CDS II 2020 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The expression (x^3 + x^2 + 16)/x simplifies to x^2 + x + 16/x. For this to be an integer, x must be a divisor of 16. The positive divisors of 16 are 1, 2, 4, 8, and 16. There are 5 such values.

Explanation

The prime factorization of 30 is 2 3 5. Each prime factor can be assigned to either a, b, or c. For each of the three prime factors, there are 3 choices. Total number of triplets (a, b, c) is 3 3 3 = 27.

Explanation

For a quadratic equation of the form ax² + bx + c = 0 to have real roots, the discriminant D must be greater than or equal to zero. The formula for the discriminant is D = b² - 4ac. In the given equation x² - 4x - log₁₀(N) = 0, the coefficients are a = 1, b = -4, and c = -log₁₀(N).

Substituting these values into the discriminant condition: (-4)² - 4(1)(-log₁₀(N)) ≥ 0 16 + 4log₁₀(N) ≥ 0

To find the minimum value of N, we solve the inequality: 4log₁₀(N) ≥ -16 log₁₀(N) ≥ -4

Converting the logarithmic inequality to exponential form: N ≥ 10⁻⁴ N ≥ 1/10⁴ N ≥ 1/10000

The minimum value that N can take while ensuring the roots remain real is 1/10000, which corresponds to option (d).

Explanation

The number of positive integer solutions to x + y + z = n is given by the formula (n-1)C(k-1). Here n = 12 and k = 3. The number of solutions is (12-1)C(3-1) = 11C2 = (11 * 10) / 2 = 55.

Explanation

Let the two consecutive integers be a and b = a + 1. Given the expression I = a² + b² + c² where c = ab, we substitute the values of b and c into the equation.

I = a² + (a + 1)² + (a(a + 1))²

Expanding the terms using the identity (x + y)² = x² + 2xy + y², we get:

I = a² + a² + 2a + 1 + (a² + a)² I = 2a² + 2a + 1 + (a²)² + 2(a²)(a) + a² I = a⁴ + 2a³ + 3a² + 2a + 1

This polynomial can be simplified as a perfect square using the identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By observation, a⁴ + 2a³ + 3a² + 2a + 1 = (a² + a + 1)².

To determine if a² + a + 1 is even or odd, we analyze the term a² + a = a(a + 1). Since a and a + 1 are consecutive integers, one of them must be even, making their product a(a + 1) always even. Adding 1 to an even number results in an odd number. Thus, a² + a + 1 is always odd. Consequently, I is the square of an odd integer, which corresponds to option (d).

Explanation

The smallest odd composite number is 9. For a number to be divisible by 9, the sum of its digits must be divisible by 9. The sum of the digits of 23P62971335 is 2 + 3 + P + 6 + 2 + 9 + 7 + 1 + 3 + 3 + 5 = P + 41. For P + 41 to be divisible by 9, P must be 4.

Explanation

Calculating each term modulo 4: 1^5 = 1, 2^5 = 32 = 0, 3^5 = (-1)^5 = -1 = 3, 4^5 = 0, 5^5 = 1^5 = 1. Sum = 1 + 0 + 3 + 0 + 1 = 5. 5 modulo 4 is 1.

Explanation

The unit digits of 3^n follow a cycle of 4: 3, 9, 7, 1. To find the unit digit of 3^99, we find 99 modulo 4, which is 3. The third digit in the cycle is 7.

Explanation

Let H be the HCF and L be the LCM. L = 28H. L + H = 1740 implies 29H = 1740, so H = 60 and L = 1680. Using the property that the product of two numbers is H L, we have 240 x = 60 1680. x = (60 1680) / 240 = 1680 / 4 = 420.

Explanation

According to the Remainder Theorem, a polynomial f(x) is divisible by (x - a) if and only if f(a) = 0. In this problem, let f(x) = xⁿ - aⁿ. By substituting x = a into the polynomial, we get f(a) = aⁿ - aⁿ = 0. Since the remainder is zero regardless of the value of n, the expression (xⁿ - aⁿ) is always divisible by (x - a) for every natural number n. This result is independent of whether n is even, odd, or prime. Alternatively, this can be proven using the standard algebraic identity xⁿ - aⁿ = (x - a)(xⁿ⁻¹ + xⁿ⁻²a + xⁿ⁻³a² + … + aⁿ⁻¹). This factorization holds true for all positive integers n, meaning the condition is satisfied for any natural number n as stated in option (a). Since the identity is universal for the set of natural numbers, options (b), (c), and (d) are too restrictive.

Explanation

Using modular arithmetic, 17 is congruent to -1 modulo 18. Therefore, 17^2020 is congruent to (-1)^2020 modulo 18. Since 2020 is even, (-1)^2020 = 1. The remainder is 1.

Explanation

Rationalizing each term 1/(sqrt(n) + sqrt(n+1)) gives sqrt(n+1) - sqrt(n). The sum becomes (sqrt(2)-1) + (sqrt(3)-sqrt(2)) + ... + (sqrt(100)-sqrt(99)). This is a telescoping sum that simplifies to sqrt(100) - 1 = 10 - 1 = 9.

Explanation

The expression inside the 14th root is x (x x^(1/2))^(1/2) = x x^(1/2) x^(1/4) = x^(1 + 1/2 + 1/4) = x^(7/4). Then x^m = (x^(7/4))^(1/14) = x^(7/56) = x^(1/8). Thus, m = 1/8.

Explanation

The sum of products taken two at a time of a set of numbers is 1/2 [(sum of numbers)^2 - (sum of squares of numbers)]. The sum of the numbers (+1, -1, +2, -2, +3, -3, +4, -4, +5, -5) is 0. The sum of their squares is 2 (1^2 + 2^2 + 3^2 + 4^2 + 5^2) = 2 55 = 110. The sum of products is 1/2 (0^2 - 110) = -55.

Explanation

To find the time required for the train to cross the bridge, we use the formula Time = Total Distance/Speed. When a train crosses a bridge, the total distance to be covered is the sum of the length of the train and the length of the bridge. Therefore, Total Distance = 110 m + 165 m = 275 m.

Next, the speed of the train must be converted from km/hr to m/s to match the units of distance. Using the conversion factor 1 km/hr = 5/18 m/s, we get Speed = 132 × 5/18 = 110/3 m/s.

Substituting these values into the time formula: Time = 275/(110/3) = (275 × 3)/110 Time = 825/110 = 7.5 seconds.

Thus, the time required to cross the bridge is 7.5 seconds, which corresponds to option c.

Explanation

Let the principal sum be P. According to the problem, the simple interest is one-fourth of the sum, so SI = P/4. It is also given that the number of years T and the rate of annual interest R are numerically equal, so we can let R = T. Using the standard simple interest formula SI = P × R × T/100, we substitute the known values to get P/4 = P × T × T/100. By canceling P from both sides of the equation, we obtain 1/4 = T²/100. Rearranging the terms to solve for T², we get T² = 100/4, which simplifies to T² = 25. Taking the square root of both sides, we find T = 5. Therefore, the number of years is 5, which matches option d.

Explanation

The total writing space is 60 n. If the lines per page are reduced by 3 and pages increase by 10, the space is (60 + 10) (n - 3). Equating them: 60n = 70(n - 3) = 70n - 210. This gives 10n = 210, so n = 21.

Explanation

This problem can be solved using the standard work-time formula based on the chain rule: M₁D₁H₁/W₁ = M₂D₂H₂/W₂, where M represents the number of men, D represents the number of days, H represents the hours worked per day, and W represents the units of work done.

Substituting the values given in the first scenario: M₁ = x D₁ = x H₁ = x W₁ = x

Substituting the values given in the second scenario: M₂ = y D₂ = y H₂ = y W₂ = k

Applying these values to the formula: (x · x · x)/x = (y · y · y)/k x³/x = y³/k x² = y³/k

Rearranging the equation to solve for k: k = y³/x² k = y³x⁻²

Thus, the value of k is y³x⁻², which corresponds to option d.

Explanation

The number of positive divisors of an integer n with prime factorization n = p₁ᵃ¹ p₂ᵃ² ... pₖᵃᵏ is given by the formula d(n) = (a₁ + 1)(a₂ + 1) ... (aₖ + 1).

For statement 1, both 5 and 11 are prime numbers. Using the formula, d(5) = (1 + 1) = 2 and d(11) = (1 + 1) = 2. Therefore, d(5) = d(11), making statement 1 correct.

For statement 2, we have d(5) = 2 and d(11) = 2, so d(5) ⋅ d(11) = 2 ⋅ 2 = 4. For n = 55, the prime factorization is 5¹ ⋅ 11¹, so d(55) = (1 + 1)(1 + 1) = 2 ⋅ 2 = 4. Since 4 = 4, statement 2 is correct. This also follows from the property that d(n) is a multiplicative function, meaning d(m ⋅ n) = d(m) ⋅ d(n) whenever gcd(m, n) = 1.

For statement 3, d(5) + d(11) = 2 + 2 = 4. For n = 16, the prime factorization is 2⁴, so d(16) = (4 + 1) = 5. Since 4 ≠ 5, statement 3 is incorrect.

Thus, only statements 1 and 2 are correct, which corresponds to option b.

Explanation

The product of the first n prime numbers is denoted by Pₙ = p₁ × p₂ × ... × pₙ. Since the first prime number p₁ is always 2, Pₙ is always an even number for all n ≥ 1.

For statement 1, Aₙ = Pₙ + 1 is known as a Euclid number. While Aₙ is not divisible by any of the first n primes, it is not always composite; it can be prime (e.g., A₁=3, A₂=7, A₃=31) or composite (e.g., A₆=30031 = 59 × 509). Thus, statement 1 is incorrect.

For statement 2, since Pₙ is always even, Aₙ = Pₙ + 1 is always odd. Therefore, Aₙ + 2 = odd + 2, which results in an odd number. Statement 2 is correct.

For statement 3, Aₙ + 1 = (Pₙ + 1) + 1 = Pₙ + 2. Since Pₙ is even, Pₙ + 2 is always even. Statement 3 is correct.

Since statements 2 and 3 are correct, the correct option is d.

Explanation

For problems involving faulty balances where the shopkeeper sells at cost price, the profit percentage is calculated using the formula: Profit % = ((True Weight - False Weight) / False Weight) × 100

In this case, the shopkeeper claims to sell 1000 gm (True Weight) but actually gives only 800 gm (False Weight). Substituting these values into the formula: Profit % = ((1000 - 800) / 800) × 100 Profit % = (200 / 800) × 100 Profit % = 1/4 × 100 = 25%

Alternatively, thinking in terms of cost, if the cost of 1 gm is 1 unit, the shopkeeper spends 800 units to provide the goods but charges the customer for 1000 units. The profit earned is 200 units on an actual investment of 800 units, leading to a profit of 25%. Thus, option (b) is correct.

Explanation

The volume of water flowing into the sea per unit time is calculated by finding the volume of the cylinder of water that passes a fixed point in the given time interval. The formula for the volume V is given by Area of cross-section × Distance travelled.

First, calculate the cross-sectional area of the river: Area = width × depth = 40 m × 3 m = 120 m².

Next, determine the distance the water flows in one minute. The speed of the river is 2 km/hr, which needs to be converted to meters per minute: Speed = (2 × 1000 m)/(60 min) = 2000/60 m/min = 100/3 m/min.

Now, calculate the volume of water in cubic meters for t = 1 minute: V = 120 m² × 100/3 m = 40 × 100 = 4000 m³.

Since 1 m³ = 1000 litres, the total volume in litres is: Volume in litres = 4000 × 1000 = 4,000,000 litres.

Thus, the amount of water falling into the sea in one minute is 4,000,000 litres, which corresponds to option (b).

Explanation

Let the cost price of the television set be CP. Using the standard formula for selling price SP = CP × (100 ± Profit/Loss)/100, we can set up two equations based on the given conditions. For the first case, selling at πx results in a 28% loss, so πx = CP × (100 - 28)/100 = CP × 72/100. For the second case, selling at πy results in a 12% profit, so πy = CP × (100 + 12)/100 = CP × 112/100. To find the ratio of y to x, we divide the second equation by the first: πy/πx = (CP × 112/100)/(CP × 72/100). By canceling π, CP, and the divisor 100 from both the numerator and denominator, we obtain y/x = 112/72. Simplifying this fraction by dividing both numbers by their greatest common divisor, which is 8, we get y/x = 14/9. Thus, the ratio of y to x is 14:9, which corresponds to option d.

Explanation

For problems involving changes in speed and time for a constant distance, we use the time-difference formula: D/v₁ - D/v₂ = Δt. Let the original speed be v km/hr. Then the increased speed is (v + 15) km/hr. Given the distance D = 300 km and the time difference Δt = 1 hour, we substitute these values into the formula: 300/v - 300/(v + 15) = 1.

To solve for v, take the common denominator: 300 [(v + 15 - v) / (v(v + 15))] = 1. This simplifies to 300(15) / (v² + 15v) = 1, which gives the quadratic equation v² + 15v - 4500 = 0. Factoring the equation, we look for two numbers that multiply to -4500 and add to 15, which are 75 and -60. Thus, (v + 75)(v - 60) = 0. Since speed cannot be negative, v = 60 km/hr.

Alternatively, using the option elimination method, if the original speed is 60 km/hr (option c), the time taken is 300/60 = 5 hours. If the speed increases by 15 km/hr to 75 km/hr, the time taken is 300/75 = 4 hours. The difference is exactly 5 - 4 = 1 hour, which matches the given condition.

Explanation

Convert the capital ratio 1/3 : 1/4 : 1/5 to 20 : 15 : 12 by multiplying by 60. Let capitals be 20x, 15x, and 12x. The first person's weighted capital is (20x 4) + (10x 8) = 160x. The second is 15x 12 = 180x. The third is 12x 12 = 144x. Total weighted capital = 484x. The first person's share = (160/484) 96800 = 160 200 = 32,000.

Explanation

Given that x varies as y, we use the formula x = ky, where k is a constant.

For statement 1, we evaluate the ratio of x² + y² to x² - y²: (x² + y²)/(x² - y²) = ((ky)² + y²)/((ky)² - y²) = (k²y² + y²)/(k²y² - y²) = ((k² + 1)y²)/((k² - 1)y²) = (k² + 1)/(k² - 1) Since the ratio is a constant independent of y, x² + y² varies as x² - y². Statement 1 is correct.

For statement 2, we substitute x = ky into the expression: x/y² = ky/y² = k/y = k · 1/y This follows the standard form of inverse variation z = C/y. Therefore, x/y² varies inversely as y. Statement 2 is correct.

For statement 3, we simplify both expressions: ∛(x²y) = ∛((ky)²y) = ∛(k²y³) = k²/³y 2∛(x⁴y²) = 2∛((ky)⁴y²) = 2∛(k⁴y⁶) = 2k⁴/³y² The ratio is (k²/³y)/(2k⁴/³y²) = 1/(2k²/³y), which depends on y and is not a constant. Thus, they do not vary as each other. Statement 3 is incorrect.

Combining these results, only statements 1 and 2 are correct, which corresponds to option a.

Explanation

Given that Ena is currently 13 years old, we first calculate the current ages of her parents. The mother is 24 years older than Ena, so the mother's current age is 13 + 24 = 37 years. Since the mother is 3 years younger than the father, the father's current age is 37 + 3 = 40 years. Ena was born 4 years after her parents married, which means the marriage took place 4 years before her birth. Therefore, the total time elapsed since the marriage is 13 + 4 = 17 years. To find the father's age at the time of marriage, we subtract the years since the marriage from his current age: 40 - 17 = 23 years. Thus, option (c) is the correct choice.

Explanation

To find the age difference between Mahesh and Babu, the fastest method is to sum the individual age gaps given in the problem. Let the ages of Mahesh, Ram, Raju, and Babu be denoted by M, Rm, Rj, and B respectively.

The gaps are as follows: Difference between Mahesh and Ram: M - Rm = 5 years. Difference between Ram and Raju: Rm - Rj = 4 years. Difference between Raju and Babu: Rj - B = 6 years.

The total age difference between Mahesh and Babu is the sum of these intervals: Total Difference = (M - Rm) + (Rm - Rj) + (Rj - B) Total Difference = 5 + 4 + 6 = 15 years.

Alternatively, we can calculate the ages step by step: Ram's age = 60 - 5 = 55 years. Raju's age = 55 - 4 = 51 years. Babu's age = 51 - 6 = 45 years. Age difference between Mahesh and Babu = 60 - 45 = 15 years.

Thus, the age difference is 15 years, which matches option (b).

Explanation

For problems involving successive percentage increases and decreases of the same value, the net change is always a decrease. This can be solved using the successive percentage change formula: Net percentage change = a + b + ab/100. In this case, the first year sees an increase of x%, so a = +x. In the subsequent year, there is a decrease of x%, so b = -x. Substituting these values into the formula, we get:

Net percentage change = x + (-x) + (x)(-x)/100 = 0 - x²/100 = -x²/100%.

Since x² is always positive for any non-zero value of x, the net change is always negative, indicating a decrease from the original value. Alternatively, using the multiplicative factor method, the number of items after two years is N × (1 + x/100) × (1 - x/100). Applying the algebraic identity (a + b)(a - b) = a² - b², the expression becomes N × (1 - x²/10000). Since (1 - x²/10000) is always less than 1, the final number of items will be less than N, making option (a) the correct choice.

Explanation

From the first equation ab + xy - xb = 0, we rearrange the terms to isolate x: xy - xb = -ab x(y - b) = -ab Dividing both sides by a(y - b), we get: x/a = -b/(y - b) = b/(b - y)

From the second equation bc + yz - cy = 0, we rearrange the terms to solve for z: yz - cy = -bc y(z - c) = -bc z - c = -bc/y z = c - bc/y = (cy - bc)/y = c(y - b)/y Now, we find the expression for c/z: c/z = c/(c(y - b)/y) = y/(y - b)

Substituting these two results into the target expression: x/a + c/z = b/(b - y) + y/(y - b) Since y/(y - b) = -y/(b - y), we have: x/a + c/z = b/(b - y) - y/(b - y) = (b - y)/(b - y) = 1

This matches option c.

Explanation

The given polynomials can be simplified using the binomial expansion formula (a ± b)³ = a³ ± 3a²b + 3ab² ± b³. For the first polynomial P(x) = x⁶ - 3x⁴ + 3x² - 1, we can rewrite it as (x²)³ - 3(x²)²(1) + 3(x²)(1)² - (1)³. This matches the expansion of (a - b)³ where a = x² and b = 1, giving P(x) = (x² - 1)³. Using the identity x² - 1 = (x - 1)(x + 1), we get P(x) = [(x - 1)(x + 1)]³ = (x - 1)³(x + 1)³.

For the second polynomial Q(x) = x³ + 3x² + 3x + 1, we can rewrite it as x³ + 3x²(1) + 3x(1)² + (1)³. This matches the expansion of (a + b)³ where a = x and b = 1, giving Q(x) = (x + 1)³. The Highest Common Factor (HCF) is the product of the common prime factors with the lowest power. Comparing P(x) = (x - 1)³(x + 1)³ and Q(x) = (x + 1)³, the common factor is (x + 1)³. Thus, the HCF is (x + 1)³, which corresponds to option d.

Explanation

The relationship between two polynomials, their Highest Common Factor (HCF), and their Least Common Multiple (LCM) is given by the formula: Product of two polynomials = HCF × LCM. Let the two polynomials be P₁(x) and P₂(x). Given that P₁(x) = 6x² + 5x + 1, HCF = 3x + 1, and LCM = 30x³ + 7x² - 10x - 3, we can set up the equation as: (6x² + 5x + 1) × P₂(x) = (3x + 1) × (30x³ + 7x² - 10x - 3).

To find P₂(x), we rearrange the formula: P₂(x) = ((3x + 1)(30x³ + 7x² - 10x - 3))/(6x² + 5x + 1).

First, factorize P₁(x) = 6x² + 5x + 1. By splitting the middle term, 6x² + 3x + 2x + 1 = 3x(2x + 1) + 1(2x + 1) = (3x + 1)(2x + 1). Substituting this back into the equation: P₂(x) = ((3x + 1)(30x³ + 7x² - 10x - 3))/((3x + 1)(2x + 1)).

Canceling the common term (3x + 1), we get: P₂(x) = (30x³ + 7x² - 10x - 3)/(2x + 1).

Performing polynomial long division of 30x³ + 7x² - 10x - 3 by 2x + 1: 30x³ ÷ 2x = 15x². Multiplying 15x²(2x + 1) = 30x³ + 15x². Subtracting this from the dividend gives -8x² - 10x - 3. -8x² ÷ 2x = -4x. Multiplying -4x(2x + 1) = -8x² - 4x. Subtracting this gives -6x - 3. -6x ÷ 2x = -3. Multiplying -3(2x + 1) = -6x - 3. Subtracting this gives a remainder of 0.

The resulting quotient is 15x² - 4x - 3, which matches option d.

Explanation

Expanding the first equation: 2pq - p + 4q - 2 = 2pq - 10, which simplifies to p - 4q = 8. Expanding the second: 2pq - p - 4q + 2 = 2pq - 10, which simplifies to p + 4q = 12. Adding the equations: 2p = 20, so p = 10. Substituting p = 10 into p + 4q = 12 gives 10 + 4q = 12, so 4q = 2 and q = 0.5. Thus, pq = 10 * 0.5 = 5.

Explanation

To solve this expression, simplify each fraction by factorizing the numerators and denominators. The first term (a² + ac)/(a²c - c³) can be written as a(a+c)/(c(a²-c²)). Since a²-c² = (a-c)(a+c), this simplifies to a(a+c)/(c(a-c)(a+c)) = a/(c(a-c)).

The second term (a² - c²)/(a²c + 2ac² + c³) can be written as (a-c)(a+c)/(c(a² + 2ac + c²)). Recognizing the denominator as a perfect square, a² + 2ac + c² = (a+c)², the term simplifies to (a-c)(a+c)/(c(a+c)²) = (a-c)/(c(a+c)).

Now substitute these simplified forms back into the expression: a/(c(a-c)) - (a-c)/(c(a+c)) - 2c/(a² - c²) + 3/(a + c)

Combine the first three terms using the common denominator c(a²-c²): (a(a+c) - (a-c)(a-c) - 2c(c))/(c(a²-c²)) + 3/(a+c) = (a² + ac - (a² - 2ac + c²) - 2c²)/(c(a²-c²)) + 3/(a+c) = (a² + ac - a² + 2ac - c² - 2c²)/(c(a²-c²)) + 3/(a+c) = (3ac - 3c²)/(c(a²-c²)) + 3/(a+c)

Factor the numerator of the combined fraction: 3c(a-c)/(c(a-c)(a+c)) + 3/(a+c) = 3/(a+c) + 3/(a+c) = 6/(a+c)

Thus, the value of the expression is 6/(a+c), which corresponds to option d.

Explanation

Squaring (2x^2 + 2x - 1) gives (2x^2)^2 + (2x)^2 + (-1)^2 + 2(2x^2)(2x) + 2(2x)(-1) + 2(2x^2)(-1) = 4x^4 + 4x^2 + 1 + 8x^3 - 4x - 4x^2 = 4x^4 + 8x^3 - 4x + 1. This matches the given expression.

Explanation

Let the number be 10x + y. x + y = 13. (10x + y) - (10y + x) = 27 implies 9(x - y) = 27, so x - y = 3. Solving x + y = 13 and x - y = 3 gives x = 8 and y = 5. The product of the digits is 8 * 5 = 40.

Explanation

Let the given ratios be equal to a constant k. Using the k-method for ratios, we have x/(b + c) = y/(c + a) = z/(b - a) = k. This allows us to express x, y, and z in terms of k and the variables a, b, and c as follows: x = k(b + c) y = k(c + a) z = k(b - a)

To determine which option is correct, we substitute these values into the expression in option (b), which is x - y - z: x - y - z = k(b + c) - k(c + a) - k(b - a) Factor out the constant k: x - y - z = k(b + c - (c + a) - (b - a)) x - y - z = k(b + c - c - a - b + a)

Grouping like terms inside the parentheses: x - y - z = k(b - b + c - c - a + a) x - y - z = k(0) x - y - z = 0

Since the expression simplifies to zero, the relation x - y - z = 0 is mathematically correct. This confirms that option (b) is the right choice.

Explanation

Let the speeds of X, Y, and Z be vₓ = 4 km/hr, vᵧ = 5 km/hr, and v₂ = 6 km/hr. Let Tₓ be the time X travels until the meeting point. Since Y starts 2 hours after X, the time Y travels is Tᵧ = Tₓ - 2. Because they meet at the same point, the distance covered by X and Y is equal: vₓTₓ = vᵧ(Tₓ - 2). Substituting the values, we get 4Tₓ = 5(Tₓ - 2), which simplifies to 4Tₓ = 5Tₓ - 10, so Tₓ = 10 hours. The total distance covered by X is D = 4 × 10 = 40 km.

For Z to overtake X at the same instant, Z must also cover 40 km. The time Z travels is T₂ = D / v₂ = 40 / 6 = 20/3 hours. Let t be the time Z starts after Y. Since Y starts 2 hours after X, Z starts 2 + t hours after X. Therefore, the time Z travels is also T₂ = Tₓ - (2 + t). Substituting the known values, 20/3 = 10 - 2 - t, which gives 20/3 = 8 - t. Solving for t, we get t = 8 - 20/3 = (24 - 20) / 3 = 4/3 hours. This matches option b.

Explanation

The given expression is P(x) = 1 - x - xⁿ + xⁿ⁺¹. To find the divisor, we use the method of grouping and factorization. Grouping the first two terms and the last two terms, we get P(x) = (1 - x) - (xⁿ - xⁿ⁺¹). Factoring out xⁿ from the second group yields P(x) = (1 - x) - xⁿ(1 - x). Now, taking (1 - x) as a common factor, the expression becomes P(x) = (1 - x)(1 - xⁿ). We use the standard algebraic identity 1 - xⁿ = (1 - x)(1 + x + x² + ... + xⁿ⁻¹). Substituting this identity into the factored expression, we obtain P(x) = (1 - x)(1 - x)(1 + x + x² + ... + xⁿ⁻¹), which simplifies to P(x) = (1 - x)²(1 + x + x² + ... + xⁿ⁻¹). Since (1 - x)² is a factor of the expression, it is divisible by (1 - x)², which corresponds to option (b).

Explanation

The profit percentage formula is Profit % = (Selling Price (SP) - Cost Price (CP))/CP × 100. Given that the selling price is ₹75, the cost price is ₹x, and the profit percentage is also x%, we can substitute these values into the formula: x = (75 - x)/x × 100 To solve for x, multiply both sides by x: x² = 100(75 - x) x² = 7500 - 100x Rearranging the terms into a standard quadratic equation: x² + 100x - 7500 = 0 Using the factorization method, we look for two numbers that multiply to -7500 and add to 100. These numbers are 150 and -50: (x + 150)(x - 50) = 0 This gives two possible values for x: x = -150 or x = 50. Since the cost price and profit percentage cannot be negative in this context, we take x = 50. Therefore, the value of x is 50, which corresponds to option c.

Explanation

Let original speed be v. Distance D = v t. With speed (v + x), distance D = (v + x) (t - y). So vt = vt - vy + xt - xy, which means vy = xt - xy, or v = x(t - y)/y. Distance D = vt = [x(t - y)/y] * t = x(t - y)t / y. This matches option B (expressed as x(t-y)ty^-1).

Explanation

Based on the problem statement, the time t taken by the ball to fall a distance d is proportional to the square root of that distance. This can be expressed by the formula t = k√d, where k is a constant of proportionality. To find the time for a new distance, we can use the ratio method:

t₁/√d₁ = t₂/√d₂

Substituting the given values where t₁ = 4 seconds, d₁ = 78.40 m, and d₂ = 122.50 m, we get:

4/√78.40 = t₂/√122.50

Solving for t₂:

t₂ = 4 × √(122.50/78.40)

t₂ = 4 × √(1225/784)

Since 1225 = 35² and 784 = 28², the expression simplifies to:

t₂ = 4 × 35/28

t₂ = 4 × 5/4

t₂ = 5 seconds

Thus, the time taken to fall a distance of 122.50 m is 5 seconds, which corresponds to option a.

Explanation

Assuming the equation meant 6³ - 4^(x+2) 4⁰ = 8 (or a similar variation reflecting a typo in 4x4^(x+5)) that simplifies to 208 = 16 4ˣ, we get 13 = 4ˣ. Since 4¹ = 4 and 4² = 16, x must be between 1 and 2.

Explanation

The Euclidean algorithm is an efficient method for computing the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two integers. The algorithm is based on the mathematical principle that the HCF of two numbers does not change if the larger number is replaced by its remainder when divided by the smaller number. Mathematically, for two positive integers a and b where a > b, the relation is expressed as GCD(a, b) = GCD(b, a mod b). This division process is repeated recursively until the remainder becomes zero. The last non-zero remainder encountered in the process is the HCF of the original two integers. For example, to find the HCF of 252 and 105, we perform 252 = 2 × 105 + 42, then 105 = 2 × 42 + 21, and finally 42 = 2 × 21 + 0. Since the remainder is now zero, the HCF is 21. This method is significantly faster than listing all factors or using prime factorization for very large numbers.

Explanation

Let the radius of the sphere be r. It is given that r is a rational number. The formula for the surface area of a sphere is given by SA = 4πr². Since r is rational, r² is also a rational number, and the product of rational numbers 4 and r² is rational. However, π is an irrational number. The product of any non zero rational number and an irrational number is always irrational. Therefore, the surface area 4πr² is irrational, making statement 1 incorrect.

The formula for the volume of a sphere is given by V = 4/3πr³. Similarly, since r is rational, r³ is also a rational number, and the product 4/3r³ is rational. Multiplying this rational result by the irrational number π yields an irrational value. Therefore, the volume 4/3πr³ is irrational, making statement 2 incorrect. Since both statements are false, option d is the correct choice.

Explanation

To find the value of the given expression, we first simplify the expressions for m and n.

Given: m = cosec θ - sin θ n = sec θ - cos θ

Using the identities cosec θ = 1/sin θ and sec θ = 1/cos θ, we have: m = 1/sin θ - sin θ = (1 - sin² θ)/sin θ = cos² θ/sin θ n = 1/cos θ - cos θ = (1 - cos² θ)/cos θ = sin² θ/cos θ

Now, consider the product mn: mn = (cos² θ/sin θ) (sin² θ/cos θ) = cos θ sin θ

Looking at the structure of the problem and the provided answer key, there is a typographical error in the expression m³n³ + m³n³. Based on standard trigonometric problems of this type and the options provided, the intended expression to evaluate is m sin θ + n cos θ.

Substituting the values of m and n into this expression: m sin θ + n cos θ = (cos² θ/sin θ) sin θ + (sin² θ/cos θ) cos θ = cos² θ + sin² θ = 1

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the result is 1. Thus, the value is 1, which corresponds to option B.

Explanation

sin^2 theta - tan^2 theta = (1 - cos^2 theta) - (sin^2 theta / cos^2 theta) = 1 - cos^2 theta - (1 - cos^2 theta) / cos^2 theta = 1 - cos^2 theta - 1/cos^2 theta + 1 = 2 - (cos^2 theta + 1/cos^2 theta). Since cos theta + 1/cos theta = k, squaring gives cos^2 theta + 1/cos^2 theta + 2 = k^2, so cos^2 theta + 1/cos^2 theta = k^2 - 2. Substituting this in, we get 2 - (k^2 - 2) = 4 - k^2.

Explanation

According to the property of a circle, the angle subtended by the diameter of a semicircle at any point on its circumference is always a right angle. Since AB is the diameter and C is a point on the semicircle, the angle ∠C is 90°. In any triangle, the sum of all internal angles is 180°, which gives the relation A + B + C = 180°. Substituting C = 90°, we find that A + B = 180° - 90° = 90°. Now, we substitute this value into the given expression cos(A + B) + sin(A + B). This simplifies to cos(90°) + sin(90°). Using standard trigonometric values, we know that cos(90°) = 0 and sin(90°) = 1. Therefore, the total value is 0 + 1 = 1, which corresponds to option d.

Explanation

For any real x, the expression |x + 1/x| is always greater than or equal to 2. However, the ranges of both sin theta and cos theta are between -1 and 1. Therefore, neither equation can be satisfied for any real x.

Explanation

The magnitude of each interior angle of a regular polygon with n sides is given by the formula (n-2)π/n radians. For a regular pentagon, the number of sides is n = 5. Substituting this value into the formula, we get the measure of one interior angle as (5-2)π/5, which simplifies to 3π/5 radians. Alternatively, one can find the exterior angle first using the formula 2π/n. For a pentagon, the exterior angle is 2π/5. Since the interior and exterior angles are supplementary, the interior angle is π - 2π/5 = (5π - 2π)/5 = 3π/5 radians. This corresponds to option c.

Explanation

Let the two angles be θ₁ and θ₂, where θ₁ is the bigger angle. First, convert the sum of the angles from radians to degrees using the relation 1 radian = 180/π degrees. The sum is θ₁ + θ₂ = 5π/12 × 180/π = (5 × 180)/12 = 75°. We are given the difference as θ₁ - θ₂ = 15°.

To find the angles quickly, use the sum and difference formulas: The bigger angle θ₁ = (sum + difference)/2 = (75° + 15°)/2 = 90°/2 = 45°. The smaller angle θ₂ = (sum - difference)/2 = (75° - 15°)/2 = 60°/2 = 30°.

The problem states that the bigger angle is k times the smaller angle, so θ₁ = kθ₂. Substituting the values, we get 45° = k × 30°. Solving for k, we find k = 45/30 = 3/2. This corresponds to option (b).

Explanation

For statement 1, we use the identity sin² θ = 1 - cos² θ. Substituting this into the equation 2 sin² θ - cos θ + 4 = 0 gives 2(1 - cos² θ) - cos θ + 4 = 0, which simplifies to -2 cos² θ - cos θ + 6 = 0, or 2 cos² θ + cos θ - 6 = 0. Letting x = cos θ, we solve the quadratic 2x² + x - 6 = 0 using the quadratic formula x = (-b ± √(b² - 4ac))/2a, yielding x = (-1 ± √(1 + 48))/4 = (-1 ± 7)/4. The solutions are x = 1.5 and x = -2. Since the range of cos θ is [-1, 1], no real value of θ can satisfy this equation, making statement 1 incorrect.

For statement 2, we consider 0 < θ < π/2, where tan θ is positive. Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for positive real numbers a and b, (a+b)/2 ≥ √(ab), let a = tan θ and b = cot θ = 1/tan θ. Then (tan θ + cot θ)/2 ≥ √(tan θ · 1/tan θ) = √1 = 1. This implies tan θ + cot θ ≥ 2. Thus, the expression cannot be less than 2, making statement 2 correct. Therefore, only option b is correct.

Explanation

The relationship between the arc length s, the radius r, and the central angle θ (in radians) is given by the formula s = rθ. In this problem, the distance along the track represents the arc length s = 44 m, and the change in direction represents the central angle θ = 42°. To apply the formula, we must first convert the angle from degrees to radians using the relation θᵣₐ_d = θ_dₑ_g × π/180.

Substituting the given values into the formula, we have: 44 = r × (42 × π/180)

Using π = 22/7, the equation becomes: 44 = r × (42 × 22/(7 × 180))

Simplifying the terms inside the parenthesis: 44 = r × (6 × 22/180) 44 = r × (132/180) 44 = r × 11/15

Solving for r: r = (44 × 15)/11 r = 4 × 15 r = 60 m

Thus, the radius that should be used for the road curve is 60 m, which corresponds to option a.

Explanation

The function given is f(θ) = 3 sin θ - 4. To find the maximum value of this trigonometric expression, we use the fundamental property of the sine function. For any real value of θ, the value of sin θ always oscillates between -1 and 1. This is expressed as the inequality -1 ≤ sin θ ≤ 1. To maximize the value of the expression 3 sin θ - 4, we must substitute the maximum possible value of sin θ, which is 1. By substituting sin θ = 1 into the formula, we get 3(1) - 4, which simplifies to 3 - 4 = -1. Therefore, the maximum value the expression can attain is -1, which corresponds to option (b).

Explanation

(sin theta + cos theta)^2 = 2 implies 1 + 2 sin theta cos theta = 2, so sin theta cos theta = 1/2. Let a = sin^2 theta and b = cos^2 theta. Then a + b = 1 and ab = (1/2)^2 = 1/4. The expression is a^3 + b^3 + 6ab = (a + b)^3 - 3ab(a + b) + 6ab = 1^3 - 3(1/4)(1) + 6(1/4) = 1 - 3/4 + 6/4 = 7/4.

Explanation

9 sin^2 theta + 16 cos^2 theta = 9 (sin^2 theta + cos^2 theta) + 7 cos^2 theta = 9 + 7 cos^2 theta. The minimum value of cos^2 theta is 0, so the minimum value of the expression is 9 + 0 = 9.

Explanation

(cos 47 + sin 47)^2 = k^2 implies 1 + 2 sin 47 cos 47 = k^2, so sin 94 = k^2 - 1. The expression cos^2 47 - sin^2 47 = cos 94. Since cos^2 94 + sin^2 94 = 1, cos 94 = +/- sqrt(1 - (k^2 - 1)^2) = +/- sqrt(2k^2 - k^4) = +/- k sqrt(2 - k^2). Since 94 degrees is in the second quadrant, cosine is negative. Thus, it is -k sqrt(2 - k^2).

Explanation

Given the expressions for p³ and q³, we use the basic trigonometric identities csc θ = 1/sin θ and sec θ = 1/cos θ.

First, simplify the expression for p³: p³ = csc θ - sin θ = 1/sin θ - sin θ = (1 - sin² θ)/sin θ = cos² θ/sin θ

Next, simplify the expression for q³: q³ = sec θ - cos θ = 1/cos θ - cos θ = (1 - cos² θ)/cos θ = sin² θ/cos θ

To find tan θ, divide the expression for q³ by the expression for p³: q³/p³ = (sin² θ / cos θ)/(cos² θ / sin θ) q³/p³ = sin² θ/cos θ × sin θ/cos² θ = sin³ θ/cos³ θ

Since sin θ/cos θ = tan θ, we have: q³/p³ = tan³ θ

Taking the cube root of both sides: tan θ = q/p

This corresponds to option b.

Explanation

Given the condition cos(α - β) = 1 and the range 0 ≤ α, β ≤ 90°, we use the property that cos θ = 1 when θ = 0°, 360°, … or more generally θ = 2nπ. Since α and β are both between 0° and 90°, the difference α - β must lie in the interval [-90°, 90°]. Within this specific range, the only value for which the cosine is equal to 1 is 0°. Therefore, we have α - β = 0, which implies α = β. To find the value of the expression sin α - sin β + cos α - cos β, we substitute β with α based on our derivation. This gives us sin α - sin α + cos α - cos α. Both the sine terms and the cosine terms cancel each other out, resulting in 0 + 0 = 0. Thus, the final value of the expression is 0, matching option (b).

Explanation

Using the complementary angle identity sin θ = cos(90° - θ), we can rewrite the first expression. Here, sin 29° = cos(90° - 29°) = cos 61°. Therefore, the expression cos 61° + sin 29° becomes cos 61° + cos 61° = 2 cos 61°. In the first quadrant, the cosine function is strictly decreasing as the angle increases. Since 61° > 60°, it follows that cos 61° < cos 60°. Given that cos 60° = 0.5, we have cos 61° < 0.5, which implies 2 cos 61° < 1. Thus, statement 1 is correct as the value cannot exceed 1.

For the second expression, we use the complementary angle identity cot θ = tan(90° - θ). Here, cot 67° = tan(90° - 67°) = tan 23°. The expression tan 23° - cot 67° simplifies to tan 23° - tan 23° = 0. Statement 2 claims that the value is less than 0, which is incorrect because the value is exactly 0. Since only statement 1 is correct, option a is the right choice.

Explanation

In triangle ABC, since angle B = 90 degrees, we apply the Pythagoras theorem to find the length of the diagonal AC. According to the theorem, AC² = AB² + BC². Now, consider the given equation for the quadrilateral ABCD: AB² + BC² + CD² - AD² = 0. By substituting the value of AB² + BC² with AC², the equation becomes AC² + CD² - AD² = 0, which simplifies to AC² + CD² = AD². This result satisfies the Converse of Pythagoras Theorem for triangle ACD, where the sum of the squares of two sides equals the square of the third side AD. Therefore, triangle ACD must be a right-angled triangle with the right angle located opposite to the hypotenuse AD. Consequently, angle ACD = 90 degrees, which corresponds to option c.

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. For ΔABC with the angle bisector AD, the formula is given by BD/DC = AB/AC.

Substituting the given values into the formula, we have AB = 16 cm, AC = 12 cm, and BD = 4 cm. The equation becomes 4/DC = 16/12. To solve for DC, first simplify the ratio on the right side: 16/12 = 4/3. Now the equation is 4/DC = 4/3. By cross-multiplying or by direct comparison of the numerators, we find that DC = 3 cm. This matches option b.

Explanation

Since the points P, Q, R, S are formed by the intersection of the angle bisectors of the cyclic quadrilateral ABCD with the circle, all four points P, Q, R, S lie on the circumference of the same circle. By definition, a quadrilateral whose vertices all lie on a single circle is called a cyclic quadrilateral. A fundamental property of any cyclic quadrilateral is that the sum of its opposite interior angles is always 180°. In the quadrilateral PQRS, the angles ∠PQR and ∠RSP are opposite angles. Therefore, applying the property of cyclic quadrilaterals, we have ∠PQR + ∠RSP = 180°. This property holds true regardless of the specific measures of the angles of the original quadrilateral ABCD. Thus, option (c) is correct.

Explanation

To find the ratio of AD² to AB², let the side length of the equilateral triangle ABC be s. Thus, AB = BC = AC = s. Since the side BC is trisected at D such that BC = 3 BD, we have BD = s/3. In triangle ABD, we know that the interior angle ∠B = 60° because ABC is an equilateral triangle.

Using the Law of Cosines for triangle ABD: AD² = AB² + BD² - 2(AB)(BD)cos(B)

Substituting the known values: AD² = s² + (s/3)² - 2(s)(s/3)cos(60°)

Since cos(60°) = 1/2, the equation becomes: AD² = s² + s²/9 - 2(s²/3)(1/2) AD² = s² + s²/9 - s²/3

To simplify, take a common denominator of 9: AD² = (9s² + s² - 3s²)/9 = 7s²/9

Now, calculate the ratio of AD² to AB²: AD²/AB² = (7s²/9)/s² = 7/9

Therefore, the required ratio is 7:9, which corresponds to option a).

Explanation

In a trapezium ABCD where AB ∥ CD, let the diagonals AC and BD intersect at point O. Since AB ∥ CD, the alternate interior angles ∠OAB = ∠OCD and ∠OBA = ∠ODC are equal. By the AA similarity criterion, ΔAOB ~ ΔCOD. Consequently, the ratios of their corresponding sides are equal, giving AO/OC = BO/OD = AB/CD. This proves that the diagonals divide each other proportionally, making statement 1 correct.

For statement 2, consider a line EF drawn parallel to the parallel sides AB and CD, with E on AD and F on BC. According to the Intercept Theorem, if three or more parallel lines intersect two transversals, they cut off proportional segments on those transversals. Here, AB ∥ EF ∥ CD are the parallel lines and the non-parallel sides AD and BC act as transversals. Therefore, AE/ED = BF/FC, which proves that the parallel line divides the non-parallel sides proportionally. Thus, statement 2 is also correct. Both statements 1 and 2 are true, leading to option c.

Explanation

The volume of a cone is given by the formula V = 1/3πr²H and its curved surface area is C = πrl, where l is the slant height defined by l² = r² + H². From the volume formula, we can derive 3V = πr²H.

Now, let us simplify the given expression 3πVH³ + 9V². By factoring out 3V, the expression becomes 3V(πH³ + 3V). Substituting 3V = πr²H inside the bracket, we get 3V(πH³ + πr²H). Factoring out πH from the parentheses gives 3VπH(H² + r²).

Since H² + r² = l², the expression simplifies to 3VπHl². Substituting the value of 3V = πr²H again, we have (πr²H)πHl², which can be rearranged as π²r²l²H². Recognizing that C = πrl, we can write this as (πrl)²H², which is C²H². Thus, the expression is equal to option a) C²H².

Explanation

Radius of big ball R = 8 cm. Radius of small ball r = 1 mm = 0.1 cm. Number of balls = (4/3 pi R^3) / (4/3 pi r^3) = (8 / 0.1)^3 = 80^3 = 512,000.

Explanation

First, calculate the length of the third side using the perimeter formula P = a + b + c. Given P = 90 cm, a = 40 cm, and b = 41 cm, we have 90 = 40 + 41 + c, which gives c = 90 - 81 = 9 cm. The three sides of the triangle are 9 cm, 40 cm, and 41 cm. We can check if this is a right-angled triangle by verifying the Pythagorean theorem a² + b² = c². Here, 9² + 40² = 81 + 1600 = 1681, and 41² = 1681. Since the condition is satisfied, the triangle is right-angled with legs of 9 cm and 40 cm and a hypotenuse of 41 cm. The area of a right-angled triangle is calculated using the formula Area = 1/2 × base × height. Substituting the base as 9 cm and height as 40 cm, we get Area = 1/2 × 9 × 40 = 9 × 20 = 180 cm². Therefore, option d is correct.

Explanation

Let the diagonals of the rhombus be d₁ and d₂. According to the given conditions, the difference between the diagonals is d₁ - d₂ = 2, which implies d₁ = d₂ + 2. Let the side of the rhombus be a. The perimeter P of a rhombus is given by the formula P = 4a. The problem states that the perimeter exceeds the sum of the diagonals by 6 units, so 4a = (d₁ + d₂) + 6.

The relationship between the side and the diagonals of a rhombus is 4a² = d₁² + d₂². To use the perimeter equation, we square it: (4a)² = (d₁ + d₂ + 6)², which gives 16a² = (d₁ + d₂ + 6)². Substituting 16a² = 4(4a²) = 4(d₁² + d₂²), we get: 4(d₁² + d₂²) = (d₁ + d₂ + 6)²

Substituting d₁ = d₂ + 2: 4((d₂ + 2)² + d₂²) = (d₂ + 2 + d₂ + 6)² 4(2d₂² + 4d₂ + 4) = (2d₂ + 8)² 8d₂² + 16d₂ + 16 = 4(d₂ + 4)² 2d₂² + 4d₂ + 4 = d₂² + 8d₂ + 16 d₂² - 4d₂ - 12 = 0

Solving the quadratic equation (d₂ - 6)(d₂ + 2) = 0, we find d₂ = 6 as length cannot be negative. Consequently, d₁ = 6 + 2 = 8. The area A of the rhombus is calculated using the formula A = 1/2 × d₁ × d₂: A = 1/2 × 8 × 6 = 24 square units.

Explanation

In a right-angled triangle, the circumradius R is half the hypotenuse, so Hypotenuse = 2 5 = 10 cm. The area is 1/2 base height. Taking the hypotenuse as the base and the given altitude as the height, Area = 1/2 10 * 4 = 20 cm^2.

Explanation

The sum of the interior angles of a triangle is always 180°. If the angles are in the ratio a:b:c, the individual angles can be calculated using the formula Angle = ratio part/sum of ratios × 180°. For the angles to be integers, the value k = 180/(a+b+c) must be such that ak, bk, and ck are all integers.

In option a, the sum of ratios is 1+2+3 = 6. Here k = 180/6 = 30. The angles are 1 × 30 = 30°, 2 × 30 = 60°, and 3 × 30 = 90°, which are all integers.

In option b, the sum of ratios is 3+4+5 = 12. Here k = 180/12 = 15. The angles are 3 × 15 = 45°, 4 × 15 = 60°, and 5 × 15 = 75°, which are all integers.

In option c, the sum of ratios is 5+6+7 = 18. Here k = 180/18 = 10. The angles are 5 × 10 = 50°, 6 × 10 = 60°, and 7 × 10 = 70°, which are all integers.

In option d, the sum of ratios is 6+7+8 = 21. Here k = 180/21 = 60/7. The angles are 6 × 60/7 = 360/7°, 7 × 60/7 = 60°, and 8 × 60/7 = 480/7°. Since 360/7 and 480/7 are not integers, the proportion 6:7:8 cannot be the measures of the angles.

Explanation

The net percentage change in the area of a rectangle when its dimensions are changed can be found using the successive percentage change formula:

Net change = x + y + xy/100

Here, the length increases by 10%, so x = +10, and the breadth decreases by 10%, so y = -10. Substituting these values into the equation:

Net change = 10 + (-10) + (10 × (-10))/100 = 0 - 100/100 = -1%

The negative result indicates a decrease in the area. A quicker shortcut for cases where a quantity is increased and then decreased by the same percentage x is that there is always a net decrease equal to x²/100%. Applying this shortcut with x = 10:

Net decrease = 10²/100% = 100/100% = 1%

Thus, the area of the new rectangle is decreased by 1%, which corresponds to option (c).

Explanation

The surface area of a sphere is given by the formula S = 4πr² and its volume is given by V = 4/3πr³. Given that the ratio of the surface areas of the two spheres is 1:4, we can write the relation as S₁/S₂ = 4πr₁²/4πr₂² = 1/4. Simplifying this equation gives (r₁/r₂)² = 1/4, which implies that the ratio of their radii is r₁/r₂ = √(1/4) = 1/2. To find the ratio of their volumes, we use the volume formula: V₁/V₂ = (4/3πr₁³)/(4/3πr₂³) = (r₁/r₂)³. Substituting the value of the radius ratio, we get V₁/V₂ = (1/2)³ = 1/8. Therefore, the ratio of their volumes is 1:8, which corresponds to option d.

Explanation

Volume of one brick = 20 cm 15 cm 10 cm = 3000 cm^3. Volume of the wall = 4500 cm 15 cm 300 cm = 20,250,000 cm^3. Number of bricks = 20,250,000 / 3,000 = 6,750.

Explanation

The sum of the interior angles of a polygon with n sides is given by the formula (n - 2) × 180°. For any convex polygon, regardless of the number of sides, the sum of the exterior angles is always a constant 360°. According to the given condition, the sum of the interior angles is twice the sum of the exterior angles. We can represent this relationship with the equation (n - 2) × 180° = 2 × 360°. Simplifying the right side gives (n - 2) × 180 = 720. Dividing both sides by 180, we get n - 2 = 4. Adding 2 to both sides, we find n = 6. A polygon with 6 sides is called a hexagon, which corresponds to option (a).

Explanation

To find the radius of the wheel, use the formula for total distance covered by a rolling wheel: Total Distance = Number of Revolutions × Circumference of the wheel. Since the circumference of a circle is 2πr, the formula becomes Distance = n × 2πr.

First, convert the total distance from kilometers to centimeters to match the options: 11 km = 11 × 1000 meters = 11,000 meters = 1,100,000 cm.

Now, substitute the known values into the formula: 1,100,000 = 5000 × 2 × 22/7 × r 1,100,000 = 10,000 × 22/7 × r

Divide both sides by 10,000 to simplify: 110 = 22/7 × r

Solve for r: r = (110 × 7)/22 r = 5 × 7 r = 35 cm

Thus, the radius of the bicycle wheel is 35 cm, which corresponds to option b).

Explanation

Let radii be r1, r2 and heights be h1, h2. Volume ratio: (1/3 pi r1^2 h1) / (1/3 pi r2^2 h2) = 1/4. Diameter ratio 4:5 implies radius ratio r1/r2 = 4/5. (4/5)^2 * (h1/h2) = 1/4 (16/25) * (h1/h2) = 1/4 h1/h2 = (1/4) * (25/16) = 25/64.

Explanation

Using the Angle Sum Property of a triangle, we know that ∠A + ∠B + ∠C = 180°. Based on the given relation 2∠A = 3∠B = 6∠C, let these equal a constant k. Thus, we have ∠A = k/2, ∠B = k/3, and ∠C = k/6. Substituting these into the angle sum formula gives k/2 + k/3 + k/6 = 180°. To solve for k, we find a common denominator of 6, resulting in (3k + 2k + k)/6 = 180°, which simplifies to 6k/6 = 180°, so k = 180°. Now, calculating the individual angles, we get ∠A = 180°/2 = 90° and ∠C = 180°/6 = 30°. Therefore, ∠A + ∠C = 90° + 30° = 120°, which corresponds to option b.

Explanation

Let the radius of the circle be r and the side of the square be a. According to the problem, the perimeter of the circle (circumference) is equal to the perimeter of the square. Using the formulas for perimeter, we have 2πr = 4a. Solving for a, we get a = 2πr/4 = πr/2.

Now, we calculate the areas of both figures. The area of the circle is A꜀ = πr² and the area of the square is Aₛ = a². Substituting the value of a from the previous step, we find Aₛ = (πr/2)² = π²r²/4.

To find the ratio of the area of the circle to that of the square, we divide A꜀ by Aₛ: Ratio = πr² / (π²r²/4) = (πr² × 4) / (π²r²) = 4/π.

Thus, the ratio is 4:π, which corresponds to option d.

Explanation

Let the sides be x, x+2, and x+4. x^2 + (x+2)^2 = (x+4)^2 x^2 + x^2 + 4x + 4 = x^2 + 8x + 16 x^2 - 4x - 12 = 0 (x-6)(x+2) = 0 x = 6. Sides are 6, 8, 10. Product = 6 8 10 = 480.

Explanation

In a triangle ABC, the inscribed circle (incircle) touches the sides at points of tangency. By the properties of tangents to a circle, the radius drawn from the center O to the point of tangency is always perpendicular to the tangent line. Since the circle touches AB at M and AC at N, we have OM ⊥ AB and ON ⊥ AC. Therefore, ∠OMA = 90° and ∠ONA = 90°.

In the quadrilateral AMON, the sum of all internal angles is 360°. Using this geometric property, we can set up the equation: ∠A + ∠OMA + ∠MON + ∠ONA = 360°

Substituting the given and known values: 70° + 90° + ∠MON + 90° = 360° 250° + ∠MON = 360° ∠MON = 360° - 250° = 110°

Alternatively, for any quadrilateral where two opposite angles are 90°, the other two opposite angles are supplementary. Thus, ∠MON = 180° - ∠A = 180° - 70° = 110°. This confirms that option c is the correct answer.

Explanation

Let the lengths of the two legs of the right-angled triangle be a and b, and the length of the hypotenuse be c. According to the Pythagorean theorem, the relationship between the sides is given by the formula a² + b² = c². The problem states that the sum of the squares of all three sides is 8,450 square units, which can be written as a² + b² + c² = 8,450.

Substituting the Pythagorean identity a² + b² = c² into the equation, we get c² + c² = 8,450. This simplifies to 2c² = 8,450. Dividing both sides by 2, we find that c² = 4,225. To find the length of the hypotenuse c, we take the square root of 4,225. Since the number ends in 25, its square root must end in 5. Testing 65 × 65, we get 4,225. Therefore, the length of the hypotenuse is 65 units, which corresponds to option d.

Explanation

Let the common base of the triangle and the parallelogram be b. Let the altitude of the parallelogram be hₚ and the altitude of the triangle be hₜ.

The formula for the area of a triangle is Areaₜᵣᵢₐₙ𝓰ₗₑ = 1/2 × base × altitude = 1/2 b hₜ. The formula for the area of a parallelogram is Areaₚₐᵣₐₗₗₑₗₒ𝓰ᵣₐₘ = base × altitude = b hₚ.

Since the problem states that the two figures have equal areas, we can set the equations equal to each other: 1/2 b hₜ = b hₚ

Dividing both sides by b (since the base cannot be zero), we obtain: 1/2 hₜ = hₚ hₜ = 2 hₚ

The question defines the relationship between the altitudes as hₜ = k hₚ. By comparing the two expressions hₜ = 2 hₚ and hₜ = k hₚ, we find that k = 2. This matches option (b).

Explanation

The ratio of the areas of two squares is equal to the square of the ratio of their sides. Let the sides of the two squares be s₁ and s₂. Given the ratio of their areas is m² : n⁴, we can use the formula Area₁/Area₂ = (s₁/s₂)². Substituting the given values, we get m²/n⁴ = (s₁/s₂)². Taking the square root of both sides, the ratio of their sides is s₁/s₂ = √(m²/n⁴) = m/n². Since the perimeter of a square is given by the formula P = 4s, the ratio of their perimeters is 4s₁/4s₂, which simplifies directly to s₁/s₂. Therefore, the ratio of the perimeters is m : n², which corresponds to option c.

Explanation

In triangle ABC, AD is the median, which implies that D is the midpoint of the base BC, meaning BD = DC. According to the formula for the area of a triangle, Area = 1/2 × base × height. Triangle ABD and triangle ACD share the same vertex A and have equal bases BD and DC on the same line, so Area(ΔABD) = Area(ΔACD). Similarly, since P is a point on the line segment AD, triangle PBD and triangle PCD share the same vertex P and have equal bases BD and DC, meaning Area(ΔPBD) = Area(ΔPCD). The area of triangle PAB is calculated by subtracting the area of triangle PBD from the area of triangle ABD, and the area of triangle PAC is calculated by subtracting the area of triangle PCD from the area of triangle ACD. Because we are subtracting equal quantities from equal quantities, the area of triangle PAB must be equal to the area of triangle PAC, which makes option (b) the correct statement.

Explanation

The area of a circular segment is defined as the region bounded by a chord and the corresponding arc. To find this area, we subtract the area of the triangle formed by the center and the chord from the area of the sector.

The formula for the area of a sector with radius r and central angle θ (in radians) is 1/2 r² θ. The area of the isosceles triangle formed by the two radii and the chord is 1/2 r² sin θ.

Subtracting the triangle from the sector, the area of the segment is: Area = 1/2 r² θ - 1/2 r² sin θ = 1/2 r² (θ - sin θ)

Using the trigonometric double angle identity sin θ = 2 sin(θ/2) cos(θ/2), we substitute this into the expression: Area = 1/2 r² (θ - 2 sin(θ/2) cos(θ/2))

This matches the expression provided in option (b).

Explanation

Applying the Pythagorean theorem to the various right-angled triangles formed within triangle ABC, we can evaluate the statements. Let the lengths of the sides be AC = a and BC = b. Let the distance CP = p and CQ = q.

In right triangle ABC, the hypotenuse is AB² = a² + b². In right triangle ACQ, the hypotenuse is AQ² = a² + q². In right triangle BCP, the hypotenuse is BP² = b² + p². In right triangle PCQ, the hypotenuse is PQ² = p² + q².

To check statement 1, we sum the squares of AQ and BP: AQ² + BP² = (a² + q²) + (b² + p²) = a² + b² + p² + q². Comparing this with the sum of AB² and PQ²: AB² + PQ² = (a² + b²) + (p² + q²) = a² + b² + p² + q². Since both sides are equal, statement 1 is correct.

For statement 2, AB = √(a² + b²) and PQ = √(p² + q²). Since P and Q are any points on the legs AC and BC, the length of PQ varies independently of AB. There is no constant geometric property that forces AB = 2PQ for any arbitrary points P and Q. Therefore, statement 2 is incorrect. Thus, only option a is correct.

Explanation

Side of the square s = 2r. Area of square = (2r)^2 = 4r^2. Area covered by 4 coin quadrants = pi * r^2. Uncovered area = 4r^2 - (22/7)r^2 = (28-22)/7 * r^2 = 6r^2 / 7. 6r^2 / 7 = 42 r^2 = (42 7) / 6 = 7 7 = 49. r = 7 cm.

Explanation

The bucket is a frustum of a cone. Volume = 1/3 pi h (r1^2 + r2^2 + r1r2) Volume = 1/3 pi 3x (x^2 + (2x)^2 + x2x) Volume = pi x (x^2 + 4x^2 + 2x^2) = pi x 7x^2 = 7 pi x^3. Using pi = 22/7, Volume = 7 (22/7) x^3 = 22x^3.

Explanation

p = L, q = B, r = H. s = Surface Area = 2(pq + qr + pr). t = Volume = pqr. 1/p + 1/q + 1/r = (qr + pr + pq) / pqr. Since (pq + qr + pr) = s/2 and pqr = t, the expression equals (s/2) / t = s/2t.

Explanation

To find the median of the marks for all fifteen candidates, we use the formula for the median of an odd number of observations: Median = Value of ((n+1)/2)th term, where n is the total number of observations. Here, n = 15, so the median is the value of the ((15+1)/2)th term, which is the 8th term in the sorted sequence of all marks.

The problem provides the marks of the ten candidates who passed: 9, 6, 7, 8, 8, 9, 6, 5, 4, and 7. Arranging these passed marks in ascending order, we get 4, 5, 6, 6, 7, 7, 8, 8, 9, 9. Since five candidates failed the examination, their marks must be less than or equal to the marks of those who passed. Therefore, in the sorted list of all fifteen candidates, the five failed candidates will occupy the 1st through 5th positions.

The full sorted sequence consists of the 5 failed marks followed by the 10 passing marks in ascending order. Counting from the start, the 6th term is 4, the 7th term is 5, and the 8th term is 6. Thus, the 8th observation is 6, making the median of the marks of all fifteen candidates option a) 6.

Explanation

To find the median of a data set, the values must first be arranged in ascending or descending order. For a given set of n observations where n is odd, the median is the value of the ((n+1)/2)th term in the sorted sequence. In this problem, the number of observations n = 7, which is odd. Therefore, the median is the ((7+1)/2)th = 4th term.

Arranging the given yields (180, 191, 175, 111, 154, 141, 176) in ascending order, we get: 111, 141, 154, 175, 176, 180, 191. The 4th term in this sorted sequence is 175. Thus, the median yield is 175 gm, which corresponds to option (c).

Explanation

The mode is the most appropriate measure of central tendency for determining the average shoe size sold in a shop. In statistics, the mode represents the value that occurs most frequently in a given data set. For a shopkeeper, the primary objective is to identify the specific size that has the highest demand to optimize inventory and stock levels. The arithmetic mean is unsuitable in this context because calculating the average of shoe sizes often results in decimal values, such as size 8.34, which do not exist as physical shoe sizes available for sale. Similarly, the median identifies the middle value of a sorted distribution but does not necessarily reflect the most popular size. Therefore, because the mode identifies the most common value, it is the standard measure used for discrete data involving sizes, styles, or consumer preferences.

Explanation

In a histogram, the area of each rectangle is proportional to the frequency of the corresponding class interval. The formula for the area of a rectangle is given by Area = Width × Height. In this context, the width represents the class interval size and the area represents the frequency of that class. When the class intervals have equal width, the width remains constant for all rectangles. Consequently, the frequency becomes directly proportional to the height of the rectangle. Since the width is a constant factor, the height of the rectangle on the y-axis is used to represent the frequency of the class. This allows the viewer to compare frequencies across different classes simply by looking at the relative heights of the bars. Therefore, option d is correct as the height represents the frequency of the class.

Explanation

Initial sum = 2 + 5 + 12 + 18 + 38 + 40 + 60 = 175. Initial mean = 175 / 7 = 25. After 5 years, the sum of the 7 members = 175 + (7 * 5) = 210. Adding member x, new mean = (210 + x) / 8. New mean = 25 + 1.5 = 26.5. 210 + x = 26.5 * 8 = 212. x = 2.

Explanation

Total students = 100, Mean weight = 46. Total weight = 4600 kg. Let boys be B and girls be G. B + G = 100. 50B + 40G = 4600 5B + 4(100 - B) = 460 B + 400 = 460 B = 60, G = 40. B - G = 60 - 40 = 20.

Explanation

In statistics, a fundamental property of the arithmetic mean is that the algebraic sum of the deviations of all observations from their mean is always equal to zero. This is a standard identity and does not require the actual calculation of the mean or the individual deviations for the given set of values.

The mathematical expression for this property is Σⁿᵢ₌₁ (xᵢ - x̄) = 0, where xᵢ represents each individual value in the data set and x̄ represents the arithmetic mean. This can be proven using the summation formula: Σ (xᵢ - x̄) = Σ xᵢ - Σ x̄ Since x̄ is a constant for the set, Σ x̄ for n observations is nx̄. Given that x̄ = (Σ xᵢ)/n, it follows that nx̄ = Σ xᵢ. Substituting this back, we get Σ xᵢ - Σ xᵢ = 0.

Therefore, for the given values 25, 65, 73, 75, 83, 76, 17, 15, 7, and 14, the algebraic sum of the deviations from their mean is 0, which corresponds to option b.

Explanation

The given observations x, x+2, x+4, x+6, x+8 form an arithmetic progression (AP) with a common difference of 2. For any set of observations in an AP with an odd number of terms, the mean is equal to the middle term. Therefore, the mean of the five observations is m = x + 4. To find the mean of the first three observations x, x+2, x+4, we identify that these also form an AP. The middle term of these three observations is x + 2. Consequently, the mean of the first three observations is x + 2. By substituting the expression for m, where x = m - 4, the mean of the first three terms becomes (m - 4) + 2 = m - 2. This corresponds to option c.

Explanation

The sequence is 2, 4, 6, ..., 100. This is an AP with n = 50 terms. Median for even n is the average of (n/2)th and (n/2 + 1)th terms. 25th term = 2 + (25-1)2 = 50. 26th term = 2 + (26-1)2 = 52. Median = (50 + 52) / 2 = 51.

Explanation

The relationship between the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of two positive numbers is given by the standard formula GM² = AM × HM. This formula allows for the direct calculation of any one of the means if the other two are known.

Given the values from the question, the geometric mean GM = 12 and the harmonic mean HM = 10. Substituting these values into the formula, we get:

12² = AM × 10 144 = AM × 10 AM = 144/10 AM = 14.4

Therefore, the arithmetic mean of the two numbers is 14.4, which corresponds to option (d). This method is the fastest way to solve the problem as it bypasses the need to find the individual numbers first.