SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

If y + (1)/(y) = k, then the value of y³ + (1)/(y³) = k³ - 3k and the value of y² + (1)/(y²) = k² - 2. To find y⁶ + (1)/(y⁶), we can use the relation (y³ + (1)/(y³))² - 2.

Given y + (1)/(y) = 4, we first calculate the cubic sum: y³ + (1)/(y³) = 4³ - 3(4) = 64 - 12 = 52

Next, we square this value to find the sixth power: y⁶ + (1)/(y⁶) = (y³ + (1)/(y³))² - 2(y³)((1)/(y³)) y⁶ + (1)/(y⁶) = 52² - 2 y⁶ + (1)/(y⁶) = 2704 - 2 = 2702

Alternatively, calculating the square first gives y² + (1)/(y²) = 4² - 2 = 14. Cubing this result yields 14³ - 3(14) = 2744 - 42 = 2702.

Explanation

Let the efficiency of B be 2 units per day. Since A is 50% more efficient than B, the efficiency of A is 2 + (0.50 × 2) = 3 units per day. Thus, the ratio of efficiencies of A to B is 3:2.

The total work is the product of B's efficiency and the time B takes to complete the work: Total Work = Efficiency of B × Time taken by B = 2 × 20 = 40 units.

When A and B work together, their combined efficiency is the sum of their individual efficiencies: Combined Efficiency = 3 + 2 = 5 units per day.

The time taken by A and B together to finish the same work is: Time = Total Work / Combined Efficiency = 40 / 5 = 8 days.

Explanation

The area A of a sector of a circle with radius r and arc length l is given by the formula A = ½rl. This formula is derived from the relationship between the sector area and the arc length as fractions of the total area and circumference of the circle respectively.

By substituting the given values r = 8 cm and l = 12 cm into the formula:

Area = ½ × 8 × 12

Area = 4 × 12

Area = 48 cm²

Explanation

Let the original cost price of the article be x. The price is first decreased by 10%, which means the price becomes 90% of its original value. This resulting price is then increased by 20%, making it 120% of the intermediate value. The relationship between the original price and the final price can be expressed as:

x × (1 - ¹⁰⁄₁₀₀) × (1 + ²⁰⁄₁₀₀) = 540

x × 0.9 × 1.2 = 540

x × 1.08 = 540

To find the original cost price x, we divide the final price by the net multiplier:

x = (540)/(1.08)

x = ⁵⁴⁰⁰⁰⁄₁₀₈

x = 500

By calculating 540 ÷ 1.08, we find that the original cost price was £500. Alternatively, using the successive percentage change concept, the net change is -10 + 20 + (-10 × 20)/(100) = 10 - 2 = 8% increase. Thus, 108% of the original price equals 540, which yields 500 as the base value.

Explanation

The curved surface area (CSA) of a cylinder is given by the formula 2π rh, where r is the radius and h is the height. Substituting the given values CSA = 2640 cm and h = 12 cm into the formula:

2 × ²²⁄₇ × r × 12 = 2640

r = (2640 × 7)/(2 × 22 × 12)

r = (2640 × 7)/(528) = 5 × 7 = 35 cm

The volume (V) of a cylinder is calculated using the formula V = π r² h. Substituting the radius r = 35 cm and height h = 12 cm:

V = ²²⁄₇ × 35 × 35 × 12

V = 22 × 5 × 35 × 12

V = 110 × 420 = 46200 cm³

Explanation

Following the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), the expression is solved by first prioritizing the innermost brackets. Within the parentheses, the operation is performed: -6 ÷ 2 = -3. Next, the terms inside the curly brackets are evaluated by performing the multiplication 3 × (-3) = -9, followed by the addition: {-9 + (-9)} = -18.

Moving to the square brackets, the expression simplifies to [-2 + (-18)] = -20. Meanwhile, the division at the beginning of the expression is calculated: 100 ÷ 10 = 10. Finally, the subtraction is performed: 10 - (-20), which simplifies to 10 + 20 = 30.

Explanation

The fourth proportional x to numbers a, b, and c is calculated using the formula a/b = c/x. Substituting the given values, 8/32 = 13/x, which simplifies to 1/4 = 13/x, resulting in x = 52.

Explanation

To find the rate of simple interest, first calculate the interest earned: £10,200 - £9,000 = £1,200. Using the formula Rate = (Simple Interest × 100) / (Principal × Time), we get (1,200 × 100) / (9,000 × 4) = 120,000 / 36,000 = 10/3 = 3 1/3 %.

Explanation

According to the Remainder Theorem, the remainder when a polynomial P(x) is divided by (x + 1) is given by P(-1). Substituting x = -1 into the expression x¹⁷ + 1 gives (-1)¹⁷ + 1, which equals -1 + 1 = 0.

Explanation

Let u be the speed of the boat in still water and v be the speed of the water current. The speed of the boat while running downstream is u + v, and the speed while running upstream is u - v. Since the distance covered in both cases is the same, the speed is inversely proportional to the time taken. This relationship is expressed as:

(u + v)/(u - v) = (Time taken upstream)/(Time taken downstream)

Substituting the given values of 10 hours for the upstream journey and 7 hours for the downstream journey:

(u + v)/(u - v) = ¹⁰⁄₇

Applying the principle of componendo and dividendo, which states that if a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d):

((u + v) + (u - v))/((u + v) - (u - v)) = (10 + 7)/(10 - 7)

Simplifying the terms in the numerator and denominator:

²ᵘ⁄₂ᵥ = ¹⁷⁄₃

ᵘ⁄ᵥ = ¹⁷⁄₃

The ratio between the speed of the boat and the speed of the water current is 17:3.

Explanation

Using the weighted average formula (2 × 35 + 1 × 38 + 2 × x) / 5 = 42, we calculate the total cost as 108 + 2x = 210. Solving for x, we get 2x = 102, which results in the price of the third variety being ₹51 per kilogram.

Explanation

When two circles with radii r₁ and r₂ touch each other externally, the distance d between their centers O and P is the sum of their radii, d = r₁ + r₂. The length of the direct common tangent AB is calculated using the formula √(d² - (r₁ - r₂)²). By substituting d = r₁ + r₂ into the equation, the expression simplifies to AB = √((r₁ + r₂)² - (r₁ - r₂)²) = √(4r₁r₂) = 2√(r₁r₂).

Given the radii r₁ = 6 cm and r₂ = 3 cm, the calculation is AB = 2√(6 × 3) = 2√(18). Since 18 can be factored into 9 × 2, its square root is 3√2. Therefore, the length of the common tangent AB is 2 × 3√2 = 6√2 cm.

Explanation

Substitute the trigonometric values into the expression: 5√3(1/2)(1/√3) - 3(1) + 3(1/2) + 2(3/4). This simplifies to 2.5 - 3 + 1.5 + 1.5, which equals 2.5.

Explanation

The speed of the stream vₛ is determined by the formula vₛ = (D - U)/(2), where D is the downstream speed (along the stream) and U is the upstream speed (against the stream). This formula is derived from the definitions of downstream speed (v_b + vₛ) and upstream speed (v_b - vₛ), where v_b is the speed of the boat in still water.

By substituting the given values into the formula:

vₛ = (20.8 - 15.2)/(2)

vₛ = (5.6)/(2)

vₛ = 2.8 km/h

The difference between the two effective speeds represents twice the speed of the stream, as the stream assists the boat in one direction and hinders it by the same magnitude in the other. Dividing this difference by two provides the exact speed of the current.

Explanation

Total work can be seen as 12 man-days (2 men * 6 days) or 12 woman-days (3 women * 4 days), meaning 1 man and 1 woman have equal efficiency. Since 1 man performs 1/12 of the work per day and 2 women perform 2/12 of the work per day, their combined daily work is 3/12 (or 1/4), resulting in the work being finished in 4 days.

Explanation

The net percentage change for a successive increase of 30% and decrease of 30% is calculated using the formula x + y + (xy/100), resulting in 30 - 30 + (30 * -30 / 100) = -9%. A negative result indicates a 9% decrease in total income.

Explanation

Since cos²θ = 3/4 and θ is acute, cos θ = √3/2, which means θ = 30°. Substituting this into the expression, sin(30° + 30°) = sin 60° = √3/2.

Explanation

B's daily work rate is found by subtracting A's rate (1/14) from the combined rate of A and B (2/7), which results in 3/14. Taking the reciprocal of this rate, B alone takes 14/3 days to complete the work.

Explanation

From the given equations, we can isolate the trigonometric terms as ʳ⁄₁₅ = sin θ + cos θ and ˢ⁄₁₆ = sin θ - cos θ. To find the required value, we square both equations:

(ʳ⁄₁₅)² = (sin θ + cos θ)² = sin² θ + cos² θ + 2sin θ cos θ (ˢ⁄₁₆)² = (sin θ - cos θ)² = sin² θ + cos² θ - 2sin θ cos θ

Using the fundamental identity sin² θ + cos² θ = 1, these expressions simplify to:

(r²)/(15²) = 1 + 2sin θ cos θ (s²)/(16²) = 1 - 2sin θ cos θ

Adding these two equations eliminates the 2sin θ cos θ term:

(r²)/(15²) + (s²)/(16²) = (1 + 2sin θ cos θ) + (1 - 2sin θ cos θ) = 1 + 1 = 2

Explanation

The population below the poverty line in State B is 25% of 7,200, which is 1,800. According to the ratio of 3:5 for males and females below the poverty line, the number of females is calculated as (5/8) * 1,800, which equals 1,125.

Explanation

For a cube with side length a, the length of the longest diagonal is given by the formula d = a√3. Given that the length of the diagonal is 7√3 cm, we can determine the side length by equating the formula to the given value:

a√3 = 7√3

By cancelling √3 from both sides, we find that the side length a = 7 cm. The volume V of a cube is calculated using the formula V = a³. Substituting the side length into this formula:

V = 7³ = 343 cm³

Explanation

The mean proportional M of two expressions x and y is defined as the square root of their product, given by the formula M = √(xy).

To find the mean proportional of (a + b)(a - b)³ and (a + b)³(a - b), we substitute these expressions into the formula:

M = √((a + b)(a - b)³ × (a + b)³(a - b))

Using the algebraic rule of exponents xᵐ × xⁿ = xᵐ⁺ⁿ, we group the terms with common bases:

M = √((a + b)¹⁺³⁾^×⁽ᵃ⁻ᵇ⁾^^^³^⁺^¹ M = √((a + b)⁴ × (a - b)⁴)

Applying the property √(xⁿ) = xⁿ^/², we simplify the square root:

M = (a + b)⁴^/² × (a - b)⁴^/² M = (a + b)² (a - b)²

Explanation

Let the initial number of students be n. If the project was originally intended to be completed in 10 days by these students, the total work required for the project is 10n student-days.

The actual work done over 15 days, with 2 students dropping out each day, forms an arithmetic progression where the first term a = n, the common difference d = -2, and the number of terms k = 15. Using the sum formula for an arithmetic progression Sₖ = ᵏ⁄₂ [2a + (k - 1)d], the total work completed is:

S₁₅ = ¹⁵⁄₂ [2n + (15 - 1)(-2)]

S₁₅ = ¹⁵⁄₂ [2n - 28]

S₁₅ = 15(n - 14)

Since the project is completed, the work done must equal the total work required:

10n = 15(n - 14)

10n = 15n - 210

5n = 210

n = 42

Explanation

The relationship between the number of men and the time taken to complete a piece of work is governed by the inverse variation formula M₁ D₁ = M₂ D₂, where M represents the number of men and D represents the number of days. Initially, we have M₁ = 4 men and D₁ = 18 days. The total work capacity required is 4 × 18 = 72 man-days.

To complete the same work in D₂ = 12 days, the required total number of men M₂ is calculated as M₂ = (M₁ × D₁) / D₂. Substituting the values, we get M₂ = 72 / 12 = 6 men. The question asks for how many more men are required beyond the original group. Therefore, we subtract the initial number of men from the total required number: 6 - 4 = 2.

Explanation

Using the trigonometric identity sin θ = cos(90° - θ), the given equation cos (40° + x) = sin 30° can be rewritten by expressing the right-hand side in terms of cosine. Since sin 30° = cos(90° - 30°) = cos 60°, the equation becomes:

cos (40° + x) = cos 60°

By equating the angles, we have: 40° + x = 60° x = 60° - 40° x = 20°

Alternatively, substituting the standard trigonometric value sin 30° = 1/2 into the equation gives cos (40° + x) = 1/2. Since cos 60° = 1/2 for acute angles, setting 40° + x = 60° results in x = 20°.

Explanation

The time T taken for the first meeting of two runners moving in opposite directions from the same point on a circular track is given by the formula T = (L)/(V_P + V_Q), where L is the track length and V_P, V_Q are their respective speeds. Substituting the given values, T = (500)/(6 + 10) = ⁵⁰⁰⁄₁₆ = 31.25 seconds.

At the moment of meeting, P has covered a distance d_P = V_P × T = 6 × 31.25 = 187.5 m. To complete the lap and reach the starting point while continuing in the same direction, P needs to cover the remaining distance D_P = L - d_P = 500 - 187.5 = 312.5 m. Similarly, Q has covered d_Q = V_Q × T = 10 × 31.25 = 312.5 m and needs to cover the remaining distance D_Q = L - d_Q = 500 - 312.5 = 187.5 m to reach the start.

After the exchange, P moves at 10 m/s and Q moves at 6 m/s. The time P takes to reach the start is (D_P)/(10) = (312.5)/(10) = 31.25 seconds. The time Q takes to reach the start is (D_Q)/(6) = (187.5)/(6) = 31.25 seconds. Because the time taken by both after the exchange is identical, they reach the starting point at the same time.

Explanation

The area of the paper used to cover the cuboid is equal to its total surface area (TSA). The formula for the TSA of a cuboid with dimensions l, b, and h is TSA = 2(lb + bh + hl). Substituting the given dimensions l = 4, b = 6, and h = 8, we calculate the area as TSA = 2(4 × 6 + 6 × 8 + 8 × 4) = 2(24 + 48 + 32) = 2 × 104 = 208 square units.

A square is then formed using this same paper, meaning the area of the square is 208 square units. If s represents the side of the square, then s² = 208. To find the side length, we take the square root: s = √(208). Evaluating this, we find 14.4² = 207.36 and 14.5² = 210.25, which indicates the side is approximately 14.422 units. Rounded to two decimal places, the side of the square is 14.42.

Explanation

The weighted average rate of interest obtained on the total investment of ₹50,000 is given by R = (Total Interest)/(Total Principal) × 100 = (5,200)/(50,000) × 100 = 10.4%.

Using the rule of allegation to find the ratio of amounts invested at 8% and 12%: The difference between the higher rate and the mean rate is 12 - 10.4 = 1.6. The difference between the mean rate and the lower rate is 10.4 - 8 = 2.4.

The ratio of the principal amounts is 1.6 : 2.4, which simplifies to 2 : 3.

The total investment is divided into 2 + 3 = 5 equal parts. The amount deposited at the 12% rate corresponds to 3 parts of the total investment. Amount at 12% = ⅗ × 50,000 = 3 × 10,000 = 30,000.

Explanation

The average amount of salary is calculated by dividing the total salary paid over the four years by the number of years. Based on the table, the annual salary expenditures in thousand rupees for the years 2018, 2019, 2020, and 2021 are 200, 250, 300, and 350, respectively.

Summing these values gives the total salary expenditure: Total Salary = 200 + 250 + 300 + 350 = 1100 thousand rupees.

To find the average, divide this total by the number of years: Average Salary = ¹¹⁰⁰⁄₄ = 275 thousand rupees.

Converting this into numerical form by multiplying by 1,000, the average annual salary expenditure is 275 × 1000 = 2,75,000.

Explanation

Let the area of the plot owned by Basab be B, the area owned by Tarun be T, and the area owned by Nakul be N. The problem states that T is 10% more than B and N is 40% more than T. These relationships can be expressed as: T = B × (1 + 0.10) = 1.1B N = T × (1 + 0.40) = 1.4T

By substituting the value of T in the equation for N, we get: N = 1.4 × (1.1B) = 1.54B

The area owned by Nakul is given as 2695 square feet. Substituting this value into the equation: 2695 = 1.54B B = (2695)/(1.54)

To simplify the calculation: B = (2695 × 100)/(154) B = ²⁶⁹⁵⁰⁰⁄₁₅₄ B = 1750

The area of the plot owned by Basab is 1750 square feet.

Explanation

The expression is simplified using the perfect square identity x² + 2xy + y² = (x + y)² and the difference of squares identity x² - y² = (x - y)(x + y).

The numerator s² + t² + 2st - u² can be grouped as (s² + 2st + t²) - u². This simplifies to (s + t)² - u². Applying the difference of squares identity, the numerator factors into (s + t - u)(s + t + u).

The denominator s² - t² - 2tu - u² can be grouped as s² - (t² + 2tu + u²). This simplifies to s² - (t + u)². Applying the difference of squares identity, the denominator factors into (s - (t + u))(s + (t + u)), which simplifies to (s - t - u)(s + t + u).

The entire expression becomes: ((s + t - u)(s + t + u))/((s - t - u)(s + t + u))

Given the condition (s + t + u) ≠ 0, the common term (s + t + u) in the numerator and denominator cancels out, leaving: (s + t - u)/(s - t - u)

Explanation

To simplify the expression (1 / sin θ) - sin θ, we first find a common denominator:

(1 / sin θ) - sin θ = (1 - sin² θ)/(sin θ)

By applying the fundamental Pythagorean identity sin² θ + cos² θ = 1, we can substitute the numerator 1 - sin² θ with cos² θ. The expression then becomes:

(cos² θ)/(sin θ)

This fraction can be factored into a product of two terms:

cos θ · (cos θ)/(sin θ)

Using the trigonometric definition cot θ = cos θ / sin θ, we substitute this into the equation to arrive at the simplified result:

cos θ cot θ

Explanation

The cost price (CP) of the mangoes is calculated based on the dozen rate. Since there are 12 mangoes in a dozen, the CP per mango is 120 / 12 = ₹10. For 1600 mangoes, the total cost price is 1600 × 10 = ₹16,000.

The total selling price (SP) is the sum of the two sales. First, 900 mangoes were sold at ₹15 each, yielding 900 × 15 = ₹13,500. The remaining 1600 - 900 = 700 mangoes were sold at ₹14 each, yielding 700 × 14 = ₹9,800. Adding these gives a total SP of 13,500 + 9,800 = ₹23,300.

The total gain is the difference between the total SP and total CP, which is 23,300 - 16,000 = ₹7,300. The gain percentage is determined by the formula Gain Percentage = (Total Gain / Total CP) × 100. Substituting the values, we get (7,300 / 16,000) × 100 = 730 / 16 = 45.625%.

Explanation

The thief runs for a total of 20 minutes (4 minutes before the chase plus 16 minutes during the chase). The distance covered by the thief in this time is dₜ = speed × time = 15 × ²⁰⁄₆₀ = 5 km . At the 16-minute mark of the chase, there is still a 200 m (0.2 km) gap between them, which means the policeman has covered 5 - 0.2 = 4.8 km in 16 minutes. The speed of the policeman is Vₚ = (Distance)/(Time) = (4.8)/(16/60) = (4.8 × 60)/(16) = 18 km/h .

To find the catching distance, we first determine the initial lead of the thief when the chase began after 4 minutes: Lead = 15 × ⁴⁄₆₀ = 1 km . The relative speed at which the policeman closes this lead is Vᵣₑₗ = Vₚ - Vₜ = 18 - 15 = 3 km/h . The time required to catch the thief is t = (Lead)/(Vᵣₑₗ₎ = ⅓ hours . Thus, the distance from the crime spot where the policeman catches the thief is D = Vₚ × t = 18 × ⅓ = 6 km .

Explanation

The expression a³ + b³ - c³ + 3abc follows the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx)) where x = a, y = b, and z = -c.

First, determine the value of the combined product term using the square of the trinomial: (a + b - c)² = a² + b² + (-c)² + 2(ab + b(-c) + (-c)a) 20² = a² + b² + c² + 2(ab - bc - ca)

Substitute the given value a² + b² + c² = 152: 400 = 152 + 2(ab - bc - ca) 248 = 2(ab - bc - ca) ab - bc - ca = 124

Now, substitute these values into the cubic identity: a³ + b³ - c³ + 3abc = (a + b - c)(a² + b² + c² - (ab - bc - ca)) a³ + b³ - c³ + 3abc = 20 × (152 - 124) a³ + b³ - c³ + 3abc = 20 × 28 a³ + b³ - c³ + 3abc = 560

Explanation

The initial amount of alcohol in the 40-litre mixture is calculated as 40 × ²⁵⁄₁₀₀ = 10 litres. When 10 litres of water are added, the new total volume of the mixture becomes 40 + 10 = 50 litres. The quantity of alcohol remains unchanged at 10 litres. Therefore, the percentage of alcohol in the new 50-litre mixture is ¹⁰⁄₅₀ × 100 = 20%.

Explanation

The rate of discount is calculated using the formula Discount% = (Marked Price - Selling Price)/(Marked Price) × 100.

Given the Marked Price (MP) as ₹15,990 and the Selling Price (SP) as ₹12,792, first find the absolute discount: Discount = 15,990 - 12,792 = 3,198

Now, substitute the values into the percentage formula: Rate of discount = (3,198)/(15,990) × 100

Simplifying the fraction, notice that 1,599 × 2 = 3,198. Therefore, 3,198 is exactly 20% of 15,990: Rate of discount = 0.2 × 100 = 20%

Explanation

To solve the expression -7p - [3q - {8p - (4q - 10p)}], we simplify from the innermost bracket outward following the standard order of operations for algebraic expressions.

First, we remove the innermost parentheses by distributing the negative sign into (4q - 10p): -7p - [3q - {8p - 4q + 10p}]

Next, we combine the like terms inside the curly braces, specifically 8p and 10p: -7p - [3q - {18p - 4q}]

We then remove the curly braces by distributing the negative sign into the terms {18p - 4q}: -7p - [3q - 18p + 4q]

Now, we combine the like terms within the square brackets, specifically 3q and 4q: -7p - [7q - 18p]

Finally, we remove the square brackets by distributing the negative sign: -7p - 7q + 18p

By combining the remaining like terms -7p and 18p, we obtain: 11p - 7q

Explanation

Let the total work be 1 unit. The daily work rate of a single woman is ¹⁄₁₅₀ because she completes the work in 150 days. The combined daily work rate of 20 women and 15 men is ⅙ as they complete the work in 6 days.

Substituting the woman's rate into the combined equation: 20 × (¹⁄₁₅₀) + 15 × (Rate of 1 man) = ⅙

Simplifying the term for 20 women: ²⁰⁄₁₅₀ = ²⁄₁₅

Now, isolate the rate of 15 men: 15 × (Rate of 1 man) = ⅙ - ²⁄₁₅

Finding a common denominator of 30 for the subtraction: 15 × (Rate of 1 man) = ⁵⁄₃₀ - ⁴⁄₃₀ = ¹⁄₃₀

The daily work rate of a single man is: Rate of 1 man = (1)/(30 × 15) = ¹⁄₄₅₀

Since a single man completes ¹⁄₄₅₀ of the work in one day, the time required for a single man to complete the entire work is 450 days.

Explanation

A number is divisible by 44 if it satisfies the divisibility criteria for both 4 and 11, as 44 = 4 × 11 and these factors are co-prime.

To check for divisibility by 4, the last two digits of the number must form a number divisible by 4. For the number 54736, the last two digits are 36. Since 36 = 4 × 9, the number is divisible by 4.

To check for divisibility by 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be either 0 or a multiple of 11. For the number 54736, the digits at odd positions are 5, 7, and 6, while the digits at even positions are 4 and 3.

Sum of digits at odd positions: 5 + 7 + 6 = 18 Sum of digits at even positions: 4 + 3 = 7 Difference: 18 - 7 = 11

Since the difference 11 is a multiple of 11, the number 54736 is divisible by 11. Because the number is divisible by both 4 and 11, it is divisible by 44.

Explanation

To simplify the expression √x - √y{√x + √y} with the given values x = 2 and y = 3, substitute the variables to obtain the fraction √2 - √3{√2 + √3}. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is √2 - √3.

The denominator is simplified using the difference of squares identity (a + b)(a - b) = a² - b². Applying this, we get (√2 + √3)(√2 - √3) = (√2)² - (√3)² = 2 - 3 = -1.

The numerator is simplified using the square of a binomial identity (a - b)² = a² + b² - 2ab. This yields (√2 - √3)² = (√2)² + (√3)² - 2(√2)(√3) = 2 + 3 - 2√6 = 5 - 2√6.

Combining these results, the expression becomes 5 - 2√6{-1}. Dividing by -1 reverses the signs of the terms in the numerator, resulting in -5 + 2√6, which is rearranged as 2√6 - 5.

Explanation

Let the total initial volume of the mixture be 16 units, consisting of 5 units of water and 11 units of syrup. When a part of the mixture is drawn off and replaced with water, the total volume remains constant at 16 units, but the quantity of syrup only decreases during the removal process.

In the final mixture, the ratio of syrup to water is 3:2. This means the quantity of syrup in the final 16 units is ³⁄₃₊₂ × 16 = ⅗ × 16 = ⁴⁸⁄₅ units.

Let x be the fraction of the mixture that is drawn off and replaced. The quantity of syrup removed is 11x. Therefore, the remaining syrup is 11 - 11x, which must equal the final quantity of syrup:

11(1 - x) = ⁴⁸⁄₅

1 - x = ⁴⁸⁄₅₅

x = 1 - ⁴⁸⁄₅₅ = ⁷⁄₅₅

Thus, ⁷⁄₅₅ part of the mixture must be drawn off and replaced with water.

Explanation

The length of the direct common tangent between two circles that touch each other externally is given by the formula 2√(r₁ r₂), where r₁ and r₂ are the radii of the two circles. This formula is derived from the general length of a direct common tangent d = √(D² - (r₁ - r₂)²), where D is the distance between the centers. When circles touch externally, D = r₁ + r₂.

Substituting the given radii r₁ = 12 cm and r₂ = 8 cm into the simplified formula: BC = 2√(12 × 8) BC = 2√(96)

Simplifying the radical by factoring out the perfect square 16: BC = 2√(16 × 6) BC = 2 × 4√6 BC = 8√6 cm

Explanation

In △ ABC, since DE ∥ BC, △ ADE is similar to △ ABC by the AA similarity criterion because ∠ ADE = ∠ ABC and ∠ AED = ∠ ACB are corresponding angles. For similar triangles, the ratio of corresponding sides is proportional:

(AD)/(AB) = (DE)/(BC)

The problem provides the ratio (AD)/(BD) = ⅘. Since AB = AD + BD, the ratio of the side of the smaller triangle to the side of the larger triangle is:

(AD)/(AB) = (AD)/(AD + BD) = (4)/(4 + 5) = ⁴⁄₉

Substituting the side lengths and the similarity ratio into the proportionality equation:

⁴⁄₉ = (DE)/(BC)

Given DE = 12 cm, we substitute this value into the equation:

⁴⁄₉ = (12)/(BC)

Solving for BC:

4 · BC = 12 · 9

BC = ¹⁰⁸⁄₄ = 27 cm

Explanation

Starting from the given quadratic equation x² + 8x - 1 = 0, dividing every term by x results in x + 8 - ¹⁄ₓ = 0, which simplifies to x - ¹⁄ₓ = -8. To find the value of x² + (1)/(x²), we use the standard algebraic relation where if x - ¹⁄ₓ = k, then x² + (1)/(x²) = k² + 2. This relation is derived from the expansion of the square of a binomial: (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2. Substituting k = -8 into the formula gives (-8)² + 2, which results in 64 + 2 = 66.

Explanation

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For the four-digit number abba, the last two digits form the number 10b + a. Since it is a four-digit number, the leading digit a cannot be zero, so a ∈ {1, 2, …, 9}. The problem also specifies the constraint a < b.

A two-digit number 10b + a is divisible by 4 under the following conditions: If b is odd, a must be 2 or 6. If b is even, a must be 0, 4, or 8.

We evaluate possible values for b from 1 to 9, applying the condition 1 ≤ a < b: For b=1, 2: No values of a satisfy both 1 ≤ a < b and the divisibility rule. For b=3: a=2 (Number: 2332) For b=4: No valid a (since a ∈ {0, 4, 8} and 1 ≤ a < 4 has no intersection). For b=5: a=2 (Number: 2552) For b=6: a=4 (Number: 4664) For b=7: a=2 and a=6 (Numbers: 2772, 6776) For b=8: a=4 (Number: 4884) For b=9: a=2 and a=6 (Numbers: 2992, 6996)

Summing these cases, there are 1 + 1 + 1 + 2 + 1 + 2 = 8 such numbers.

Explanation

The expression x⁴ - (1)/(x⁴⁾ can be factored using the difference of squares identity as (x² + (1)/(x²⁾)(x² - (1)/(x²⁾). We are given that x² + (1)/(x²⁾ = 6. To find the value of the second factor, we use the algebraic relation (a - b)² = (a + b)² - 4ab. Applying this to x² and (1)/(x²⁾, we get:

(x² - (1)/(x²⁾)² = (x² + (1)/(x²⁾)² - 4(x²)((1)/(x²⁾) (x² - (1)/(x²⁾)² = 6² - 4 = 36 - 4 = 32 x² - (1)/(x²⁾ = ± √(32) = ± 4√2

The condition 0 < x < 1 implies that x² < 1. Since x² is a fraction between 0 and 1, its reciprocal (1)/(x²⁾ must be greater than 1. This means x² < (1)/(x²⁾, so the difference x² - (1)/(x²⁾ must be negative. Thus, we take the negative root: x² - (1)/(x²⁾ = - 4√2.

Finally, substituting these values into the factored form: x⁴ - (1)/(x⁴⁾ = (6) × (- 4√2) = - 24√2

Explanation

Let the upstream speed be u km/h and the downstream speed be v km/h. According to the time-distance relationship Time = Distance / Speed, the given conditions provide two equations: 20/u + 44/v = 8 and 15/u + 22/v = 5. To solve this system efficiently, multiply the second equation by 2 to get 30/u + 44/v = 10. Subtracting the first equation from this result yields (30 - 20)/u = 10 - 8, which simplifies to 10/u = 2, or u = 5 km/h.

Substituting u = 5 back into the second equation gives 15/5 + 22/v = 5, which simplifies to 3 + 22/v = 5, meaning 22/v = 2, or v = 11 km/h. The speed of the boat in still water is the arithmetic mean of the downstream and upstream speeds, given by the formula Boat Speed = (v + u) / 2. Substituting the values, we find the speed is (11 + 5) / 2 = 16 / 2 = 8 km/h.

Explanation

The problem establishes work equivalence ratios between men (M), women (W), and boys (B). From 4M = 6W, we find the efficiency of a man relative to a woman as 1M = ⁶⁄₄W = 1.5W. Similarly, from 4W = 6B, the efficiency of a woman relative to a boy is 1W = ⁶⁄₄B = 1.5B.

By substitution, the efficiency of a man in terms of boys is 1M = 1.5 × (1.5B) = 2.25B. The combined work rate of one man and one woman is 1M + 1W = 2.25B + 1.5B = 3.75B.

Using the work formula M₁ D₁ = M₂ D₂, where M represents the number of workers (or their equivalent efficiency) and D represents days: 1B × 60 = 3.75B × D D = (60)/(3.75) D = ⁶⁰⁰⁰⁄₃₇₅ D = 16 days

Explanation

When two objects move in the same direction, their relative speed is the difference between their individual speeds. In this case, the relative speed of the policeman with respect to the thief is 12 km/h - 10 km/h = 2 km/h. The initial distance separating them is 400 m, which is equivalent to 0.4 km. The time required for the policeman to close this gap and catch the thief is calculated using the formula t = (Distance)/(Relative Speed). Substituting the values, we get t = (0.4 km)/(2 km/h) = 0.2 hours.

To determine the distance from the starting point where the catch occurs, we calculate the total distance covered by the policeman during this time. Using the formula Distance = Speed × Time, the distance is 12 km/h × 0.2 hours = 2.4 km. This can also be solved using the ratio of speeds. Since time is constant, the ratio of distances covered is the same as the ratio of their speeds, which is 12:10 or 6:5. The difference of 1 unit in the ratio corresponds to the initial 400 m lead. Thus, 6 units represent the policeman's distance: 6 × 400 m = 2400 m, or 2.4 km.