SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

In a triangle, the point where the internal bisectors of two angles meet is known as the incenter. The angle formed at this incenter by the bisectors of ∠B and ∠C is related to the third angle ∠A by the following geometric property:

∠ BOC = 90° + (∠ A)/2

In the given triangle △ ABC, the value of ∠A is 72°. By substituting this value into the formula, we can calculate the measure of ∠BOC:

∠ BOC = 90° + (72°)/2

∠ BOC = 90° + 36°

∠ BOC = 126°

This formula is derived from the angle sum property of triangles. In △ OBC, the sum of angles is 180°, so ∠ BOC = 180° - ((∠ B + ∠ C)/2). Since ∠ B + ∠ C = 180° - ∠ A, the expression simplifies to 180° - ((180° - ∠ A)/2), which yields 90° + (∠ A)/2.

Explanation

The lateral surface area of a cylinder is given by the formula 2π rh, where r is the radius and h is the height. Given that the lateral area is 3168 cm² and the height is 48 cm, we can determine the radius as follows:

2 × ²²⁄₇ × r × 48 = 3168

Solving for r: r = (3168 × 7)/(2 × 22 × 48) = (66 × 7)/(44) = (3 × 7)/(2) = 10.5 cm

The volume V of a cylinder is calculated using the formula V = π r² h. Substituting the values for r and h:

V = ²²⁄₇ × (10.5)² × 48 V = ²²⁄₇ × 110.25 × 48 V = 22 × 15.75 × 48 V = 16632 cm³

Explanation

The relationship between the Cost Price (CP) and the Printed Price (MP) can be directly determined using the standard formula:

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

Substituting the given values, where the discount is 10% and the profit is 20%:

(CP)/(MP) = (100 - 10)/(100 + 20)

(CP)/(MP) = ⁹⁰⁄₁₂₀

By simplifying the fraction by dividing both the numerator and the denominator by 30, we get:

(CP)/(MP) = ¾

Thus, the ratio between the cost price and the printed price is 3:4.

Explanation

The total work is calculated as the LCM of 4 and 12, which is 12 units. The efficiency of A is 12/4 = 3 units per day, and the efficiency of B is 12/12 = 1 unit per day. When they work together, their combined efficiency is 3 + 1 = 4 units per day. Over a period of 2 days, the total work completed is 4 × 2 = 8 units. The work remaining is the total work minus the work completed, which is 12 - 8 = 4 units. To find the fraction of the work left, the remaining work is divided by the total work, resulting in ⁴⁄₁₂, which simplifies to ⅓.

Explanation

Let the filling rates of pipes A, B, and C be a, b, and c respectively, where c is negative as it is an outlet pipe. Based on the given times, we establish the work rates per hour: Rate of (A + B) = (1)/(1.5) = ⅔ Rate of C = -(1)/(4.5) = -²⁄₉ Rate of (A + C) = ⅙

Using the rates for A and C: a + c = ⅙ a - ²⁄₉ = ⅙ a = ⅙ + ²⁄₉ = (3 + 4)/(18) = ⁷⁄₁₈ units per hour.

Now, substitute the rate of A into the combined rate of A and B: a + b = ⅔ ⁷⁄₁₈ + b = ⅔ b = ⅔ - ⁷⁄₁₈ = (12 - 7)/(18) = ⁵⁄₁₈ units per hour.

The time taken by Pipe B alone to fill the tank is the reciprocal of its rate: Time = (1)/(b) = ¹⁸⁄₅ = 3.6 hours. To convert 0.6 hours into minutes, multiply by 60: 0.6 × 60 = 36 minutes. Therefore, the total time is 3 hours 36 minutes.

Explanation

Starting with the given equation 3y - (3)/(y) = 5, factor out the common coefficient 3 to get 3(y - (1)/(y)) = 5. Dividing both sides by 3 provides the simplified relation y - (1)/(y) = ⁵⁄₃.

To determine the value of y² + (1)/(y²⁾, apply the algebraic identity (a - b)² = a² + b² - 2ab. By substituting a = y and b = (1)/(y), the identity yields: (y - (1)/(y))² = y² + (1)/(y²⁾ - 2(y)((1)/(y)) (y - (1)/(y))² = y² + (1)/(y²⁾ - 2

Now, substitute the value of y - (1)/(y) into this expression: (⁵⁄₃)² = y² + (1)/(y²⁾ - 2 ²⁵⁄₉ = y² + (1)/(y²⁾ - 2

Solving for y² + (1)/(y²⁾ gives: y² + (1)/(y²⁾ = ²⁵⁄₉ + 2 y² + (1)/(y²⁾ = (25 + 18)/(9) y² + (1)/(y²⁾ = ⁴³⁄₉

Explanation

When the security guard catches the thief, they have both covered the same total distance from the starting point. The thief ran for a total of 40 minutes (20 minutes before the guard started plus 20 minutes during the chase). The security guard ran for 20 minutes to cover that same distance.

Using the distance formula D = speed × time, and noting that distance is constant: vₜₕᵢₑ_f × tₜₕᵢₑ_f = v_gᵤₐᵣ_d × t_gᵤₐᵣ_d

Substituting the given values: 12 m/h × 40 minutes = v_gᵤₐᵣ_d × 20 minutes

v_gᵤₐᵣ_d = (12 × 40)/(20) v_gᵤₐᵣ_d = 12 × 2 v_gᵤₐᵣ_d = 24 m/h

Because the security guard covers the same distance as the thief in exactly half the time (20 minutes compared to 40 minutes), the guard's speed must be double the thief's speed.

Explanation

The simple interest formula is SI = (P × R × T)/(100). Given the principal P = 32,400 and the simple interest SI = 11,664, the problem states that the rate of interest R is one-fourth of the number of years T. This gives the relationship R = (T)/(4), or T = 4R.

Substituting these values into the simple interest formula: 11,664 = (32,400 × R × 4R)/(100) 11,664 = 324 × 4R² 11,664 = 1,296R² R² = (11,664)/(1,296)

By calculation, 1,296 × 9 = 11,664, so: R² = 9 R = √9 = 3

Explanation

Let the initial number of seats for Mathematics, English, and Chemistry be 6x, 4x, and 7x respectively. The new seats after the specified percentage increases are calculated as follows:

Increased Mathematics seats = 6x × (1 + ²⁰⁄₁₀₀) = 6x × 1.2 = 7.2x Increased English seats = 4x × (1 + ⁴⁰⁄₁₀₀) = 4x × 1.4 = 5.6x Increased Chemistry seats = 7x × (1 + ⁶⁰⁄₁₀₀) = 7x × 1.6 = 11.2x

The ratio of the increased seats is 7.2x : 5.6x : 11.2x. Multiplying the entire ratio by 10 to eliminate decimals gives 72 : 56 : 112. Dividing each part of the ratio by the common factor 8 simplifies the ratio to 9 : 7 : 14.

Explanation

The speed of the lorry is given as 66 km/h. To find the number of revolutions per minute, we first convert the speed into centimeters per minute: v = (66 × 100,000 cm)/(60 min) = 110,000 cm/min.

The distance covered by the wheel in one full revolution is equal to its circumference C, which is calculated using the formula C = 2π r. Given the radius r = 182 cm and using π = ²²⁄₇: C = 2 × ²²⁄₇ × 182 = 44 × 26 = 1144 cm.

The number of revolutions per minute N is the total distance covered in one minute divided by the distance covered in one revolution: N = (Speed in cm/min)/(Circumference in cm) = (110,000)/(1144) ≈ 96.153.

Rounding to the nearest whole number, the wheel makes approximately 96 revolutions per minute.

Explanation

Initial savings are calculated as the difference between monthly income and expenditure: S = I - E = 80,000 - 45,000 = 35,000. The absolute increase in income is 16% of 80,000, which is 80,000 × 0.16 = 12,800. The absolute increase in expenditure is 8% of 45,000, which is 45,000 × 0.08 = 3,600.

The increase in savings Δ S is the difference between the increase in income and the increase in expenditure: Δ S = 12,800 - 3,600 = 9,200. To find the percentage increase in savings, we divide the change in savings by the initial savings and multiply by 100:

Percentage Increase in Savings = (9,200)/(35,000) × 100 = ⁹²⁰⁄₃₅ = ¹⁸⁴⁄₇ ≈ 26.2857%

Rounding to two decimal places, the percentage increase in savings is 26.29%.

Explanation

Let the present age of the son be x years. Since the woman is 28 years older than her son, her present age is x + 28 years.

In three years, the son's age will be x + 3 and the woman's age will be (x + 28) + 3 = x + 31. According to the problem, the woman's age in three years will be twice the age of her son at that time. This relationship can be expressed by the following linear equation:

x + 31 = 2(x + 3) x + 31 = 2x + 6 31 - 6 = 2x - x 25 = x

Solving for x gives the son's present age as 25 years.

Explanation

To find the time taken by Anil alone, let the total work be the least common multiple (LCM) of 15 and 20, which is 60 units. The combined rate of work for Anil and Bipin is 60 / 15 = 4 units per day. The individual rate of work for Bipin is 60 / 20 = 3 units per day. Subtracting Bipin's rate from the combined rate gives Anil's rate: 4 - 3 = 1 unit per day. The time taken by Anil to complete the work alone is the total work divided by his rate, calculated as 60 / 1 = 60 days. Alternatively, using the formula T = (xy) / (y - x) where x = 15 and y = 20, the time is (20 × 15) / (20 - 15) = 300 / 5 = 60 days.

Explanation

The molar mass of Sulfurous acid (H₂SO₃) is calculated by summing the atomic masses of its constituent elements: H ≈ 1.008 u, S ≈ 32.06 u, and O ≈ 16.00 u. Molar Mass of H₂SO₃ = (2 × 1.008) + (1 × 32.06) + (3 × 16.00) = 2.016 + 32.06 + 48.00 = 82.076 g/mol. The percentage composition of an element is determined by the formula: Mass % = ((Atomic weight × number of atoms)/(Molar mass of compound)) × 100. The percentage of Hydrogen (H) is ((2.016)/(82.076)) × 100 ≈ 2.46%, which rounds to 2.5%. The percentage of Sulfur (S) is ((32.06)/(82.076)) × 100 ≈ 39.06%, which is approximately 39%. The percentage of Oxygen (O) is ((48.00)/(82.076)) × 100 ≈ 58.48%, which is approximately 58.5%.

Explanation

Let the two numbers be x and y. Given the sum x + y = 98 and the difference x - y = 28, the two numbers are: x = (98 + 28)/(2) = ¹²⁶⁄₂ = 63 y = (98 - 28)/(2) = ⁷⁰⁄₂ = 35

Explanation

The speed of the boat in still water u is calculated using the given distance and time: u = (20 km)/(0.5 hours) = 40 km/h. The downstream speed D is 125% of the speed in still water, so D = 1.25 × 40 = 50 km/h.

The downstream speed is also the sum of the boat speed and stream speed v, which gives u + v = 50. Substituting u = 40, we find the stream speed v = 10 km/h. The upstream speed U is the difference between the boat speed and stream speed: U = u - v = 40 - 10 = 30 km/h. The time taken to cover 15 km upstream is T = (Distance)/(U) = ¹⁵⁄₃₀ = ½ hours.

Explanation

The number of days for which the interest is calculated is determined by counting from January 24, 2012, to June 18, 2012. Excluding the day the loan was taken, the days are: January (7 days), February (29 days, as 2012 is a leap year), March (31 days), April (30 days), May (31 days), and June (18 days). Total time T = 7 + 29 + 31 + 30 + 31 + 18 = 146 days.

Since 2012 is a leap year, the time in years is T = ¹⁴⁶⁄₃₆₆. The rate of interest R = 8 ¾% = ³⁵⁄₄%.

Simple Interest SI = (P × R × T)/(100) SI = (78000 × 35)/(4) × ¹⁴⁶⁄₃₆₆{100} SI = (780 × 35 × 146)/(4 × 366) SI = (195 × 35 × 146)/(366) SI = ⁹⁹⁶⁴⁵⁰⁄₃₆₆ ≈ 2722.54

Total Amount A = P + SI A = 78000 + 2722.54 = 80722.54

Rounding to the nearest integer, the total amount paid by Archana is ₹80,723.

Explanation

Let M and D be the present ages of the mother and daughter. Based on the first condition, seven years ago the mother's age was M - 7 and the daughter's age was D - 7. The relationship is given by M - 7 = 7(D - 7), which simplifies to M - 7D = -42. From the second condition, three years from now, their ages will be M + 3 and D + 3. The relationship is M + 3 = 3(D + 3), which simplifies to M - 3D = 6.

By subtracting the first equation from the second, we eliminate M: (M - 3D) - (M - 7D) = 6 - (-42), which gives 4D = 48, resulting in D = 12. Substituting D = 12 back into the second equation gives M - 3(12) = 6, so M - 36 = 6, yielding M = 42. These ages satisfy both conditions provided in the problem statement. Seven years ago, the ages were 35 and 5 (35 = 7 × 5), and three years from now, the ages will be 45 and 15 (45 = 3 × 15).

Explanation

When two circles touch each other externally, the distance between their centers d is exactly equal to the sum of their radii r₁ and r₂, expressed as d = r₁ + r₂. In this configuration, the circles share three common tangents. Two of these are direct common tangents, which are lines that touch both circles while keeping both circles on the same side of the line. The third is a transverse common tangent that passes through the specific point of contact where the two circles meet. This transverse tangent is perpendicular to the line segment joining the centers of the two circles. Since the circles touch at only one point and do not overlap, no other transverse tangents can be formed, resulting in a total of 3 common tangents.

Explanation

The divisibility rule for 11 states that 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 number 9592330, we calculate the sums as follows: Sum of digits at odd positions (1st, 3rd, 5th, and 7th digits): 9 + 9 + 3 + 0 = 21 Sum of digits at even positions (2nd, 4th, and 6th digits): 5 + 2 + 3 = 10

Now, we find the difference between these sums: Difference = 21 - 10 = 11

Since the difference is 11, which is a multiple of 11, the number 9592330 is divisible by 11.

Explanation

The given equation is sin θ + sin² θ = 1. By rearranging the terms, we get sin θ = 1 - sin² θ. Using the fundamental trigonometric identity sin² θ + cos² θ = 1, we know that 1 - sin² θ = cos² θ. Therefore, we can establish the relation sin θ = cos² θ.

By squaring both sides of this relation, we obtain sin² θ = (cos² θ)² = cos⁴ θ. We need to find the value of the expression cos² θ + cos⁴ θ. Substituting the derived relations into this expression, we replace cos² θ with sin θ and cos⁴ θ with sin² θ. This transforms the expression into sin θ + sin² θ. Since the problem states that sin θ + sin² θ = 1, the value of cos² θ + cos⁴ θ is also 1.

Explanation

The number of distinct meeting points for two objects moving on a circular track is determined by the ratio of their speeds. When two persons run in opposite directions with speeds v₁ and v₂, the number of distinct crossing points is given by the sum of the terms of their speed ratio in its simplest form.

First, we find the simplest ratio of the given speeds: v₁ : v₂ = 20 : 30 = 2 : 3

In this ratio, let x = 2 and y = 3. For motion in opposite directions, the formula for the number of distinct meeting points is x + y. Substituting the values, we get: Number of points = 2 + 3 = 5

This result remains constant regardless of the actual length of the track, as the meeting points are determined by the relative displacement segments. Every time they meet, they have collectively covered exactly one circumference. Given the relative speed of 20 + 30 = 50 m/s and a track of 100 m, they meet every 2 seconds. In 10 seconds (the time taken for the faster runner to complete 3 laps and the slower to complete 2 laps), they will have met 5 times at distinct locations before returning to the starting point and repeating the cycle.

Explanation

Susan's typing rate is 12 / 6 = 2 pages per minute. Mary's typing rate is 6 / 12 = 0.5 pages per minute. When working together, their combined typing speed is the sum of their individual rates: Rₜₒₜₐₗ = 2 + 0.5 = 2.5 pages per minute. The total number of pages they can type in 32 minutes is calculated by multiplying this combined rate by the total time: 2.5 × 32 = 80 pages.

Explanation

To ensure that S does not incur any loss, the interest earned by S from lending the money to T must be at least equal to the interest S pays to R. Let the principal amount borrowed by S be P. According to the problem, S pays a fixed interest of ₹17,000 to R. When S lends the same principal P to T, T pays 8.5% more than the principal, which constitutes the interest income for S.

The condition for S to break even or make a profit is: Interest earned from T ≥ Interest paid to R 8.5% of P ≥ 17,000 (8.5)/(100) × P ≥ 17,000

To find the minimum principal P, we equate the two values: P = (17,000 × 100)/(8.5) P = (1,700,000)/(8.5) P = 2,00,000

Thus, S must borrow a minimum of ₹2,00,000 so that the 8.5% interest received from T exactly covers the ₹17,000 interest obligation to R.

Explanation

The total work is the product of the number of workers, the time taken, and their efficiency. Let M represent the efficiency of one man and W represent the efficiency of one woman. Based on the given data, the total work can be expressed as 8 × 10 × M = 10 × 12 × W. This simplifies to 80M = 120W, or M = 1.5W, meaning one man is as efficient as 1.5 women.

Using the total work in terms of woman-days, we have 10 × 12 = 120 units of work. The question asks for the time taken by 4 men and 4 women. Their combined efficiency is 4M + 4W. Substituting M = 1.5W into this expression gives 4(1.5W) + 4W = 6W + 4W = 10W.

The number of days required is calculated by dividing the total work by the combined efficiency: 120W / 10W = 12 days.

Explanation

Dividing the given equation x² - 7x + 1 = 0 by x, we obtain x - 7 + ¹⁄ₓ = 0, which simplifies to x + ¹⁄ₓ = 7. The expression to be evaluated is x² - (1)/(x²⁾, which can be factored as (x + ¹⁄ₓ)(x - ¹⁄ₓ).

To find the value of x - ¹⁄ₓ, we use the identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4. Substituting the known value gives (x - ¹⁄ₓ)² = 7² - 4 = 49 - 4 = 45. This implies x - ¹⁄ₓ = ±√(45) = 3√5.

The condition 0 < x < 1 dictates the sign of the result. When x is between 0 and 1, its reciprocal ¹⁄ₓ is greater than 1. Consequently, x < ¹⁄ₓ, making the difference x - ¹⁄ₓ negative. Thus, we must take x - ¹⁄ₓ = -3√5.

Substituting these values into the factored expression yields x² - (1)/(x²⁾ = (7)(-3√5) = -21√5.

Explanation

The selling price (SP) is calculated by subtracting the discount from the marked price (MP). Using the formula SP = MP × (100 - Discount%)/100:

SP = 9,400 × (100 - 4) / 100 SP = 9,400 × 96 / 100 SP = 94 × 96

To calculate 94 × 96 quickly, use the identity (a - b)(a + b) = a² - b² by expressing the numbers as (95 - 1)(95 + 1):

SP = 95² - 1² SP = 9,025 - 1 SP = 9,024

Alternatively, calculate the discount amount first. Since 1% of 9,400 is 94, a 4% discount is 4 × 94 = 376. Subtracting this discount from the marked price gives 9,400 - 376 = 9,024.

Explanation

The total expenditure is 150. The cost of 10 pens at 8 each is 10 × 8 = 80. The cost of 10 erasers at 5 each is 10 × 5 = 50. The combined cost for pens and erasers is 80 + 50 = 130.

The remaining amount spent on sharpeners is 150 - 130 = 20. Since each sharpener costs 4, the number of sharpeners purchased is 20 / 4 = 5.

The total number of items bought is 10 pens + 10 erasers + 5 sharpeners = 25 items.

The average price per item is calculated as: Average Price = (Total Cost)/(Total Items) = ¹⁵⁰⁄₂₅ = 6

Explanation

The combined rate of work is the sum of the individual rates of the groups. Since 10 women can complete the work in 6 days, their combined rate of work is ⅙ units per day. Similarly, as 6 men can complete the same work in 5 days, their combined rate of work is ⅕ units per day.

When both groups work together, the total rate of work per day is calculated as: Combined Rate = ⅙ + ⅕ = (5 + 6)/(30) = ¹¹⁄₃₀

The number of days required to finish the work is the reciprocal of the combined daily rate: Total Days = ³⁰⁄₁₁ = 2 ⁸⁄₁₁

Explanation

When two triangles are similar, the ratio of their perimeters is equal to the ratio of their corresponding sides. Given △ ABC ∼ △ PQR, the ratio of similarity is determined by the corresponding sides AB and PQ:

Ratio = (AB)/(PQ) = ⁴⁄₆ = ⅔

First, calculate the perimeter of △ PQR by summing its side lengths:

Perimeter of △ PQR = PQ + QR + RP = 6 + 9 + 12 = 27 cm

Let P_A_B_C be the perimeter of △ ABC. According to the property of similar triangles:

P_A_B_C{Perimeter of △ PQR} = ⅔

Substituting the known value of the perimeter of △ PQR:

P_A_B_C = 27 × ⅔ = 9 × 2 = 18 cm

Explanation

For two numbers a and b, the third proportional c is defined by the property of continued proportion where a:b = b:c. This relationship is represented by the formula:

c = (b²)/(a)

Given the values a = 25 and b = 35, the calculation for the third proportional is:

c = (35²)/(25)

c = (35 × 35)/(25)

Dividing both the numerator and denominator by 5 twice:

c = 7 × 7

c = 49

Explanation

To find the new ratio, we apply the percentage increases to each part of the initial ratio 2:3:5. Let the initial salaries be 200, 300, and 500.

The new salary of the father after a 15% increase is 200 × (1 + ¹⁵⁄₁₀₀) = 200 × 1.15 = 230. The new salary of the mother after a 10% increase is 300 × (1 + ¹⁰⁄₁₀₀) = 300 × 1.10 = 330. The new salary of the daughter after a 20% increase is 500 × (1 + ²⁰⁄₁₀₀) = 500 × 1.20 = 600.

The resulting ratio of their new salaries is 230 : 330 : 600. Dividing each term by the common factor of 10 simplifies the ratio to 23 : 33 : 60.

Explanation

The surface area of a sphere is given by the formula 4πr². Setting 4πr² equal to 9π gives r² = 2.25, which means the radius r is 1.5 units. Since the diameter is twice the radius, the diameter of the ball is 3 units.

Explanation

The area of a sector of a circle with radius r and central angle θ is given by the formula A = (θ)/(360°) × π r².

Given the radius r = 7 cm, the central angle θ = 60°, and π = ²²⁄₇, we substitute these values into the formula:

A = (60°)/(360°) × ²²⁄₇ × 7²

A = ⅙ × ²²⁄₇ × 7 × 7

Simplifying the expression by canceling out the common factor of 7 in the numerator and denominator, we get:

A = ⅙ × 22 × 7

A = ¹⁵⁴⁄₆

Dividing both the numerator and denominator by 2 results in:

A = ⁷⁷⁄₃ cm²

Explanation

The given equation is 65% × 350 - x% × 250 + 40% × 120 = 158. To find the value of x, first calculate the known percentage terms. The value of 65% of 350 is calculated as 0.65 × 350 = 227.5. The value of 40% of 120 is 0.4 × 120 = 48. Substituting these values into the equation, we get 227.5 - 2.5x + 48 = 158, where 2.5x represents ˣ⁄₁₀₀ × 250.

Combining the numerical terms on the left side, 227.5 + 48 = 275.5. The equation now simplifies to 275.5 - 2.5x = 158. Rearranging the terms to solve for x, we subtract 158 from 275.5 to get 2.5x = 117.5. Dividing 117.5 by 2.5 yields x = 47.

Explanation

Given the equation 2cot x = 5, we find the value of the cotangent function as cot x = ⁵⁄₂. To evaluate the expression (2cos x - sin x)/(2cos x + sin x), we divide both the numerator and the denominator by sin x. Since cot x = (cos x)/(sin x), the expression can be rewritten as:

(2(cos x)/(sin x)) - (sin x)/(sin x){2((cos x)/(sin x)) + (sin x)/(sin x)} = (2cot x - 1)/(2cot x + 1)

By substituting the given value 2cot x = 5 directly into this transformed formula, we get:

(5 - 1)/(5 + 1) = ⁴⁄₆

Reducing the fraction to its simplest form results in ⅔. Alternatively, substituting 2cos x = 5sin x directly into the original expression yields (5sin x - sin x)/(5sin x + sin x), which simplifies to (4sin x)/(6sin x) = ⅔.

Explanation

The SAS (Side-Angle-Side) similarity criterion states that two triangles are similar if two pairs of corresponding sides are in the same ratio and the angles included between these sides are equal. In △ ABC and △ DEF, the given proportionality is (AB)/(DE) = (BC)/(FD). The sides AB and BC in the first triangle share the common vertex B, which means ∠ B is the angle included between them. Similarly, in the second triangle, the sides DE and FD share the common vertex D, making ∠ D the included angle. To satisfy the SAS similarity condition, the included angles must be equal, requiring ∠ B = ∠ D.

Explanation

To find the time when the two cyclists meet at the starting point, we must calculate the time each takes to complete one full lap and then find the Least Common Multiple (LCM) of those times.

The time taken by the first cyclist to complete one round is T₁ = (Distance)/(Speed) = ⁷⁵⁄₃₀ = 2.5 hours, which can be expressed as ⁵⁄₂ hours. The time taken by the second cyclist to complete one round is T₂ = ⁷⁵⁄₂₅ = 3 hours, or ³⁄₁ hours.

The cyclists will meet at the starting point at a time that is a multiple of both T₁ and T₂. The first such instance occurs at the LCM(T₁, T₂). For fractions, the LCM is calculated as:

LCM(⁵⁄₂, ³⁄₁) = (LCM(5, 3))/(HCF(2, 1))

Calculating these values gives LCM(5, 3) = 15 and HCF(2, 1) = 1. Therefore, the meeting time at the starting point is ¹⁵⁄₁ = 15 hours.

Explanation

The efficiency ratio of P to Q is 2 : 1. Since time taken is inversely proportional to efficiency, the ratio of time taken by P and Q is 1 : 2. Let the time taken by P be x days and the time taken by Q be 2x days. Given that P takes 40 days less than Q:

2x - x = 40 x = 40

Thus, P takes 40 days and Q takes 80 days to complete the work individually. To find the time taken working together, we use the formula for combined work:

Time = (Work)/(Total Efficiency)

Let the total work be the LCM of 40 and 80, which is 80 units. Efficiency of P = ⁸⁰⁄₄₀ = 2 units/day Efficiency of Q = ⁸⁰⁄₈₀ = 1 unit/day Combined efficiency = 2 + 1 = 3 units/day

Time taken together = ⁸⁰⁄₃ = 26.66... days.

Rounding to the nearest integer, the whole number of days is 27.

Explanation

Let the total number of persons enrolled in the voter list be x. The number of votes cast is 98% of x, or 0.98x, because 2% of the enrolled persons did not participate. According to the problem, candidate A received 43% of the total enrolled voters, so A's votes are 0.43x. Since candidate A defeated candidate B by 200 votes, B's votes are 0.43x - 200.

The total number of cast votes is the sum of valid votes and invalid votes. The valid votes are the sum of votes received by A and B. Therefore, we can set up the following equation: 0.98x = (0.43x) + (0.43x - 200) + 500

Simplifying the equation: 0.98x = 0.86x + 300 0.12x = 300 x = (300)/(0.12) = 2500

The total number of enrolled voters is 2500. The question asks for the total number of casted votes, which is 98% of the enrolled voters: Total casted votes = 0.98 × 2500 = 2450

Explanation

The given expression is sin 25° sin 65° - cos 25° cos 65°. This can be rewritten by factoring out a negative sign as -(cos 25° cos 65° - sin 25° sin 65°). Using the trigonometric identity cos(A + B) = cos A cos B - sin A sin B, where A = 25° and B = 65°, the expression simplifies to -cos(25° + 65°). Summing the angles gives -cos 90°. Since the value of cos 90° is 0, the result is 0. Alternatively, by using complementary angle relations where sin θ = cos(90° - θ), we have sin 65° = cos 25° and cos 65° = sin 25°. Substituting these into the original expression yields sin 25° cos 25° - cos 25° sin 25° = 0.

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if its last three digits are divisible by 8. For the number 98998, the last three digits are 998. Dividing 998 by 8 gives 998 ÷ 8 = 124 with a remainder of 6, since 8 × 124 = 992. Because the last three digits are not divisible by 8, the entire number 98998 is not divisible by 8.

Explanation

Let the third number be 100. The first number is 17% more than the third number, which gives 100 + 17 = 117. The second number is 50% more than the third number, which gives 100 + 50 = 150. The ratio of the first number to the second number is expressed as 117 : 150. Simplifying this ratio by dividing both terms by their greatest common divisor, 3, results in: ¹¹⁷⁄₃ : ¹⁵⁰⁄₃ = 39 : 50

Explanation

The expression (cos x - sin x)² + (cos x + sin x)² simplifies using the standard algebraic identity (a - b)² + (a + b)² = 2(a² + b²). By setting a = cos x and b = sin x, the expression becomes 2(cos² x + sin² x). Applying the fundamental trigonometric identity cos² x + sin² x = 1, the value of the expression is 2(1) = 2. The given condition cos x + sin x = √2cos x is specific to certain values of x, but the result of the identity remains constant for all values of x.

Explanation

The relationship between the number of workers and the time required to complete a fixed amount of work is inversely proportional, governed by the formula M₁ D₁ = M₂ D₂, where M is the number of persons and D is the number of days.

Given the initial conditions: M₁ = 14 persons D₁ = 60 days

For the second scenario: M₂ = 30 persons D₂ = ?

Substituting these values into the formula: 14 × 60 = 30 × D₂ 840 = 30 × D₂ D₂ = ⁸⁴⁰⁄₃₀ D₂ = 28

Since the total work (man-days) remains constant at 840, increasing the workforce to 30 persons reduces the time required to 28 days.

Explanation

The excluded number can be found by calculating the difference between the sum of the original eight numbers and the sum of the remaining seven numbers. The sum of a set of numbers is given by the formula Sum = Average × Count.

Initial sum of 8 numbers = 8 × 18 = 144 Sum of the remaining 7 numbers = 7 × 20 = 140

The excluded number is the difference between these two sums: Excluded number = 144 - 140 = 4

Alternatively, using the concept of deviations, the average of the remaining numbers increased from 18 to 20, which is an increase of 2 per number. For 7 numbers, this represents a total increase of 7 × 2 = 14. This increase occurred because the excluded number was 14 units less than the original average of 18. Thus, the excluded number is 18 - 14 = 4.

Explanation

Let the two numbers be x and y. According to the problem, the cube of their sum is (x + y)³ = 1728. Taking the cube root of both sides gives x + y = ³√(1728) = 12. We are also given that the product of the numbers is xy = 32.

To find the sum of their cubes, x³ + y³, we use the algebraic identity: x³ + y³ = (x + y)³ - 3xy(x + y)

Substituting the known values into the identity: x³ + y³ = 1728 - 3(32)(12) x³ + y³ = 1728 - 3(384) x³ + y³ = 1728 - 1152 x³ + y³ = 576

Alternatively, identifying two numbers whose sum is 12 and product is 32 leads to 8 and 4. Calculating the sum of their cubes directly gives 8³ + 4³ = 512 + 64 = 576.

Explanation

The ratio of the speeds of Ashok (S_A) and Bharat (S_B) is given by S_A : S_B = 2⅔ : 1 = ⁸⁄₃ : 1, which simplifies to 8 : 3. Since both runners reach the winning post at the same time, the time taken is constant. For a constant time, the distance covered is directly proportional to the speed (D ∝ S). Therefore, the ratio of the distance covered by Ashok (D_A) to the distance covered by Bharat (D_B) is also 8 : 3.

Let the distance to the winning post be D. Ashok covers the entire distance D, while Bharat, having a head start of 160m, covers a distance of D - 160. Setting up the ratio based on their speeds: (D)/(D - 160) = ⁸⁄₃

Cross-multiplying gives 3D = 8(D - 160), which simplifies to 3D = 8D - 1280. Rearranging the terms, we get 5D = 1280. Solving for D yields D = ¹²⁸⁰⁄₅ = 256m.

Alternatively, using the ratio method, the difference between the distance units is 8 - 3 = 5 units. This difference represents the head start given to Bharat, so 5 units is equal to 160m. One unit is therefore ¹⁶⁰⁄₅ = 32m. Since Ashok covers 8 units to reach the winning post, the total distance is 8 × 32 = 256m.

Explanation

The value of the expression is determined using the trigonometric identity for complementary angles, which states that sin(90° - θ) = cos θ. Since 32° + 58° = 90°, the angle 58° can be written as 90° - 32°.

Applying the identity: sin 58° = sin(90° - 32°) = cos 32°

Squaring both sides, we get: sin² 58° = cos² 32°

Substituting this relationship into the given expression: cos² 32° - sin² 58° = cos² 32° - cos² 32° = 0

Explanation

To find the value of the expression 64x³ + 38x²y + 20xy² + y³, substitute the given values x = 3 and y = -4 into each term individually.

The first term is 64x³ = 64 × 3³ = 64 × 27 = 1728. The second term is 38x²y = 38 × 3² × (-4) = 38 × 9 × (-4) = -1368. The third term is 20xy² = 20 × 3 × (-4)² = 20 × 3 × 16 = 960. The fourth term is y³ = (-4)³ = -64.

Adding these values together gives the total value of the expression: 1728 - 1368 + 960 - 64 = 360 + 960 - 64 = 1320 - 64 = 1256.