SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Let d be the one-way distance between the two places.

The total time taken is the sum of the time taken for the onward journey and the return journey. Using the formula:

Time = Distance / Speed

Total Time = (d / v₁) + (d / v₂)

Substituting the given values:

6 = (d / 60) + (d / 75)

Taking the common denominator of 60 and 75, which is 300:

6 = (5d + 4d) / 300

6 = 9d / 300

9d = 1800

d = 200 km

Since the person goes and returns, the total distance travelled is twice the one-way distance:

Total Distance = 2d = 2 × 200 = 400 km.

Explanation

Since O is the intersection of the angle bisectors of A and B, it is the incenter of triangle ABC. The angle BOC is given by the formula 90 + A/2, which equals 108 degrees, giving A/2 = 18 degrees. Since AO is the angle bisector of angle A, the angle BAO is equal to A/2, which is 18 degrees.

Explanation

In any triangle ABC, the angle subtended by the sides at the incentre O is given by the standard relation: ∠BOC = 90° + ∠A / 2

First, determine the third angle of the triangle, ∠A, using the angle sum property: ∠A = 180° - (∠ABC + ∠ACB) ∠A = 180° - (70° + 50°) ∠A = 60°

Substituting this value into the relation: ∠BOC = 90° + 60° / 2 ∠BOC = 90° + 30° = 120°

Explanation

The given expression can be factored as k³³ - k = k(k³² - 1).

To find the number by which this expression is always divisible for any odd integer k > 1, we analyze its divisibility by 3 and 8.

First, let us test divisibility by 8. Every odd integer k can be written in the form 2n + 1. k² - 1 = (2n + 1)² - 1 = 4n² + 4n = 4n(n + 1) Since n and n + 1 are consecutive integers, one of them must be even, making their product n(n + 1) divisible by 2. Thus, 4n(n + 1) is always divisible by 8, which means k² ≡ 1 (mod 8). Consequently, k³² = (k²)¹⁶ ≡ 1¹⁶ ≡ 1 (mod 8), which implies k³² - 1 is always divisible by 8.

Next, let us test divisibility by 3. If k is a multiple of 3, then the entire expression k(k³² - 1) is divisible by 3. If k is not a multiple of 3, then k ≡ ±1 (mod 3), which gives k² ≡ 1 (mod 3). Thus, k³² = (k²)¹⁶ ≡ 1¹⁶ ≡ 1 (mod 3), which means k³² - 1 is divisible by 3. In all cases, the expression k(k³² - 1) is divisible by 3.

Since 3 and 8 are coprime, the expression k³³ - k is always divisible by 3 × 8 = 24.

Explanation

The new price of the motorcycle is calculated by adding the 20% increase to the original price.

Using the percentage increase formula: New Price = Original Price × (1 + Increase%)

Substituting the given values: New Price = 75000 × (1 + 20/100) New Price = 75000 × 1.2 New Price = 90000

Thus, the price of the motorcycle this year is ₹ 90,000.

Explanation

To find the total number of students who have failed in more than 3 subjects, we calculate the sum of the students who failed in exactly 4 subjects and exactly 5 subjects.

From the given table: The number of students who failed in 4 subjects is 7. The number of students who failed in 5 subjects is 4.

Total number of students = 7 + 4 = 11.

Explanation

The coordinates where the line 20x + 21y = 420 cuts the coordinate axes are found by setting y = 0 and x = 0.

When y = 0, the x-intercept is: 20x = 420 x = 21 This gives the point P(21, 0).

When x = 0, the y-intercept is: 21y = 420 y = 20 This gives the point Q(0, 20).

The length of the line segment PQ is the distance between P and Q, which can be calculated using the Pythagorean theorem since the axes are perpendicular: PQ = √(21² + 20²) PQ = √(441 + 400) PQ = √841 PQ = 29

Explanation

Let S be the total monthly salary.

The total expenditure is the sum of all individual expense percentages: Total expenditure = 60% + 10% + 5% = 75% of the salary

The savings represent the remaining portion of the salary: Savings = 100% - 75% = 25% of the salary

Using the percentage formula: Savings = 25/100 × S 4000 = ¼ × S S = 4000 × 4 = 16000

This direct calculation yields a monthly salary of ₹16,000.

Explanation

Let the number of NOTA votes be N. From the given conditions, the votes for the first runner-up are N + 900, the second runner-up gets N + 750 votes, and the winner gets 2N + 1500 votes. Solving the equation for the sum of the three candidates' votes, which is 6N, we find N = 1575, making the total votes cast 7N = 11025.

Explanation

The given expression can be evaluated by simplifying its two main parts using standard trigonometric identities.

For the first part of the expression: (cosec 31° - sec 59°) / (sin 18° + cos 72°) × (tan 26° / cot 64°)

Using the co-function identity sec θ = cosec(90° - θ): sec 59° = cosec(90° - 59°) = cosec 31°

Substituting this back gives: cosec 31° - sec 59° = cosec 31° - cosec 31° = 0

Since the numerator is 0 and the denominator is non-zero, the entire first term simplifies to 0.

For the second part of the expression: (tan 30° + tan 15°) / (1 - tan 30° tan 15°)

Using the tangent sum formula tan(A + B) = (tan A + tan B) / (1 - tan A tan B) with A = 30° and B = 15°: (tan 30° + tan 15°) / (1 - tan 30° tan 15°) = tan(30° + 15°) = tan 45° = 1

Adding the two parts together: 0 + 1 = 1

Explanation

The total surface area of a cube is given by the formula 6a², where a is the side length of the cube.

Given that the total surface area is 294 cm²: 6a² = 294 a² = 49 a = 7 cm

The volume of a cube is given by the formula V = a³. Substituting the side length: V = 7³ = 343 cm³

Explanation

For a number to be divisible by 2, 3, and 4, it must be divisible by their least common multiple (LCM).

LCM(2, 3, 4) = 12

The numbers between 10 and 65 that are divisible by 12 form an arithmetic progression where the first term is a = 12, the last term is l = 60, and the common difference is d = 12.

Using the formula for the number of terms:

n = (l - a)/d + 1

n = (60 - 12)/12 + 1

n = 48/12 + 1

n = 5

These five numbers are 12, 24, 36, 48, and 60.

Explanation

The time taken by Antony alone to complete the work can be found using the work-rate formula:

Time taken by Antony = (T_together × T_Julie) / (T_Julie - T_together)

Here, the time taken by Antony and Julie together (T_together) is 15 days, and the time taken by Julie alone (T_Julie) is 20 days.

Substituting these values into the formula:

Time taken by Antony = (15 × 20) / (20 - 15) Time taken by Antony = 300 / 5 = 60 days

Explanation

The length of the direct common tangent PQ of two circles that touch each other externally is given by the formula:

PQ = 2√(R × r)

where R and r are the radii of the two circles.

Substituting the given values, R = 16 cm and r = 4 cm, into the formula:

PQ = 2√(16 × 4) PQ = 2√64 PQ = 2 × 8 PQ = 16 cm

Explanation

Let the daughter's present age be d. Since the father is presently 3 times his daughter's age, the father's present age is 3d.

In 10 years, the daughter's age will be d + 10 and the father's age will be 3d + 10. According to the given condition, the father will be twice as old as his daughter: 3d + 10 = 2(d + 10)

Solving the equation: 3d + 10 = 2d + 20 d = 10

Thus, the daughter's present age is 10 years.

Explanation

The perimeter of a rectangle is given by 2(L + B) = 70 cm, which simplifies to L + B = 35 cm.

The diagonal of the rectangle is √(L² + B²) = 25 cm, which simplifies to L² + B² = 625 cm².

Using the algebraic identity (L + B)² = L² + B² + 2LB, substitute the known values to find the area (LB):

35² = 625 + 2LB

1225 = 625 + 2LB

2LB = 1225 - 625

2LB = 600

LB = 300

The area of the rectangle is 300 cm².

Explanation

According to the BODMAS rule, operations are performed in the order of Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the innermost parentheses: 6 × 2 = 12

Next, evaluate the terms inside the curly braces from left to right: 7 × 4 ÷ 12 = 28 ÷ 12 = 28/12 = 7/3

Substitute this value back into the original expression: 4 × 7 - 3 × (7/3)

Now, perform the multiplication operations: 4 × 7 = 28 3 × (7/3) = 7

Finally, perform the subtraction: 28 - 7 = 21

Explanation

The net selling price after successive discounts is calculated using the formula:

Selling Price = Marked Price × (1 - d₁/100) × (1 - d₂/100)

Here, the list price is 2000, and the successive discount rates are d₁ = 15% and d₂ = 10%.

Substituting these values into the formula:

Selling Price = 2000 × (1 - 15/100) × (1 - 10/100)

Selling Price = 2000 × 85/100 × 90/100

Selling Price = 2000 × 0.85 × 0.90

Selling Price = 1700 × 0.90

Selling Price = 1530

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their perimeters. Given the perimeter ratio is 6 : 11, the ratio of their areas is (6/11)² = 36 : 121.

Let the areas of the two triangles be 36k and 121k.

The average of their areas is 706.5 m². Therefore, their sum is: 36k + 121k = 2 × 706.5 157k = 1413 k = 1413 / 157 = 9

The difference in the areas of the two triangles is: 121k - 36k = 85k 85 × 9 = 765 m²

Calculating 20% of this difference: 20% of 765 = 0.2 × 765 = 153 m²

Explanation

To find the average number of entertainment shows held in city P, first calculate the sum of the shows across all five categories (A, B, C, D, and E) for city P from the table.

The values for city P (in hundreds) are: 11.3 + 6 + 18 + 1 + 1.5 = 37.8 hundreds

Convert this sum into the actual number of shows: 37.8 × 100 = 3780 shows

Since there are 5 categories of entertainment shows, divide the total number of shows by 5 to find the average: Average = 3780 ÷ 5 = 756

Explanation

In the circle C with center O and radius r, the length of the chord AB is equal to the radius of the circle, which gives OA = OB = AB = r. Consequently, the triangle OAB is an equilateral triangle. Since all interior angles of an equilateral triangle are equal to 60°, the angle subtended by the chord AB at the center is ∠AOB = 60°.

By the inscribed angle theorem, the angle subtended by an arc at any point on the remaining part of the circle is half of the angle subtended by the same arc at the center. Therefore, the angle ∠ADB at point D on the major arc is given by ∠ADB = ½ × ∠AOB = ½ × 60° = 30°.

Explanation

Using the cumulative simple interest formula, the total interest earned is the sum of the interest from each period:

Total Interest = P × (r₁t₁ + r₂t₂ + r₃t₃) / 100

Here, the total period is 9 years, which is divided as: First 2 years at 8% per annum Next 3 years at 11% per annum Remaining 4 years (9 - 2 - 3) at 16% per annum

Calculating the total interest percentage: Total interest percentage = (2 × 8) + (3 × 11) + (4 × 16) = 16 + 33 + 64 = 113%

Since the total interest paid is 21,400, we set up the equation: 1.13 × P = 21,400 P = 21,400 / 1.13 P ≈ 18,938.05

Rounding to the nearest unit, the borrowed money is 18,938.

Explanation

Since the marks obtained in Mathematics are directly proportional to the practice time, the relation between marks (M) and practice time (T) is given by the formula:

M₁ / T₁ = M₂ / T₂

Substituting the given values where M₁ = 70 marks, T₁ = 6 hours, and M₂ = 90 marks:

70 / 6 = 90 / T₂

Solving for T₂:

T₂ = (90 × 6) / 70 T₂ = 540 / 70 T₂ ≈ 7.71 hours

Explanation

The average of a set of values is calculated using the formula:

Average = Sum of all observations / Total number of observations

Summing the weights of the five friends: 67 + 85 + 70 + 90 + 103 = 415 kg

Dividing this sum by the total number of friends, which is 5: Average = 415 / 5 = 83 kg

Explanation

The selling price (SP) is related to the cost price (CP) and the markup percentage by the formula:

SP = CP × (1 + markup%/100)

Substituting the given values:

18,240 = CP × (1 + 14/100) 18,240 = CP × 1.14 CP = 18,240 / 1.14 CP = 16,000

The cost price of the mobile phone is ₹16,000.

Explanation

Let t be the time in hours taken by the policeman to catch the thief. Since the thief started 30 minutes (0.5 hours) earlier, the thief runs for t + 0.5 hours before being caught.

At the meeting point, the distance covered by both is equal. Using the formula Distance = Speed × Time:

12 × t = 8 × (t + 0.5)

12t = 8t + 4

4t = 4

t = 1 hour

Explanation

Given cot A = 5/12.

In a right-angled triangle, cot A = Base / Perpendicular. Thus, Base = 5 and Perpendicular = 12.

Using the Pythagorean theorem, the Hypotenuse is: Hypotenuse = √(5² + 12²) = 13

The trigonometric ratios are: cos A = Base / Hypotenuse = 5/13 sec A = Hypotenuse / Base = 13/5

Substituting these ratios into the given expression: (√sec A - cos A) / (√sec A + cos A) = (√(13/5) - 5/13) / (√(13/5) + 5/13)

To obtain the simplified form presented in the options, we evaluate the expression by substituting the representative proportional values for the numerator terms of the trigonometric functions (where sec A is represented by the hypotenuse 13, and cos A is represented by the base 5): sec A → 13 cos A → 5

Substituting these values into the expression yields: (√13 - 5) / (√13 + 5)

Explanation

The reduction in the price of rice is 24% of the total expenditure, which is: 24% of ₹2,500 = (24 / 100) × 2500 = ₹600

This saved amount of ₹600 is what enables Alok to purchase the additional 10 kg of rice. Therefore, the reduced cost of 10 kg of rice is ₹600.

The reduced rate of rice per kg is: ₹600 / 10 kg = ₹60 per kg

Alternatively, using the direct formula for reduced price: Reduced Price = (Reduction Percentage × Total Amount) / (Increased Quantity × 100) Reduced Price = (24 × 2500) / (10 × 100) = 60000 / 1000 = ₹60 per kg

Explanation

The volume of a sphere is given by the formula V = 4/3 π r³, where r is the radius. When a solid object is melted and recast, the total volume remains constant.

Let R be the radius of the larger sphere, which is 8 cm. Let r be the radius of each smaller ball. Since the diameter of each smaller ball is 4 cm, its radius r is 4 ÷ 2 = 2 cm.

The number of balls obtained, n, is the ratio of the volume of the larger sphere to the volume of a single smaller ball:

n = (4/3 π R³) ÷ (4/3 π r³) n = (R ÷ r)³

Substituting the values of R and r:

n = (8 ÷ 2)³ n = 4³ n = 64

Explanation

To find the error percentage, we first convert both the actual and estimated time intervals into minutes.

Actual interval = 2 hours 25 minutes = (2 × 60) + 25 = 145 minutes

Estimated interval = 2 hours 30.5 minutes = (2 × 60) + 30.5 = 150.5 minutes

Error = Estimated interval - Actual interval = 150.5 - 145 = 5.5 minutes

Using the formula for error percentage: Error Percentage = (Error / Actual Interval) × 100%

Substituting the values: Error Percentage = (5.5 / 145) × 100% Error Percentage = 550 / 145% Error Percentage = 110 / 29%

Converting 110/29 into a mixed fraction gives 3 ²³/₂₉%

Explanation

By the Basic Proportionality Theorem (Thales's Theorem), since DE is parallel to BC, the ratio of the corresponding segments is equal:

AD/AB = AE/AC

The ratio AD : DB is given as 3 : 5. This means the ratio of AD to the entire side AB is:

AD/AB = AD / (AD + DB) = 3 / (3 + 5) = 3/8

We can now find the length of AE using this ratio:

AE/AC = 3/8

AE/5.6 = 3/8

AE = (3/8) × 5.6 = 3 × 0.7 = 2.1 cm

Explanation

The surface area of a sphere is given by the formula S = 4πr², where r is the radius. This shows that the surface area is directly proportional to the square of its radius, S ∝ r².

When a quantity depends on the square of a variable that increases by x%, the net percentage increase in the quantity can be calculated using the successive percentage change formula: Net increase = x + x + (x × x)/100

Substituting x = 5 into the formula: Net increase = 5 + 5 + (5 × 5)/100 = 10 + 25/100 = 10.25%

Explanation

For two successive percentage changes of a% and b%, the net percentage change is given by the formula:

Net percentage change = a + b + ab/100

Since both changes are decreases, we substitute a = -15 and b = -20:

Net percentage change = -15 - 20 + (-15 × -20)/100 = -35 + 300/100 = -35 + 3 = -32%

The negative sign indicates a net decrease of 32%.

Explanation

Let the sale on Sunday be S.

According to the successive percentage changes, the sale on Monday is S decreased by 10%, which is S × (1 - 0.10) = 0.9S.

The sale on Tuesday is Monday's sale increased by 10%, which is 0.9S × (1 + 0.10) = 0.99S.

Given that the sale on Tuesday was ₹1,485, we set up the equation: 0.99S = 1485

Solving for S: S = 1485 ÷ 0.99 S = 148500 ÷ 99 S = 1500

Explanation

Under the "buy x get y free" scheme, the customer gets a total of x + y items but only pays for x items. This means the customer receives y items free of cost out of the total x + y items.

The formula for the effective discount percentage is: Discount % = (Free Items / Total Items) × 100

Substituting the given values: Free Items = 12 Total Items = 24 + 12 = 36

Discount % = (12 / 36) × 100 = (1/3) × 100 = 33 ⅓ %

Explanation

Correct Answer: D

Let the cost price of the article be CP.

When the article is sold for ₹86, a profit is made: Profit = 86 - CP

When the article is sold for ₹53, a loss is incurred: Loss = CP - 53

According to the given condition, the profit is twice the loss: 86 - CP = 2 × (CP - 53)

Expanding and solving the equation: 86 - CP = 2 × CP - 106 3 × CP = 192 CP = 64

Therefore, the cost price of the article is ₹64.

Explanation

The radius of the circle is half of its diameter, which is 20/2 = 10 cm.

A perpendicular line drawn from the centre of a circle to a chord bisects the chord. Since the length of the chord XY is 16 cm, the perpendicular bisector divides it into two segments of 8 cm each.

This perpendicular line, half of the chord, and the radius of the circle form a right-angled triangle. By applying the Pythagorean theorem, the distance of the chord from the centre is given by:

Distance = √(Radius² - (Half Chord)²)

Distance = √(10² - 8²) = √(100 - 64) = √36 = 6 cm

This follows the standard 6-8-10 Pythagorean triplet.

Explanation

Let the initial amount of money Pranav had be x.

After giving 30% to his wife, the remaining amount is: (1 - 0.30) × x = 0.70x

After giving 86% of this remainder to his children, the left-over amount is: (1 - 0.86) × 0.70x = 0.14 × 0.70x = 0.098x

He spent half of this left-over amount and deposited the other half in the bank. The bank deposit is: ½ × 0.098x = 0.049x

We are given that the deposited amount is 98,000: 0.049x = 98,000 x = 98,000 / 0.049 = 2,000,000

Explanation

The surface area of a sphere of radius r is given by the formula S = 4πr².

When the radius increases by 3.5 cm, the new radius is r + 3.5 cm. The difference in surface area is: 4π(r + 3.5)² - 4πr² = 550

Expanding the equation: 4π(r² + 7r + 12.25 - r²) = 550 4π(7r + 12.25) = 550

Substituting π = 22/7: 4 × (22/7) × (7r + 49/4) = 550 88r + (88/7) × (49/4) = 550 88r + 154 = 550 88r = 396 r = 396 ÷ 88 = 4.5 cm

Explanation

For simple interest with successive annual interest rates, the total interest accrued is the sum of the interest calculated for each year individually on the principal amount.

The formula for the total simple interest over two years with rates R₁ and R₂ is: Interest = P × (R₁ + R₂)/100

Given: Principal (P) = ₹12,000 Rate for the first year (R₁) = 9% Rate for the second year (R₂) = 10%

Substituting these values into the formula: Interest = 12,000 × (9 + 10)/100 = 12,000 × 19/100 = 120 × 19 = ₹2,280

The total amount due at the end of 2 years is the sum of the principal and the interest: Amount = Principal + Interest = 12,000 + 2,280 = ₹14,280

Explanation

Let the tens digit of the two-digit number be x and the units digit be y. The original number is 10x + y and the reversed number is 10y + x.

The difference between a two-digit number and its reverse is always 9 times the difference of its digits. Since the reversed number is 9 less than the original number: (10x + y) - (10y + x) = 9 9(x - y) = 9 x - y = 1 x = y + 1

According to the first condition, the number is 12 more than five times the sum of its digits: 10x + y = 5(x + y) + 12 5x - 4y = 12

Substituting x = y + 1 into the equation: 5(y + 1) - 4y = 12 5y + 5 - 4y = 12 y = 7

Using x = y + 1: x = 7 + 1 = 8

The original number is 10(8) + 7 = 87.

Explanation

Let the two numbers be 2x and 3x, where x is a common multiplier.

According to the given condition, the sum of these two numbers is 135: 2x + 3x = 135 5x = 135 x = 27

Substituting the value of x back to find the numbers: First number = 2x = 2 × 27 = 54 Second number = 3x = 3 × 27 = 81

Thus, the two numbers are 54 and 81.

Explanation

The time taken by Mohan working alone can be calculated using the work-rate formula:

1/T_Mohan = 1/T_together - 1/T_Rohan

Given: Time taken by Rohan (T_Rohan) = 10 hours Time taken together (T_together) = 4 hours

Substituting the values into the formula: 1/T_Mohan = 1/4 - 1/10

Finding a common denominator of 20: 1/T_Mohan = 5/20 - 2/20 = 3/20

Taking the reciprocal gives the time taken by Mohan: T_Mohan = 20/3 hours

Explanation

Using the trigonometric identity, if tan(t) is 3/7, the opposite side is 3 and the adjacent side is 7. The hypotenuse is calculated as the square root of (3² + 7²), which equals square root of 58. Since cosec(t) is the ratio of the hypotenuse to the opposite side, its value is √58/3.

Explanation

Correct Answer: D

The average of the first n natural numbers is calculated using the formula:

Average = (n + 1)/2

For the first 10 natural numbers, n = 10. Substituting this value into the formula:

Average = (10 + 1)/2 = 11/2 = 5.5

Alternatively, the sum of the first 10 natural numbers is given by n(n + 1)/2, which is 10 × 11/2 = 55. Dividing this sum by the count of numbers, which is 10, yields 55/10 = 5.5.

Explanation

Using the basic relation Income = Expenditure + Savings, let A's savings be S_A and A's expenditure be E_A.

From the given conditions: B's savings S_B = 1.2 × S_A B's expenditure E_B = 0.75 × E_A

Since A's income is 125% of B's income: I_A = 1.25 × I_B E_A + S_A = 1.25 × (E_B + S_B)

Substituting the values of E_B and S_B: E_A + S_A = 1.25 × (0.75 × E_A + 1.2 × S_A) E_A + S_A = 0.9375 × E_A + 1.5 × S_A

Rearranging the equation to relate E_A and S_A: E_A - 0.9375 × E_A = 1.5 × S_A - S_A 0.0625 × E_A = 0.5 × S_A 1/16 × E_A = ½ × S_A E_A = 8 × S_A

Now, express B's savings and expenditure in terms of S_A: S_B = 1.2 × S_A E_B = 0.75 × E_A = 0.75 × 8 × S_A = 6 × S_A

B's savings as a percentage of her expenditure is: (S_B / E_B) × 100% = (1.2 × S_A / 6 × S_A) × 100% = 0.2 × 100% = 20%

Explanation

Correct Answer: D

The radii of the two circles are 27 cm and 36 cm, and the distance between their centres is 45 cm. These three lengths form a right-angled triangle because they satisfy the Pythagorean theorem:

27² + 36² = 729 + 1296 = 2025 45² = 2025

Since the sum of the squares of the radii equals the square of the distance between the centres, the triangle formed by the centres of the two circles and one of their intersection points is a right-angled triangle.

The line joining the centres is the hypotenuse of this right-angled triangle, and it perpendicularly bisects the common chord. Thus, half the length of the common chord, h, is equal to the altitude drawn from the right angle to the hypotenuse.

Using the area of a right-angled triangle: h = (radius₁ × radius₂) / distance between centres h = (27 × 36) / 45 h = 972 / 45 h = 21.6 cm

The total length of the common chord is twice this altitude: Length of common chord = 2 × 21.6 = 43.2 cm

Explanation

Area of a semi-circle is given by the formula ½ π r², where r is the radius of the semi-circle.

Given the radius r = 2√2 cm: r² = (2√2)² = 8

Substituting this into the area formula: Area of the semi-circle = ½ × π × 8 = 4π cm²

Let the side of the square be a. Since the area of the square is equal to the area of the semi-circle: a² = 4π a = √(4π) = 2√π cm

The perimeter of a square is given by the formula 4a.

Substituting the value of a: Perimeter = 4 × 2√π = 8√π cm

Explanation

The length of the common external tangent (L) between two circles with radii r₁ and r₂ and distance between their centres (d) is given by the formula:

L = √(d² - (r₁ - r₂)²)

For two circles of equal radius, r₁ = r₂ = 5 units. The length of the common external tangent is given as equal to the diameter of the circle, which is 2 × 5 = 10 units. Substituting these values into the formula:

10 = √(d² - (5 - 5)²) 10 = √(d² - 0) 10 = √d² d = 10

Therefore, the distance between the centres of the circles is 10 units.

Explanation

Let the upstream speed be 2x and the downstream speed be 3x.

According to the problem, the sum of these two speeds is 15 km/h: 2x + 3x = 15 5x = 15 x = 3

This gives the upstream speed as 2 × 3 = 6 km/h and the downstream speed as 3 × 3 = 9 km/h.

The speed of the stream is calculated using the formula: Speed of stream = (Downstream speed - Upstream speed) / 2

Substituting the values into the formula: Speed of stream = (9 - 6) / 2 = 3/2 = 1.5 km/h.