SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if the number formed by its last three digits is divisible by 8. In the given 8-digit number 123456xy, the last three digits form the number 6xy. This can be expressed mathematically as 600 + 10x + y.

To check for divisibility, we divide 600 by 8 and find that 600 = 8 × 75, which leaves a remainder of 0. Since 600 is perfectly divisible by 8, the divisibility of the entire number depends solely on the 2-digit number formed by xy, which is 10x + y. Therefore, 10x + y must be a multiple of 8.

The digits x and y can range from 0 to 9. We need to identify all multiples of 8 from 0 to 99. These multiples are 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and 96. In terms of (x, y) pairs, these are written as (0,0), (0,8), (1,6), (2,4), (3,2), (4,0), (4,8), (5,6), (6,4), (7,2), (8,0), (8,8), and (9,6).

To find the total number of such pairs without listing them, we can use the formula for the number of terms in an arithmetic progression: n = (l - a)/(d) + 1. Here, the first multiple a = 0, the last multiple less than 100 is l = 96, and the common difference d = 8. Substituting these values gives n = (96 - 0)/(8) + 1 = 12 + 1 = 13. Thus, there are 13 possible pairs of (x, y).

Explanation

The area of a sector of a circle is determined by the formula A = (θ)/(360) π r², where θ is the central angle and r is the radius. Given the area A = 308 cm² and the central angle θ = 45°, we substitute these values along with π = ²²⁄₇ into the formula:

308 = ⁴⁵⁄₃₆₀ × ²²⁄₇ × r²

The fraction ⁴⁵⁄₃₆₀ simplifies to ⅛. Substituting this back into the equation:

308 = ⅛ × ²²⁄₇ × r² 308 = ¹¹⁄₂₈ × r²

To solve for r², rearrange the equation as follows:

r² = (308 × 28)/(11)

Dividing 308 by 11 gives 28. Therefore:

r² = 28 × 28 r = √(28²) r = 28

Explanation

When a quantity undergoes a successive increase and decrease of the same percentage x%, the net change is always a decrease given by the formula (x²)/(100)%. Here, the value of x is 20. Plugging this into the formula, the net percentage decrease is (20²)/(100) = ⁴⁰⁰⁄₁₀₀ = 4%. Alternatively, using the net percentage change formula a + b + ᵃᵇ⁄₁₀₀ where a = 20 and b = -20, the result is 20 - 20 + (20 × (-20))/(100) = -⁴⁰⁰⁄₁₀₀ = -4%. The negative sign represents a net decrease.

Explanation

The condition C₁C₂ = r₁ + r₂ states that the distance between the centers of two circles is equal to the sum of their radii. Geometrically, this occurs when two circles touch each other externally at exactly one point. In this configuration, there are two direct common tangents that lie on the outer sides of both circles and one transverse common tangent that passes through the point of contact where the circles meet. Summing these together, the total number of common tangents is 2 + 1 = 3.

Explanation

For a 6-digit number 674pq0 to be divisible by 3, the sum of its digits must be a multiple of 3. The sum is 6 + 7 + 4 + p + q + 0 = 17 + p + q. Substituting the values p = 5 and q = 2, the sum becomes 17 + 5 + 2 = 24. Since 24 is divisible by 3, the condition for divisibility by 3 is satisfied.

For a number to be divisible by 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11. For the number 674pq0, the sum of digits at odd positions is 6 + 4 + q = 10 + q, and the sum of digits at even positions is 7 + p + 0 = 7 + p. The difference is (10 + q) - (7 + p) = 3 + q - p. Substituting p = 5 and q = 2, the difference becomes 3 + 2 - 5 = 0. Since 0 is divisible by 11, the condition for divisibility by 11 is satisfied. Both p and q are non-zero integers as required.

Explanation

In a chase scenario where both individuals start running at the same time, the time t until the policeman catches the thief is constant for both. Using the relation Distance = Speed × Time, when time is constant, distance is directly proportional to speed (D ∝ S). Therefore, the ratio of the distances covered is equal to the ratio of their speeds.

Given the ratio of speeds of the thief to the policeman is 7:8, the ratio of the distances covered will also be 7:8. Let the distance covered by the thief be 7x and the distance covered by the policeman be 8x. The difference between these distances must account for the initial gap of 450 m between them.

8x - 7x = 450 m x = 450 m

The distance covered by the policeman is 8x: Distance = 8 × 450 = 3600 m

To convert the distance into kilometers, divide by 1000: 3600 / 1000 = 3.6 km

Explanation

The diameter d of the largest sphere that can be taken out of a cube is equal to the side length a of the cube. Given a = 5.6 cm, the radius r of the sphere is: r = ᵃ⁄₂ = (5.6)/(2) = 2.8 cm

The volume V of a sphere is calculated using the formula: V = ⁴⁄₃ π r³

Substituting the values with π ≈ ²²⁄₇: V = ⁴⁄₃ × ²²⁄₇ × (2.8)³ V = ⁴⁄₃ × ²²⁄₇ × 2.8 × 2.8 × 2.8 V = ⁴⁄₃ × 22 × 0.4 × 7.84 V = (88 × 3.136)/(3) V = (275.968)/(3) V ≈ 91.9893

Rounding to two decimal places, the volume is 91.98 cm³.

Explanation

The surface area of a sphere is calculated using the formula A = 4π r². Given that the curved surface area is 154 cm², we can find the radius r by substituting the known values: 4 × ²²⁄₇ × r² = 154 r² = (154 × 7)/(4 × 22) r² = (7 × 7)/(4) r = √(⁴⁹⁄₄) = 3.5 cm

The volume of a sphere is given by the formula V = ⁴⁄₃π r³. Substituting r = ⁷⁄₂ cm into the formula: V = ⁴⁄₃ × ²²⁄₇ × ⁷⁄₂ × ⁷⁄₂ × ⁷⁄₂ V = (11 × 49)/(3) V = ⁵³⁹⁄₃ ≈ 179.666... cm³

Rounding the result to one decimal place, the volume of the sphere is 179.7 cm³.

Explanation

Given the expression x + ¹⁄ₓ = -14, we use the algebraic identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4 to find the value of x - ¹⁄ₓ.

(x - ¹⁄ₓ)² = (-14)² - 4 = 196 - 4 = 192

Taking the square root on both sides: x - ¹⁄ₓ = ± √(192) = ± 8√3

The required value is x² - (1)/(x²⁾, which can be factored using the difference of squares identity: x² - (1)/(x²⁾ = (x + ¹⁄ₓ)(x - ¹⁄ₓ)

For x < -1 (specifically x = -7 - 4√3), the value of x - ¹⁄ₓ is negative: x - ¹⁄ₓ = -8√3

Substituting the known values into the factored expression: x² - (1)/(x²⁾ = (-14) × (-8√3) = 112√3

Explanation

To find the required ratio, first calculate the sum of donuts sold by stores P and R in the month of October. Based on the data provided in the table, Store P sold 125 units and Store R sold 960 units in October. Sum₍_P₊_R₎ = 125 + 960 = 1085

Next, calculate the sum of donuts sold by stores Q and R in the same month. Store Q sold 905 units and Store R sold 960 units in October. Sum₍_Q₊_R₎ = 905 + 960 = 1865

The ratio of the total number of donuts sold by stores P and R to the total number of donuts sold by stores Q and R is expressed as: Ratio = 1085 : 1865

To simplify this ratio, divide both terms by their common factor of 5: 1085 ÷ 5 = 217 1865 ÷ 5 = 373

Applying this simplification, the final ratio is 217 : 373.

Explanation

To find what percentage one number is of another, the formula used is Percentage = (Value / Total) × 100. In this calculation, the value is 0.5 and the total is 20. Substituting these into the formula gives (0.5 / 20) × 100. Simplifying the expression, 100 divided by 20 equals 5, and multiplying 0.5 by 5 results in 2.5. Therefore, 0.5 is 2.5% of 20.

Explanation

According to the Internal Angle Bisector Theorem, the angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. In triangle ABC, since AD is the internal bisector of ∠ A, the ratio of the sides is given by:

(AB)/(AC) = (BD)/(DC)

Substituting the given values into the formula:

(8.4)/(5.6) = (BD)/(2.8)

To find BD, we rearrange the equation:

BD = (8.4 × 2.8)/(5.6)

Since 5.6 is exactly twice 2.8, the expression simplifies to:

BD = (8.4)/(2) = 4.2 cm

The total length of the side BC is the sum of the segments BD and DC:

BC = BD + DC BC = 4.2 + 2.8 = 7 cm

Explanation

The total base cost for booking 2 rooms at Rs. 1,500 each is Rs. 3,000 (1,500 × 2). Since Rakesh booked 2 rooms, he is eligible for the 12% discount. The price after the first discount is 3,000 × (1 - 0.12) = 3,000 × 0.88 = Rs. 2,640. Additionally, because he paid using an SBI Silver Card, a 5% discount is applied to the already discounted amount. The final amount payable is 2,640 × (1 - 0.05) = 2,640 × 0.95 = Rs. 2,508.

Explanation

Let the present ages of Ram and Ramesh be 3x and 5x respectively. According to the problem, after 7 years, their ages will be 3x + 7 and 5x + 7. The ratio of these ages is given as 4 : 5.

The equation representing this relationship is: (3x + 7)/(5x + 7) = ⅘

Cross-multiplying the terms: 5(3x + 7) = 4(5x + 7) 15x + 35 = 20x + 28

Rearranging the terms to solve for x: 20x - 15x = 35 - 28 5x = 7 x = ⁷⁄₅

The present age of Ramesh is represented by 5x. Substituting the value of x: Ramesh age = 5 × ⁷⁄₅ = 7 years

Explanation

Given k⁴ + (1)/(k⁴⁾ = 47, we use the algebraic identity x² + (1)/(x²⁾ = (x + ¹⁄ₓ)² - 2 to reduce the powers. Adding 2 to both sides of the given equation results in k⁴ + (1)/(k⁴⁾ + 2 = 49, which can be written as (k² + (1)/(k²⁾)² = 7². This gives k² + (1)/(k²⁾ = 7.

Repeating the process, we add 2 to both sides again: k² + (1)/(k²⁾ + 2 = 9, which yields (k + ¹⁄ₖ)² = 3². Thus, k + ¹⁄ₖ = 3. To find the value of k³ + (1)/(k³⁾, we use the cubic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting k + ¹⁄ₖ = 3 into the formula k³ + (1)/(k³⁾ = (k + ¹⁄ₖ)³ - 3(k + ¹⁄ₖ), we calculate 3³ - 3(3) = 27 - 9 = 18.

Explanation

The relationship between Selling Price (SP), Cost Price (CP), and Profit/Loss percentage can be expressed using the formula SP = CP × (1 ± Percentage). To find the new selling price directly from the initial selling price, we use the ratio of the target percentage to the initial percentage:

New Selling Price = Initial SP × (100 + Gain%)/(100 - Loss%)

Substituting the given values into the formula:

New Selling Price = 4,00,000 × (100 + 5)/(100 - 20)

New Selling Price = 4,00,000 × ¹⁰⁵⁄₈₀

Dividing 4,00,000 by 80 gives 5,000.

New Selling Price = 5,000 × 105 = 5,25,000

Thus, to realize a profit of 5%, the plot must be sold for Rs. 5,25,000.

Explanation

The Highest Common Factor (HCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. A useful rule for finding the HCF is that if the larger number is a multiple of the smaller number, the smaller number itself is the HCF. To check this, we divide 1980 by 110: 1980 ÷ 110 = ¹⁹⁸⁰⁄₁₁₀ = ¹⁹⁸⁄₁₁ = 18 Since 1980 is exactly divisible by 110, it follows that 1980 = 110 × 18. Because 110 is a factor of 1980 and the largest possible factor of 110 is itself, the HCF of 110 and 1980 must be 110. Alternatively, using the prime factorization method: 110 = 2¹ × 5¹ × 11¹ 1980 = 2² × 3² × 5¹ × 11¹ The HCF is calculated by taking the product of the lowest powers of the common prime factors, which are 2, 5, and 11: HCF = 2¹ × 5¹ × 11¹ = 110

Explanation

The expression is in the form (a+b)³ + (a-b)³, which follows the algebraic identity: (a+b)³ + (a-b)³ = 2(a³ + 3ab²)

In the given expression (x + 2y)³ + (x - 2y)³, we can set a = x and b = 2y. Substituting these values into the identity: 2(x³ + 3(x)(2y)²) = 2(x³ + 3x(4y²)) = 2(x³ + 12xy²) = 2x³ + 24xy²

Alternatively, expanding both binomial cubes individually: (x + 2y)³ = x³ + 6x²y + 12xy² + 8y³ (x - 2y)³ = x³ - 6x²y + 12xy² - 8y³

Summing the two expanded forms: (x³ + x³) + (6x²y - 6x²y) + (12xy² + 12xy²) + (8y³ - 8y³) The terms 6x²y and 8y³ cancel out because they have opposite signs, resulting in: 2x³ + 24xy²

Explanation

To find the time period for simple interest, we first calculate the interest earned by subtracting the principal from the total amount. Given the principal P = 5250 and the amount A = 9870, the simple interest SI is calculated as: SI = A - P = 9870 - 5250 = 4620

The formula for simple interest is SI = (P × R × T)/(100). Rearranging this formula to solve for time T, we get: T = (SI × 100)/(P × R)

Substituting the known values SI = 4620, P = 5250, and R = 11: T = (4620 × 100)/(5250 × 11) T = ⁴⁶²⁰⁰⁰⁄₅₇₇₅₀ T = 8

By dividing the total interest by the interest earned per year, we find the duration to be 8 years.

Explanation

To find the required ratio, first calculate the total production of trucks by companies H and I. From the table, company H produces 400 trucks and company I produces 300 trucks. The combined production of trucks for H and I is 400 + 300 = 700.

Next, calculate the total production of bikes by companies J and K. The table shows that company J produces 350 bikes and company K produces 400 bikes. The combined production of bikes for J and K is 350 + 400 = 750.

The ratio of the production of trucks by H and I to the production of bikes by J and K is given by: Ratio = 700 : 750

Dividing both terms by their greatest common divisor, which is 50, simplifies the ratio: 700 ÷ 50 = 14 750 ÷ 50 = 15

This results in a simplified ratio of 14 : 15.

Explanation

By squaring the given equation (sin θ + cos θ)² = (√3-1{2√2})², we get 1 + sin 2θ = 4 - 2√3{8} = 2 - √3{4}, which leads to sin 2θ = -(2 + √3){4}. Assuming a common typographical error in the expression to find (intended as tan θ + cot θ rather than tan θ + cos θ), we calculate tan θ + cot θ = (1)/(sin θ cos θ) = (2)/(sin 2θ) = (-8)/(2+√(3)), which rationalizes to 8(√3-2).

Explanation

The trigonometric identity relating secant and cosecant functions of complementary angles is sec A = csc(90° - A). Given the equation sec 2θ = csc(θ - 36°), we can equate the arguments by expressing both sides in terms of the same function or by using the property that if sec A = csc B, then A + B = 90° for acute angles.

Setting the sum of the angles to 90°: 2θ + (θ - 36°) = 90° 3θ - 36° = 90° 3θ = 90° + 36° 3θ = 126° θ = (126°)/(3) = 42°

Since 2θ = 2(42°) = 84°, which is less than 90°, the condition that 2θ is an acute angle is satisfied.

Explanation

To determine the number of states that witnessed a continuous increase in rainfall for two consecutive periods (three immediate years), we identify instances where Rainfall_Yₑₐᵣ₁ < Rainfall_Yₑₐᵣ₂ < Rainfall_Yₑₐᵣ₃.

State T: Rainfall increased from 80 in 2015 to 85 in 2016, and then to 86 in 2017. It also increased from 80 in 2019 to 81 in 2020 and 82 in 2021. State A: Rainfall increased from 70 in 2018 to 74 in 2019, and then to 75 in 2020. State M: Rainfall increased from 96 in 2018 to 97 in 2019, and then to 98 in 2020. State K: Rainfall increased from 67 in 2019 to 68 in 2020, and then to 70 in 2021. State R: Rainfall increased from 59 in 2016 to 60 in 2017, and then to 61 in 2018.

Comparing the remaining states: State N: Only one increase occurred from 2019 to 2020 (74 to 75). State P: Increases were not continuous (60 to 67 and 64 to 65). State D: Increases were not continuous (70 to 71 and 69 to 75).

A total of 5 states (T, A, M, K, and R) witnessed a continuous increase in rainfall for at least one two-year immediate period.

Explanation

The expression is evaluated using trigonometric identities and the properties of complementary angles. The identity cos(90° - θ) = sin θ allows cos 55° to be rewritten as cos(90° - 35°) = sin 35°. Substituting this into the second term gives sin 35° · sin 35° = sin² 35°. By the Pythagorean identity sin² θ + cos² θ = 1, the sum of the first two terms becomes cos² 35° + sin² 35° = 1. The final part of the expression typically involves the ratio (tan 34°)/(cot 56°). Since cot θ = tan(90° - θ), we find cot 56° = tan(90° - 56°) = tan 34°. This makes the ratio (tan 34°)/(tan 34°) = 1. Combining these values, the result is 1 + 1 = 2.

Explanation

The total surface area of a cube is given by the formula 6a², where a represents the length of one side. Given the total surface area is 1536 m², we set up the equation 6a² = 1536. Dividing both sides by 6 yields a² = 256. Taking the square root of 256 gives the side length a = 16 m. The volume of a cube is calculated using the formula V = a³. Substituting the side length into this formula results in V = 16³ = 4096 m³.

Explanation

In simple interest, the interest earned is directly proportional to the time period when the principal and interest rate are constant. If a sum P becomes n times itself, the interest earned is SI = (n - 1)P. The relationship between time T and the multiple n is expressed as (T₁)/(n₁ - 1) = (T₂)/(n₂ - 1).

Given that the sum triples (n₁ = 3) in T₁ = 26 years, we substitute the values to find the time T₂ required for the sum to become five times (n₂ = 5):

(26)/(3 - 1) = (T₂)/(5 - 1)

²⁶⁄₂ = (T₂)/(4)

13 = (T₂)/(4)

T₂ = 13 × 4

T₂ = 52 years

Explanation

The problem follows the remainder theorem principle where if a number N gives a remainder R₁ when divided by D₁, and D₂ is a factor of D₁, then the remainder R₂ when N is divided by D₂ is equal to R₁ D₂. Here, the number N can be expressed as N = 363k + 17. Since 11 is a factor of 363 (363 = 11 × 33), the part 363k is perfectly divisible by 11. The remainder is determined solely by dividing the initial remainder 17 by 11. Performing the division 17 = (11 × 1) + 6 yields a remainder of 6.

Explanation

The relationship between the difference of cubes and the difference of the terms is given by the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b).

By substituting the given values a³ - b³ = 80 and a - b = 2 into this formula, we get: 80 = (2)³ + 3ab(2) 80 = 8 + 6ab

Rearranging the equation to solve for ab: 6ab = 80 - 8 6ab = 72 ab = 12

Explanation

To find the average of the high rates for Kolgate shares across the four stock exchanges, we identify the values listed under the High column for Kolgate in each region. According to the table, the high rates are 34 for Mumbai, 60 for Uttarakhand, 25 for Rajasthan, and 20 for Uttar Pradesh.

The formula for the arithmetic mean is:

Average = (Σ High Rates)/(Number of Exchanges)

Substituting the given values into the formula:

Sum of High Rates = 34 + 60 + 25 + 20 = 139

Number of Exchanges = 4

Average = ¹³⁹⁄₄ = 34.75

The resulting value represents the mean high trading price for Kolgate across the specified markets.

Explanation

Let u be the speed of the boat in still water and v be the speed of the river stream. The speed of the boat downstream is given by (u + v) and the speed upstream is (u - v).

Since the distance D covered in both cases is the same, and the time taken to row upstream is thrice the time taken to row downstream (tᵤₚ = 3 × t_dₒ_wₙ), we use the inverse relationship between speed and time: (u + v)/(u - v) = tᵤₚ{t_dₒ_wₙ} (u + v)/(u - v) = ³⁄₁

Applying cross-multiplication: u + v = 3(u - v) u + v = 3u - 3v 4v = 2u v = ᵘ⁄₂

Given the speed of the boat in still water u = 40 km/h, the speed of the river stream is: v = ⁴⁰⁄₂ = 20 km/h

Explanation

The required number must be of the form k × LCM(7, 9, 11) + 3, where k is an integer. Since 7, 9, and 11 are pairwise coprime, their LCM = 7 × 9 × 11 = 693. To find the largest five-digit number satisfying this condition, we first identify the largest five-digit multiple of 693. We divide the largest five-digit number, 99999, by the LCM:

99999 ÷ 693 = 144 with a remainder of 187

By subtracting the remainder from 99999, we find the largest five-digit multiple of 693: 99999 - 187 = 99792

To ensure the number leaves a remainder of 3 when divided by 7, 9, and 11, we add 3 to this multiple: 99792 + 3 = 99795

Explanation

According to the theorem of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If A₁ and A₂ are the areas of two similar triangles and s₁ and s₂ are their corresponding sides, the relationship is expressed as:

A₁{A₂} = (s₁{s₂})²

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

s₁{s₂} = √(A₁₎{A₂}

Given that the ratio of the areas is √3 : √2, we substitute these values into the formula:

s₁{s₂} = √({3){√2}} = √({3)}{√(2)}

Applying the property of radicals where √(x) = ⁴√x, we get:

s₁{s₂} = ⁴√3{⁴√2}

This simplifies to the ratio ⁴√3 : ⁴√2.

Explanation

The total marks obtained by student A across all subjects is 180 + 240 + 360 + 300 + 60 + 330 = 1470. The total marks obtained by student B is 90 + 420 + 390 + 210 + 270 + 120 = 1500. The difference between their total marks is 1500 - 1470 = 30.

Explanation

Since triangle ABC and triangle DEF are congruent (△ ABC ≅ △ DEF), their corresponding angles are equal by the property of Corresponding Parts of Congruent Triangles (CPCT). This implies that m∠ A = m∠ D, m∠ B = m∠ E, and m∠ C = m∠ F.

In △ ABC, the sum of the interior angles is always 180°. Therefore: m∠ A + m∠ B + m∠ C = 180°

Given that m∠ B = 30°, we substitute this value into the equation: m∠ A + 30° + m∠ C = 180° m∠ A + m∠ C = 180° - 30° m∠ A + m∠ C = 150°

Because m∠ A = m∠ D from the congruence property, we can substitute m∠ D for m∠ A in the equation: m∠ D + m∠ C = 150°

Explanation

To find the number of students who obtained 60% marks and above in all subjects, each student's performance across all six subjects (English, Science, Mathematics, Social Science, Hindi, and Computer Science) must be checked against the threshold of 60. Since the maximum marks for each subject are 100, a percentage of 60% corresponds to 60 marks.

Student A scored 58 in Science, which is below 60. Student B scored 36 in Social Science and 52 in Hindi, both of which are below 60. Student C scored 58 in Science, which is below 60. Student E scored 59 in Mathematics, which is below 60. These four students do not meet the criteria.

Student D scored 84, 78, 72, 73, 65, and 68 across the six subjects. Every score is greater than or equal to 60. Student F scored 74, 68, 75, 82, 88, and 86 across the six subjects. Every score is greater than or equal to 60. Therefore, only Student D and Student F satisfy the condition of obtaining 60% or more in all subjects. The total number of such students is 2.

Explanation

By definition, a right-angled triangle contains one angle of 90°. According to the angle sum property of a triangle, the sum of all three interior angles is always 180°. If one angle is 35° and another is 90°, the third angle x can be found using the formula x + 90° + 35° = 180°. Solving for x gives x = 180° - 125° = 55°. Therefore, the two angles other than the given 35° angle are 90° and 55°.

Explanation

To calculate the overall profit or loss percentage, we use the effective ratio method. Let the cost price (CP) of 100 units of weight be 100. Since the dealer sells at an 11% loss, the selling price (SP) for the nominal weight is 100 - 11 = 89.

However, the dealer uses 22% less weight, meaning he only gives 78 units of weight while charging for 100. The cost price of the goods actually delivered is therefore 78.

The overall profit or loss is determined by the ratio of the total selling price to the actual cost price: Overall Ratio = (SP)/(CPₐ_cₜᵤₐₗ₎ = ⁸⁹⁄₇₈

Profit Percentage = SP - CPₐ_cₜᵤₐₗ{CPₐ_cₜᵤₐₗ} × 100 Profit Percentage = (89 - 78)/(78) × 100 = ¹¹⁄₇₈ × 100 Profit Percentage = ¹¹⁰⁰⁄₇₈ ≈ 14.102%

Since the value is positive, the dealer makes a gain of approximately 14.1%.

Explanation

Statement I can be simplified using the identity a² - b² = (a - b)(a + b). Each pair of consecutive terms results in: (100² - 99²) = (100 - 99)(100 + 99) = 199 (98² - 97²) = (98 - 97)(98 + 97) = 195 Continuing this pattern down to (22² - 21²) = 43, the expression becomes the sum of an arithmetic progression: 199 + 195 + 191 + … + 43 The number of terms n is (100 - 22)/(2) + 1 = 40. Using the sum formula S = ⁿ⁄₂(first term + last term): S = ⁴⁰⁄₂(199 + 43) = 20 × 242 = 4840 Statement I is correct.

Statement II involves the product (k² + (1)/(k²⁾)(k - ¹⁄ₖ)(k⁴ + (1)/(k⁴⁾)(k + ¹⁄ₖ)(k⁴ - (1)/(k⁴⁾). Rearranging and applying (a - b)(a + b) = a² - b²: [(k - ¹⁄ₖ)(k + ¹⁄ₖ)] = k² - (1)/(k²⁾ (k² - (1)/(k²⁾)(k² + (1)/(k²⁾) = k⁴ - (1)/(k⁴⁾ The expression now is (k⁴ - (1)/(k⁴⁾)(k⁴ + (1)/(k⁴⁾)(k⁴ - (1)/(k⁴⁾). Simplifying the first two terms gives (k⁸ - (1)/(k⁸⁾). The final product is (k⁸ - (1)/(k⁸⁾)(k⁴ - (1)/(k⁴⁾), which results in k¹² - k⁴ - (1)/(k⁴⁾ + (1)/(k¹²⁾. This does not equal k¹⁶ - (1)/(k¹⁶⁾, making Statement II incorrect.

Explanation

The total work required to complete the project can be calculated using the concept of man-days. In the first 20 days, 60 men were employed, totaling 60 × 20 = 1200 man-days. For the remaining 60 days of the 80-day schedule, the workforce was increased to 80 men (60 + 20), totaling 80 × 60 = 4800 man-days. The total work performed to finish the project on time is 1200 + 4800 = 6000 man-days.

If the additional men had not been hired, the original 60 men would have been responsible for the entire 6000 man-days of work. The total time required for 60 men to complete this work is T = ⁶⁰⁰⁰⁄₆₀ = 100 days. Given that the original schedule was 80 days, the number of extra days needed would be 100 - 80 = 20 days.

Another way to calculate this is by focusing on the work remaining after the first 20 days. The work that 80 men finished in 60 days would take 60 men a longer period. Using the inverse proportion relationship M₁ D₁ = M₂ D₂, we have 80 × 60 = 60 × D₂, resulting in D₂ = 80 days. Since there were only 60 days remaining in the schedule, the additional time required is 80 - 60 = 20 days.

Explanation

For the largest sphere to be carved out of a cube, the diameter of the sphere must be equal to the side length of the cube. Given the side of the cube is 4.2 cm, the diameter of the sphere is 4.2 cm and its radius r is 2.1 cm.

The volume V of a sphere is calculated using the formula V = ⁴⁄₃ π r³. Substituting the values:

V = ⁴⁄₃ × ²²⁄₇ × (2.1)³

V = ⁴⁄₃ × ²²⁄₇ × 2.1 × 2.1 × 2.1

By simplifying the terms, V = 4 × 22 × 0.1 × 2.1 × 2.1.

V = 8.8 × 4.41 = 38.808 cm³

Rounding to two decimal places, the volume is 38.80 cm³.

Explanation

The net percentage change resulting from a markup followed by a discount is given by the successive percentage change formula: Net % Change = x + y + ˣʸ⁄₁₀₀. In this scenario, the markup x is +20 and the discount y is -20. Substituting these values into the formula:

Net % Change = 20 - 20 + (20 × (-20))/(100)

Net % Change = 0 - ⁴⁰⁰⁄₁₀₀ = -4%

A negative result indicates a loss, specifically a 4% loss. Alternatively, if the cost price is 100, a 20% markup results in a marked price of 120. A 20% discount on 120 equals 24 (0.20 × 120 = 24), making the selling price 120 - 24 = 96. Comparing the selling price of 96 to the original cost price of 100 yields a 4% loss.

Explanation

The expression is cos x cosec x - sin x sec x. Using the reciprocal identities cosec x = (1)/(sin x) and sec x = (1)/(cos x), the expression becomes:

(cos x)/(sin x) - (sin x)/(cos x)

Taking the common denominator sin x cos x, we get:

(cos² x - sin² x)/(sin x cos x)

Applying the double angle formula for cosine, cos 2x = cos² x - sin² x, and the double angle formula for sine, sin 2x = 2 sin x cos x (which implies sin x cos x = (sin 2x)/(2)), the expression simplifies to:

(cos 2x)/(sin 2x){2} = 2 (cos 2x)/(sin 2x) = 2 cot 2x

Explanation

The relationship between the variables is given by the equation A = k · √B{C³}, where k is the constant of proportionality.

Substituting the initial values A = 15, B = 27, and C = 2 into the equation allows us to solve for k: 15 = k · √(27){2³} 15 = k · 3√3{8} k = (15 · 8)/(3√(3)) = (40)/(√(3))

Now, use the value of k and the second set of conditions where A = 9 and C = 2 to find B: 9 = (40)/(√(3)) · √B{2³} 9 = (40)/(√(3)) · √B{8} 9 = 5√B{√3} √B = 9√3{5}

Squaring both sides of the equation gives: B = (81 · 3)/(25) = ²⁴³⁄₂₅

Explanation

The weight of the new person can be calculated using the direct relationship between the change in average and the total weight change. Since the average weight of 5 men decreases by 3 kg, the total weight of the group must decrease by the product of the number of people and the change in the average.

Total weight decrease = Number of persons × Decrease in average weight Total weight decrease = 5 × 3 = 15 kg

The new person is replacing a man weighing 150 kg. Because the total weight of the group has dropped by 15 kg, the new person must weigh 15 kg less than the person who was replaced.

Weight of the new person = Weight of the replaced person - Total weight decrease Weight of the new person = 150 - 15 = 135 kg

Explanation

The expression is of the form a³ + b³ + c³ where a = 25, b = -75, and c = 50. According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Checking the condition for the given values: a + b + c = 25 + (-75) + 50 = 0

Since the sum of the terms is zero, the value of the expression is: 3 × 25 × (-75) × 50 75 × (-75) × 50 -5625 × 50 = -281250

Explanation

To find the greatest 5-digit number divisible by 11, 33, 99, and 121, we first determine the Least Common Multiple (LCM) of these numbers. The prime factorizations are 11 = 11¹, 33 = 3 × 11, 99 = 3² × 11, and 121 = 11². The LCM is calculated by taking the highest power of each prime factor present in the numbers: LCM = 3² × 11² = 9 × 121 = 1089.

The greatest 5-digit number is 99999. To find the largest multiple of 1089 that does not exceed 99999, we divide 99999 by 1089. Performing the division 99999 ÷ 1089 gives a quotient of 91 and a remainder of 900. By subtracting this remainder from the largest 5-digit number, we get 99999 - 900 = 99099. This is the largest 5-digit number that is exactly divisible by 1089, and consequently, it is divisible by 11, 33, 99, and 121.

Explanation

Given tan A = ⅔, we evaluate the expression by treating it as a ratio 5 sin² A - 2 cos² A{15 sin² A + 3 cos² A}. To simplify the expression using the given trigonometric identity, we divide both the numerator and the denominator by cos² A:

5 {sin² A{cos² A} - 2 cos² A{cos² A}}{15 sin² A{cos² A} + 3 cos² A{cos² A}} = 5 tan² A - 2{15 tan² A + 3}

Substituting the value tan A = ⅔, we have tan² A = ⁴⁄₉. Plugging this into the simplified expression:

(5(4)/(9)) - 2{15(⁴⁄₉) + 3} = (20)/(9) - 2{⁶⁰⁄₉ + 3}

To solve the numerator and denominator, find a common denominator of 9:

Numerator: (20 - 18)/(9) = ²⁄₉

Denominator: (60 + 27)/(9) = ⁸⁷⁄₉

The final value is obtained by dividing the numerator by the denominator:

²⁄₉ ÷ ⁸⁷⁄₉ = ²⁄₈₇

Explanation

The fourth proportion to three numbers a, b, and c is defined as x such that a : b = c : x, which simplifies to x = ᵇᶜ⁄ₐ. Substituting the given values 12, 24, and 27: x = (24 × 27)/(12) = 2 × 27 = 54

The third proportion to two numbers A and B is defined as y such that A : B = B : y, which simplifies to y = B²{A}. We are given that this third proportion is equal to the fourth proportion calculated above (y = 54) for the numbers A and 36: 54 = 36²{A} A = (36 × 36)/(54)

Dividing both 36 and 54 by their greatest common divisor 18: A = (2 × 36)/(3) A = 2 × 12 = 24

Explanation

Given the equation tan θ + cot θ = 2, we can substitute cot θ = (1)/(tan θ) to get tan θ + (1)/(tan θ) = 2. For any variable x, the relation x + ¹⁄ₓ = 2 implies x² - 2x + 1 = 0, which simplifies to (x - 1)² = 0. Thus, x = 1. In this context, tan θ = 1. Since θ is an acute angle, θ = 45°, and consequently cot θ = 1.

Substituting these values into the given expression: tan² θ + cot² θ + 2 tan⁵ θ cot⁴ θ = (1)² + (1)² + 2(1)⁵(1)⁴ = 1 + 1 + 2(1)(1) = 2 + 2 = 4

Explanation

The net discount for two successive discount rates a% and b% is given by the formula a + b - ᵃᵇ⁄₁₀₀. For scheme I, the net discount is 10 + 10 - (10 × 10)/(100) = 20 - 1 = 19%. For scheme II, the net discount is 15 + 5 - (15 × 5)/(100) = 20 - 0.75 = 19.25%. Scheme III provides a single flat discount of 20%. Since 20% is greater than both 19.25% and 19%, scheme III offers the maximum discount percentage.