SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

In the right-angled triangle △ ABC, where the right angle is at B, the side AB is the adjacent side to ∠ CAB and the side BC is the opposite side. The trigonometric ratio that relates the opposite side to the adjacent side is the tangent function.

According to the definition of tangent for ∠ CAB:

tan(∠ CAB) = (Opposite)/(Adjacent) = (BC)/(AB)

Given that AB = 12 cm and ∠ CAB = 60°, we substitute these values into the equation:

tan(60°) = (BC)/(12)

Since the value of tan(60°) is known to be √3, the equation becomes:

√3 = (BC)/(12)

Solving for BC by multiplying both sides by 12 gives:

BC = 12√3 cm

Explanation

The expression (3²)² + 3 * 3 / 3 - 3 simplifies to 81 + 3 - 3, which equals 81. Since 9² is 81, option B is the correct answer.

Explanation

The ratio of efficiency of A to B is 5.4:1, which is 27:5, so the ratio of time taken is 5:27. The difference in time units is 22, which corresponds to 22 days, meaning A takes 5 days and B takes 27 days. Working together, they complete the job in (5 * 27) / (5 + 27) = 135/32 = 4 7/32 days.

Explanation

To find the time taken after pipe S3 is opened, we first determine the efficiencies of the pipes. Given the tank capacity is the LCM of 15, 20, and 40, which is 120 units, the efficiencies are: S1 = 8 units/hr, S2 = 6 units/hr, and S3 = -3 units/hr (emptying). In the first 2 hours, S1 and S2 fill (8 + 6) * 2 = 28 units, leaving 120 - 28 = 92 units to be filled. When all three pipes are open, the net efficiency is 8 + 6 - 3 = 11 units/hr, so the remaining time required is 92/11 hours.

Explanation

Amit's hourly work rate is 1/5. Amit and Paul together have a rate of 1/3, so Paul's rate is 1/3 - 1/5 = 2/15. Since Jeevan and Paul together have a rate of 1/4, Jeevan's rate is 1/4 - 2/15 = 7/60. Jeevan takes 60/7 hours, which is approximately 8.57 hours, rounding to the nearest integer 9.

Explanation

The given equation (2a - 5)/a - (4b - 5)/b + (6c + 5)/c = 0 simplifies to (2 - 5/a) - (4 - 5/b) + (6 + 5/c) = 0. Rearranging the constants gives 2 - 4 + 6 = 5/a - 5/b - 5/c, which further simplifies to 4 = 5(1/a - 1/b - 1/c), resulting in (1/a - 1/b - 1/c) = 4/5.

Explanation

Let O be the center of the circle and R be its radius. For any chord of length L at a distance d from the center, the relationship is given by the Pythagorean theorem: R² = ((L)/(2))² + d².

Let the distance of the chord AB (length 28 cm) from the center be x. Since the chord CD (length 22 cm) is parallel and on the same side of the center at a distance of 4 cm from AB, its distance from the center is x + 4.

For chord AB: R² = 14² + x² = 196 + x²

For chord CD: R² = 11² + (x + 4)² = 121 + x² + 8x + 16

Equating the two expressions for R²: 196 + x² = 137 + x² + 8x 8x = 196 - 137 8x = 59 x = 7.375 cm

Substitute x back into the equation for R²: R² = 196 + (7.375)² R² = 196 + 54.390625 R² = 250.390625 R = √(250.390625) ≈ 15.8237 cm

Rounding to two decimal places, the radius of the circle is 15.82 cm.

Explanation

First, square the given equation x - 1/x = 2√2 to get x² + 1/x² - 2 = 8, which simplifies to x² + 1/x² = 10. Then, cube this result using the identity (a + b)³ = a³ + b³ + 3ab(a + b), which gives x⁶ + 1/x⁶ + 3(10) = 10³, resulting in 1000 - 30 = 970.

Explanation

First, calculate the cost price using the initial selling price and profit percentage: Cost Price = 22,400 / 1.12 = 20,000. To double the profit percentage from 12% to 24%, the new selling price is calculated as 20,000 * 1.24 = 24,800. Thus, option B is the correct answer.

Explanation

The semi-perimeter of a triangle is defined as half of its total perimeter. Since the perimeter is given as 100 cm, the semi-perimeter is 100 / 2 = 50 cm.

Explanation

The single equivalent discount for three successive discounts of r₁%, r₂%, and r₃% can be calculated by determining the final selling price relative to the initial marked price. Using the successive discount formula for a marked price of 100, the final selling price SP is given by 100 × (1 - (r₁)/(100)) × (1 - (r₂)/(100)) × (1 - (r₃)/(100)).

Substituting the given values of 25% for each discount, the expression becomes 100 × (1 - 0.25)³, which simplifies to 100 × (0.75)³. Since 0.75 is equal to ¾, the calculation is 100 × ²⁷⁄₆₄. This results in 100 × 0.421875 = 42.1875. The single equivalent discount is the total reduction from the original price, calculated as 100 - 42.1875 = 57.8125%. When rounded to two decimal places, this yields 57.81%.

Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given the side ratio of 5:7, the area ratio is 5 squared to 7 squared, which is 25:49.

Explanation

By rationalizing the expression inside the square root, we multiply the numerator and denominator by (1 - sin 3theta), resulting in the square root of ((1 - sin 3theta)² / cos² 3theta). Taking the square root yields (1 - sin 3theta) / cos 3theta, which simplifies to sec 3theta - tan 3theta.

Explanation

The expression can be simplified using the trigonometric identity tan(90 - x) = cot(x). Here, tan(65) = cot(25) and tan(55) = cot(35), so tan(25)tan(65) = 1 and tan(35)tan(55) = 1. Since tan(45) = 1, the product of all terms is 1 * 1 * 1 = 1.

Explanation

By substituting the values x = 5 and y = -7 into the expression 27x³ + 58x²y + 31xy² + 8y³, we get 27(125) + 58(25)(-7) + 31(5)(49) + 8(-343). This simplifies to 3375 - 10150 + 7595 - 2744, which equals -1924.

Explanation

The expression sec x - cos x can be rewritten as (1/cos x) - cos x, which simplifies to (1 - cos² x) / cos x. Since 1 - cos² x = sin² x, the expression becomes sin² x / cos x, which is equal to (sin x / cos x) * sin x, or tan x sin x.

Explanation

The expression is simplified by using the algebraic identity (a - b)(a + b) = a² - b². By grouping the terms inside the brackets, we can apply this identity sequentially starting with the terms (1 - p) and (1 + p).

The product (1 - p)(1 + p) simplifies to 1 - p². Multiplying this result by the next term (1 + p²) gives (1 - p²)(1 + p²) = 1 - (p²)² = 1 - p⁴. Following this pattern through the remaining factors, (1 - p⁴)(1 + p⁴) = 1 - p⁸, then (1 - p⁸)(1 + p⁸) = 1 - p¹⁶, and finally (1 - p¹⁶)(1 + p¹⁶) = 1 - p³².

Substituting the simplified product back into the original expression yields [(1 - p³²) - 1]. Subtracting the constant term results in 1 - p³² - 1, which leaves -p³².

Explanation

Train 1 travels alone from 4:00 p.m. to 5:00 p.m. During this 1 hour, it covers a distance of 65 km/h × 1 h = 65 km. At 5:00 p.m., when Train 2 starts, the remaining distance between the two trains is 195 - 65 = 130 km. Since the trains are moving toward each other, their relative speed is the sum of their individual speeds: Vᵣₑₗ = 65 + 35 = 100 km/h. The time taken to cover the remaining distance is T = Distance / Vᵣₑₗ = 130 / 100 = 1.3 hours. To convert the decimal part to minutes, 0.3 × 60 = 18 minutes. Therefore, the trains meet 1 hour and 18 minutes after 5:00 p.m., which is at 6:18 p.m.

Explanation

According to the Intersecting Chords Theorem, if two chords AB and CD of a circle intersect at an internal point P, the product of the segments of one chord is equal to the product of the segments of the other chord. This is expressed by the formula AP · PB = CP · PD.

Since P lies on the chord AB, the segment PB can be found by subtracting AP from the total length of the chord AB, such that PB = AB - AP = 20cm - 12cm = 8cm.

Substituting the given values into the theorem formula, we get 12 · 8 = 16 · PD. Solving for PD gives 96 = 16 · PD, which simplifies to PD = 6cm.

The total length of chord CD is the sum of its segments CP and PD. Therefore, CD = CP + PD = 16cm + 6cm = 22cm.

Explanation

The weight of the student who left is determined by the difference between the total weight of the class before and after the departure. The initial total weight is the product of the number of students and their average weight, which is 15 × 40 = 600kg. After one student leaves, the class has 14 students remaining with a new average weight of 39.5kg. The new total weight is 14 × 39.5 = 553kg. By subtracting the new total weight from the initial total weight, 600 - 553 = 47kg, we find the weight of the student who left.

Alternatively, using the deviation method, the weight of the student who left is equal to the initial average plus the total weight lost by the remaining students due to the decrease in the average. The formula is: Weight = Initial Average + [Remaining Students × (Initial Average - New Average)] . Substituting the given values: Weight = 40 + [14 × (40 - 39.5)] = 40 + (14 × 0.5) = 40 + 7 = 47kg .

Explanation

In a triangle XYZ right-angled at Y, the sum of the acute angles is X + Z = 90°. Given sin X = ½, the angle X is 30°. Consequently, Z = 90° - 30° = 60°. The expression cos X cos Z + sin X sin Z represents the trigonometric identity for the cosine of the difference of two angles, cos(Z - X).

Substituting the values of the angles into the identity: cos(60° - 30°) = cos 30° = √3/2

Alternatively, substituting the individual trigonometric values: cos 30° = √3/2 cos 60° = ½ sin 30° = ½ sin 60° = √3/2

The calculation is: (√3/2 × ½) + (½ × √3/2) = √3/4 + √3/4 = 2√3/4 = √3/2

Explanation

To find the total time taken, first determine the total work by taking the Least Common Multiple (LCM) of the individual time periods. The LCM of 16 and 12 is 48, so the total work is 48 units.

The efficiency of Rekha is 48 / 16 = 3 units per day, and the efficiency of Bina is 48 / 12 = 4 units per day. Since they work on alternate days starting with Rekha, in one 2-day cycle, the total work completed is 3 + 4 = 7 units.

In 6 such cycles (which equals 12 days), the work completed is 6 × 7 = 42 units. The remaining work is 48 - 42 = 6 units. On the 13th day, it is Rekha's turn. She completes 3 units of work, leaving 6 - 3 = 3 units still to be done.

On the 14th day, it is Bina's turn. Since her efficiency is 4 units per day, the time she takes to complete the remaining 3 units is 3 / 4 of a day. Therefore, the total time taken to complete the work is 13 + 3/4 = 13¾ days.

Explanation

The simple interest formula is SI = (P × R × T)/(100), where P is the principal, R is the rate per annum, and T is the time in years.

In the first scenario, the interest is SI₁ = (P × x × 3)/(100).

In the second scenario, the rate becomes 3x% and the time is y more than the original 3 years, which is 3 + y years. The interest is SI₂ = (P × 3x × (3 + y))/(100).

Given that the new interest is seven times the earlier interest, we set up the equation SI₂ = 7 × SI₁:

(P × 3x × (3 + y))/(100) = 7 × (P × x × 3)/(100)

By cancelling the common terms P, x, and the denominator 100 from both sides, the equation simplifies to:

3(3 + y) = 7 × 3

Dividing both sides by 3:

3 + y = 7

y = 4

Explanation

To evaluate the expression (16 + 14) ÷ 2 - 12 + 16 × 4 - 28 + 13(-16 + 13), apply the BODMAS rule, which establishes the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

First, solve the terms inside the brackets: 16 + 14 = 30 -16 + 13 = -3

Substituting these values back into the expression results in: 30 ÷ 2 - 12 + 16 × 4 - 28 + 13(-3)

Next, perform the division and multiplication operations: 30 ÷ 2 = 15 16 × 4 = 64 13 × (-3) = -39

The expression now simplifies to: 15 - 12 + 64 - 28 - 39

Finally, perform the addition and subtraction from left to right: 15 - 12 = 3 3 + 64 = 67 67 - 28 = 39 39 - 39 = 0

Explanation

The ratio of the averages of two data sets over the same number of years is equivalent to the ratio of their respective sums. For college D, the total number of students enrolled from 2010 to 2014 is calculated as 710 + 750 + 680 + 720 + 740 = 3600. The average number of students enrolled is 3600 / 5 = 720.

The total number of students who passed from college D during the same period is 650 + 710 + 640 + 690 + 710 = 3400. The average number of students who passed is 3400 / 5 = 680.

The ratio of the average enrolled students to the average passed students is 720 : 680. Dividing both sides by their greatest common divisor, 40, simplifies the ratio to 18 : 17.

Explanation

The relationship between the sum and difference of two numbers is given by the algebraic identity: (x + y)² = (x - y)² + 4xy

Substituting the given values x - y = 5 and xy = 150 into the formula: (x + y)² = (5)² + 4(150) (x + y)² = 25 + 600 (x + y)² = 625

Taking the square root of both sides to find (x + y): x + y = √(625) x + y = 25

Since x and y are specified as positive numbers, their sum (x + y) must be positive. Alternatively, the numbers can be identified by finding factors of 150 that differ by 5, which are 15 and 10. Their sum is 15 + 10 = 25.

Explanation

To find the value of the expression ³⁄₂ ÷ ⅐ × [(½ - ⅓) ÷ ¹⁄₄₂], we apply the BODMAS rule, starting with the innermost parentheses.

First, solve the subtraction inside the round brackets: ½ - ⅓ = (3 - 2)/(6) = ⅙

Next, solve the division inside the square brackets: ⅙ ÷ ¹⁄₄₂ = ⅙ × 42 = 7

Now the expression simplifies to: ³⁄₂ ÷ ⅐ × 7

Following the order of operations from left to right for division and multiplication: ³⁄₂ ÷ ⅐ = ³⁄₂ × 7 = ²¹⁄₂

Finally, multiply the result by 7: ²¹⁄₂ × 7 = ¹⁴⁷⁄₂

Converting the improper fraction ¹⁴⁷⁄₂ into a mixed fraction gives: 147 ÷ 2 = 73 with a remainder of 1, which is 73 ½

Explanation

To find the time when A and B meet for the first time on a circular track while running in the same direction, we use the formula T = (L)/(vᵣₑₗ₎, where L is the length of the track and vᵣₑₗ is the relative speed of the two runners. First, convert the speeds from km/h to m/s by multiplying with ⁵⁄₁₈. The speed of A is 27 × ⁵⁄₁₈ = 7.5 m/s and the speed of B is 45 × ⁵⁄₁₈ = 12.5 m/s. Since they are moving in the same direction, the relative speed is the difference between their individual speeds, calculated as 12.5 - 7.5 = 5 m/s. Substituting the values into the meeting time formula gives T = ¹⁶⁰⁰⁄₅ = 320 seconds.

Explanation

A completes the full work in 4 × ¹⁰⁰⁄₂₀ = 20 days, and B completes it in 10 × 3 = 30 days. Taking the total work as the Least Common Multiple of 20 and 30, let the work be 60 units. The efficiency of A is ⁶⁰⁄₂₀ = 3 units per day, and the efficiency of B is ⁶⁰⁄₃₀ = 2 units per day. In 9 days, A and B together complete (3 + 2) × 9 = 45 units. The remaining work is 60 - 45 = 15 units, which C completes in 6 days, giving C an efficiency of ¹⁵⁄₆ = 2.5 units per day. The combined efficiency of B and C is 2 + 2.5 = 4.5 units per day. To complete 75% of the total work, which is 0.75 × 60 = 45 units, B and C together will take (45)/(4.5) = 10 days.

Explanation

The equivalent discount for two successive discounts of a% and b% is calculated using the formula D = (a + b - ᵃᵇ⁄₁₀₀)%. Given the individual discount rates are 9% and 17%, substituting these values into the formula gives D = (9 + 17 - (9 × 17)/(100))%. Summing the two percentages yields 26%, and the product term results in ¹⁵³⁄₁₀₀, which is 1.53%. Subtracting this value from the sum results in 26 - 1.53, which equals a single equivalent discount of 24.47%.

Explanation

Given the equations (cos α)/(sin β) = 10 and (cos α)/(cos β) = 11, we can equate the values of cos α such that 10 sin β = 11 cos β. This relation simplifies to tan β = ¹¹⁄₁₀.

To find cos² β, we use the identity cos² β = (1)/(sec² β) = (1)/(1 + tan² β). Substituting the value of tan β, we get: cos² β = (1)/(1 + (11){10)²} = (1)/(1 + 121){100} = (1)/(221){100} = ¹⁰⁰⁄₂₂₁ This result matches the ratio presented in the second option.

Explanation

When multiple solid metallic spheres are melted and recast into a single sphere, the total volume remains conserved. The volume of a sphere with radius r is defined as V = ⁴⁄₃ π r³. Therefore, if R is the radius of the new sphere and r₁, r₂, r₃ are the radii of the three original spheres, the relationship is established as:

⁴⁄₃ π R³ = ⁴⁄₃ π r₁³ + ⁴⁄₃ π r₂³ + ⁴⁄₃ π r₃³

By dividing both sides by the constant factor ⁴⁄₃ π, the equation simplifies to:

R³ = r₁³ + r₂³ + r₃³

Substituting the given values r₁ = 1 cm, r₂ = 6 cm, and r₃ = 8 cm into the equation:

R³ = 1³ + 6³ + 8³ R³ = 1 + 216 + 512 R³ = 729

Taking the cube root of both sides to find R:

R = ³√(729) R = 9 cm

Explanation

Factor out sinθ from the numerator and cosθ from the denominator: (sinθ(1 - 2sin²θ))/(cosθ(2cos²θ - 1))

Apply the trigonometric double angle identities for cos 2θ: cos 2θ = 1 - 2sin²θ cos 2θ = 2cos²θ - 1

Substituting these identities into the expression results in: (sinθ(cos 2θ))/(cosθ(cos 2θ))

The common factor cos 2θ in the numerator and denominator cancels out, leaving: (sinθ)/(cosθ) = tanθ

Explanation

To find the time taken when they work together, we use the combined rate formula or the LCM method. Let the total work be the Least Common Multiple (LCM) of the individual times taken by Richa, Rita, and Reena.

The LCM of 8, 12, and 24 is 24 units. This represents the total work to be completed.

The individual efficiencies (work done per hour) are: Efficiency of Richa = 24 / 8 = 3 units/hour Efficiency of Rita = 24 / 12 = 2 units/hour Efficiency of Reena = 24 / 24 = 1 unit/hour

Total combined efficiency = 3 + 2 + 1 = 6 units/hour.

The total time taken by all three to complete the work together is calculated as: Time = Total Work / Total Efficiency = 24 / 6 = 4 hours.

Explanation

The area of a sector of a circle is related to its arc length and radius by the formula A = ½ × l × r, where A is the area of the sector, l is the length of the arc, and r is the radius of the circle. This formula is derived from the relationship between the sector area A = (θ)/(360) × π r² and the arc length l = (θ)/(360) × 2π r. By substituting the given values into the formula, we get 112 = ½ × l × 28. Simplifying the equation, we have 112 = 14 × l. Dividing both sides by 14, the length of the arc l is found to be 8 cm.

Explanation

Let the total marks in the examination be T. The difference between the percentage of marks obtained by Jack and John is 35% - 20% = 15%. The numerical difference in their marks, considering one is below the passing mark and one is above, is 20 - (-10) = 30 marks.

By equating the percentage difference to the marks difference: 15% of T = 30 T = (30 × 100) / 15 = 200

To find the passing percentage, we can convert the marks John lacked into a percentage. Since 15% = 30 marks, 1% = 30 / 15 = 2 marks. John failed by 10 marks, which is equivalent to 10 / 2 = 5%. Adding this to his score gives the minimum passing percentage: 20% + 5% = 25%

Explanation

In a sequence of four numbers a, b, c, and d, if they are in proportion, then the ratio of the first two is equal to the ratio of the last two, represented as a : b = c : d or a/b = c/d. To find the fourth proportional d, the formula is d = (b × c) / a.

Given the numbers a = 8, b = 12, and c = 14, we substitute them into the formula: d = (12 × 14) / 8 d = 168 / 8 d = 21

Alternatively, using the ratio method, 8/12 simplifies to 2/3. For the second pair to maintain the same ratio, 14/d = 2/3. Solving for d gives 2d = 42, which results in d = 21.

Explanation

The relationship between the Marked Price (MP) and the Cost Price (CP) when both the profit percentage and discount percentage are known is given by the formula:

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

Substituting the given values, where CP = £650, Gain% = 20%, and Discount% = 22%:

(MP)/(650) = (100 + 20)/(100 - 22)

(MP)/(650) = (120)/(78)

To find MP, simplify the ratio on the right side. Both 120 and 78 are divisible by 6, which reduces the fraction to:

(MP)/(650) = (20)/(13)

Now, calculate MP by multiplying both sides by 650:

MP = (20/13) × 650

Since 650 ÷ 13 = 50, the equation becomes:

MP = 20 × 50 = 1000

The marked price of the article is £1,000.

Explanation

To find the value of x³ - (1)/(x³⁾, we first determine the value of x - ¹⁄ₓ. Using the identity (x² + (1)/(x²⁾)² = (x² - (1)/(x²⁾)² + 4, we substitute the given value:

(x² + (1)/(x²⁾)² = (4√6)² + 4 = 96 + 4 = 100

Taking the square root, since x > 1, we have x² + (1)/(x²⁾ = 10. Next, we use the relationship (x - ¹⁄ₓ)² = x² + (1)/(x²⁾ - 2:

(x - ¹⁄ₓ)² = 10 - 2 = 8

Given x > 1, x - ¹⁄ₓ must be positive, so x - ¹⁄ₓ = √8 = 2√2. The expression for x³ - (1)/(x³⁾ is derived from the algebraic identity (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ). Substituting the value 2√2:

x³ - (1)/(x³⁾ = (2√2)³ + 3(2√2)

x³ - (1)/(x³⁾ = 16√2 + 6√2 = 22√2

Explanation

The total volume of the wall is calculated by multiplying its length, breadth, and height. Using the formula V_wₐₗₗ = L × B × H, we substitute the given values: V_wₐₗₗ = 14 m × 0.125 m × 5 m = 8.75 m³

The number of bricks required is determined by dividing the total volume of the wall by the volume of a single brick. Using the formula N = V_wₐₗₗ{V_bᵣᵢ_cₖ}: N = (8.75)/(0.0014)

To simplify the calculation, multiply both the numerator and the denominator by 10,000 to remove the decimals: N = ⁸⁷⁵⁰⁰⁄₁₄ = 6250

Explanation

Given the relation 8 cot A = 7, it follows that cot A = ⅞. In a right-angled triangle, cot A is defined as the ratio of the base to the perpendicular, meaning we can take the perpendicular p = 8 and the base b = 7.

The hypotenuse h is determined by the Pythagorean formula: h = √(p² + b²) = √(8² + 7²) = √(64 + 49) = √(113)

The value of sin A is defined as the ratio of the perpendicular to the hypotenuse: sin A = ᵖ⁄ₕ = (8)/(√(113))

This result is achieved by substituting the side lengths into the standard trigonometric ratio for sine.

Explanation

In a proportion of the form a : b :: c : d, the product of the extreme terms equals the product of the mean terms, expressed as ad = bc. For the given proportion 28 : 98 :: 98 : y, we substitute the values into the equation:

28 × y = 98 × 98

To solve for y, we rearrange the equation:

y = (98 × 98)/(28)

Simplifying the fraction by dividing both 98 and 28 by their common factor 14:

y = (7 × 98)/(2)

Further dividing 98 by 2:

y = 7 × 49

y = 343

Explanation

Efficiency is inversely proportional to the time taken to complete a task. Let the efficiency of B be 100. Since A is 40% more efficient than B, the efficiency of A is 140. The ratio of the efficiencies of A and B is 140 : 100, which simplifies to 7 : 5.

The total work is defined as the product of efficiency and time. Using the data for A: Total Work = Efficiency of A × Time taken by A = 7 × 31 = 217 units

When A and B work together, their combined efficiency is the sum of their individual efficiencies: Combined Efficiency = 7 + 5 = 12 units/day

The time taken by both to complete the job together is: Time = (Total Work)/(Combined Efficiency) = ²¹⁷⁄₁₂ days

Explanation

The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9. For the number 6234a6, the sum of the digits is 6 + 2 + 3 + 4 + a + 6 = 21 + a. To satisfy the divisibility condition, 21 + a must be a multiple of 9. The smallest multiple of 9 greater than 21 is 27. Setting 21 + a = 27 results in a = 6. Since 6 is a single-digit number, it correctly completes the number 623466 such that the sum of its digits is 27, which is perfectly divisible by 9.

Explanation

The algebraic identity for the difference of two cubes is expressed as a³ - b³ = (a - b)(a² + ab + b²). By substituting the given condition (a - b) = 1 into this formula, the expression becomes a³ - b³ = 1 · (a² + ab + b²). This simplifies directly to a² + ab + b², which represents the value of the difference of the cubes when the linear difference is one.

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual days taken by A and B. Total Work = LCM(8, 18) = 72 units.

The efficiency (work done per day) of each person is: Efficiency of A (E_A) = 72 / 8 = 9 units/day Efficiency of B (E_B) = 72 / 18 = 4 units/day

Since they work on alternate days starting with B, we calculate the work done in a 2-day cycle: Work in 1 cycle (Day 1: B, Day 2: A) = 4 + 9 = 13 units.

To find the number of full cycles: Number of cycles = 72 / 13 = 5 cycles (with a remainder) Work done in 5 cycles (10 days) = 13 × 5 = 65 units. Remaining work = 72 - 65 = 7 units.

On the 11th day, it is B's turn: Work done by B = 4 units. Remaining work after 11 days = 7 - 4 = 3 units.

On the 12th day, it is A's turn. A completes the remaining 3 units: Time taken by A = (Remaining Work)/(Efficiency of A) = ³⁄₉ = ⅓ day.

Total time taken = 11 + ⅓ = 11 ⅓ days.

Explanation

The given expression is (1 + sin t)/(4(1 - sin t)) - (1 - sin t)/(4(1 + sin t)). Factor out the common term ¼ to get ¼ [ (1 + sin t)/(1 - sin t) - (1 - sin t)/(1 + sin t) ].

To subtract the fractions, find a common denominator by multiplying the denominators: (1 - sin t)(1 + sin t) = 1 - sin² t = cos² t. The numerator becomes (1 + sin t)² - (1 - sin t)².

Using the algebraic identity (a + b)² - (a - b)² = 4ab, where a = 1 and b = sin t, the numerator simplifies to 4(1)(sin t) = 4 sin t.

Substitute these back into the expression: ¼ [ (4 sin t)/(cos² t) ] = (sin t)/(cos² t)

This can be rewritten using the definitions tan t = (sin t)/(cos t) and sec t = (1)/(cos t): (sin t)/(cos t) · (1)/(cos t) = tan t sec t

Explanation

The net profit percentage is determined by the formula for successive percentage changes: Net % = a + b + ᵃᵇ⁄₁₀₀, where a is the markup percentage and b is the discount percentage. Given a = 40 and b = -10, the calculation is 40 - 10 + (40 × (-10))/(100) = 30 - 4 = 26%.

Alternatively, using the base value method, let the cost price be 100. A 40% markup makes the marked price 140. A 10% discount is then applied to the marked price, which is 10% of 140 = 14. The final selling price is 140 - 14 = 126. The profit percentage is the difference between the selling price and the cost price relative to the cost price, which is 126 - 100 = 26%.

Explanation

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11. To determine the remainder of the number 9454351626 when divided by 11, we calculate these sums starting from the units place:

Sum of digits at odd positions: 6 + 6 + 5 + 4 + 4 = 25 Sum of digits at even positions: 2 + 1 + 3 + 5 + 9 = 20

The difference between the sum of digits at odd positions and the sum of digits at even positions is 25 - 20 = 5. This result indicates that 9454351626 is not divisible by 11 and leaves a remainder of 5. To make the number divisible by 11, we must add a value that increases the number to the next multiple of 11. This required value is calculated by subtracting the current remainder from the divisor: 11 - 5 = 6.

Explanation

The efficiency of a worker is inversely proportional to the time taken to complete a task. Let the efficiency of Sohan be E_S = 1 unit/day. Since Ajeet works 4 times as fast as Sohan, Ajeet's efficiency is E_A = 4 × 1 = 4 units/day.

The total work is the product of Sohan's efficiency and the time he takes to complete it: Total Work = E_S × Days taken by Sohan = 1 × 20 = 20 units

When Ajeet and Sohan work together, their combined efficiency is: E_A₊_S = E_A + E_S = 4 + 1 = 5 units/day

The time taken by both to finish the work together is: Time = (Total Work)/(E_A₊_S₎ = ²⁰⁄₅ = 4 days