SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To simplify the expression (4x + 1)² - (4x + 3)(4x - 1), we expand each component using standard algebraic identities.

Applying the identity (a + b)² = a² + 2ab + b² to the first part of the expression: (4x + 1)² = (4x)² + 2(4x)(1) + (1)² = 16x² + 8x + 1

Next, we expand the product of the two binomials in the second part using the distributive property: (4x + 3)(4x - 1) = (4x · 4x) + (4x · -1) + (3 · 4x) + (3 · -1) = 16x² - 4x + 12x - 3 = 16x² + 8x - 3

Now, we substitute these expanded forms back into the original expression and subtract: (16x² + 8x + 1) - (16x² + 8x - 3) = 16x² + 8x + 1 - 16x² - 8x + 3

By grouping and canceling like terms: (16x² - 16x²) + (8x - 8x) + (1 + 3) = 0 + 0 + 4 = 4

Explanation

The centroid G of a triangle divides each median in the ratio 2:1, measured from the vertex to the midpoint of the opposite side. In △ ABC, since D is the mid-point of BC, AD is the median. The relationship between the segments of the median is given by the formula AG : GD = 2 : 1.

The total length of the median AD can be expressed as: AD = AG + GD

Substituting the ratio AG = 2 × GD: AD = 2 × GD + GD = 3 × GD

Given that GD = 10 cm, the length of AD is: AD = 3 × 10 cm = 30 cm

Explanation

To find the value of x⁶ + (1)/(x⁶) from x + ¹⁄ₓ = 2√2, we use the algebraic identity where if x + ¹⁄ₓ = k, then x² + (1)/(x²) = k² - 2. Substituting k = 2√2:

x² + (1)/(x²) = (2√2)² - 2 = 8 - 2 = 6

Now, we treat x² as a single variable to find its cube, using the identity where if a + ¹⁄ₐ = m, then a³ + (1)/(a³) = m³ - 3m. Here, let a = x² and m = 6:

(x²)³ + (1)/((x²)³) = 6³ - 3(6)

x⁶ + (1)/(x⁶) = 216 - 18 = 198

Alternatively, cubing the initial term gives x³ + (1)/(x³) = (2√2)³ - 3(2√2) = 16√2 - 6√2 = 10√2. Squaring this result yields (10√2)² - 2 = 200 - 2 = 198.

Explanation

Let the capacity of the tank be the Least Common Multiple (LCM) of the filling times, which is LCM(30, 50) = 150 units. The efficiency of the filling pipe is 150 / 30 = 5 units per hour. When the leakage is also functioning, the combined efficiency of the pipe and the leak is 150 / 50 = 3 units per hour.

The efficiency of the leakage is calculated by subtracting the filling pipe's efficiency from the combined efficiency: 3 - 5 = -2 units per hour. The negative sign denotes that the leakage empties the tank. To find the time required for the leakage to empty a full tank of 150 units, we divide the total capacity by the leakage efficiency: 150 / 2 = 75 hours.

Explanation

When two people cover the same distance at different speeds, the distance can be calculated using the formula D = (s₁ × s₂)/(s₂ - s₁) × Δ t, where s₁ and s₂ are the speeds and Δ t is the difference in time taken.

Given s₁ = 9 km/h and s₂ = 12 km/h, the time difference is 20 minutes. Since speeds are in km/h, convert time to hours: Δ t = ²⁰⁄₆₀ = ⅓ hour.

Substituting these values into the formula: D = (9 × 12)/(12 - 9) × ⅓ D = ¹⁰⁸⁄₃ × ⅓ D = 36 × ⅓ D = 12 km

Alternatively, using the concept that time is inversely proportional to speed for a constant distance, the ratio of speeds is 9:12 or 3:4. Therefore, the ratio of time taken is 4:3. The difference in ratio units is 1 unit, which corresponds to 20 minutes. Thus, the second man takes 3 × 20 = 60 minutes (1 hour) to complete the journey. Distance = Speed × Time = 12 km/h × 1 hour = 12 km.

Explanation

The time taken for two people to meet for the first time on a circular track when running in opposite directions is given by the formula T = (L)/(v₁ + v₂), where L is the length of the track and v₁, v₂ are their respective speeds.

First, convert the speeds from km/h to m/s using the conversion factor ⁵⁄₁₈: Speed of X: v₁ = 54 × ⁵⁄₁₈ = 15 m/s Speed of Y: v₂ = 63 × ⁵⁄₁₈ = 17.5 m/s

The relative speed in the opposite direction is the sum of the individual speeds: vᵣₑₗ = 15 + 17.5 = 32.5 m/s

Substitute the values into the time formula: T = (4225)/(32.5) = ⁴²²⁵⁰⁄₃₂₅

Calculating the division: 325 × 100 = 32500 42250 - 32500 = 9750 325 × 30 = 9750 T = 100 + 30 = 130 seconds

Explanation

The total time T taken by Monika, Anita, and Manju to finish the work together is calculated using the formula for combined work rates: 1/T = 1/T₁ + 1/T₂ + 1/T₃, where T₁, T₂, and T₃ are the individual times taken by each person working alone. Substituting the given values into the formula:

1/T = 1/6 + 1/8 + 1/5

To solve this, find the least common multiple (LCM) of the denominators 6, 8, and 5, which is 120. Convert the fractions to have this common denominator:

1/T = (20 + 15 + 24) / 120 = 59 / 120

To find the time T, take the reciprocal of the total rate:

T = 120 / 59 hours

Converting this improper fraction into a mixed fraction:

120 = 59 × 2 + 2

T = 2 ²⁄₅₉ hours

Explanation

The discount offered is the difference between the marked price and the selling price. Using the formula:

Discount = Marked Price - Selling Price Discount = 15000 - 13500 = 1500

The rate of discount is calculated as a percentage of the marked price:

Discount Rate = ((Discount)/(Marked Price)) × 100 Discount Rate = (¹⁵⁰⁰⁄₁₅₀₀₀) × 100 = ⅒ × 100 = 10%

Explanation

The time taken by Q to finish the work can be determined using the work rate formula (1)/(T_P₊_Q₎ = (1)/(T_P₎ + (1)/(T_Q₎, where T_P₊_Q is the time for both together and T_P and T_Q are their individual times. Substituting the given values T_P₊_Q = 6 and T_P = 18 results in ⅙ = ¹⁄₁₈ + (1)/(T_Q₎. Solving for T_Q gives (1)/(T_Q₎ = ⅙ - ¹⁄₁₈. Using a common denominator of 18, the equation becomes (1)/(T_Q₎ = (3 - 1)/(18) = ²⁄₁₈ = ⅑. Taking the reciprocal, T_Q = 9 days.

Alternatively, using the efficiency method, let the total work be the Least Common Multiple (LCM) of the given time durations, LCM(6, 18) = 18 units. The combined efficiency of P and Q is 18 / 6 = 3 units per day, and the individual efficiency of P is 18 / 18 = 1 unit per day. The efficiency of Q is calculated as the difference between the combined efficiency and P's efficiency, which is 3 - 1 = 2 units per day. The time required for Q to complete the total work of 18 units alone is 18 / 2 = 9 days.

Explanation

The expression tan 5° tan 25° tan 45° tan 65° tan 85° can be solved using the trigonometric identity tan θ · tan(90° - θ) = 1, which is derived from tan(90° - θ) = cot θ and the fact that tan θ · cot θ = 1.

By pairing the terms with complementary angles, we see that 5° + 85° = 90° and 25° + 65° = 90°. This gives us tan 5° · tan 85° = 1 and tan 25° · tan 65° = 1. Substituting these values into the product along with the standard value tan 45° = 1, the expression becomes (1) · (1) · (1) = 1.

Explanation

Let the total number of enrolled voters be x. According to the problem, 10% did not vote, so 90% of voters cast their votes. Out of these, 10% were invalid, making 90% of the cast votes valid.

Total valid votes = x × (90/100) × (90/100) = 0.81x

The winner received 56% of the valid votes, which implies the loser received 100% - 56% = 44% of the valid votes. The margin of victory is the difference between these shares:

Margin = (56% - 44%) of valid votes = 12% of 0.81x

Setting this equal to the given margin of 1,458 votes: (12/100) × 0.81x = 1,458 x = (1,458 × 100 × 100) / (12 × 81)

Dividing 1,458 by 81 gives 18. x = (18 × 10,000) / 12 x = 1.5 × 10,000 = 15,000

Explanation

The total area to be white washed consists of the area of the four walls and the area of the ceiling. The area of the four walls is given by the lateral surface area formula 2h(l + b), where l, b, and h are the length, breadth, and height of the room respectively. The area of the ceiling is l × b.

Substituting the given dimensions: Total area = 2 × 6 × (10 + 8) + (10 × 8) Total area = 12 × 18 + 80 Total area = 216 + 80 = 296 m²

The cost of white washing is calculated by multiplying the total area by the rate of ₹7.50 per m²: Total cost = 296 × 7.50 Total cost = 296 × ¹⁵⁄₂ Total cost = 148 × 15 = 2,220

The calculation 148 × 10 = 1,480 and 148 × 5 = 740 sums up to ₹2,220.

Explanation

The expression sin (A + B) = sin A cos B + cos A sin B is a fundamental compound angle identity in trigonometry. This formula allows the sine of the sum of two angles to be expanded into terms involving the sine and cosine of the individual angles. It is derived using the geometric properties of right-angled triangles or through Euler's formula, which states that eⁱ⁽^A⁺^B⁾ = cos(A+B) + isin(A+B). By expanding eⁱ^A · eⁱ^B = (cos A + i sin A)(cos B + i sin B) and equating the imaginary parts, we obtain the identity sin (A + B) = sin A cos B + cos A sin B.

Explanation

A tangent to a circle is defined as a line that touches the circle at exactly one point. In contrast, a line that intersects a circle at two distinct points is known as a secant. Mathematically, as the two points of intersection of a secant move closer together and eventually coincide, the secant becomes a tangent. Because the statement identifies a line passing through two distinct points as a tangent instead of a secant, it is geometrically false.

Explanation

Total work is calculated using the formula W = E × H × D, where E represents efficiency, H is the number of hours worked per day, and D is the total number of days. Given that A's efficiency is twice that of B's, we can let A's efficiency be 2 and B's efficiency be 1.

The total work is derived from A's parameters: W = 2 × 8 × 12 = 192 units

Since B must complete the same 192 units of work, we substitute B's efficiency and daily hours into the work formula to find the number of days D: 1 × 12 × D = 192 12D = 192 D = ¹⁹²⁄₁₂ = 16

Explanation

When two people P and Q run around a circular track of length L in the same direction with speeds vₚ and v_q, the number of distinct meeting points on the track is given by the formula N = |x - y|, where x:y is the simplified ratio of their speeds such that gcd(x, y) = 1.

In this problem, the speed of P is 6 m/sec and the speed of Q is b m/sec. We are given that they cross each other at exactly two points, so |x - y| = 2. Since b is a natural number less than 6 (b ∈ {1, 2, 3, 4, 5}), we evaluate each case:

If b = 1, the speed ratio is 6:1. The number of meeting points is |6 - 1| = 5. If b = 2, the speed ratio is 6:2, which simplifies to 3:1. The number of meeting points is |3 - 1| = 2. If b = 3, the speed ratio is 6:3, which simplifies to 2:1. The number of meeting points is |2 - 1| = 1. If b = 4, the speed ratio is 6:4, which simplifies to 3:2. The number of meeting points is |3 - 2| = 1. If b = 5, the speed ratio is 6:5. The number of meeting points is |6 - 5| = 1.

Only the value b = 2 satisfies the condition of having exactly two meeting points. Therefore, there is only 1 such value that b can take.

Explanation

In problems involving a dishonest dealer who sells at cost price but uses a false weight, the gain percentage is calculated using the formula:

Gain % = (Error)/(True Value - Error) × 100

Alternatively, this can be expressed as the ratio of the weight saved to the actual weight given to the customer:

Gain % = (Weight Saved)/(Actual Weight Sold) × 100

In this case, the true weight is 1 kg (1000 grams) and the dealer uses 950 grams. The weight saved (error) is 1000 - 950 = 50 grams. Substituting these values into the formula:

Gain % = ⁵⁰⁄₉₅₀ × 100

Gain % = ⁵⁄₉₅ × 100 = ¹⁄₁₉ × 100

Gain % = 5.2631...

Rounding to two decimal places, the gain percentage is 5.26%.

Explanation

The selling price is determined by applying successive discounts to the list price. The formula for the final selling price after multiple discounts is SP = LP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)).

Given the list price LP = 1200, the trade discount d₁ = 20%, and the cash discount d₂ = 30%, the calculation is as follows:

SP = 1200 × (1 - ²⁰⁄₁₀₀) × (1 - ³⁰⁄₁₀₀) SP = 1200 × ⁸⁰⁄₁₀₀ × ⁷⁰⁄₁₀₀ SP = 1200 × 0.8 × 0.7 SP = 960 × 0.7 SP = 672

Alternatively, the effective single discount can be calculated using the formula D = d₁ + d₂ - (d₁ × d₂)/(100). Substituting the values gives 20 + 30 - (20 × 30)/(100) = 50 - 6 = 44%. The selling price is then 1200 × (100% - 44%) = 1200 × 56% = 12 × 56 = 672.

Explanation

To find the value of x⁶ + (1)/(x⁶⁾, we first calculate the value of x³ + (1)/(x³⁾ using the algebraic identity x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). Given x + ¹⁄ₓ = 7, we substitute this into the formula:

x³ + (1)/(x³⁾ = 7³ - 3(7) = 343 - 21 = 322

Next, we square this result to reach the sixth power using the identity (a + b)² = a² + b² + 2ab. Let a = x³ and b = (1)/(x³⁾:

(x³ + (1)/(x³⁾)² = (x³)² + ((1)/(x³⁾)² + 2(x³)((1)/(x³⁾)

322² = x⁶ + (1)/(x⁶⁾ + 2

Calculating 322²: 322 × 322 = 103684

Substituting this back into the equation: 103684 = x⁶ + (1)/(x⁶⁾ + 2 x⁶ + (1)/(x⁶⁾ = 103684 - 2 = 103682

Explanation

Let the current ages of Sudhir and Ashish be 5x and 7x respectively. According to the problem, twelve years ago, the ratio of their ages was 1:2. This relationship can be expressed by the equation:

(5x - 12)/(7x - 12) = ½

By cross-multiplying to solve for x, we get: 2(5x - 12) = 1(7x - 12) 10x - 24 = 7x - 12 3x = 12 x = 4

Using the value of x, Sudhir's current age is 5x = 5 × 4 = 20 years. The question asks for Sudhir's age five years from now, which is calculated by adding 5 to his current age: 20 + 5 = 25 years.

Explanation

The total interest for two years is ₹2,400, which implies the interest for one year is 2,400 / 2 = 1,200. The overall annual interest rate earned on the total investment is (1,200 / 10,000) × 100 = 12%.

Using the method of alligation, the ratio of the two parts of the investment can be determined by comparing the individual interest rates with the mean interest rate. The difference between the higher rate (15%) and the mean rate (12%) is 15 - 12 = 3. The difference between the mean rate (12%) and the lower rate (10%) is 12 - 10 = 2.

The ratio of the sum invested at 10% to the sum invested at 15% is therefore 3 : 2. The total sum of ₹10,000 is divided into 3 + 2 = 5 units. Each unit corresponds to 10,000 / 5 = 2,000. Consequently, the sum invested at 15% interest is 2 × 2,000 = 4,000.

Explanation

Average speed is calculated using the formula Average Speed = (Total Distance)/(Total Time).

The total distance covered by the car is the sum of the two individual distances: Total Distance = 350 + 450 = 800 km

The total time is the sum of the time taken for each part of the journey. The time for the first part is given as 5 hours. For the second part, the time is calculated by dividing the distance by the speed: Time₂ = ⁴⁵⁰⁄₉₀ = 5 hours

Thus, the total time for the entire journey is: Total Time = 5 + 5 = 10 hours

Applying these values to the average speed formula: Average Speed = ⁸⁰⁰⁄₁₀ = 80 km/h

Explanation

To determine which scheme offers the maximum discount, we calculate the effective discount percentage for each. Scheme 1 is a flat 30%. Scheme 2 provides two successive discounts of 15%, which is calculated using the formula x + y - ˣʸ⁄₁₀₀ as 15 + 15 - (15 × 15)/(100) = 27.75%. For schemes where items are given free, the discount percentage is determined by the formula (Free items)/(Total items) × 100. In Scheme 3, "Buy 2, get 1 free" results in a discount of ⅓ × 100 ≈ 33.33%. In Scheme 4, "Buy 5, get 3 free" results in a discount of ³⁄₅₊₃ × 100 = ⅜ × 100 = 37.5%. Comparing the effective rates of 30%, 27.75%, 33.33%, and 37.5%, Scheme 4 provides the highest savings for the customer.

Explanation

The length of the direct common tangent L between two circles with radii r₁ and r₂ is given by the formula L = √(d² - (r₁ - r₂)²), where d is the distance between the centers of the circles. When two circles touch each other externally, the distance between their centers is equal to the sum of their radii, such that d = r₁ + r₂.

Substituting d = r₁ + r₂ into the tangent length formula, we get L = √((r₁ + r₂)² - (r₁ - r₂)²). Applying the algebraic identity (a+b)² - (a-b)² = 4ab, the expression simplifies to L = √(4r₁r₂), or L = 2√(r₁r₂).

Using the given radii r₁ = 22 cm and r₂ = 16 cm: L = 2√(22 × 16) L = 2 × 4 × √(22) L = 8√(22) cm

Explanation

In congruent triangles △ ABC and △ DEF, the property of Corresponding Parts of Congruent Triangles (CPCT) ensures that all corresponding sides are equal in length. Specifically, the side AC corresponds to DF, meaning AC = DF. This equality necessitates that the ratio of the side AC to the side DF is exactly 1:1. A ratio of 2:1 would imply that the triangles have different sizes, which is a characteristic of similar triangles with a scale factor of 2, rather than congruent triangles which must be identical in size. Consequently, the assertion that the ratio of AC to DF is 2:1 is logically and geometrically incorrect.

Explanation

To solve the expression, divide both the numerator and the denominator by cos x. This converts the trigonometric terms into tan x since (sin x)/(cos x) = tan x and (cos x)/(cos x) = 1. The expression becomes:

(9tan x - 42)/(5){15tan x + 21}

Given that tan x = ⁷⁄₅, substitute this value into the equation:

Numerator: 9 (⁷⁄₅) - ⁴²⁄₅ = ⁶³⁄₅ - ⁴²⁄₅ = ²¹⁄₅

Denominator: 15 (⁷⁄₅) + 21 = 3(7) + 21 = 21 + 21 = 42

Now, calculate the final value by dividing the numerator by the denominator:

(21)/(5){42} = (21)/(5 × 42) = ²¹⁄₂₁₀ = ⅒ = 0.1

Explanation

Let the two natural numbers be x and y. We are given the difference of their cubes as x³ - y³ = 6272 and the difference between the numbers as x - y = 8. We can use the algebraic identity x³ - y³ = (x - y)³ + 3xy(x - y) to find the product xy directly.

Substituting the known values into the identity: 6272 = 8³ + 3xy(8) 6272 = 512 + 24xy

Subtracting 512 from both sides of the equation: 6272 - 512 = 24xy 5760 = 24xy

Dividing by 24 to isolate the product xy: xy = 5760 / 24 xy = 240

Explanation

Given the ratio (a)/(b) = ⁷⁄₆, we first determine the ratio (6a)/(13b) by multiplying the given equation by ⁶⁄₁₃: ⁶⁄₁₃ × (a)/(b) = ⁶⁄₁₃ × ⁷⁄₆ ⟹ (6a)/(13b) = ⁷⁄₁₃

Using the property of componendo and dividendo, which states that if (x)/(y) = (p)/(q) then (x+y)/(x-y) = (p+q)/(p-q), we substitute x = 6a, y = 13b, p = 7, and q = 13: (6a + 13b)/(6a - 13b) = (7 + 13)/(7 - 13) = ²⁰⁄₋₆

Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor, 2, yields: ¹⁰⁄₋₃ = -¹⁰⁄₃

Explanation

In the second scenario, A gives B a start of 45 m and the race ends in a dead heat. This means A covers 200 m in the same time B covers 200 - 45 = 155 m. Let T_A be the time taken by A to cover 200 m. B's speed v_B is therefore 155 / T_A m/s.

In the first scenario, A gives B a start of 25 m, meaning B must cover 200 - 25 = 175 m. Since A wins by 10 seconds, B takes 10 seconds longer than A to reach the finish line. The time taken by B to run 175 m is 175 / v_B. Substituting v_B = 155 / T_A, B's time is 175 / (155 / T_A) = (175 / 155) T_A = (35 / 31) T_A.

The difference between B's time and A's time is given as 10 seconds: (35 / 31) T_A - T_A = 10 (4 / 31) T_A = 10 T_A = 310 / 4 = 77.5 seconds.

Explanation

In a circle, the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle. Here, ∠ AOP is the central angle subtended by the arc AP. The point B lies on the circumference of the circle, making ∠ ABP the inscribed angle subtended by the same arc AP. Since the line segment AB is the diameter passing through the center O, the angle ∠ OBP is identical to ∠ ABP. Using the inscribed angle theorem:

∠ OBP = ½ × ∠ AOP

Substituting the given value:

∠ OBP = ½ × 95° = 47.5°

Alternatively, considering △ OBP, the exterior angle ∠ AOP is equal to the sum of the two interior opposite angles ∠ OBP and ∠ OPB. Since OB = OP (radii of the same circle), △ OBP is an isosceles triangle, which implies ∠ OBP = ∠ OPB. Therefore, 95° = 2 × ∠ OBP, leading to 47.5°.

Explanation

Ram's copying rate is calculated as R_Rₐₘ = ⁶⁰⁄₁₅ = 4 pages per hour. The combined copying rate of Ram and Riya is R_Tₒₜₐₗ = ¹⁸⁰⁄₃₀ = 6 pages per hour. Riya's individual copying rate is the difference between the combined rate and Ram's rate, which is R_Rᵢ_yₐ = 6 - 4 = 2 pages per hour. The time required for Riya to copy 20 pages is determined by dividing the total pages by her rate: Time = ²⁰⁄₂ = 10 hours.

Explanation

Daily earnings for 4 women and 7 men is 11480 / 7 = 1640. Daily earnings for 10 women and 17 men is 36360 / 9 = 4040. Let w be the daily wage of a woman and m be the daily wage of a man. 4w + 7m = 1640 10w + 17m = 4040 Multiply the first equation by 5 and the second by 2 to solve by elimination: 20w + 35m = 8200 20w + 34m = 8080 Subtracting the equations gives m = 120. Substitute m = 120 into the first equation: 4w + 7(120) = 1640 4w = 1640 - 840 = 800 w = 200 The combined daily earnings for 11 women and 9 men is 11(200) + 9(120) = 2200 + 1080 = 3280. Total earnings for 13 days are 3280 × 13 = 42640.

Explanation

The cost price (CP) is found using the relation CP = (100 × SP)/(100 + P%). For a selling price of Rs. 28 and a profit of 12%, the calculation is: CP = (100 × 28)/(112) = ²⁸⁰⁰⁄₁₁₂ = 25

With a cost price of Rs. 25 and a new selling price of Rs. 26.25, the profit earned is 26.25 - 25 = 1.25. The profit percentage is determined by the formula: Profit% = (Profit)/(CP) × 100 = (1.25)/(25) × 100 = 5%

Explanation

To find the fraction of work left, we first determine the combined work rate of A and B. A completes the work in 9 days, so A's rate is ⅑ work per day. B completes the work in 12 days, so B's rate is ¹⁄₁₂ work per day.

The combined rate of work for A and B working together is: ⅑ + ¹⁄₁₂ = (4 + 3)/(36) = ⁷⁄₃₆ work per day

In 5 days, the total amount of work completed by both is: 5 × ⁷⁄₃₆ = ³⁵⁄₃₆

Taking the total work to be 1 unit, the fraction of work remaining is calculated by subtracting the work completed from the total: 1 - ³⁵⁄₃₆ = (36 - 35)/(36) = ¹⁄₃₆

Explanation

The speed of the bus is determined by the ratio of distance to time: v_bᵤₛ = (60 km)/(80 min) = 0.75 km/min. Given that the airplane travels five times as fast as the bus, its speed is vₐᵢᵣₚₗₐₙₑ = 5 × 0.75 km/min = 3.75 km/min. The distance covered by the airplane in 25 minutes is calculated by multiplying its speed by the duration: d = 3.75 km/min × 25 min = 93.75 km.

Explanation

For a fixed principal and time period, the simple interest is directly proportional to the rate of interest. The problem asks for the additional interest generated by increasing the rate from 4.5% to 5%, which is an increase of 0.5%.

Since 4.5% interest corresponds to ₹135, the interest for a 1% rate can be calculated as: Interest per 1% = (135)/(4.5) = 30

The additional interest for the 0.5% increase is: Additional Interest = 30 × 0.5 = 15

Alternatively, the principal P is calculated using I = (P × R × T)/(100): 135 = (P × 4.5 × 1)/(100) ⟹ P = (135 × 100)/(4.5) = 3000 The additional interest on ₹3000 at the incremental rate of 0.5% (5% - 4.5%) for one year is: Additional Interest = (3000 × 0.5 × 1)/(100) = 30 × 0.5 = 15

Explanation

The expression is simplified using the BODMAS rule, which dictates the order of operations as Brackets, Orders (powers and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

First, simplify the terms within the parentheses and the exponent: (30 - 26) = 4 3 - 2 = 1

Substitute these values back into the expression: 2 + 4² ÷ 8¹ × 0.5

Next, evaluate the powers: 4² = 16 8¹ = 8

The expression becomes: 2 + 16 ÷ 8 × 0.5

Now, perform the division and multiplication from left to right: 16 ÷ 8 = 2 2 × 0.5 = 1

Finally, perform the addition: 2 + 1 = 3

Explanation

In an equilateral triangle, the centroid divides the median in the ratio 2:1. Since the median is also the altitude in an equilateral triangle, the distance from a vertex to the centroid PL represents two-thirds of the height h. The height of an equilateral triangle with side s is calculated using the formula h = √3{2}s. Substituting the given side length s = 6 cm, the height is h = √3{2} × 6 = 3√3 cm. Therefore, the length PL = ⅔h = ⅔ × 3√3 = 2√3 cm. This distance is also equivalent to the circumradius R of the triangle, defined by the formula R = (s)/(√(3)). Applying this directly gives PL = (6)/(√(3)) = 6√3{3} = 2√3 cm.

Explanation

The expression is simplified using the BODMAS rule, which prioritizes operations in the order of Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Beginning with the innermost parentheses and curly brackets: The term {4 - (7 - 3)} simplifies to 4 - 4 = 0. The term within the square brackets {5 + (-3) × (-6)} simplifies to 5 + 18 = 23.

The expression now becomes: 15 - (-5) × 0 ÷ [3 × 23] 15 - (-5) × 0 ÷ 69

Following the order of operations for multiplication and division from left to right: The product (-5) × 0 equals 0. The division 0 ÷ 69 equals 0.

The final step is the subtraction: 15 - 0 = 15

Explanation

The given equations are (cos β)/(sec α) = 15 and (sin β)/(sec α) = 16. Since (1)/(sec α) = cos α, these can be rewritten as:

cos β cos α = 15 sin β cos α = 16

Dividing the second equation by the first eliminates the term cos α:

(sin β cos α)/(cos β cos α) = ¹⁶⁄₁₅ tan β = ¹⁶⁄₁₅

To find sin² β, use the trigonometric identity sin² β = (tan² β)/(1 + tan² β). Substituting the value of tan β:

sin² β = ((16)/(15))²{1 + (¹⁶⁄₁₅)²} = (256)/(225){1 + ²⁵⁶⁄₂₂₅}

Multiplying the numerator and denominator by 225:

sin² β = (256)/(225 + 256) = ²⁵⁶⁄₄₈₁

Explanation

Let the total salary of the person be x. The problem involves successive percentage deductions from the remaining balance at each step.

First, the person spends 25% on daily uses, which leaves 100% - 25% = 75% of his income. Remaining amount = x × ⁷⁵⁄₁₀₀

Next, he spends 18% of the rest on house rent, leaving 100% - 18% = 82% of the previous remainder. Remaining amount = x × ⁷⁵⁄₁₀₀ × ⁸²⁄₁₀₀

Finally, he spends 16% of the rest on travel, leaving 100% - 16% = 84% of the current remainder. Final remaining amount = x × ⁷⁵⁄₁₀₀ × ⁸²⁄₁₀₀ × ⁸⁴⁄₁₀₀

We are given that this final amount is 861. Setting up the equation: x × ¾ × ⁴¹⁄₅₀ × ²¹⁄₂₅ = 861

Notice that 41 × 21 = 861. Substituting this into the equation: x × (3 × 861)/(4 × 50 × 25) = 861

Divide both sides by 861: x × ³⁄₅₀₀₀ = 1 x = ⁵⁰⁰⁰⁄₃ x = 1666.66...

The resulting value 1666 ⅔ matches the numerical value provided in the correct option.

Explanation

Using the algebraic identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x), let x = sinθ, so 1/x = cosecθ. Substituting the given value, we get (√5)³ - 3(√5) = 5√5 - 3√5 = 2√5.

Explanation

For any angle θ, the trigonometric identity csc² θ - cot² θ = 1 allows us to relate the sum and difference of csc θ and cot θ. Given the expression csc θ + cot θ = x, it follows from the identity (csc θ + cot θ)(csc θ - cot θ) = 1 that csc θ - cot θ = ¹⁄ₓ.

To find the value of csc θ, we use the formula: csc θ = (x + 1)/(x){2} = (x² + 1)/(2x)

Substituting the given value x = ³⁄₂: csc θ = ((3)/(2))² + 1{2(³⁄₂)} csc θ = (9)/(4) + 1{3} csc θ = (13)/(4){3} csc θ = ¹³⁄₁₂

The value ¹³⁄₁₂ is the result of this standard trigonometric relationship.

Explanation

Let the total work be the Least Common Multiple (LCM) of the time taken by the group and the individuals, which is LCM(12, 36, 36) = 36 units. The combined efficiency of A, B, and C is ³⁶⁄₁₂ = 3 units per day. Since A and B can each complete the work in 36 days, their individual efficiencies are ³⁶⁄₃₆ = 1 unit per day each.

The efficiency of C is found by subtracting the efficiencies of A and B from the total combined efficiency: 3 - 1 - 1 = 1 unit per day. The time required for C to complete the work alone is the total work divided by C's efficiency, which is ³⁶⁄₁ = 36 days.

Explanation

The volume V of a sphere is calculated using the formula V = ⁴⁄₃ π r³, where r is the radius of the sphere. Given that the diameter is 7 cm, the radius r is half of the diameter, which is ⁷⁄₂ cm.

Substituting the values into the formula: V = ⁴⁄₃ × ²²⁄₇ × (⁷⁄₂)³ V = ⁴⁄₃ × ²²⁄₇ × (7 × 7 × 7)/(2 × 2 × 2)

By simplifying the expression, we can cancel out common factors: V = ⅓ × (11 × 7 × 7)/(1) V = (11 × 49)/(3) V = ⁵³⁹⁄₃

Performing the division: V = 179.666...

Rounding the result to two decimal places, we get 179.67 cm³.

Explanation

The standard trigonometric values for the angle 30 degrees are sin 30° = ½, cos 30° = √3{2}, and tan 30° = (1)/(√(3)). Substituting these values into the expression, we get:

sin 30° + cos 30° + tan 30° = ½ + √3{2} + (1)/(√(3))

To simplify the expression, find the least common multiple of the denominators 2 and √3, which is 2√3. Adjusting each term to have this common denominator:

1 × √3{2√3} + √3 × √3{2√3} + (1 × 2)/(√(3) × 2) = √3 + 3 + 2{2√3}

Combining the constants in the numerator results in:

5 + √3{2√3}

Explanation

The expression sec⁴ θ - tan⁴ θ can be factored using the algebraic identity a² - b² = (a - b)(a + b). By letting a = sec² θ and b = tan² θ, the expression becomes:

sec⁴ θ - tan⁴ θ = (sec² θ - tan² θ)(sec² θ + tan² θ)

Using the fundamental trigonometric identity sec² θ - tan² θ = 1, the expression simplifies to:

1 × (sec² θ + tan² θ)

Given that sec² θ + tan² θ = ²⁵⁄₁₈, we substitute this value into the simplified equation:

sec⁴ θ - tan⁴ θ = 1 × ²⁵⁄₁₈ = ²⁵⁄₁₈

Explanation

The volume of a cone is calculated using the formula V = ⅓ π r² h. Given the volume V = 73920 cm³ and height h = 160 cm, we substitute these values into the formula to find the radius r:

73920 = ⅓ × ²²⁄₇ × r² × 160

To isolate r², rearrange the equation:

r² = (73920 × 3 × 7)/(22 × 160)

First, divide 73920 by 160 to get 462. Then, the equation becomes:

r² = (462 × 21)/(22)

Dividing 462 by 22 gives 21. Thus:

r² = 21 × 21 = 441

Taking the square root of both sides, the radius r = 21 cm. The diameter of the base is defined as d = 2r. Therefore, the diameter is 2 × 21 = 42 cm.

Explanation

Let A and B represent the number of toffees held by A and B respectively. According to the first condition, if A gives one toffee to B, they have an equal number of toffees: A - 1 = B + 1 A - B = 2

According to the second condition, if B gives one toffee to A, then the number of toffees with A is double those with B: A + 1 = 2(B - 1) A + 1 = 2B - 2 A - 2B = -3

Subtracting the second equation from the first equation to eliminate A: (A - B) - (A - 2B) = 2 - (-3) B = 5

Substituting B = 5 into the first equation: A - 5 = 2 A = 7

The total number of toffees with A and B is A + B = 7 + 5 = 12.

Explanation

The total number of students across all five colleges is given as 2080. First, calculate the sum of students in the three colleges for which the numbers are explicitly provided: 430 (IIT Delhi) + 350 (IIT Kanpur) + 400 (IIT Guwahati) = 1180

The combined number of students in IIT Bombay (B) and IIT Madras (M) is the difference between the total and this sum: B + M = 2080 - 1180 = 900

The question states that the number of students in IIT Bombay is 20% less than the number of students in IIT Madras. This relationship can be expressed using the formula: B = M × (1 - 0.20) = 0.8M

Substitute the expression for B into the combined sum equation: 0.8M + M = 900 1.8M = 900 M = (900)/(1.8) = 500

By substituting the value of M back into the relationship for B: B = 0.8 × 500 = 400