SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The total capital Ramesh possesses is ₹18,000. The capital is divided into three parts. The first part is ₹7,000, and the second part is ₹6,000. Therefore, the remaining capital is calculated as 18,000 - (7,000 + 6,000) = 5,000.

The simple interest for one year is calculated using the formula SI = (P · R · T)/(100). Since the time period is one year (T = 1), the interest from each part is: Interest from the first part: (7,000 · 5 · 1)/(100) = 350 Interest from the second part: (6,000 · 6 · 1)/(100) = 360

The sum of interest from these two parts is 350 + 360 = 710. Given that the total interest received at the end of the year is ₹1,160, the interest earned from the remaining capital of ₹5,000 must be 1,160 - 710 = 450.

To find the rate of interest (R) on the remaining capital, substitute the values into the simple interest formula: 450 = (5,000 · R · 1)/(100) 450 = 50 · R R = ⁴⁵⁰⁄₅₀ = 9%

Explanation

The volume V of a cuboid is determined by the formula V = l × b × h, where l represents the length, b represents the breadth, and h represents the height. Given the dimensions l = 8 cm, b = 4 cm, and h = 6 cm, the volume is calculated as:

V = 8 × 4 × 6 = 192 cm³

Explanation

The length of the longest diagonal of a cuboid with dimensions l, b, and h is given by the space diagonal formula d = √(l² + b² + h²).

Given the dimensions l = 1.2 cm, b = 1.3 cm, and h = 1.5 cm, we substitute these into the formula:

d = √(1.2² + 1.3² + 1.5²) d = √(1.44 + 1.69 + 2.25) d = √(5.38)

To determine the square root of 5.38, observe that 2.3² = 5.29 and 2.4² = 5.76. Since 5.38 is very close to 5.29, we check 2.32²: 2.32 × 2.32 = 5.3824

The value √(5.38) is approximately 2.31948, which when rounded to two decimal places is 2.32 cm.

Explanation

The given equation is x + 1/x = 65/8. By expressing the fraction 65/8 as a mixed fraction, we get 65/8 = 8 + 1/8. Comparing this with the structure of the equation x + 1/x, and knowing that x must be an integer, we can directly identify that x = 8.

To find the value of x - 1/x, we substitute x = 8 into the expression: x - 1/x = 8 - 1/8 x - 1/x = (8 × 8 - 1) / 8 x - 1/x = (64 - 1) / 8 = 63/8

Alternatively, using the algebraic identity (x - 1/x)² = (x + 1/x)² - 4, we substitute the given value: (x - 1/x)² = (65/8)² - 4 (x - 1/x)² = 4225/64 - 4 (x - 1/x)² = (4225 - 256) / 64 (x - 1/x)² = 3969/64 Taking the square root of both sides, we find x - 1/x = ± √(3969/64) = ± 63/8. Since x is the integer 8, the positive result 63/8 is obtained.

Explanation

The Triangle Inequality Theorem states that for any triangle with sides a, b, and c, the length of any side must be greater than the absolute difference of the other two sides and less than their sum. This is expressed as |a - b| < c < a + b.

Given the sides are 6 and 12, let the third side be x. Substituting the known values into the inequality gives |12 - 6| < x < 12 + 6, which simplifies to 6 < x < 18.

Since x must be an integer, the possible values are the integers strictly between 6 and 18, which are {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. To find the count of integers in an open interval (m, n), we use the formula (n - m) - 1. Applying this, we get (18 - 6) - 1 = 11.

Explanation

To find the time when two runners meet for the first time at the starting point, we calculate the Least Common Multiple (LCM) of the time each individual takes to complete one full round. Arvind takes 180 seconds and Chetan takes 420 seconds to complete one round. The LCM of 180 and 420 is determined using their prime factorizations: 180 = 2² × 3² × 5 420 = 2² × 3 × 5 × 7 LCM(180, 420) = 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1260 seconds To convert the time from seconds to minutes, we divide by 60: (1260 seconds)/(60 seconds/minute) = 21 minutes

Explanation

To find the average of the low rates for Company C across all stock exchanges, we sum the low rates from Hyderabad, Mumbai, and Bengaluru and divide by three. The low rates for Company C are 175 in Hyderabad, 180 in Mumbai, and 185 in Bengaluru.

Average Low Rate = (175 + 180 + 185)/(3) = ⁵⁴⁰⁄₃ = 180

The question asks to identify the stock exchange(s) where the rates of shares for Company C are less than this average of 180. Comparing the rates of Company C at each exchange to 180: at the Hyderabad exchange, the rates are 180 (High) and 175 (Low). Since the low rate of 175 is strictly less than the average of 180, the condition is met at the Hyderabad stock exchange. At Mumbai, the rates are 195 and 180, neither of which is less than 180. At Bengaluru, the rates are 190 and 185, both of which are greater than 180.

Explanation

According to the divisibility rule for 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be either 0 or a multiple of 11. For the number 6954P, the sum of digits at odd positions is 6 + 5 + P = 11 + P. The sum of digits at even positions is 9 + 4 = 13.

To find the value of P, we calculate the difference between these sums: (11 + P) - 13 = P - 2. For the number to be divisible by 11, this difference must satisfy P - 2 = 0 or be a multiple of 11. Solving P - 2 = 0 yields P = 2. Since P must be a single digit, 2 is the correct value that satisfies the divisibility criteria.

Explanation

The bank's final earning is the difference between the income generated from stock market investments and the interest paid to its depositors.

First, calculate the rate of return the bank earns from its investment. The bank earns ₹30,000 for every ₹3,00,000 invested. Investment Earning Rate = (30,000)/(3,00,000) × 100 = 10%

The bank pays 6% interest to the depositors. The net profit rate for the bank is the difference between the earning rate and the interest paid: Net Profit Rate = 10% - 6% = 4%

Given the total deposited amount is ₹60,00,000, the final earning is calculated by applying the net profit rate to the total deposit: Final Earning = 4% of 60,00,000 = ⁴⁄₁₀₀ × 60,00,000 = 2,40,000

Alternatively, total income from the investment is 10% of 60,00,000 = 6,00,000, and the total interest paid out is 6% of 60,00,000 = 3,60,000. Subtracting the interest paid from the investment income yields 6,00,000 - 3,60,000 = 2,40,000.

Explanation

The average expenditure on transport per year is calculated by finding the sum of the expenditures in that category across all specified years and dividing by the total number of years. According to the table, the transport expenditures for the years 1998, 1999, 2000, and 2001 are 38, 45, 57, and 63 lakh respectively.

The sum of these expenditures is calculated as: Sum = 38 + 45 + 57 + 63 = 203

The period covers 4 years. The average expenditure is determined using the formula: Average = (Total Sum)/(Number of Years)

Substituting the values: Average = ²⁰³⁄₄ = 50.75

Explanation

Given csc x = 5/4, we know that sin x = 1/csc x = 4/5. Using the Pythagorean identity sin² x + cos² x = 1, or recognizing the (3, 4, 5) Pythagorean triplet where the opposite side is 4 and the hypotenuse is 5, the adjacent side is √(5² - 4²) = 3.

From this, we derive the necessary trigonometric values: sin² x = (4/5)² = 16/25 cos² x = (3/5)² = 9/25 tan² x = (4/3)² = 16/9 cot² x = (3/4)² = 9/16

Substitute these values into the given expression (5sin² x + 8cot² x)/(27tan² x - 125cos² x):

Numerator calculation: 5(16/25) + 8(9/16) = 16/5 + 9/2 = (32 + 45) / 10 = 77/10

Denominator calculation: 27(16/9) - 125(9/25) = 3(16) - 5(9) = 48 - 45 = 3

Final value: (77/10)/(3) = ⁷⁷⁄₃₀

Explanation

The average price of a mixture is calculated using the weighted average formula Pₐᵥ_g = (Q₁ P₁ + Q₂ P₂)/(Q₁ + Q₂), where Q represents the quantity and P represents the price per unit.

Given: Q₁ = 42 litres, P₁ = 25 per litre Q₂ = 28 litres, P₂ = 40 per litre

To simplify the calculation, find the ratio of the quantities: Q₁ : Q₂ = 42 : 28 = 3 : 2

Using the simplified ratio in the weighted average formula: Average Price = ((3 × 25) + (2 = × 40))/(3 + 2) Average Price = (75 + 80)/(5) Average Price = ¹⁵⁵⁄₅ = 31

Alternatively, using the total values: Total Cost = (42 × 25) + (28 × 40) = 1050 + 1120 = 2170 Total Quantity = 42 + 28 = 70 litres Average Price = ²¹⁷⁰⁄₇₀ = 31

Explanation

In triangle ABC, let D be the midpoint of BC, which implies BD = DC. We are given that DL ⊥ AB and DM ⊥ AC such that DL = DM. Consider the right-angled triangles △ DLB and △ DMC. In these triangles, the hypotenuses BD and DC are equal, and the sides DL and DM are equal. According to the RHS (Right angle-Hypotenuse-Side) congruence criterion, △ DLB ≅ △ DMC. By the property of Corresponding Parts of Congruent Triangles (CPCT), the angles opposite the equal sides must be equal, so ∠ B = ∠ C. In triangle ABC, since the angles opposite to sides AC and AB are equal, the sides AB and AC must also be equal. A triangle with at least two equal sides is classified as an isosceles triangle.

Explanation

The selling price of an item after successive discounts can be calculated using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)), where MP is the marked price and d₁, d₂ are the percentage discounts. Given the marked price of the TV set as ₹18,500 and the successive discounts as 20% and 5%, the calculation is as follows:

SP = 18,500 × (1 - ²⁰⁄₁₀₀) × (1 - ⁵⁄₁₀₀) SP = 18,500 × ⁸⁰⁄₁₀₀ × ⁹⁵⁄₁₀₀ SP = 18,500 × 0.8 × 0.95

First, applying the 20% discount: 18,500 × 0.8 = 14,800

Next, applying the 5% discount to the remaining amount: 14,800 × 0.95 = 14,800 × (1 - 0.05) = 14,800 - 740 = 14,060

Alternatively, using the effective discount formula D = d₁ + d₂ - (d₁ × d₂)/(100), the total discount is 20 + 5 - (20 × 5)/(100) = 24%. The selling price is then 18,500 × (100% - 24%) = 18,500 × 0.76 = 14,060.

Explanation

Let the initial income be 300 units. Since 66⅔% is equivalent to the fraction ⅔, the initial expenditure is ⅔ × 300 = 200 units. This results in initial savings of 300 - 200 = 100 units.

When the income increases by 25%, the new income becomes 300 × 1.25 = 375 units. When the expenditure increases by 20%, the new expenditure becomes 200 × 1.20 = 240 units. The new savings are calculated as the difference between the new income and new expenditure: 375 - 240 = 135 units.

The percentage effect on savings is calculated using the formula: Percentage change = (New Savings - Initial Savings)/(Initial Savings) × 100 Substituting the values: (135 - 100)/(100) × 100 = 35%

Explanation

The description of the function states it varies directly with x and inversely with y. This implies a linear combination of these two variations, expressed by the general form f(x, y) = mx + (n)/(y), where m and n are constants.

To find the constants, substitute the provided values. Using the first condition where x = 0, y = 1, and f = 15: 15 = m(0) + ⁿ⁄₁ n = 15

Next, use the second condition where x = 1, y = 15, and f = 2: 2 = m(1) + ¹⁵⁄₁₅ 2 = m + 1 m = 1

The specific function is defined as f(x, y) = x + (15)/(y). To determine the value of the function when x = 5 and y = 3, substitute these coordinates into the formula: f(5, 3) = 5 + ¹⁵⁄₃ f(5, 3) = 5 + 5 = 10

Explanation

The Simple Interest (SI) is calculated as the difference between the total Amount (A) and the Principal (P). In this case, SI = 8,928 - 7,200 = 1,728. Using the standard Simple Interest formula SI = (P × R × T)/(100), where R is the rate of interest and T is the time in years, we substitute the given values: 1,728 = (7,200 × 8 × T)/(100). This simplifies to 1,728 = 72 × 8 × T, or 1,728 = 576 × T. Dividing 1,728 by 576 yields T = 3 years.

Explanation

To find the ratio of the total sales of manufacturer Q to manufacturer S, calculate the sum of sales for each manufacturer over the entire period provided in the table from 2016 to 2020.

Total sales for manufacturer Q = 280 + 290 + 310 + 310 + 330 = 1520 (in thousands) Total sales for manufacturer S = 400 + 420 + 430 + 440 + 450 = 2140 (in thousands)

The required ratio is 1520 : 2140. To simplify this ratio, divide both terms by their greatest common divisor, which is 20:

1520 ÷ 20 = 76 2140 ÷ 20 = 107

The simplified ratio is 76 : 107.

Explanation

Given the trigonometric equation sinθ + cosθ = √3cosθ, we rearrange the terms to isolate sinθ on one side: sinθ = √3cosθ - cosθ sinθ = (√3 - 1)cosθ

To find the value of cotθ, which is defined as (cosθ)/(sinθ), we divide both sides by (√3 - 1)sinθ: cotθ = (cosθ)/(sinθ) = (1)/(√(3) - 1)

To rationalize the denominator, multiply the numerator and denominator by the conjugate expression (√3 + 1): cotθ = 1(√3 + 1){(√3 - 1)(√3 + 1)}

Applying the algebraic identity (a - b)(a + b) = a² - b² to the denominator: cotθ = √3 + 1{(√3)² - (1)²} cotθ = √3 + 1{3 - 1} cotθ = √3 + 1{2}

Explanation

The expression -7/6 + 4/9 ÷ 5/3 × 1/16 is simplified using the BODMAS rule, which prioritizes Division and Multiplication (from left to right) before Addition.

First, perform the division operation: 4/9 ÷ 5/3 = 4/9 × 3/5 = (4 × 3) / (9 × 5) = 12/45 = 4/15

Next, perform the multiplication operation using the result from the previous step: 4/15 × 1/16 = 4 / (15 × 16) = 1 / (15 × 4) = 1/60

Now, substitute this value back into the original expression to perform the addition: -7/6 + 1/60

To add these fractions, find the least common multiple of the denominators 6 and 60, which is 60. Convert -7/6 into an equivalent fraction with a denominator of 60: -7/6 = (-7 × 10) / (6 × 10) = -70/60

Combine the fractions: -70/60 + 1/60 = (-70 + 1) / 60 = -69/60

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3: -69 ÷ 3 = -23 60 ÷ 3 = 20

The simplified value is -23/20.

Explanation

The perimeter of a sector of a circle is calculated by adding the length of the circular arc to the two radii that form the sector. The formula for the arc length l is (θ)/(360°) × 2π r, where θ is the central angle and r is the radius.

Substituting the given values θ = 15°, r = 14 cm, and π = 3.14: l = ¹⁵⁄₃₆₀ × 2 × 3.14 × 14 l = ¹⁄₂₄ × 87.92 l ≈ 3.6633 cm

The perimeter P of the sector is given by P = l + 2r. P = 3.6633 + 2(14) P = 3.6633 + 28 = 31.6633 cm

Rounding to two decimal places, the perimeter is 31.66 cm.

Explanation

The perimeter of a sector of a circle is calculated by adding the lengths of the two radii to the length of the arc. The formula for the perimeter P is P = 2r + l, where l is the arc length. The arc length l is determined by the formula l = (θ)/(360) × 2π r.

Given the radius r = 10 cm, the central angle θ = 30°, and π = 3.14, the calculation is as follows:

l = ³⁰⁄₃₆₀ × 2 × 3.14 × 10 l = ¹⁄₁₂ × 62.8 l ≈ 5.233 cm

Substituting the arc length back into the perimeter formula:

P = 2(10) + 5.233 P = 20 + 5.233 P = 25.233 cm

Rounding to one decimal place, the perimeter of the sector is 25.2 cm.

Explanation

According to the Remainder Theorem, when a polynomial f(m) is divided by a linear divisor of the form (m - a), the remainder is equal to f(a). In this case, the divisor is (m - 2), so we substitute a = 2 into the polynomial f(m) = m⁵ + 5m⁴ - 3m + 7.

Substituting m = 2: f(2) = (2)⁵ + 5(2)⁴ - 3(2) + 7

Calculating each term: 2⁵ = 32 5(2⁴) = 5(16) = 80 3(2) = 6

Now, combine the values: f(2) = 32 + 80 - 6 + 7 f(2) = 112 - 6 + 7 f(2) = 106 + 7 f(2) = 113

The remainder obtained after dividing the polynomial by (m - 2) is 113.

Explanation

From the problem statement, we can derive two linear equations. The first condition states that 18% of the yellow marbles added to 24% of the blue marbles equals 30: 0.18y + 0.24b = 30 Multiplying the entire equation by 100 gives 18y + 24b = 3000. Dividing by 6 simplifies this to: 3y + 4b = 500

The second condition states that three times the yellow marbles exceeds two times the blue marbles by 200: 3y - 2b = 200

To eliminate y and solve for b, subtract the second equation from the simplified first equation: (3y + 4b) - (3y - 2b) = 500 - 200 6b = 300 b = 50

Now, substitute b = 50 into the equation 3y - 2b = 200 to find y: 3y - 2(50) = 200 3y - 100 = 200 3y = 300 y = 100

The ratio of y² to b² is: y² : b² = 100² : 50² Using the property (a² / b²) = (a / b)²: (100 / 50)² = 2² = 4 Thus, the ratio is 4 : 1.

Explanation

The Triangle Inequality Theorem states that for any triangle with sides a, b, and c, the sum of the lengths of any two sides must be strictly greater than the length of the third side, which is expressed as a + b > c. Given that the perimeter of the triangle is a + b + c = 52, we can rewrite the sum of two sides as a + b = 52 - c. Substituting this value into the inequality yields 52 - c > c. By adding c to both sides, we get 52 > 2c, and dividing by 2 results in c < 26. Since this logic applies to all three sides of the triangle, every side must have a length less than half of the perimeter.

Explanation

The net rate of filling the tank is the sum of the rates of the individual pipes, with the emptying pipe having a negative rate. Let the total capacity of the tank be the Least Common Multiple (LCM) of the given times 12, 18, and 8 hours.

LCM(12, 18, 8) = 72 units.

Efficiency of the first filling pipe = 72 / 12 = 6 units/hour. Efficiency of the second filling pipe = 72 / 18 = 4 units/hour. Efficiency of the third emptying pipe = 72 / 8 = 9 units/hour.

When all three pipes are opened simultaneously, the net efficiency is: Net Efficiency = 6 + 4 - 9 = 1 unit/hour.

The time taken to fill the tank is given by the formula: Time = Total Capacity / Net Efficiency Time = 72 / 1 = 72 hours.

Explanation

According to the divisibility rule for 11, 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. For the given number 7845K854, the digits at the odd positions (starting from the left) are 7, 4, K, and 5. Their sum is S₁ = 7 + 4 + K + 5 = 16 + K. The digits at the even positions are 8, 5, 8, and 4. Their sum is S₂ = 8 + 5 + 8 + 4 = 25. The difference between these sums is D = S₂ - S₁ = 25 - (16 + K) = 9 - K. For the number to be divisible by 11, this difference must be equal to 0 or a multiple of 11. Since K is a single-digit integer, the only value that results in a difference of 0 is K = 9.

Explanation

To simplify the expression (sin A)/(1 + cos A), we multiply both the numerator and the denominator by the conjugate of the denominator, which is 1 - cos A:

(sin A)/(1 + cos A) · (1 - cos A)/(1 - cos A) = (sin A(1 - cos A))/(1 - cos² A)

Using the Pythagorean identity sin² A + cos² A = 1, we know that 1 - cos² A = sin² A. Substituting this into the denominator yields:

(sin A(1 - cos A))/(sin² A)

Dividing both the numerator and the denominator by sin A results in:

(1 - cos A)/(sin A)

Alternatively, using half-angle identities where sin A = 2 sin(A/2) cos(A/2) and 1 + cos A = 2 cos²(A/2), the expression simplifies as follows:

(2 sin(A/2) cos(A/2))/(2 cos²(A/2)) = (sin(A/2))/(cos(A/2)) = tan(A/2)

Since (1 - cos A)/(sin A) = (2 sin²(A/2))/(2 sin(A/2) cos(A/2)) = tan(A/2), the two expressions are identical.

Explanation

In any triangle, the point where the internal angle bisectors meet is the incenter. The angle formed at the incenter by two vertex angles is given by the formula ∠ QOR = 90° + ½ ∠ P. By substituting the given value of ∠ QOR = 107° into the formula, we have 107° = 90° + (∠ P)/(2). Rearranging the equation to solve for ∠ P, we subtract 90° from both sides to get 17° = (∠ P)/(2). Multiplying both sides by 2, we find that ∠ P = 34°.

Explanation

To find tan θ, divide the numerator and denominator of the expression (7sin θ + 4cos θ)/(9sin θ - 2cos θ) = ⁵⁄₄ by cos θ. This gives:

(7tan θ + 4)/(9tan θ - 2) = ⁵⁄₄

Cross-multiplying the equation results in:

4(7tan θ + 4) = 5(9tan θ - 2) 28tan θ + 16 = 45tan θ - 10 17tan θ = 26 tan θ = ²⁶⁄₁₇

The value of tan² θ is (²⁶⁄₁₇)² = ⁶⁷⁶⁄₂₈₉. Now substitute this into the target expression:

(tan²θ + 5)/(tan²θ - 5) = (676)/(289) + 5{⁶⁷⁶⁄₂₈₉ - 5}

Simplify the numerator and denominator by using the common denominator 289:

Numerator: (676 + 5(289))/(289) = (676 + 1445)/(289) = ²¹²¹⁄₂₈₉ Denominator: (676 - 5(289))/(289) = (676 - 1445)/(289) = ⁻⁷⁶⁹⁄₂₈₉

The ratio becomes:

(2121 / 289)/(-769 / 289) = -²¹²¹⁄₇₆₉

Explanation

To find the net percentage change, we use the formula for successive percentage changes x and y, which is x + y + ˣʸ⁄₁₀₀. First, calculate the combined effect of the two initial increases of 12% and 23%.

Net increase = 12 + 23 + (12 × 23)/(100) = 35 + ²⁷⁶⁄₁₀₀ = 37.76%

Next, apply the 34% decrease to this combined value by setting x = 37.76 and y = -34 in the same formula:

Net change = 37.76 - 34 + (37.76 × (-34))/(100) Net change = 3.76 - (1283.84)/(100) Net change = 3.76 - 12.8384 = -9.0784%

The negative sign indicates a net decrease. When rounded to the nearest integer, 9.0784% becomes 9%. Alternatively, using the multiplier method, the final value is calculated as 1.12 × 1.23 × 0.66 = 0.909216. The total decrease is 1 - 0.909216 = 0.090784, which equals a 9.0784% decrease.

Explanation

The shopkeeper sells the salt at its cost price, meaning the profit is derived entirely from the weight manipulation. When a trader uses a false weight, the gain percentage is calculated using the relation between the weight saved and the weight actually delivered to the customer.

The weight promised to the customer is 1000 gm (1 kg), but the shopkeeper removes 200 gm, resulting in an actual weight given of 1000 - 200 = 800 gm. The formula for the gain percentage in such cases is:

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

Substituting the given values: Gain % = (200)/(1000 - 200) × 100 Gain % = ²⁰⁰⁄₈₀₀ × 100 Gain % = ¼ × 100 = 25%

This calculation shows that by providing only 800 gm while charging for 1000 gm, the shopkeeper effectively earns a profit of 25% on his actual cost.

Explanation

The policeman catches the thief by covering the initial gap of 600 metres using the relative speed between them. Since both are running in the same direction, the relative speed is the difference of their speeds: Relative Speed = 15 km/h - 12 km/h = 3 km/h

To convert this speed into metres per minute for easier calculation: 3 km/h = (3 × 1000)/(60) m/min = 50 m/min

The time required to bridge the initial distance of 600 metres is: Time = (Initial Distance)/(Relative Speed) Time = (600 metres)/(50 m/min) = 12 minutes

Explanation

The increase in simple interest is directly proportional to the increase in the rate of interest over the given period. The formula for the difference in simple interest is Δ SI = (P × Δ R × T)/(100), where P is the principal, Δ R is the increase in rate, and T is the time in years.

Given that Δ SI = 1,512, Δ R = 7%, and T = 6 years, the equation becomes:

1,512 = (P × 7 × 6)/(100)

1,512 = (P × 42)/(100)

P = (1,512 × 100)/(42)

Dividing 1,512 by 42 yields 36. Therefore:

P = 36 × 100 = 3,600

Explanation

The perimeter of a sector is given by the formula P = 2r + l, where r is the radius and l is the arc length. The arc length l for a central angle θ is calculated as l = (θ)/(360) × 2π r.

Given the central angle θ = 60° and π = 22/7, the perimeter equation becomes: 64/3 = 2r + ⁶⁰⁄₃₆₀ × 2 × ²²⁄₇ × r 64/3 = 2r + ⅙ × ⁴⁴⁄₇ × r 64/3 = 2r + ²²⁄₂₁r 64/3 = (42r + 22r)/(21) 64/3 = ⁶⁴ʳ⁄₂₁

Solving for r: 1/3 = r/21 r = 7 cm

The area A of the sector is given by the formula A = (θ)/(360) × π r². Substituting the known values: A = ⁶⁰⁄₃₆₀ × ²²⁄₇ × 7² A = ⅙ × ²²⁄₇ × 49 A = ⅙ × 22 × 7 A = ¹⁵⁴⁄₆ = ⁷⁷⁄₃ cm²

Explanation

The perpendicular from the centre of a circle to a chord bisects the chord. This geometry forms a right-angled triangle where the radius r is the hypotenuse, the perpendicular distance from the centre d = 12 cm is one leg, and half the chord length s = 32 / 2 = 16 cm is the other leg. Using the Pythagoras theorem r² = d² + s², we substitute the given values to get r² = 12² + 16². Calculating the squares gives r² = 144 + 256 = 400. Taking the square root of both sides yields r = √(400) = 20 cm.

Explanation

The selling price is minimum when the total discount percentage offered is at its maximum. To compare the schemes, we calculate the effective discount for successive rates a% and b% using the formula a + b - ᵃᵇ⁄₁₀₀.

In Scheme I, the effective discount is 15 + 20 - (15 × 20)/(100) = 35 - 3 = 32%. In Scheme II, the effective discount is 25 + 10 - (25 × 10)/(100) = 35 - 2.5 = 32.5%. In Scheme III, the discount is a single value of 33%.

By comparing the three results, Scheme III provides the highest total discount of 33%. A higher discount percentage directly results in a lower effective selling price for the buyer.

Explanation

Let the breadth of the rectangular garden be b meters and its length be l meters. The perimeter P of a rectangle is given by the formula P = 2(l + b). Given that the perimeter is 48 meters, we have 2(l + b) = 48, which simplifies to l + b = 24.

The problem states that the length is 6 meters more than the breadth, so l = b + 6. Substituting this expression for l into the simplified perimeter equation gives (b + 6) + b = 24. Solving for b, we get 2b + 6 = 24, which leads to 2b = 18, and thus b = 9 meters.

The length is l = 9 + 6 = 15 meters. The area A of the garden is calculated using the formula A = l × b. Therefore, A = 15 × 9 = 135 square meters.

Explanation

The volume of each cube is V = 512 cm³. Since the volume of a cube is a³, where a is the side length, a = ³√(512) = 8 cm. When three cubes are joined end to end, the length L of the resulting cuboid becomes 3 × 8 = 24 cm, while the breadth B and height H remain 8 cm. The surface area of the resulting cuboid is calculated using the formula 2(LB + BH + HL). Substituting the dimensions, the surface area is 2(24 × 8 + 8 × 8 + 8 × 24) = 2(192 + 64 + 192) = 2(448) = 896 cm². Alternatively, each cube has 6 faces, making a total of 18 faces for three separate cubes. When joined at two interfaces, 4 faces (2 at each interface) are hidden from the surface. The remaining surface area consists of 18 - 4 = 14 square faces, each with an area of a² = 64 cm², giving 14 × 64 = 896 cm².

Explanation

Given that r leaves a remainder of 3 when divided by 8, we can use the property of modular arithmetic where r ≡ 3 8. To find the remainder of the expression r² + 6r + 7 when divided by 8, we substitute the value 3 for r in the polynomial. This results in 3² + 6(3) + 7, which simplifies to 9 + 18 + 7 = 34. The remainder of the original expression is equivalent to the remainder of 34 when divided by 8. Performing the division, 34 = 8 × 4 + 2, which identifies 2 as the remainder.

Explanation

The trigonometric ratio sec(t) is defined as the ratio of the hypotenuse to the adjacent side in a right-angled triangle. Given sec(t) = (x)/(y), we designate the hypotenuse as x and the adjacent side as y. Using the Pythagorean theorem, the square of the opposite side (perpendicular) is calculated as the difference between the square of the hypotenuse and the square of the adjacent side. If p represents the opposite side, then p² = x² - y², leading to p = √(x² - y²). Since cot(t) is the ratio of the adjacent side to the opposite side, it follows that cot(t) = (y)/(√(x² - y²)).

Explanation

The mean proportional between two numbers a and b is defined as the square root of their product, expressed by the formula √(a × b). Substituting the given values a = 36 and b = 100, the calculation becomes √(36 × 100). This simplifies to √(3600), which equals 60. Alternatively, using the property of square roots, √(36 × 100) = √(36) × √(100) = 6 × 10 = 60.

Explanation

Let the total number of voters in the constituency be 100. According to the problem, the distribution is as follows:

Number of males = 45% of 100 = 45 Number of females = 100 - 45 = 55

To find the number of literate males, we use the fact that 60% of males are illiterate, meaning 40% are literate: Literate males = 40% of 45 = 0.40 × 45 = 18

To find the number of illiterate females, we use the fact that 60% of females are literate, meaning 40% are illiterate: Illiterate females = 40% of 55 = 0.40 × 55 = 22

The question asks for the percentage by which the number of illiterate females is more than the number of literate males. We use the percentage increase formula: Percentage increase = (Illiterate females - Literate males)/(Literate males) × 100 Percentage increase = (22 - 18)/(18) × 100 = ⁴⁄₁₈ × 100 = ²⁄₉ × 100

Since ⅑ ≈ 11.11%, then ²⁄₉ ≈ 22.22%.

Explanation

For a sequence of n consecutive numbers, the average (arithmetic mean) is equal to the middle term when n is odd. Given that the sum of five consecutive numbers is 80, the average is calculated as 80 / 5 = 16. This average represents the middle (third) number in the sequence. To determine the largest (fifth) number, we simply add 1 for each of the two subsequent positions in the sequence. Therefore, the largest number is 16 + 1 + 1 = 18.

Using an algebraic approach, if the five consecutive numbers are represented as x, x+1, x+2, x+3, and x+4, their sum is 5x + 10 = 80. Solving for x yields 5x = 70, which means x = 14. The largest number in this set is x+4, which equals 14 + 4 = 18.

Explanation

The overall gain percentage in a transaction involving different portions sold at different profit rates can be calculated using the weighted average method. Let the total number of articles be 1. The shopkeeper sells ⅝ of the articles at a 20% gain and the remaining 1 - ⅝ = ⅜ at the cost price, which implies a 0% gain.

The total gain percentage is given by the formula: Total Gain % = (Fraction 1 × Profit Rate 1) + (Fraction 2 × Profit Rate 2)

Substituting the given values into the formula: Total Gain % = (⅝ × 20%) + (⅜ × 0%) Total Gain % = ¹⁰⁰⁄₈% + 0% Total Gain % = 12.5%

Converting the decimal to a mixed fraction, we get 12 ½%. Alternatively, if we assume the total number of articles is 8 and the cost price of each article is 100, the total cost price is 800. The selling price of 5 articles at a 20% profit is 5 × 120 = 600, and the selling price of the remaining 3 articles at cost price is 3 × 100 = 300. The total selling price is 900. The total profit is 900 - 800 = 100, and the profit percentage is (¹⁰⁰⁄₈₀₀) × 100 = 12.5%.

Explanation

The work rate of pipe X is 1/60 of the tank per hour, and the work rate of pipe Y is 1/72 of the tank per hour. When both pipes are opened together, their combined rate per hour is calculated as follows:

Combined Rate = ¹⁄₆₀ + ¹⁄₇₂

To add these fractions, find the least common multiple (LCM) of 60 and 72, which is 360.

Combined Rate = ⁶⁄₃₆₀ + ⁵⁄₃₆₀ = ¹¹⁄₃₆₀

In 20 hours, the total portion of the tank filled by both pipes is:

Portion Filled = 20 × ¹¹⁄₃₆₀ = ¹¹⁄₁₈

The portion of the tank that remains empty is the total capacity (1) minus the portion filled:

Empty Portion = 1 - ¹¹⁄₁₈ = (18 - 11)/(18) = ⁷⁄₁₈

Explanation

Given x = 7 - 4√3, we find its reciprocal ¹⁄ₓ by rationalizing the denominator. ¹⁄ₓ = (1)/(7 - 4√(3)) × 7 + 4√3{7 + 4√3} = 7 + 4√3{7² - (4√3)²} = 7 + 4√3{49 - 48} = 7 + 4√3

Now, find the sum of x and its reciprocal: x + ¹⁄ₓ = (7 - 4√3) + (7 + 4√3) = 14

To find the value of x² + (1)/(x²⁾, we use the algebraic identity a² + b² = (a + b)² - 2ab. Setting a = x and b = ¹⁄ₓ: x² + (1)/(x²⁾ = (x + ¹⁄ₓ)² - 2

Substitute the value of (x + ¹⁄ₓ) into the equation: x² + (1)/(x²⁾ = 14² - 2 = 196 - 2 = 194

Explanation

By testing the value a = 1, we solve for x, y, and z using the given system. When a = 1, the equations are x - y + z = 2, x + 4y - 2z = 9, and 2x - 3y + 4z = 4. Solving this system yields x = 3, y = 2, and z = 1, which are three consecutive integers (1, 2, 3), thus satisfying the condition. Testing other options like a = 2 results in non-consecutive values for the variables.

Explanation

To find the angle where sin(θ) = cos(θ), we use the trigonometric identity tan(θ) = (sin(θ))/(cos(θ)). Setting the two functions equal to each other, sin(θ) = cos(θ), and dividing both sides by cos(θ) yields tan(θ) = 1. In the first quadrant, the tangent function equals 1 when θ = 45°. From the standard trigonometric table, both sin(45°) and cos(45°) are equal to (1)/(√(2)), approximately 0.7071.

Explanation

Since AOC is the diameter of the circle, the points A, O, and C lie on a straight line, and the central angle ∠ AOC is 180°. This means that the sum of the angles subtended by arc AB and arc BC at the center O must be 180°. In any circle, the length of an arc is directly proportional to the angle it subtends at the center.

Given that arc AB = ⅕ of arc BC, let the degree measure of the angle subtended by arc BC at the center be ∠ BOC = x. Consequently, the angle subtended by arc AB is ∠ AOB = ⅕x. Setting up the linear equation for the diameter:

∠ AOB + ∠ BOC = 180° ⅕x + x = 180° ⁶⁄₅x = 180° x = 180° × ⅚ x = 30° × 5 = 150°