SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Using the Law of Cosines, the cosine of angle ACB (or angle C) in triangle ABC with sides a = BC = 1, b = AC = 2, and c = AB = √3 is given by the formula:

cos(C) = (a² + b² - c²) ÷ (2ab)

Substituting the given side lengths into the formula:

cos(C) = (1² + 2² - (√3)²) ÷ (2 × 1 × 2) cos(C) = (1 + 4 - 3) ÷ 4 cos(C) = 2 ÷ 4 cos(C) = ½

Since cos(60°) = ½, the angle ACB is equal to 60°.

Explanation

In an isosceles triangle, the two base angles are equal. Let the measure of each base angle be x.

The vertex angle, say y, is given as four times the sum of the base angles: y = 4(x + x) = 8x

Since the sum of all three angles in a triangle is 180°: x + x + y = 180° 2x + 8x = 180° 10x = 180° x = 18°

The measure of the vertex angle is: y = 8 × 18° = 144°

The square root of the vertex angle is: √144° = 12°

Explanation

Let the third number be 100.

The first number is 61% less than the third number, which is 100 - 61 = 39.

The second number is 35% less than the third number, which is 100 - 35 = 65.

To find what percentage the first number is of the second number, use the percentage formula:

Percentage = (First Number / Second Number) × 100

Substituting the values:

Percentage = (39 / 65) × 100

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 13:

39 / 65 = 3 / 5

Percentage = (3 / 5) × 100 = 60%

Explanation

To find the required ratio, first calculate the sum of books for Maths and Computers, and then the sum of books for Circuit Theory and Graph Theory.

Sum of Maths and Computers = 17,462 + 14,308 = 31,770

Sum of Circuit Theory and Graph Theory = 9948 + 642 = 10,590

The ratio of the two sums is: 31,770 / 10,590 = 3 / 1 = 3 : 1

Explanation

Let the initial quantities of milk and water in the bucket be 7x and 5x respectively.

When 9 litres of milk is replaced by water, 9 litres is subtracted from the milk and 9 litres is added to the water. The new ratio is: (7x - 9) / (5x + 9) = 7/9

Cross-multiplying to solve for x: 9 × (7x - 9) = 7 × (5x + 9) 63x - 81 = 35x + 63 63x - 35x = 63 + 81 28x = 144 x = 144 / 28 = 36/7

The initial quantity of milk in the bucket is 7x: Initial milk = 7 × (36/7) = 36 litres

Explanation

According to the intersecting secant theorem, when two secants of a circle intersect at an external point P, the products of their respective segments are equal:

PA × PB = PC × PD

The length of PA is the sum of AB and BP: PA = AB + BP = 7 + 4.2 = 11.2 cm

Let CD be x. The length of PC is: PC = CD + PD = x + 2.8

Substituting the given values into the formula: 11.2 × 4.2 = (x + 2.8) × 2.8

Dividing both sides by 2.8: 4 × 4.2 = x + 2.8 16.8 = x + 2.8 x = 16.8 - 2.8 x = 14 cm

Explanation

According to the divisibility rule for 11, a number is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

Mathematically, any integer N with digits aₙaₙ₋₁...a₁a₀ can be written as N = a₀ + 10a₁ + 10²a₂ + ... + 10ⁿaₙ. Since 10 ≡ -1 (mod 11), any power of 10 satisfies 10ᵏ ≡ (-1)ᵏ (mod 11). Substituting this into the expansion of N gives N ≡ a₀ - a₁ + a₂ - a₃ + ... (mod 11).

Grouping the positive and negative terms, N ≡ (a₀ + a₂ + a₄ + ...) - (a₁ + a₃ + a₅ + ...) (mod 11). This expression represents the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions. For N to be divisible by 11, N must be congruent to 0 modulo 11, which requires this difference to be 0 or an integer multiple of 11.

Explanation

The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Area(△ABC) / Area(△PQR) = (AC / PR)²

Substituting the given values into the formula:

289 / 576 = (AC / 12)²

Taking the square root on both sides:

17 / 24 = AC / 12

AC = (17 × 12) / 24

AC = 17 / 2

AC = 8.5 cm

Explanation

The principal sum P is determined from the first investment scenario where the sum grows to 8200 at 8% simple interest per annum for 8 years:

Amount = P × (1 + (R₁ × T₁) / 100) 8200 = P × (1 + (8 × 8) / 100) 8200 = 1.64P P = 5000

In the second scenario, the product of the rate of interest and time is 120% (for instance, 12% simple interest per annum for 10 years, or 20% simple interest per annum for 6 years):

Simple Interest = (P × R₂ × T₂) / 100 Simple Interest = (5000 × 120) / 100 = 6000

The final amount is the sum of the principal and the simple interest:

Amount = 5000 + 6000 = 11,000

Explanation

The surface area of a sphere of radius r is given by S = 4πr², and its volume is given by V = (4/3)πr³.

Let the radii of the two spheres be r₁ and r₂. The ratio of their surface areas is: S₁/S₂ = (r₁/r₂)² = 49/81

Taking the square root of both sides gives the ratio of their radii: r₁/r₂ = √(49/81) = 7/9

The ratio of their volumes is the cube of the ratio of their radii: V₁/V₂ = (r₁/r₂)³ = (7/9)³ = 343/729

Thus, the ratio of their volumes is 343:729.

Explanation

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

For the larger cube, the surface area is 54 cm², so: 6a₁² = 54 a₁² = 9 a₁ = 3 cm

The volume of the larger cube is V₁ = a₁³ = 3³ = 27 cm³.

For each smaller cube, the surface area is 8/12 cm², which simplifies to 2/3 cm², so: 6a₂² = 2/3 a₂² = 2 / (3 × 6) = 2/18 = 1/9 a₂ = 1/3 cm

The volume of each smaller cube is V₂ = a₂³ = (1/3)³ = 1/27 cm³.

The number of smaller cubes formed is the ratio of the volume of the larger cube to the volume of a smaller cube: Number of cubes = V₁ / V₂ = 27 / (1/27) = 27 × 27 = 729.

Explanation

The total number of votes is 8000. Since 15% of the votes were invalid, the percentage of valid votes is 100% - 15% = 85%.

The total number of valid votes is: 8000 × 0.85 = 6800

One candidate received 52% of the valid votes. Therefore, the other candidate received: 100% - 52% = 48% of the valid votes

The number of valid votes that the other candidate received is: 6800 × 48% = 6800 × 0.48 = 3264

Explanation

To find the average marks obtained by all six students in Hindi, first calculate the sum of the percentages obtained by each student in Hindi.

Sum of percentages = 90% + 90% + 80% + 100% + 100% + 70% = 530%

The average percentage of marks obtained by the six students in Hindi is: Average percentage = 530% ÷ 6 = 88.333%

Since the maximum marks for Hindi is 120, the average marks obtained is calculated by finding this average percentage of the maximum marks: Average marks = (530 ÷ 6) × (120 ÷ 100) = (530 × 1.2) ÷ 6 = 636 ÷ 6 = 106

Explanation

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

Given that the surface area is 616 cm² and taking π as 22/7, the equation can be written as:

616 = 4 × (22/7) × r²

616 = (88/7) × r²

To solve for r², multiply both sides by 7/88:

r² = (616 × 7) / 88

r² = 7 × 7

r² = 49

Taking the square root of both sides gives:

r = √49

r = 7 cm

Explanation

Distance is calculated using the formula: Distance = Speed × Time

The walking speed is 4 km/h and the time taken is 6 hours and 15 minutes. Converting this time into hours: 6 hours 15 minutes = 6 + 15/60 = 6 + ¼ = 25/4 hours

Using the formula, the distance covered is: Distance = 4 × 25/4 = 25 km

The cycle covers the same distance of 25 km in 5 hours and 10 minutes. Converting the cycling time into hours: 5 hours 10 minutes = 5 + 10/60 = 5 + 1/6 = 31/6 hours

Using the relation Speed = Distance / Time, the speed of the cycle is: Speed = 25 / (31/6) = 150/31 km/h

Converting 150/31 into a mixed number: 150/31 = 4 26/31 km/h

Explanation

By Fermat's Little Theorem, for any prime number p and an integer a not divisible by p, the relation is aᵖ⁻¹ ≡ 1 (mod p).

Here, a = 2 and p = 17. Since 17 is a prime number, we have: 2¹⁶ ≡ 1 (mod 17)

The exponent 256 is a multiple of 16: 256 = 16 × 16

Therefore, the expression can be rewritten as: 2²⁵⁶ = (2¹⁶)¹⁶

Applying the modulo operation: 2²⁵⁶ ≡ (2¹⁶)¹⁶ ≡ 1¹⁶ ≡ 1 (mod 17)

Explanation

Correct Answer: D

Given that 3 tan θ = 4, we get: tan θ = 4/3

The expression to evaluate is: (9sin θ + 2cos θ) / (9sin θ - 2cos θ)

Dividing both the numerator and the denominator by cos θ: (9 tan θ + 2) / (9 tan θ - 2)

Substituting tan θ = 4/3 into this expression: (9 × 4/3 + 2) / (9 × 4/3 - 2) = (12 + 2) / (12 - 2) = 14 / 10 = 7/5

Explanation

Let the cost price of B be x. Since the cost price of A is thrice that of B, the cost price of A is 3x.

The selling price of A, which is sold at a loss of 10%, is: 3x × (1 - 0.10) = 2.7x

The selling price of B, which is sold at 5/4 of its cost price, is: x × 1.25 = 1.25x

The difference between the selling price of A and the selling price of B is given as ₹5,800: 2.7x - 1.25x = 5800 1.45x = 5800 x = 4000

Therefore, the cost price of A is: 3x = 3 × 4000 = ₹12,000

Explanation

The average weight of the combined group of apples is calculated using the weighted average formula:

Average = (n₁ × A₁ + n₂ × A₂) / (n₁ + n₂)

where n₁ and n₂ are the quantities, and A₁ and A₂ are their respective averages.

Given: n₁ = 40, A₁ = 33 kg n₂ = 32, A₂ = 60 kg

Simplifying the ratio of the number of apples to make the calculation faster: n₁ : n₂ = 40 : 32 = 5 : 4

Now, substitute these simplified ratio values into the formula: Average = (5 × 33 + 4 × 60) / (5 + 4) Average = (165 + 240) / 9 Average = 405 / 9 Average = 45 kg

Explanation

Total votes cast in the election are 30,000. Since 20% of the votes are invalid, the percentage of valid votes is 80%.

Total valid votes = 30,000 × 80% = 30,000 × 0.8 = 24,000

Ananth secured 55% of these valid votes. Since there are only two candidates, Aravind must have received the remaining 45% of the valid votes.

Valid votes obtained by Aravind = 24,000 × 45% = 24,000 × 0.45 = 10,800

Explanation

To find the value of the given trigonometric expression, simplify each part of the expression separately.

The first part of the expression is: (sin θ / (1 + cos θ)) + ((1 + cos θ) / sin θ)

Taking the common denominator: = (sin² θ + (1 + cos θ)²) / (sin θ (1 + cos θ)) = (sin² θ + 1 + 2 cos θ + cos² θ) / (sin θ (1 + cos θ))

Using the trigonometric identity sin² θ + cos² θ = 1: = (1 + 1 + 2 cos θ) / (sin θ (1 + cos θ)) = (2 + 2 cos θ) / (sin θ (1 + cos θ)) = 2(1 + cos θ) / (sin θ (1 + cos θ)) = 2 / sin θ

The second part of the expression is: 1 / (tan θ + cot θ)

Converting tan θ and cot θ to terms of sin θ and cos θ: = 1 / (sin θ / cos θ + cos θ / sin θ) = 1 / ((sin² θ + cos² θ) / (sin θ cos θ)) = 1 / (1 / (sin θ cos θ)) = sin θ cos θ

Now, multiply the simplified forms of both parts: (2 / sin θ) × (sin θ cos θ) = 2 cos θ

Explanation

The average marks in History are calculated as: (77 + 80 + 83) / 3 = 240 / 3 = 80

The average marks in Economics are calculated as: (74 + 82 + 86) / 3 = 242 / 3 ≈ 80.67

Rounding 80.67 to the nearest integer yields 81.

Explanation

Using the inverse relationship between efficiency and time: Efficiency is inversely proportional to the time taken to complete a work.

Let the efficiency of Nirmal be 100. Since Amit is 25% more efficient than Nirmal, Amit's efficiency is 125.

The ratio of their efficiencies is: Nirmal : Amit = 100 : 125 = 4 : 5

Therefore, the ratio of the time taken by them to complete the work is: Nirmal : Amit = 5 : 4

Given that Nirmal takes 45 days: 5 units = 45 days 1 unit = 9 days

The time taken by Amit is: 4 units = 4 × 9 = 36 days

Explanation

To find the year in which the total loan disbursement of banks A and B is exactly equal to that of banks C and D, we calculate the sum of the disbursements for both groups of banks for each year.

For the year 2019: Total disbursement of banks A and B = 22 + 18 = 40 Crores Total disbursement of banks C and D = 20 + 20 = 40 Crores

Explanation

The remaining principal after the down payment is: Principal = Cash Price - Down Payment Principal = 2540 - 1340 = Rs. 1200

The amount paid after one month is Rs. 1205. Therefore, the interest charged for one month is: Interest = 1205 - 1200 = Rs. 5

The rate of interest per annum, R, is calculated using the simple interest formula: Interest = (Principal × R × Time) / 100

Since the time is 1 month, which is 1/12 of a year, substituting the values gives: 5 = (1200 × R × 1) / (100 × 12) 5 = 10 × R / 10 5 = R

Thus, the rate of interest charged is 5% p.a.

Explanation

Correct Answer: D

The dimensions of the cuboid are determined from the given relationships: Length (l) = 60 cm Breadth (b) = l ÷ 2 = 60 ÷ 2 = 30 cm Height (h) = b ÷ 3 = 30 ÷ 3 = 10 cm

The total surface area of a cuboid is calculated using the formula: Total Surface Area = 2(lb + bh + hl)

Substituting the values into the formula: Total Surface Area = 2 × (60 × 30 + 30 × 10 + 10 × 60) Total Surface Area = 2 × (1800 + 300 + 600) Total Surface Area = 2 × 2700 Total Surface Area = 5400 cm²

Explanation

To find the value of the given expression, apply the BODMAS rule, which dictates the order of operations: Brackets, Orders, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

First, evaluate the numerator of the fraction: [1/4 ÷ 1/4 × 1/4] × 1/4

Inside the bracket, perform the division first: 1/4 ÷ 1/4 = 1

Then perform the multiplication inside the bracket: 1 × 1/4 = 1/4

Now, multiply by the term outside the bracket: 1/4 × 1/4 = 1/16

Next, evaluate the denominator of the fraction: [1/8 + 1/8 × 1/8]

Perform the multiplication first: 1/8 × 1/8 = 1/64

Then perform the addition: 1/8 + 1/64 = 8/64 + 1/64 = 9/64

Now, divide the numerator by the denominator: (1/16) / (9/64) = 1/16 × 64/9 = 4/9

Finally, evaluate the second term and add it to the first result: 1 - 4/9 = 5/9

Adding the two parts together: 4/9 + 5/9 = 9/9 = 1

Explanation

Let the ages of Ruby and Soni be R and S, respectively.

The sum of their ages is calculated from their average age using the formula: Sum of ages = Average age × Number of people Sum of ages = 40 × 2 = 80 years

The ratio of their ages is R : S = 11 : 5.

Using the ratio formula, Soni's age S is: S = [5 / (11 + 5)] × 80 S = (5 / 16) × 80 S = 5 × 5 = 25 years

Explanation

Correct Answer: D

The relationship between Marked Price (MP) and Cost Price (CP) when a profit of P% is earned after giving a discount of D% is given by the formula:

MP / CP = (100 + P) / (100 - D)

Substituting P = 28 and D = 20 into the formula:

MP / CP = (100 + 28) / (100 - 20) MP / CP = 128 / 80 = 8 / 5 = 1.6

When the article is sold without any discount, the selling price equals the marked price. The profit percentage in this case is:

Profit % = (MP / CP - 1) × 100 Profit % = (1.6 - 1) × 100 = 0.6 × 100 = 60%

Explanation

Correct Answer: D

Using the formula for successive percentage increase:

Population after two years = Initial Population × (1 + r₁/100) × (1 + r₂/100)

Here, the initial population is 4,00,000, r₁ = 20%, and r₂ = 30%.

Substituting the values into the formula:

Population after two years = 4,00,000 × (1 + 20/100) × (1 + 30/100) = 4,00,000 × 1.2 × 1.3 = 4,00,000 × 1.56 = 6,24,000

Explanation

Let the larger number be x and the smaller number be y.

Based on the first condition, the sum of the two numbers is 76: x + y = 76 y = 76 - x

Based on the second condition, three times the larger number is 46 more than four times the smaller number: 3x = 4y + 46

Substitute y = 76 - x into the second equation: 3x = 4(76 - x) + 46 3x = 304 - 4x + 46 7x = 350 x = 50

Substitute x = 50 back into the expression for y: y = 76 - 50 = 26

The two numbers are 50 and 26.

Explanation

The relative speed of the policeman with respect to the thief is the difference between their speeds because they are travelling in the same direction: Relative speed = 65 - 41 = 24 km/h

To convert the relative speed into meters per second: Speed in m/s = 24 × 5/18 = 20/3 m/s

The initial distance between the policeman and the thief is 1200 m.

Using the formula for time: Time = Distance / Relative speed Time = 1200 / (20/3) = 1200 × 3 / 20 = 180 seconds

Converting the time from seconds to minutes: Time in minutes = 180 / 60 = 3 minutes

Explanation

In the isosceles triangle PQR with PQ = PR = 40 cm and QR = 60 cm, the altitude from the vertex P to the base QR bisects the base into two equal parts of 30 cm each.

Using the Pythagorean theorem, the height of this altitude is: h = √(40² - 30²) = √700 = 10√7 cm

The area of triangle PQR is: Area = ½ × base × height = ½ × 60 × 10√7 = 300√7 cm²

The same area can also be expressed using PR as the base and the altitude from Q to PR (let this altitude be h_Q): Area = ½ × PR × h_Q 300√7 = ½ × 40 × h_Q 300√7 = 20 × h_Q h_Q = 15√7 cm

Explanation

Let the cost price of 100 grams of goods be ₹100.

Since the shopkeeper uses 25% less weight, the actual weight of goods delivered is 75 grams, which has a cost price of ₹75.

The selling price of these goods is 40% less than the cost price of the claimed 100 grams: SP = 100 × (1 - 0.40) = ₹60

Since the cost price of the delivered goods is ₹75 and the selling price is ₹60, the shopkeeper incurs a loss.

Loss percentage = ((CP - SP) / CP) × 100 Loss percentage = ((75 - 60) / 75) × 100 = (15 / 75) × 100 = 20%

Explanation

For a dishonest shopkeeper selling goods at cost price but using less weight, the gain percentage is calculated using the formula:

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

Here, the weight used is 27% less than the true weight. Let the true weight be 100 units. Error = 27 units Effective weight used = 100 - 27 = 73 units

Substituting these values into the formula:

Gain % = (27 / 73) × 100 Gain % = 2700 / 73 ≈ 36.986%

Rounding off to one decimal place, the shopkeeper's gain percent is 37.0%.

Explanation

When three cubes of side s = 5 cm are joined end to end, they form a cuboid with the following dimensions: Length (l) = 3 × 5 = 15 cm Breadth (b) = 5 cm Height (h) = 5 cm

The total surface area of a cuboid is given by the formula: Total Surface Area = 2 × (l × b + b × h + h × l)

Substituting the dimensions into the formula: Total Surface Area = 2 × (15 × 5 + 5 × 5 + 5 × 15) Total Surface Area = 2 × (75 + 25 + 75) Total Surface Area = 2 × 175 = 350 cm²

Explanation

The overall rate of simple interest earned on the total sum is calculated as:

Overall Interest Rate = (Total Simple Interest × 100) ÷ (Total Principal × Time) Overall Interest Rate = (960 × 100) ÷ (5000 × 1) = 19.2%

Applying the method of allegation to find the ratio of the parts:

Difference between the higher rate and the overall rate = 24% - 19.2% = 4.8% Difference between the overall rate and the lower rate = 19.2% - 16% = 3.2%

The ratio of the sum lent at the lower rate (16%) to the sum lent at the higher rate (24%) is 4.8 : 3.2, which simplifies to 3 : 2.

The sum lent at the higher rate is: Sum = 5000 × (2 ÷ (3 + 2)) = 5000 × 2/5 = ₹2,000

Explanation

Train 1 starts from A at 11 am and travels for 1 hour before Train 2 starts from B at 12 noon.

Distance covered by Train 1 in this 1 hour = Speed × Time = 70 km/h × 1 h = 70 km.

The remaining distance between the two trains at 12 noon is: 420 km - 70 km = 350 km.

Since the two trains are moving towards each other, their relative speed is the sum of their individual speeds: Relative Speed = 70 km/h + 30 km/h = 100 km/h.

The time taken for them to meet after 12 noon is: Time = Remaining Distance / Relative Speed = 350 km / 100 km/h = 3.5 hours.

Adding 3.5 hours (3 hours and 30 minutes) to 12 noon gives the meeting time of 3.30 pm.

Explanation

Total work can be assumed as the LCM of 20, 30, and 40, which is 120 units.

The individual efficiencies of each person are: Mira = 120 / 20 = 6 units/day Nina = 120 / 30 = 4 units/day Sima = 120 / 40 = 3 units/day

In the first 4 days, Mira and Sima work together: Work completed = (6 + 3) × 4 = 36 units

Remaining work to be done = 120 - 36 = 84 units

Next, Mira is replaced by Nina, meaning Nina and Sima work together to finish the remaining work: Combined efficiency of Nina and Sima = 4 + 3 = 7 units/day Time taken to complete the remaining work = 84 / 7 = 12 days

Total time taken to complete the entire work altogether = 4 + 12 = 16 days.

Explanation

By the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides the two sides in the same ratio. In triangle DEW, since AB is parallel to EW, the smaller triangle DAB is similar to triangle DEW.

Therefore, the ratio of their corresponding sides is equal: AD/DE = DB/DW

Substituting the given values: 4/12 = DB/24

Simplifying the fraction: 1/3 = DB/24

Solving for DB: DB = 24/3 = 8 cm

Explanation

The area of a rhombus is given by the formula: Area = side × height

By substituting the given values: 70 = side × 5 side = 70 / 5 = 14 m

Since all four sides of a rhombus are equal, the perimeter is calculated as: Perimeter = 4 × side Perimeter = 4 × 14 = 56 m

Explanation

For any equilateral triangle with side length a, the perimeter P is given by the formula: P = 3a

In an equilateral triangle, each median is also an altitude, and its length h is given by the formula: h = (√3/2)a

Since all three medians of an equilateral triangle are equal, the sum of the medians S is: S = 3 × (√3/2)a = (3√3/2)a

The ratio of the perimeter to the sum of the medians is: Ratio = P / S = 3a / ((3√3/2)a) = 2 / √3 = 2:√3

Substituting the given side length a = 16 cm: P = 3 × 16 = 48 cm S = (3√3/2) × 16 = 24√3 cm Ratio = 48 / 24√3 = 2 / √3 = 2:√3

Explanation

The volume of silver used to make the hemispherical bowl is the difference between the outer and inner volumes of the hemisphere.

The formula for the volume of a hemispherical shell is: V = (2/3) × π × (R³ - r³)

The inner diameter is 4 cm, so the inner radius r is 2 cm. The thickness of the silver is 0.5 cm, which gives the outer radius R = 2 + 0.5 = 2.5 cm.

Substituting these values into the formula with π = 3.14: V = (2/3) × 3.14 × (2.5³ - 2³) V = (2/3) × 3.14 × (15.625 - 8) V = (2/3) × 3.14 × 7.625 V = (47.885) / 3 V ≈ 15.96 cm³

Explanation

Let T be the total number of votes.

Since 30% of the voters did not cast their votes, the total votes cast is 70% of T, which is 0.7T.

Subtracting the 200 invalid votes, the number of valid votes V is: V = 0.7T - 200

The winner received 65% of the valid votes, which means the loser received 35% of the valid votes.

The difference in votes between the winner and the loser is 30% of the valid votes (65% - 35% = 30%).

Since the winner won by 1200 votes, we can set up the equation: 30% of V = 1200 0.3 × V = 1200 V = 4000

Substituting the value of V back to find the total votes: 0.7T - 200 = 4000 0.7T = 4200 T = 6000

Explanation

A number is divisible by 72 if and only if it is divisible by both 8 and 9, since 8 and 9 are coprime.

First, for divisibility by 8, the last three digits of 4645a52 must form a number divisible by 8. When a = 1, the last three digits are 152. Dividing 152 by 8 gives 19, which is an integer.

Second, for divisibility by 9, the sum of the digits of 4645a52 must be a multiple of 9. The sum of the digits is 4 + 6 + 4 + 5 + a + 5 + 2 = 26 + a. Substituting a = 1 gives 26 + 1 = 27, which is divisible by 9.

Since a = 1 satisfies both divisibility rules, the number is divisible by 72.

Explanation

Let the number be N. According to the division algorithm, N can be expressed as: N = 459k + 19, where k is a non-negative integer.

To find the remainder when N is divided by 17, we evaluate N modulo 17: N mod 17 = (459k + 19) mod 17

Since 459 is a multiple of 17 (459 = 17 × 27), the term 459k leaves a remainder of 0 when divided by 17.

Thus, the remainder of N modulo 17 is equivalent to the remainder of 19 modulo 17: 19 = 17 × 1 + 2

The remainder is 2.

Explanation

Let the original two-digit number be represented as 10x + y, where x is the tens digit and y is the units digit.

According to the given conditions, the sum of the digits is 9: x + y = 9

The number obtained by interchanging the digits is 10y + x. This new number exceeds the original number by 45: (10y + x) - (10x + y) = 45 9y - 9x = 45 y - x = 5

Solving this system of two linear equations by adding them together: 2y = 14 y = 7

Substituting y = 7 back into the first equation: x + 7 = 9 x = 2

Thus, the tens digit of the original number is 2 and the units digit is 7, which gives the number 27.

Explanation

Let α = cos⁻¹(7/25) and β = cos⁻¹(15/17).

This gives cos α = 7/25 and cos β = 15/17. Since α and β are acute angles, we can determine their sine values using the identity sin²θ + cos²θ = 1:

sin α = √(1 - (7/25)²) = 24/25 sin β = √(1 - (15/17)²) = 8/17

Now, we use the trigonometric subtraction formula for sine:

sin(α - β) = sin α cos β - cos α sin β

Substituting the values into the formula:

sin(α - β) = (24/25) × (15/17) - (7/25) × (8/17) sin(α - β) = 360/425 - 56/425 sin(α - β) = 304/425

Since cos α < cos β, we have α > β, which means α - β is a positive acute angle. Taking the inverse sine on both sides yields:

α - β = sin⁻¹(304/425)

Explanation

For any three points in a plane, a circle passing through all of them exists if and only if the three points are non-collinear. If the three points lie on the same straight line, meaning they are collinear, it is physically impossible to draw a circle that passes through all three points. Therefore, the statement that there is always a circle passing through any three points in a plane is incorrect.

Explanation

Let the total number of votes polled be T.

Since there are only two candidates and no invalid votes, the winning candidate secured: 100% - 44% = 56% of the total votes

The difference in vote percentage between the two candidates is: 56% - 44% = 12%

This percentage difference corresponds to the defeat margin of 1440 votes. We can set up the direct relation: 12% of T = 1440 12/100 × T = 1440 T = 1440 × 100 / 12 T = 12000