2x = 7x 7x = 560 x = ⁵⁶⁰⁄₇ = 80 The share of S is represented by 3x. Substituting the value of x into this expression: S = 3 × 80 = 240

Explanation

The average marks for Mathematics are found by summing the individual scores of all five students and dividing by the total number of students. From the data provided, the scores for Suresh, Ramesh, Ankith, Ranjith, and Aryan are 70, 85, 90, 95, and 75, respectively.

The sum of these marks is calculated as follows: 70 + 85 + 90 + 95 + 75 = 415

The total number of students is 5. Applying the formula for the arithmetic mean: Average = (Σ Marks)/(n) Average = ⁴¹⁵⁄₅ = 83

This calculation provides the mean score specifically for the Mathematics subject across the group of five students.