SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The area of a sector A is related to its radius r and arc length l by the formula A = ½ r l. This relationship is derived from the fact that both area and arc length are proportional to the central angle of the sector. Given that the radius r = 12 cm and the area A = 32π cm², we can substitute these values directly into the formula:

32π = ½ × 12 × l

32π = 6l

To find l, we divide both sides by 6:

l = (32π)/(6)

Simplifying the fraction by dividing both the numerator and the denominator by 2 yields:

l = ¹⁶⁄₃π cm

Explanation

A chord is defined as a straight line segment whose endpoints both lie on the circumference of a circle. The diameter is a specific type of chord that passes through the center of the circle and represents the maximum possible distance between any two points on the circumference. Therefore, the diameter is the longest chord of a circle, with a length equal to 2r, where r is the radius. A radius, by definition, connects the center of the circle to a point on the circumference. Since one of its endpoints (the center) does not lie on the circumference, a radius is not a chord at all, making the assertion that it is the longest chord fundamentally incorrect.

Explanation

Let the total work be the least common multiple of the time taken by A and B, which is LCM(18, 12) = 36 units.

The efficiency of A is the total work divided by the time taken, which is 36 / 18 = 2 units per day. Similarly, the efficiency of B is 36 / 12 = 3 units per day.

B worked for 4 days, completing a total of 3 × 4 = 12 units of work. The remaining work is the total work minus the work completed by B, which equals 36 - 12 = 24 units.

A completes the remaining 24 units of work at an efficiency of 2 units per day. Therefore, the time taken by A to finish the remaining work is 24 / 2 = 12 days.

Explanation

To find the time of the last meeting, we first determine the speeds of the two racers. Racer A completes one round of 800m in 200 sec, so the speed v_A = ⁸⁰⁰⁄₂₀₀ = 4 m/s. Racer B completes one round in 250 sec, so the speed v_B = ⁸⁰⁰⁄₂₅₀ = 3.2 m/s.

The race ends for the faster person (Racer A) when they cover the total distance of 18km (18000m). The time taken by Racer A to finish the race is T_fᵢₙᵢₛₕ = ¹⁸⁰⁰⁰⁄₄ = 4500 sec.

In a circular track, the time interval between consecutive meetings of two people starting from the same point and running in the same direction is given by t = (L)/(v_A₎₋ᵥ_____B, where L is the track length. Substituting the values, we get t = (800)/(4 - 3.2) = (800)/(0.8) = 1000 sec.

The racers meet at intervals of 1000 sec, which means they meet at 1000, 2000, 3000, and 4000 sec. Since the faster racer completes the race at 4500 sec, no further meeting can occur after this time. The last meeting before Racer A finishes the race occurs at 4000 sec.

Explanation

To find the value of the expression 27x³ - 58x² y + 31xy² - 8y³, substitute x = -5 and y = -7 into each term separately:

First term: 27x³ = 27(-5)³ = 27(-125) = -3375

Second term: -58x² y = -58(-5)² (-7) = -58(25)(-7) = 58 × 175 = 10150

Third term: 31xy² = 31(-5)(-7)² = 31(-5)(49) = -155 × 49 = -7595

Fourth term: -8y³ = -8(-7)³ = -8(-343) = 2744

Now, sum all the calculated terms: -3375 + 10150 - 7595 + 2744 = (10150 + 2744) - (3375 + 7595) = 12894 - 10970 = 1924

Explanation

Let the two natural numbers be a and b. Given (a - b)³ = 1728, we have a - b = ³√(1728) = 12. The product ab is given as 108. To find the sum of the cubes a³ + b³, we use the identity a³ + b³ = (a + b)(a² - ab + b²), which can be expressed in terms of known values as (a + b)((a - b)² + ab).

We first determine the sum of the numbers using the relation (a + b)² = (a - b)² + 4ab. Substituting the values, we get (a + b)² = 12² + 4(108) = 144 + 432 = 576, which gives a + b = √(576) = 24. Now, substituting these into the sum of cubes formula: a³ + b³ = 24 × (12² + 108) = 24 × (144 + 108) = 24 × 252 = 6048.

Alternatively, the numbers can be identified as 18 and 6, because 18 - 6 = 12 and 18 × 6 = 108. Calculating the sum of their cubes directly results in 18³ + 6³ = 5832 + 216 = 6048.

Explanation

The relative speed between the policeman and the thief is 8 km/h (40 km/h - 32 km/h). The time taken to catch the thief is the initial distance divided by the relative speed, which is 0.4 km / 8 km/h = 0.05 hours. Therefore, the distance run by the thief is 32 km/h × 0.05 hours = 1.6 km, or 1600 meters.

Explanation

Given (x - y)³ = 1728, we find x - y = 12. Using the identity x³ - y³ = (x - y)³ + 3xy(x - y) and the given product xy = 108, we calculate x³ - y³ = 1728 + 3(108)(12) = 1728 + 3888 = 5616. Alternatively, identifying the numbers as 18 and 6 (since 18 - 6 = 12 and 18 * 6 = 108), the difference of their cubes is 18³ - 6³ = 5832 - 216 = 5616.

Explanation

For school A, Biology students are 30% of 900, which is 270; dividing by the ratio 7:8 gives 126 males and 144 females. For school D, Biology students are 18% of 800, which is 144; dividing by the ratio 5:7 gives 60 males and 84 females. The combined total is 186 males to 228 females, which simplifies to 31:38.

Explanation

The curved surface area of a conical tent is given by the formula CSA = π r l, where r is the radius of the base and l is the slant height. The radius is half of the base diameter, so r = 48/2 = 24 m. The slant height l is found using the Pythagorean relation l = √(r² + h²). Substituting the given height h = 10 m and radius r = 24 m, we get l = √(24² + 10²) = √(576 + 100) = √(676) = 26 m. This follows the Pythagorean triplet (10, 24, 26), which is a multiple of the (5, 12, 13) triplet. Substituting these values into the area formula yields CSA = π × 24 × 26. Calculating the product, 24 × 26 = (25 - 1)(25 + 1) = 25² - 1 = 624. Therefore, the curved surface area of the tent is 624 π m².

Explanation

The expression can be simplified by converting the tangent terms into sine and cosine using the identity tan θ = (sin θ)/(cos θ). The term tan x + tan y simplifies as follows:

tan x + tan y = (sin x)/(cos x) + (sin y)/(cos y) = (sin x cos y + cos x sin y)/(cos x cos y) = (sin(x + y))/(cos x cos y)

Similarly, for the subtraction in the denominator:

tan x - tan y = (sin x)/(cos x) - (sin y)/(cos y) = (sin x cos y - cos x sin y)/(cos x cos y) = (sin(x - y))/(cos x cos y)

Dividing these two results, the cos x cos y terms cancel out:

(tan x + tan y)/(tan x - tan y) = (sin(x + y))/(sin(x - y))

Multiplying this by the first part of the original expression:

((sin(x - y))/(sin(x + y))) ((sin(x + y))/(sin(x - y))) = 1

Explanation

The expression follows the algebraic identity ((a + b)² - (a - b)²) / (ab) = 4ab / ab. Canceling out the variables leaves the constant 4, regardless of the values of a and b.

Explanation

The third proportional c of two quantities a and b is defined by the relation a : b = b : c, which gives the formula c = b²{a}.

In this case, the first quantity is a = x² - y² and the second quantity is b = x - y. Substituting these into the formula, the third proportional is calculated as:

c = (x - y)²{x² - y²}

Using the algebraic identity for the difference of squares, x² - y² = (x - y)(x + y), the expression becomes:

c = ((x - y)(x - y))/((x - y)(x + y))

Cancelling the common factor (x - y) from both the numerator and the denominator results in:

c = (x - y)/(x + y)

Explanation

The effective discount for Scheme A, which offers two successive discounts of 10% and 15%, is calculated using the successive discount formula x + y - ˣʸ⁄₁₀₀. Plugging in the values, the effective discount is 10 + 15 - (10 × 15)/(100) = 25 - 1.5 = 23.5%.

For Scheme B, the discount percentage for a Buy x Get y Free scheme is calculated as (y)/(x + y) × 100. With x = 5 and y = 2, the discount is (2)/(5 + 2) × 100 = ²⁄₇ × 100. Given that ⅐ ≈ 14.28%, the value of ²⁄₇ is approximately 28.57%.

Comparing the two results, Scheme B offers a discount of 28.57%, which is higher than the 23.5% discount provided by Scheme A.

Explanation

Let the total work be the least common multiple of 20 and 15, which is 60 units. The efficiency of A is 60 / 20 = 3 units per day, and the efficiency of B is 60 / 15 = 4 units per day. B worked for 9 days before leaving, completing 9 × 4 = 36 units of work. The remaining work is calculated as 60 - 36 = 24 units. Since A finishes this remaining work alone at an efficiency of 3 units per day, the time taken by A is 24 / 3 = 8 days.

Explanation

To find the time taken for the trains to cross each other, we calculate the total distance (the sum of their lengths: 320m + 540m = 860m) and divide it by their relative speed. Since the trains are moving in the same direction, the relative speed is the difference between their speeds (144 km/h - 108 km/h = 36 km/h), which is 10 m/s when converted. The time taken is 860 m / 10 m/s = 86 seconds.

Explanation

The total work is the product of the number of men and the total days required, which is 10 × 25 = 250 man-days. In the first 12 days, 10 men completed 10 × 12 = 120 man-days of work. Therefore, the remaining work is 250 - 120 = 130 man-days. When 3 more men are engaged, the total workforce becomes 10 + 3 = 13 men. The time required to finish the remaining 130 man-days of work with 13 men is calculated as 130 / 13 = 10 days. This can also be solved using the relation M₁ D₁ = M₂ D₂ for the remaining work, where M₁ = 10 men, D₁ = 13 days (the remaining time if no men were added), M₂ = 13 men, and D₂ is the unknown time. Substituting the values gives 10 × 13 = 13 × D₂, which simplifies to D₂ = 10 days.

Explanation

Simple interest is directly proportional to time when the principal and rate of interest are constant. According to the problem, the interest accrued in T₁ = 5 years is 25% of the principal. For the interest to be equal to the principal, it must reach 100% of the principal value. Since 100% is exactly 4 times 25%, the time required to accumulate that interest will also be 4 times the initial duration. Thus, the time required is 5 × 4 = 20 years.

Alternatively, using the simple interest formula SI = (P × R × T)/(100), we can first find the rate R. Substituting the given values SI = 0.25P and T = 5, we get 0.25P = (P × R × 5)/(100). Simplifying this yields R = 5%. To find the time T when the interest equals the principal (SI = P), we substitute the rate back into the formula: P = (P × 5 × T)/(100). Dividing both sides by P gives 1 = (5T)/(100), which results in T = 20 years.

Explanation

The expression can be simplified by taking 4 to the power of 12 as a common factor: 4¹²(1 + 4 + 16 + 64) = 4¹²(85). Since 85 is equal to 5 multiplied by 17, the entire expression is divisible by 17.

Explanation

The expression (sin A + cos A) × cosec A can be simplified to (sin A / sin A) + (cos A / sin A), which equals 1 + cot A. Substituting the given value of cot A = 12/5, we get 1 + 12/5 = 17/5.

Explanation

The initial saving is ₹12,000 (₹45,000 - ₹33,000). The increase in income is ₹9,000 (20% of ₹45,000) and the increase in expenditure is ₹3,960 (12% of ₹33,000), which results in a savings increase of ₹5,040 (₹9,000 - ₹3,960). The percentage increase in saving is (5,040 / 12,000) × 100 = 42%.

Explanation

A number is divisible by 24 if it is divisible by both 3 and 8. The number 49512 is divisible by 8 because its last three digits (512) are divisible by 8, and it is divisible by 3 because the sum of its digits (4+9+5+1+2=21) is divisible by 3.

Explanation

The original sum was underestimated by 9 kg because 77 kg was misread as 68 kg. Adding the 9 kg difference and dividing it across the 20 people results in an average increase of 0.45 kg, making the correct average 59.8 + 0.45 = 60.25 kg.

Explanation

The area of a triangle formed by joining the midpoints of the sides of a larger triangle is always one-fourth of the area of the original triangle. Since the area of triangle PQR is 32 cm², the area of triangle ABC is 32/4 = 8 cm².

Explanation

The length of a chord of a larger circle that is tangent to a smaller concentric circle is given by the formula 2√(R² - r²), where R and r are the radii of the larger and smaller circles respectively. Substituting the given values, we get 2√(15² - 13²) = 2√(225 - 169) = 2√56 = 2 × 2√14 = 4√14 cm.

Explanation

The total cost of the articles is ₹10,000. First, a flat discount of ₹500 is applied, reducing the price to ₹9,500 (10,000 - 500). Then, an additional 7% discount is applied to this balance for using a credit card. The additional discount is ₹665 (7% of 9,500), making the final amount to be paid ₹8,835 (9,500 - 665).

Explanation

Using the trigonometric half-angle identities, 1 - cos A = 2 sin²(A/2) and 1 + cos A = 2 cos²(A/2). Dividing these expressions yields (1 - cos A) / (1 + cos A) = tan²(A/2), so tan(A/2) = √((1 - cos A)/(1 + cos A)).

Explanation

The length of the direct common tangent to two circles touching each other externally is given by the formula 2√(r₁r₂). Substituting the given radii (r₁ = 10 cm and r₂ = 5 cm), the calculation is 2√(10 × 5) = 2√50 = 2 × 5√2 = 10√2 cm.

Explanation

According to the Area Theorem for similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Mathematically, this is expressed as:

Area₁{Area₂} = (Side₁{Side₂})²

To find the ratio of the corresponding sides, take the square root of the ratio of the areas:

Side₁{Side₂} = √(Area₁₎{Area₂}

Given that the ratio of the areas is 196 : 625, substitute these values into the formula:

Ratio of sides = √(¹⁹⁶⁄₆₂₅)

Since 14² = 196 and 25² = 625, the square root of the ratio simplifies to:

Ratio of sides = ¹⁴⁄₂₅

Explanation

Using the trigonometric identity sec² θ - tan² θ = 1, we can factorize it as (sec θ + tan θ)(sec θ - tan θ) = 1. Given that sec θ + tan θ = 5, we can substitute this value to find the conjugate expression:

5(sec θ - tan θ) = 1 ⟹ sec θ - tan θ = ⅕

To isolate tan θ, subtract the second equation from the first:

(sec θ + tan θ) - (sec θ - tan θ) = 5 - ⅕

2 tan θ = (25 - 1)/(5) = ²⁴⁄₅

Dividing both sides by 2 gives:

tan θ = ¹²⁄₅

Explanation

Let the length of the equal sides be 3x and the third side be 5x. The area of an isosceles triangle with equal sides a and third side b is given by the formula A = ᵇ⁄₄√(4a² - b²). Substituting the given values into the formula:

30√(11) = ⁵ˣ⁄₄√(4(3x)² - (5x)²)

30√(11) = ⁵ˣ⁄₄√(36x² - 25x²)

30√(11) = ⁵ˣ⁄₄√(11x²)

30√(11) = 5x²√(11){4}

Canceling √(11) from both sides results in 30 = (5x²)/(4). Multiplying by 4 and dividing by 5 gives x² = 24, which simplifies to x = √(24) = 2√6. The length of the third side is 5x = 5 × 2√6 = 10√6 cm.

Explanation

The given expression is in the form of the algebraic identity (a + b)(a - b) = a² - b². By substituting a = (1)/(sin θ) and b = (1)/(tan θ), the expression becomes:

((1)/(sin θ))² - ((1)/(tan θ))²

Using the reciprocal trigonometric identities where (1)/(sin θ) = csc θ and (1)/(tan θ) = cot θ, the expression simplifies to:

csc² θ - cot² θ

According to the fundamental Pythagorean identity 1 + cot² θ = csc² θ, it follows that csc² θ - cot² θ = 1. Therefore, the value of the entire expression is 1.

Explanation

When the percentage profit earned at one selling price (SP₁) is equal to the percentage loss incurred at another selling price (SP₂) for the same article, the cost price (CP) is the arithmetic mean of the two selling prices. This is because the absolute profit (SP₁ - CP) and absolute loss (CP - SP₂) must be equal to yield equal percentages on the same base.

CP = (SP₁ + SP₂)/(2) = (2000 + 1200)/(2) = ³²⁰⁰⁄₂ = 1600

To find the selling price required for a 20% profit, we use the formula:

New SP = CP × (1 + (Profit%)/(100))

Substituting the values:

New SP = 1600 × (1 + ²⁰⁄₁₀₀) New SP = 1600 × 1.2 = 1920

Explanation

To determine if 24983 is divisible by 7, we perform direct division. Dividing 24983 by 7, we get: 24 ÷ 7 = 3 with a remainder of 3. Bringing down the next digit, 9, we have 39. 39 ÷ 7 = 5 with a remainder of 4. Bringing down the next digit, 8, we have 48. 48 ÷ 7 = 6 with a remainder of 6. Bringing down the final digit, 3, we have 63. 63 ÷ 7 = 9 with a remainder of 0. The division yields a quotient of 3569 without any remainder. Alternatively, 24983 can be decomposed into multiples of 7: 24983 = 21000 + 3500 + 420 + 63. Since 21000 (7 × 3000), 3500 (7 × 500), 420 (7 × 60), and 63 (7 × 9) are all divisible by 7, their sum 24983 is also divisible by 7.

Explanation

The expression can be simplified by grouping the first and last terms: sin 6x + sin 2x + 2sin 4x. Applying the sum-to-product formula sin A + sin B = 2sin((A+B)/(2))cos((A-B)/(2)) to the first two terms where A=6x and B=2x:

sin 6x + sin 2x = 2sin(⁶ˣ⁺²ˣ⁄₂)cos(⁶ˣ⁻²ˣ⁄₂) = 2sin 4x cos 2x

Substituting this back into the original expression gives:

2sin 4x cos 2x + 2sin 4x = 2sin 4x (cos 2x + 1)

Using the double-angle identity cos 2x = 2cos² x - 1, we know that 1 + cos 2x = 2cos² x. Substituting this identity into the expression yields:

2sin 4x (2cos² x) = 4cos² x sin 4x

Explanation

In a mean proportion a:b::b:c, the product of the means equals the product of the extremes, expressed by the formula b² = ac. Given the proportion 405:y::y:125, we apply this relation to get y² = 405 × 125. To simplify the calculation, we can factorize the numbers: 405 = 81 × 5 and 125 = 25 × 5. Substituting these factors back into the equation, we have y² = 81 × 5 × 5 × 25, which simplifies to y² = 81 × 25 × 25. Taking the square root of both sides results in y = √(81) × √(25²) = 9 × 25. Calculating the product gives y = 225. Given that y > 0, the value is confirmed as 225.

Explanation

The time taken for the policeman to catch the thief is determined by the initial distance between them and their relative speed. Since both are running in the same direction, the relative speed is the difference between the speed of the policeman and the speed of the thief.

Relative Speed = 10 km/h - 8 km/h = 2 km/h.

To simplify the calculation for time in minutes, the relative speed can be converted to kilometers per minute by dividing by 60. Relative Speed = ²⁄₆₀ km/min = ¹⁄₃₀ km/min.

The initial distance between the policeman and the thief is 600 meters, which is 0.6 km. Using the formula Time = (Distance)/(Relative Speed):

Time = (0.6)/(1){30}

Time = 0.6 × 30 = 18 minutes.

Explanation

For a number to be divisible by 36, it must be divisible by both 4 and 9, as 36 = 4 × 9 and these factors are co-prime. The divisibility rule for 4 requires the last two digits of the number to be divisible by 4. In the number 29412, the last two digits are 12, which is divisible by 4 (12 = 4 × 3). The divisibility rule for 9 requires the sum of the digits of the number to be divisible by 9. For the number 29412, the sum of the digits is 2 + 9 + 4 + 1 + 2 = 18. Since 18 is divisible by 9 (18 = 9 × 2), the number 29412 satisfies both conditions for divisibility by 36.

Explanation

Let the total work be the Least Common Multiple (LCM) of 15 and 12, which is 60 units. The efficiency of Amar is 60 / 15 = 4 units per day, and the efficiency of Prem is 60 / 12 = 5 units per day. Since they work on alternate days starting with Amar, they complete 4 + 5 = 9 units of work every 2 days. In 6 such cycles (which equals 12 days), the work completed is 9 × 6 = 54 units. The remaining work is 60 - 54 = 6 units. On the 13th day, it is Amar's turn, and he completes 4 units of work. The remaining work after the 13th day is 6 - 4 = 2 units. On the 14th day, it is Prem's turn. Prem can complete 5 units in a full day, so he will complete the remaining 2 units in 2 / 5 of a day. Thus, the total time taken to complete the work is 12 + 1 + ⅖ = 13 ⅖ days.

Explanation

The total work is the least common multiple of 4 and 9, which is 36 units. The efficiency of A is 36 / 4 = 9 units per day, and the efficiency of B is 36 / 9 = 4 units per day. They work on alternate days starting with B, so in the first two days (one cycle), the total work done is 4 + 9 = 13 units. In two cycles, which equals 4 days, the work completed is 13 × 2 = 26 units. This leaves 36 - 26 = 10 units of work remaining. On the 5th day, B works and completes 4 units, leaving 10 - 4 = 6 units. On the 6th day, it is A's turn to work. Since A's efficiency is 9 units per day, A will take 6 / 9 = 2/3 of a day to finish the remaining 6 units. The total time taken to complete the work is 5 + 2/3 = 5 ⅔ days.

Explanation

The price of the mixture is calculated using the weighted average formula Pₘ = n₁P₁ + n₂P₂{n₁ + n₂}, where n₁ and n₂ represent the quantities (or ratios) and P₁ and P₂ represent their respective prices.

Substituting the given values: Pₘ = (7 × 100 + 20 × 150)/(7 + 20) Pₘ = (700 + 3000)/(27) Pₘ = ³⁷⁰⁰⁄₂₇

Performing the division: 3700 ÷ 27 ≈ 137.037

Rounding to the nearest Rupee, the price of the mixture is ₹137.

Explanation

The expression x⁵ - (1)/(x⁵⁾ can be derived from the product of (x² + (1)/(x²⁾) and (x³ - (1)/(x³⁾). Expanding this product, we get (x² + (1)/(x²⁾)(x³ - (1)/(x³⁾) = x⁵ - ¹⁄ₓ + x - (1)/(x⁵⁾, which simplifies to x⁵ - (1)/(x⁵⁾ = (x² + (1)/(x²⁾)(x³ - (1)/(x³⁾) - (x - ¹⁄ₓ).

Given x - ¹⁄ₓ = -6, we calculate the required components. Squaring both sides of the given equation gives (x - ¹⁄ₓ)² = (-6)², which results in x² + (1)/(x²⁾ - 2 = 36. Thus, x² + (1)/(x²⁾ = 38.

Next, we calculate x³ - (1)/(x³⁾ using the identity (x - ¹⁄ₓ)³ = x³ - (1)/(x³⁾ - 3(x - ¹⁄ₓ). Substituting the given value, we get (-6)³ = x³ - (1)/(x³⁾ - 3(-6), which simplifies to -216 = x³ - (1)/(x³⁾ + 18. Therefore, x³ - (1)/(x³⁾ = -216 - 18 = -234.

Finally, substituting these values into the initial formula: x⁵ - (1)/(x⁵⁾ = (38)(-234) - (-6). Performing the multiplication, 38 × -234 = -8892. Adding the final term, we get -8892 + 6 = -8886.

Explanation

Let the number of male workers be M and the number of female workers be F. Based on the given condition, 48% of the number of males is equal to two-third of the number of females. This can be expressed as:

⁴⁸⁄₁₀₀ M = ⅔ F

First, simplify the percentage fraction ⁴⁸⁄₁₀₀ by dividing both the numerator and the denominator by 4, which results in ¹²⁄₂₅. The equation then becomes:

¹²⁄₂₅ M = ⅔ F

To find the ratio of males to females, rearrange the equation to solve for (M)/(F):

(M)/(F) = ⅔ × ²⁵⁄₁₂

By simplifying the expression, we divide the 12 in the denominator by the 2 in the numerator:

(M)/(F) = ⅓ × ²⁵⁄₆ = ²⁵⁄₁₈

The ratio of the number of males to the number of females in the factory is 25:18.

Explanation

To find the value of (a - b)² + (b - c)² + (c - a)², we first expand the expression: (a - b)² + (b - c)² + (c - a)² = (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 2(a² + b² + c²) - 2(ab + bc + ca)

We can find the value of 2(ab + bc + ca) using the identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values a + b + c = 19 and a² + b² + c² = 155: 19² = 155 + 2(ab + bc + ca) 361 = 155 + 2(ab + bc + ca) 2(ab + bc + ca) = 361 - 155 = 206

Now, substitute a² + b² + c² = 155 and 2(ab + bc + ca) = 206 into the expanded target expression: (a - b)² + (b - c)² + (c - a)² = 2(155) - 206 = 310 - 206 = 104

Explanation

For two successive discounts d₁ and d₂, the effective single discount rate is calculated using the formula d₁ + d₂ - d₁ × d₂{100}. Applying this to the successive discounts of 30% and 5%, the effective discount is 30 + 5 - (30 × 5)/(100) = 35 - 1.5 = 33.5%.

The difference in discount percentages between the single 35% discount and the effective successive discount of 33.5% is 35% - 33.5% = 1.5%. To find the value of this difference on the principal amount of ₹3,600, we calculate 1.5% of 3,600: Difference = (1.5)/(100) × 3600 = 1.5 × 36 = 54.

Explanation

Let the total work be 20 units. Since Raja and Rocky together can complete the work in 5 days, their combined efficiency is 20 / 5 = 4 units per day. They work together for 2 days, completing 4 × 2 = 8 units. The remaining work is 20 - 8 = 12 units. Raja completes this remaining work in 4 days, meaning Raja's efficiency is 12 / 4 = 3 units per day. To find Rocky's efficiency, we subtract Raja's efficiency from the combined efficiency: 4 - 3 = 1 unit per day. The time Rocky takes to complete the total work alone is the total work divided by his efficiency, which is 20 / 1 = 20 days.

Explanation

The trigonometric identity sin² A + cos² A = 1 relates the sine and cosine of an angle. Substituting the given value sin A = -3/5 into the identity yields (-3/5)² + cos² A = 1, which simplifies to 9/25 + cos² A = 1. Solving for the cosine squared gives cos² A = 1 - 9/25 = 16/25.

Taking the square root, we find cos A = ± 4/5. Since the angle A lies in the third quadrant, where the cosine function is negative, we have cos A = -4/5. The secant function is defined as the reciprocal of the cosine function, so sec A = 1 / cos A. Substituting the value of cos A, we get sec A = 1 / (-4/5) = -5/4.

Explanation

To find how long A worked, we first determine the efficiencies of A and B. Let the total work be 10 units, which is the least common multiple of their individual completion times, 5 and 10. The efficiency of A is 10 / 5 = 2 units per day, and the efficiency of B is 10 / 10 = 1 unit per day. According to the problem, B finished the remaining work in 8 days after A left, meaning B performed 8 days × 1 unit/day = 8 units of work alone. The work done by A and B together before A left was the total work minus the work done by B alone, which is 10 - 8 = 2 units. Their combined efficiency is 2 + 1 = 3 units per day. The time they worked together is the shared work divided by their combined efficiency, which is 2 / 3 days. Thus, A left after 2/3 days.

Explanation

The speed of the rower downstream is calculated by dividing the total distance by the time taken, v_d = 144 / 9 = 16 km/h. Similarly, the speed of the rower upstream is vᵤ = 144 / 12 = 12 km/h.

In boat and stream problems, the speed of the stream s is defined as half the difference between the downstream speed and the upstream speed, represented by the formula s = (v_d - vᵤ) / 2. Substituting the values into this formula gives s = (16 - 12) / 2, which simplifies to 4 / 2 = 2 km/h.

Explanation

The perimeter of an equilateral triangle is given by P = 3a, where a is the length of each side. Given the perimeter is 48 cm, we find the side length as: a = ⁴⁸⁄₃ = 16 cm

The area A of an equilateral triangle is calculated using the formula: A = √3{4} a²

Substituting the value of a = 16 cm into the formula: A = √3{4} × (16)² A = √3{4} × 256 A = 64√3 cm²