CDS II 2025 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

According to the Remainder Theorem, if a polynomial p(x) is divided by (x - a), the remainder is p(a). Given the problem, when p(x) is divided by (x - 1), the remainder is 2, so p(1) = 2. Similarly, when p(x) is divided by (x - 2), the remainder is 1, so p(2) = 1.

When p(x) is divided by a quadratic divisor of the form (x - 1)(x - 2), the remainder must be a polynomial of degree less than 2. Let the remainder be r(x) = ax + b. The division algorithm gives the relation p(x) = (x - 1)(x - 2)q(x) + (ax + b).

Substituting the known values: For x = 1: p(1) = a(1) + b ⇒ a + b = 2 For x = 2: p(2) = a(2) + b ⇒ 2a + b = 1

Subtracting the first equation from the second: (2a + b) - (a + b) = 1 - 2 a = -1

Substituting a = -1 back into a + b = 2: -1 + b = 2 b = 3

The remainder r(x) = ax + b becomes -1x + 3, which is 3 - x. This matches option c.

Explanation

For any positive real number x, the AM-GM inequality states that (x + 1/x)/2 >= sqrt(x * 1/x) = 1, which means x + 1/x >= 2. I. Since x + 1/x >= 2, it is always greater than 1. Correct. II. Since x + 1/x >= 2, then (x + 1/x)^2 >= 4, which is always greater than 2. Correct. III. Since x + 1/x >= 2, then (x + 1/x)^4 >= 16, which is always greater than 9. Correct. All three statements are correct.

Explanation

We seek the largest natural number p such that q^2 - 5p - 4 < 0 for some natural number q > p. This inequality can be rewritten as 5p > q^2 - 4. If p = 4, then q must be at least 5. For q = 5, q^2 - 4 = 21. 5p = 20. 20 is not greater than 21. For q > 5, q^2 - 4 will be even larger. So p = 4 is not possible. If p = 3, then q must be at least 4. For q = 4, q^2 - 4 = 12. 5p = 15. 15 > 12, so 16 - 15 - 4 = -3, which is negative. The largest value of p is 3.

Explanation

To find the number of combinations, we consider the parity of the prime numbers. For x-y and x+y to be prime, x and y cannot both be odd primes because the sum and difference of two odd numbers are always even. The only even prime number is 2. If x+y and x-y were both even, they would both have to equal 2, implying x=2 and y=0, but y must be a natural number and a prime. Therefore, one of x or y must be 2. Since x-y must be a positive prime, x must be greater than y, so we must have y=2.

Substituting y=2 into the conditions, we require x, x-2, and x+2 to all be prime numbers. This forms a prime triplet of the form (p, p+2, p+4) where p = x-2. In any sequence of three odd numbers spaced by 2, exactly one of them must be divisible by 3. For all three to be prime, one of them must be equal to 3. The only such prime triplet is (3, 5, 7). Setting x-2=3 gives x=5, which means the primes are x-y=3, y=2, x=5, and x+y=7. All these values are prime and less than 20. Testing other odd primes for x such as 7, 11, 13, 17, or 19 shows that either x-2 or x+2 fails to be prime. Thus, only one combination (5, 2, 7, 3) exists, which corresponds to option a.

Explanation

The expression on the right hand side is a difference of squares, which can be factorized using the formula a² - b² = (a - b)(a + b). Applying this to x⁴ - 1, we get (x²)² - 1² = (x² - 1)(x² + 1). Further factorizing the term (x² - 1) using the same formula results in (x - 1)(x + 1)(x² + 1).

Now, we compare this factorized form with the given left hand side expression (x + 1)(x + p)(x² + p²). By matching the corresponding factors, we observe that the factor (x + 1) is present on both sides. Comparing (x + p) with (x - 1) gives p = -1. Similarly, comparing (x² + p²) with (x² + 1) gives p² = 1, which is consistent with p = -1 since (-1)² = 1. Thus, the value of p that satisfies the equation is -1, which corresponds to option a.

Explanation

Let a = (2 + √3)ˣ and b = (2 - √3)ˣ. We are given the equation a + b = 2. First, observe the relationship between the two bases: (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1. Therefore, the product ab = ((2 + √3)(2 - √3))ˣ = 1ˣ = 1. To find the value of a - b, we use the standard algebraic identity relating the sum and difference of two variables: (a - b)² = (a + b)² - 4ab. Substituting the known values into this formula, we get (a - b)² = (2)² - 4(1) = 4 - 4 = 0. Taking the square root of both sides, we find that a - b = 0. Thus, the expression (2 + √3)ˣ - (2 - √3)ˣ is equal to 0, which corresponds to option a.

Explanation

Let x = 1/a and y = 1/b. The given equations are x + y = 5/6 and x² + y² = 13/36. We need to find the value of x³ + y³. Using the algebraic identity (x + y)² = x² + y² + 2xy, we substitute the given values to find xy: (5/6)² = 13/36 + 2xy 25/36 = 13/36 + 2xy 2xy = 25/36 - 13/36 = 12/36 = 1/3 xy = 1/6

Now, we use the formula for the sum of cubes: x³ + y³ = (x + y)(x² + y² - xy). Substituting the known values into this formula: x³ + y³ = (5/6)(13/36 - 1/6) To subtract the fractions inside the bracket, find a common denominator: x³ + y³ = (5/6)(13/36 - 6/36) x³ + y³ = (5/6)(7/36) x³ + y³ = 35/216

Thus, the value of 1/a³ + 1/b³ is 35/216, which corresponds to option b.

Explanation

To find the remainder when a polynomial P(x) is divided by a divisor D(x), we can use the Remainder Theorem logic. For the divisor x² + 1, we set x² + 1 = 0, which implies x² = -1. By substituting this value of x² into the dividend P(x) = x⁶, we can directly determine the remainder.

Since x⁶ can be rewritten as (x²)³, substituting x² = -1 gives: Remainder = (-1)³ = -1

Alternatively, using the algebraic identity for the sum of cubes, we know that a³ + b³ = (a + b)(a² - ab + b²). Letting a = x² and b = 1, we have: x⁶ + 1 = (x² + 1)(x⁴ - x² + 1) Rearranging this gives: x⁶ = (x² + 1)(x⁴ - x² + 1) - 1

When x⁶ is divided by x² + 1, the term (x² + 1)(x⁴ - x² + 1) is perfectly divisible, leaving a remainder of -1. Therefore, option (a) is correct.

Explanation

According to the Factor Theorem, for a linear expression (x - a) to be a factor of a polynomial P(x), the value of the polynomial at x = a must be zero, i.e., P(a) = 0. In this case, the given factor is (x + 2), which can be rewritten as (x - (-2)). Therefore, we must substitute x = -2 into the polynomials to check which one results in zero.

For option (a), the polynomial is P(x) = x⁵ - 4x⁴ - 3x³ + 8x² - 14x + 12. Substituting x = -2: P(-2) = (-2)⁵ - 4(-2)⁴ - 3(-2)³ + 8(-2)² - 14(-2) + 12 P(-2) = -32 - 4(16) - 3(-8) + 8(4) + 28 + 12 P(-2) = -32 - 64 + 24 + 32 + 28 + 12 P(-2) = -96 + 64 + 32 P(-2) = -96 + 96 = 0

Since the result is zero, (x + 2) is a factor of the polynomial in option (a). For other options, substituting x = -2 will not yield zero. For instance, in option (b), the second term becomes positive, significantly increasing the sum, and in options (c) and (d), the signs of the coefficients of x³ or x are changed, preventing the sum from canceling out to zero.

Explanation

The number of digits in N is floor(log10(N)) + 1. log10(60^60) = 60 log10(60) = 60 (log10(6) + log10(10)) = 60 * (log10(2) + log10(3) + 1). Using given values: 60 (0.301 + 0.477 + 1) = 60 1.778 = 106.68. Number of digits = floor(106.68) + 1 = 106 + 1 = 107.

Explanation

Using modulo 18: 17 is congruent to -1 (mod 18). 19 is congruent to 1 (mod 18). (17^25 + 19^25) mod 18 = (-1)^25 + (1)^25 = -1 + 1 = 0. The remainder is 0.

Explanation

Using the algebraic identities for the sum and difference of cubes, we can simplify the expressions for p and q. Since x³ + y³ = (x + y)(x² - xy + y²), the expression for p simplifies to p = x + y. Similarly, since x³ - y³ = (x - y)(x² + xy + y²), the expression for q simplifies to q = x - y. We are given that HCF(x, y) = H. Let x = Ha and y = Hb, where a and b are coprime, meaning HCF(a, b) = 1. Then, HCF(p, q) = HCF(Ha + Hb, Ha - Hb) = H · HCF(a + b, a - b). Let d = HCF(a + b, a - b). By the properties of HCF, d must divide the sum (a + b) + (a - b) = 2a and the difference (a + b) - (a - b) = 2b. Therefore, d must divide HCF(2a, 2b) = 2 · HCF(a, b) = 2(1) = 2. This means the value of d can be either 1 or 2. If a and b have opposite parity (one even, one odd), then a + b and a - b are both odd, making d = 1 and HCF(p, q) = H. If a and b are both odd, then a + b and a - b are both even, making d = 2 and HCF(p, q) = 2H. Thus, both statements I and II are correct.

Explanation

To solve this problem, use the substitution method to transform the fourth-degree equation into a quadratic equation. Let y = x². Substituting this into the given equation x⁴ = x² + 1 gives y² = y + 1, which can be rewritten in the standard quadratic form as y² - y - 1 = 0.

Applying the quadratic formula y = (-b ± √(b² - 4ac))/2a where a = 1, b = -1, and c = -1, we get y = (1 ± √((-1)² - 4(1)(-1)))/2(1) = (1 ± √(1 + 4))/2 = (1 ± √5)/2. Since y = x² and x is a real number, y must be positive. Therefore, x² = (1 + √5)/2.

The target expression is 2x⁴. From the original equation, we know x⁴ = x² + 1. Substituting the value of x², we have x⁴ = (1 + √5)/2 + 1 = (1 + √5 + 2)/2 = (3 + √5)/2. To find 2x⁴, multiply this result by 2: 2x⁴ = 2 × (3 + √5)/2 = 3 + √5, which corresponds to option b.

Explanation

Given the equation (p + q)/(q + r) = (r + s)/(s + p), we use the method of cross-multiplication to simplify the relation. Cross-multiplying the terms gives (p + q)(s + p) = (r + s)(q + r). Expanding both sides of the equation, we get ps + p² + qs + qp = rq + r² + sq + sr.

Subtracting qs from both sides simplifies the equation to ps + p² + qp = rq + r² + sr. Rearranging all terms to the left side to form a quadratic expression, we have p² - r² + ps - rs + pq - rq = 0. This expression can be factorized by grouping terms: (p - r)(p + r) + s(p - r) + q(p - r) = 0. Taking the common factor (p - r) out, we obtain (p - r)(p + r + s + q) = 0.

According to the zero product property, for the product of two factors to be zero, at least one of the factors must be zero. Therefore, either p - r = 0, which implies p = r, or p + q + r + s = 0. This confirms that option (c) is the correct choice as it encompasses both possible mathematical conditions derived from the original equation.

Explanation

We check the gcd of (12n + 2) and (8n + 1). Using the Euclidean algorithm: gcd(12n + 2, 8n + 1) = gcd(12n + 2 - (8n + 1), 8n + 1) = gcd(4n + 1, 8n + 1). gcd(4n + 1, 8n + 1) = gcd(4n + 1, 8n + 1 - 2(4n + 1)) = gcd(4n + 1, -1) = 1. The gcd is always 1 for any natural number n. For n < 7, n can be 1, 2, 3, 4, 5, 6. All 6 values satisfy the condition.

Explanation

The expression x^3 + y^3 + 3xy - 1 can be factored using the identity a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) with a=x, b=y, c=-1. So, x^3 + y^3 - 1 + 3xy = (x + y - 1)(x^2 + y^2 + 1 - xy + x + y). The expression (x + y)^4 - 1 is a difference of squares: (x + y)^4 - 1 = ((x + y)^2 - 1)((x + y)^2 + 1) = (x + y - 1)(x + y + 1)((x + y)^2 + 1). The common factor (HCF) is (x + y - 1).

Explanation

Given that n is an even natural number, we can substitute n = 2k where k ∈ ℕ. The expression for x becomes x = 2k(2k + 1)(2k + 2). Factoring out 2 from the last term gives x = 2k(2k + 1) ⋅ 2(k + 1) = 4k(k + 1)(2k + 1). We know that the product of two consecutive integers k(k + 1) is always divisible by 2. Additionally, among the three terms k, k + 1, and 2k + 1, one must be divisible by 3. If k is not divisible by 3, then either k + 1 or 2k + 1 must be. For example, if k ≡ 1 (mod 3), then 2k + 1 ≡ 3 ≡ 0 (mod 3). If k ≡ 2 (mod 3), then k + 1 ≡ 3 ≡ 0 (mod 3). Thus, the product k(k + 1)(2k + 1) is always divisible by 2 × 3 = 6. Consequently, x = 4 × 6m = 24m for some integer m. For n = 2, x = 2(3)(4) = 24, which is not divisible by 48, proving statement I is incorrect. For statement II, we calculate x² = (24m)² = 576m². Since 576 = 144 × 4, we have x² = 144(4m²), which is always divisible by 144. Therefore, only statement II is correct, making option b the right choice.

Explanation

To find the LCM, we must first factorize each expression completely using algebraic identities. The first expression, x⁴ + x²y² + y⁴, is a classic identity that can be rewritten as (x² + xy + y²)(x² - xy + y²). The second expression, x³y + y⁴, factors as y(x³ + y³), which further decomposes into y(x + y)(x² - xy + y²) using the sum of cubes formula. The third expression, x⁴y² - x³y³, factors simply as x³y²(x - y).

The LCM is the product of the highest powers of all distinct irreducible factors present across the expressions. The distinct factors are x³, y², (x - y), (x + y), (x² + xy + y²), and (x² - xy + y²). Multiplying these together, we get LCM = x³y²(x - y)(x² + xy + y²)(x + y)(x² - xy + y²).

Applying the difference of cubes identity (x - y)(x² + xy + y²) = x³ - y³ and the sum of cubes identity (x + y)(x² - xy + y²) = x³ + y³, the expression simplifies to x³y²(x³ - y³)(x³ + y³). Using the difference of squares formula a² - b² = (a - b)(a + b), we arrive at x³y²(x⁶ - y⁶), which matches option b.

Explanation

Let the number XYZ be 100X + 10Y + Z and ZYX be 100Z + 10Y + X. The difference D = |(100X + 10Y + Z) - (100Z + 10Y + X)| = |99X - 99Z| = 99|X - Z|. D is a multiple of 99, so the remainder when D is divided by 99 is 0.

Explanation

Let k = p + q. We need k^k to be divisible by 512 = 2^9. If k = 4, 4^4 = (2^2)^4 = 2^8 (not divisible by 2^9). If k = 6, 6^6 = (2 3)^6 = 2^6 3^6 (not divisible by 2^9). If k = 8, 8^8 = (2^3)^8 = 2^24 (divisible by 2^9). The least value of p + q is 8.

Explanation

First, simplify the denominators of the first two terms. The denominator of the second term is c² + bc - a² - ab, which can be rewritten as (c² - a²) + b(c - a). Applying the difference of squares formula x² - y² = (x - y)(x + y), we get (c - a)(c + a) + b(c - a) = (c - a)(c + a + b). Now, both the first and second terms have the same common denominator (c - a)(c + a + b).

Combining the first two terms, we have ((a + b)² + (a + b)c)/((c - a)(c + a + b)). Factoring out (a + b) from the numerator gives ((a + b)(a + b + c))/((c - a)(c + a + b)). By canceling the common term (a + b + c) from the numerator and denominator, the sum simplifies to (a + b)/(c - a).

Now, subtract the third term: (a + b)/(c - a) - (a + 2b + c)/(2(c - a)). To perform the subtraction, use a common denominator of 2(c - a). This yields (2(a + b) - (a + 2b + c))/(2(c - a)) = (2a + 2b - a - 2b - c)/(2(c - a)) = (a - c)/(2(c - a)). Since a - c = -(c - a), the expression becomes -(c - a)/(2(c - a)) = -1/2. Thus, option a is correct.

Explanation

Given the equation aᵇ = bᵃ, we can find the relationship between a and b by manipulating the powers. To isolate a with a power involving b and a, we raise both sides of the equation to the power of 1/a.

Applying the power rule (xᵐ)ⁿ = xᵐⁿ, we get: (aᵇ)¹/ᵃ = (bᵃ)¹/ᵃ aᵇ/ᵃ = bᵃ ˣ ¹/ᵃ aᵇ/ᵃ = b¹ aᵇ/ᵃ = b

The expression given in the question, despite the formatting errors, represents a form that simplifies using this identity. If we consider the expression as aᵇ/ᵃ/b or aᵇ/ᵃ × b⁻¹, we can substitute the value of aᵇ/ᵃ derived above: aᵇ/ᵃ/b = b/b = 1

Thus, the result of the expression is 1, which matches option a. This is a standard result in algebra for equations of the form aᵇ = bᵃ where a ≠ b. For example, if a=2 and b=4, then 2⁴ = 16 and 4² = 16. Testing the derived identity: aᵇ/ᵃ = 2⁴/² = 2² = 4, which equals b. Consequently, aᵇ/ᵃ / b = 4/4 = 1.

Explanation

x = 2 + sqrt(2) + 2sqrt(2) = 2 + 3sqrt(2). x - 2 = 3sqrt(2). Squaring both sides: (x - 2)^2 = (3sqrt(2))^2 => x^2 - 4x + 4 = 18 => x^2 - 4x - 14 = 0. Therefore, x^2 - 4x - 10 = (x^2 - 4x - 14) + 4 = 0 + 4 = 4.

Explanation

Using the Componendo and Dividendo rule, which states that if (a+b)/(a-b) = c/d, then a/b = (c+d)/(c-d), we can simplify the given equation (√(p+x) + √(p-x))/(√(p+x) - √(p-x)) = p/1. Applying this rule, we get √(p+x)/√(p-x) = (p+1)/(p-1). Squaring both sides to remove the radicals yields (p+x)/(p-x) = (p+1)²/(p-1)² = (p² + 2p + 1)/(p² - 2p + 1). Applying Componendo and Dividendo once more to this result gives ((p+x) + (p-x))/((p+x) - (p-x)) = ((p² + 2p + 1) + (p² - 2p + 1))/((p² + 2p + 1) - (p² - 2p + 1)). This simplifies to 2p/2x = (2p² + 2)/4p, which further reduces to p/x = (p² + 1)/2p. Solving for x by cross-multiplication, we obtain x(p² + 1) = 2p², which gives x = 2p²/(p² + 1), matching option d.

Explanation

The given quadratic equation is ((a - b)/2)x² - ((a + b)/2)x + b = 0. Comparing this with the standard form Ax² + Bx + C = 0, we have A = (a - b)/2, B = -(a + b)/2, and C = b.

The fastest way to find the roots is to check if x = 1 is a solution by substituting it into the equation. For x = 1, the expression becomes (a - b)/2 - (a + b)/2 + b = (a - b - a - b + 2b)/2 = 0/2 = 0. Since the equation is satisfied, x = 1 is the first root.

Using the formula for the product of roots αβ = C/A, we substitute the known values: 1 · β = b/((a - b)/2) = 2b/(a - b). Therefore, the second root is β = 2b/(a - b).

We can verify this using the sum of roots formula α + β = -B/A: 1 + 2b/(a - b) = (a - b + 2b)/(a - b) = (a + b)/(a - b). Comparing this with -B/A = ((a + b)/2)/((a - b)/2) = (a + b)/(a - b), the results match perfectly. Thus, the roots are 1, 2b/(a - b), which corresponds to option b.

Explanation

Given x - 1/x = 2. Squaring: (x - 1/x)^2 = 4 => x^2 - 2 + 1/x^2 = 4 => x^2 + 1/x^2 = 6. We want x^2 - 1/x^2 = (x - 1/x)(x + 1/x). Since (x + 1/x)^2 = (x - 1/x)^2 + 4 = 2^2 + 4 = 8, and x > 0, x + 1/x = sqrt(8) = 2sqrt(2). Thus, x^2 - 1/x^2 = 2 2sqrt(2) = 4sqrt(2).

Explanation

(a - b)^2 + (b - c)^2 + (c - a)^2 = 6 2(a^2 + b^2 + c^2) - 2(ab + bc + ca) = 6 a^2 + b^2 + c^2 - (ab + bc + ca) = 3 Given a^2 + b^2 + c^2 = 29, then 29 - (ab + bc + ca) = 3, so ab + bc + ca = 26. (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) = 29 + 2(26) = 29 + 52 = 81. a + b + c = +/- 9.

Explanation

Given that p = (√5-2)/(√5+2) and q = (√5+2)/(√5-2), we can observe that p and q are reciprocals of each other, meaning pq = 1. The required expression is p/q + q/p, which can be simplified to (p²+q²)/pq. Since pq=1, the expression becomes p²+q².

Using the algebraic identity p²+q² = (p+q)² - 2pq, we first calculate the sum p+q: p+q = (√5-2)/(√5+2) + (√5+2)/(√5-2) To find the sum, we take the common denominator (√5+2)(√5-2) = (√5)² - 2² = 5-4 = 1. The numerator is (√5-2)² + (√5+2)². Using the formula (a-b)² + (a+b)² = 2(a²+b²), the numerator becomes 2((√5)² + 2²) = 2(5+4) = 2(9) = 18. Thus, p+q = 18/1 = 18.

Now, substitute the values of p+q and pq into the identity: p/q + q/p = (p+q)² - 2pq = 18² - 2(1) = 324 - 2 = 322. Therefore, option c is the correct value.

Explanation

We need the hundreds digit of 25^10 = 5^20. Looking at powers of 5 mod 1000: 5^3 = 125 5^4 = 625 5^5 = 3125 = 125 (mod 1000) The pattern for 5^n (n >= 3) is 125 if n is odd and 625 if n is even. Since 20 is even, 5^20 = 625 (mod 1000). The digit at the hundreds place is 6.

Explanation

N leaves remainder 3 when divided by 4, 6, 7, and 9. N = lcm(4, 6, 7, 9) * k + 3. lcm(4, 6, 7, 9) = 252. N = 252k + 3. For k = 1, 2, 3, N is 255, 507, 759. For k = 4, N = 252 * 4 + 3 = 1008 + 3 = 1011. Smallest 4-digit number is 1011.

Explanation

Let the given infinite nested radical be x = √(2 + √(2 + √(2 + ...))). To solve for x, we square both sides of the equation to get x² = 2 + √(2 + √(2 + ...)). Since the expression inside the first square root is the same infinite series, we can substitute x back into the equation, yielding x² = 2 + x. This is a standard quadratic equation x² - x - 2 = 0. Factoring the quadratic gives (x - 2)(x + 1) = 0, which implies x = 2 or x = -1. Because a square root cannot result in a negative value, we conclude that x = 2.

According to the problem, x = csc θ, so we have csc θ = 2. Using the fundamental trigonometric identity sin θ = 1/csc θ, we substitute the value of csc θ to find sin θ = 1/2. This matches option d.

Explanation

To find the value of cscθ, we start with the given equation 8 sinθ - cosθ = 4. Rearranging the terms to isolate cosθ, we get cosθ = 8 sinθ - 4. Squaring both sides of the equation, we obtain cos²θ = (8 sinθ - 4)². Using the trigonometric identity cos²θ = 1 - sin²θ, the equation becomes 1 - sin²θ = 64 sin²θ - 64 sinθ + 16.

Rearranging all terms to one side yields a quadratic equation in sinθ: 65 sin²θ - 64 sinθ + 15 = 0. Applying the quadratic formula sinθ = (-b ± √(b² - 4ac))/2a, we substitute a = 65, b = -64, and c = 15: sinθ = (64 ± √(64² - 4(65)(15)))/2(65) = (64 ± √(4096 - 3900))/130 = (64 ± √196)/130 = (64 ± 14)/130.

This gives two possible values: sinθ = 78/130 = 3/5 or sinθ = 50/130 = 5/13. Verifying these in the original equation 8 sinθ - cosθ = 4: For sinθ = 3/5, cosθ = 4/5, so 8(3/5) - 4/5 = 20/5 = 4, which is correct. For sinθ = 5/13, cosθ = 12/13, so 8(5/13) - 12/13 = 28/13 ≠ 4.

Thus, sinθ = 3/5. Since cscθ = 1/sinθ, we have cscθ = 5/3, which corresponds to option c.

Explanation

Use the trigonometric identity sec²θ = 1 + tan²θ to rewrite the given equation. Substituting this into 2 tanθ = sec²θ - 2 gives 2 tanθ = (1 + tan²θ) - 2, which simplifies to 2 tanθ = tan²θ - 1. Rearranging the terms into a standard quadratic equation, we get tan²θ - 2 tanθ - 1 = 0.

To solve for tanθ, apply the quadratic formula x = (-b ± √(b² - 4ac))/2a where x = tanθ, a = 1, b = -2, and c = -1. This yields tanθ = (2 ± √((-2)² - 4(1)(-1)))/2(1) = (2 ± √8)/2 = (2 ± 2√2)/2 = 1 ± √2. Since 0 < θ < π/2, tanθ must be positive, so we take tanθ = 1 + √2.

The value of cotθ is the reciprocal of tanθ: cotθ = 1/(1 + √2) To rationalize the denominator, multiply the numerator and denominator by √2 - 1: cotθ = 1(√2 - 1)/((√2 + 1)(√2 - 1)) = (√2 - 1)/((√2)² - 1²) = (√2 - 1)/(2 - 1) = √2 - 1. Thus, the correct option is a) √2 - 1.

Explanation

sqrt((1 - sin theta) / (1 + sin theta)) = sqrt((1 - sin theta)^2 / (1 - sin^2 theta)) = sqrt((1 - sin theta)^2 / cos^2 theta) = (1 - sin theta) / cos theta = sec theta - tan theta. The expression is (sec theta - tan theta) - (sec theta - tan theta) = 0.

Explanation

Given that cot θ = √7, we use the trigonometric identities csc²θ = 1 + cot²θ and sec²θ = 1 + tan²θ to find the required values. First, we calculate csc²θ = 1 + (√7)² = 1 + 7 = 8. Since tan θ is the reciprocal of cot θ, we have tan θ = 1/√7, which implies tan²θ = 1/7. Thus, sec²θ = 1 + 1/7 = 8/7.

Now, substitute these values into the expression (csc²θ - sec²θ)/(csc²θ + sec²θ):

(8 - 8/7)/(8 + 8/7)

To simplify, we find a common denominator for the numerator and the denominator:

((56 - 8)/7)/((56 + 8)/7) = (48/7)/(64/7)

The denominators of 7 cancel out, leaving:

48/64

Dividing both the numerator and the denominator by their greatest common divisor, 16, we get:

3/4

Therefore, the expression is equal to option d) 3 / 4.

Explanation

Let the acute angles be alpha and beta. alpha + beta = 90 degrees. Given alpha - beta = pi/12 radians = 15 degrees. Adding the equations: 2alpha = 105 degrees => alpha = 52.5 degrees. Subtracting the equations: 2beta = 75 degrees => beta = 37.5 degrees. One of the angles is 52.5 degrees.

Explanation

log10(998 + sqrt(x^2 - 18x + 76)) = 3 998 + sqrt(x^2 - 18x + 76) = 10^3 = 1000 sqrt(x^2 - 18x + 76) = 2 x^2 - 18x + 76 = 4 x^2 - 18x + 72 = 0 Roots alpha and beta satisfy alpha + beta = 18 and alpha * beta = 72. (alpha - beta)^2 = (alpha + beta)^2 - 4 alpha beta = 18^2 - 4 * 72 = 324 - 288 = 36.

Explanation

Given the equation x⁴ + y⁴ = 14x²y², we use the algebraic identities (a+b)² = a² + b² + 2ab and (a-b)² = a² + b² - 2ab to find the values of x² + y² and x² - y².

For statement I, we add 2x²y² to both sides of the given equation: x⁴ + y⁴ + 2x²y² = 14x²y² + 2x²y² (x² + y²)² = 16x²y² Taking the square root of both sides, we get x² + y² = 4xy. Applying the logarithm property log(abc) = log a + log b + log c: log₁₀(x² + y²) = log₁₀(4xy) = log₁₀4 + log₁₀x + log₁₀y Since log₁₀4 = log₁₀2² = 2log₁₀2, we have: log₁₀(x² + y²) = log₁₀x + log₁₀y + 2log₁₀2. Thus, statement I is correct.

For statement II, we subtract 2x²y² from both sides of the given equation: x⁴ + y⁴ - 2x²y² = 14x²y² - 2x²y² (x² - y²)² = 12x²y² Taking the square root of both sides, we get x² - y² = √12xy = 2√3xy. Applying the logarithm property: log₁₀(x² - y²) = log₁₀(2√3xy) = log₁₀2 + log₁₀√3 + log₁₀x + log₁₀y Since log₁₀√3 = log₁₀3⁰·⁵ = 0.5log₁₀3, we have: log₁₀(x² - y²) = log₁₀x + log₁₀y + log₁₀2 + 0.5log₁₀3. Thus, statement II is also correct.

Since both statements are correct, option c is the right choice.

Explanation

To solve this problem efficiently, use the method of substitution to simplify the quadratic expression. Let a = (3x + y) and b = (x + 5y). The given expression transforms into a² + ab - 20b². This is a quadratic form that can be factored by finding two numbers that multiply to -20 and add to 1. These numbers are 5 and -4. Therefore, the expression can be factored as (a + 5b)(a - 4b).

Now, substitute the original values of a and b back into the factors: For the first factor, (a + 5b) = (3x + y) + 5(x + 5y) = 3x + y + 5x + 25y = 8x + 26y = 2(4x + 13y). For the second factor, (a - 4b) = (3x + y) - 4(x + 5y) = 3x + y - 4x - 20y = -x - 19y = -1(x + 19y).

The complete factorization of the expression is -2(4x + 13y)(x + 19y). Since both (4x + 13y) and (x + 19y) are present in the product, both statements I and II are factors of the given expression. Thus, option c is correct.

Explanation

Let the numerator be N = x/(x-y) + y/(y-z) + z/(z-x) and the denominator be D = (x+y)/(x-y) + (y+z)/(y-z) + (z+x)/(z-x) + 3. To simplify the denominator, we use the algebraic identity (a+b)/(a-b) = (2a - (a-b))/(a-b) = 2a/(a-b) - 1. Applying this to each term of D:

(x+y)/(x-y) = 2x/(x-y) - 1 (y+z)/(y-z) = 2y/(y-z) - 1 (z+x)/(z-x) = 2z/(z-x) - 1

Substituting these expressions back into the denominator D: D = (2x/(x-y) - 1) + (2y/(y-z) - 1) + (2z/(z-x) - 1) + 3 D = 2x/(x-y) + 2y/(y-z) + 2z/(z-x) - 3 + 3 D = 2(x/(x-y) + y/(y-z) + z/(z-x))

Comparing this with the numerator, we find that D = 2N. The value of the given expression is therefore N/D = N/2N = 1/2. This matches option (b).

Explanation

To find the value of (tanα/tanβ)², we express tan²α and tan²β in terms of the given variables.

Starting with the first equation: psin²α + qcos²α = m Dividing the entire equation by cos²α: ptan²α + q = msec²α Since sec²α = 1 + tan²α, we substitute: ptan²α + q = m(1 + tan²α) ptan²α + q = m + mtan²α (p - m)tan²α = m - q tan²α = (m - q)/(p - m)

Next, we use the second equation: qsin²β + pcos²β = n Dividing the entire equation by cos²β: qtan²β + p = nsec²β Substituting sec²β = 1 + tan²β: qtan²β + p = n(1 + tan²β) qtan²β + p = n + ntan²β (q - n)tan²β = n - p tan²β = (n - p)/(q - n)

Now, calculating the required ratio: (tanα/tanβ)² = tan²α/tan²β = (m - q)/(p - m) ÷ (n - p)/(q - n) = (m - q)/(p - m) · (q - n)/(n - p) = ((m - q)(q - n))/((p - m)(n - p))

To align this with the options, we factor out negative signs from the terms (q - n) and (p - m): = ((m - q) · [-(n - q)])/([-(m - p)] · (n - p)) = (-(m - q)(n - q))/(-(m - p)(n - p)) = ((m - q)(n - q))/((m - p)(n - p))

This result matches option c.

Explanation

Two angles are complementary if their sum is 90°. Given that α and η are complementary angles, we have the relation α + η = 90°. In trigonometry, for any angle θ, the tangent of its complement is equal to its cotangent, which is expressed by the formula tan(90° - θ) = cot θ. In this problem, if we define m = tan α and n = tan η, then n = tan(90° - α) = cot α. Since the relationship between tangent and cotangent is cot α = 1/tan α, we can substitute the values to get n = 1/m. Multiplying both sides by m yields mn = 1. Rearranging this equation to set it to zero gives mn - 1 = 0, which matches option (a). This relationship is a standard property where the product of the tangents of complementary angles is always equal to unity.

Explanation

Given the expressions p = csc θ - sin θ and q = sec θ - cos θ, we can simplify them using basic trigonometric identities. Since csc θ = 1/sin θ, we have p = 1/sin θ - sin θ = (1 - sin² θ)/sin θ. Using the identity sin² θ + cos² θ = 1, we can substitute 1 - sin² θ with cos² θ, which gives p = cos² θ/sin θ. Similarly, since sec θ = 1/cos θ, we have q = 1/cos θ - cos θ = (1 - cos² θ)/cos θ = sin² θ/cos θ.

Now, we substitute these simplified forms of p and q into the expression p sin θ + q cos θ:

p sin θ + q cos θ = (cos² θ/sin θ) sin θ + (sin² θ/cos θ) cos θ

The sin θ and cos θ terms in the denominators cancel out with the multipliers, leaving:

p sin θ + q cos θ = cos² θ + sin² θ

By applying the Pythagorean identity sin² θ + cos² θ = 1, the expression simplifies to 1. Therefore, the value is 1, which corresponds to option (c).

Explanation

p = csc theta - sin theta = (1 - sin^2 theta) / sin theta = cos^2 theta / sin theta. q = sec theta - cos theta = (1 - cos^2 theta) / cos theta = sin^2 theta / cos theta. p^2 q^2 = (cos^4 theta / sin^2 theta) (sin^4 theta / cos^2 theta) = sin^2 theta * cos^2 theta. p^2 + q^2 + 3 = (cos^4 theta / sin^2 theta) + (sin^4 theta / cos^2 theta) + 3 = (cos^6 theta + sin^6 theta + 3 sin^2 theta cos^2 theta) / (sin^2 theta * cos^2 theta). Using a^3 + b^3 = (a + b)(a^2 - ab + b^2), sin^6 theta + cos^6 theta = (sin^2 theta + cos^2 theta)(sin^4 theta - sin^2 theta * cos^2 theta + cos^4 theta) = 1 ((sin^2 theta + cos^2 theta)^2 - 3 sin^2 theta cos^2 theta) = 1 - 3 sin^2 theta * cos^2 theta. So, p^2 + q^2 + 3 = (1 - 3 sin^2 theta cos^2 theta + 3 sin^2 theta cos^2 theta) / (sin^2 theta cos^2 theta) = 1 / (sin^2 theta cos^2 theta). p^2 q^2 (p^2 + q^2 + 3) = (sin^2 theta cos^2 theta) (1 / (sin^2 theta * cos^2 theta)) = 1.

Explanation

sin alpha / sin beta = 4sqrt(2)/3 cos alpha / cos beta = 2sqrt(3)/9 tan alpha / tan beta = (sin alpha / sin beta) / (cos alpha / cos beta) = (4sqrt(2)/3) / (2sqrt(3)/9) = (4sqrt(2)/3) (9 / 2sqrt(3)) = 6sqrt(2) / sqrt(3) = 2*sqrt(6). tan^2 alpha = 24 * tan^2 beta. From sin alpha = (4sqrt(2)/3) sin beta and cos alpha = (2sqrt(3)/9) cos beta, (32/9) sin^2 beta + (12/81) cos^2 beta = 1 (32/9) sin^2 beta + (4/27) (1 - sin^2 beta) = 1 (96/27 - 4/27) sin^2 beta = 1 - 4/27 = 23/27 (92/27) sin^2 beta = 23/27 => sin^2 beta = 23/92 = 1/4. tan^2 beta = (1/4) / (3/4) = 1/3. tan^2 alpha = 24 * (1/3) = 8.

Explanation

To find the value of tan² β, we use the fundamental trigonometric identity sin² α + cos² α = 1. From the given equations, we can express sin α and cos α in terms of sin β and cos β as follows: sin α = (4√2/3) sin β and cos α = (2√3/9) cos β.

Substituting these expressions into the identity sin² α + cos² α = 1 gives: ((4√2/3) sin β)² + ((2√3/9) cos β)² = 1 32/9 sin² β + 12/81 cos² β = 1 32/9 sin² β + 4/27 cos² β = 1

Dividing the entire equation by cos² β to introduce tan² β: 32/9 tan² β + 4/27 = 1/cos² β Using the identity 1/cos² β = sec² β = 1 + tan² β, we get: 32/9 tan² β + 4/27 = 1 + tan² β (32/9 - 1) tan² β = 1 - 4/27 23/9 tan² β = 23/27 tan² β = 23/27 × 9/23 = 9/27 = 1/3

Thus, the value of tan² β is 1/3, which corresponds to option c).

Explanation

The given expression (1 + sin θ)/(cos θ) can be simplified as 1/cos θ + sin θ/cos θ = sec θ + tan θ. According to the problem, sec θ + tan θ = p + √(p² + 1). We use the fundamental trigonometric identity sec² θ - tan² θ = 1, which can be factored as (sec θ + tan θ)(sec θ - tan θ) = 1. This implies that sec θ - tan θ = 1/(sec θ + tan θ).

Substituting the given value, we get sec θ - tan θ = 1/(p + √(p² + 1)). Rationalizing the denominator by multiplying the numerator and denominator by √(p² + 1) - p, we obtain sec θ - tan θ = (√(p² + 1) - p)/((√(p² + 1))² - p²) = (√(p² + 1) - p)/(p² + 1 - p²) = √(p² + 1) - p.

Now we have two equations:

1. sec θ + tan θ = √(p² + 1) + p
2. sec θ - tan θ = √(p² + 1) - p

Adding these two equations eliminates tan θ: 2 sec θ = (√(p² + 1) + p) + (√(p² + 1) - p) = 2√(p² + 1). Dividing by 2, we find sec θ = √(p² + 1), which matches option b.

Explanation

In trigonometry, the tangent of an angle θ in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Using the formula tan θ = Opposite/Adjacent, if we assume the opposite side is p and the adjacent side is 1, then tan θ = p/1 = p, which matches option (a). To verify this against the other options, we use the Pythagorean theorem to find the hypotenuse: Hypotenuse = √(Opposite² + Adjacent²) = √(p² + 1²) = √(p² + 1). Consequently, sin θ = Opposite/Hypotenuse = p/√(p² + 1) and cos θ = Adjacent/Hypotenuse = 1/√(p² + 1). Since the question asks for tan θ, the most direct and correct value based on these relations is p.

Explanation

To solve this problem, use the method of substitution to simplify the quadratic expression. Let a = (3x + y) and b = (x + 5y). The expression becomes a² + ab - 20b². This is a standard quadratic form that can be factored by splitting the middle term. We need two numbers that multiply to -20 and add to 1. These numbers are 5 and -4.

The factored form is: a² + 5ab - 4ab - 20b² = a(a + 5b) - 4b(a + 5b) = (a + 5b)(a - 4b)

Now, substitute the original values of a and b back into these factors: For the first factor (a + 5b): (3x + y) + 5(x + 5y) = 3x + y + 5x + 25y = 8x + 26y = 2(4x + 13y) This shows that (4x + 13y) is a factor, making statement I correct.

For the second factor (a - 4b): (3x + y) - 4(x + 5y) = 3x + y - 4x - 20y = -x - 19y = -1(x + 19y) This shows that (x + 19y) is also a factor, making statement II correct.

Since both (4x + 13y) and (x + 19y) are factors of the expression, the correct option is c) Both I and II.

Explanation

To find the value of tan θ + cot θ, we start by expressing it in terms of sine and cosine: tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin² θ + cos² θ)/(sin θ cos θ). Using the fundamental trigonometric identity sin² θ + cos² θ = 1, we get tan θ + cot θ = 1/(sin θ cos θ). Now, consider the given expression for q: q = sec θ + csc θ = 1/cos θ + 1/sin θ. Combining these fractions gives q = (sin θ + cos θ)/(sin θ cos θ). Substituting the given value sin θ + cos θ = p into this equation, we have q = p/(sin θ cos θ). Rearranging this formula to solve for the term 1/(sin θ cos θ), we obtain 1/(sin θ cos θ) = q/p. Since we already established that tan θ + cot θ = 1/(sin θ cos θ), it follows that tan θ + cot θ = q/p, which matches option (b).

Explanation

Since both trains are travelling in the same direction, the relative speed is the difference between their individual speeds. The relative speed is calculated as 100 km/hr - 60 km/hr = 40 km/hr. To convert this speed into meters per second, we multiply by 5/18, which gives 40 × 5/18 = 200/18 = 100/9 m/s.

When train X crosses a man in train Y, the distance covered is equal to the length of train X because the man is considered a point object. Using the distance formula Distance = Relative Speed × Time, the length of train X is 100/9 m/s × 9 seconds = 100 meters. Thus, option (b) is the correct answer.

Explanation

The time difference between the departures of X and Y is 10 minutes, which is equal to 10/60 = 1/6 hour. In this time, X covers an initial distance gap of 4.8 × 1/6 = 0.8 km. Since Y is moving in the same direction at a higher speed, the relative speed of Y with respect to X is calculated as 6 - 4.8 = 1.2 km/hr.

Using the relative speed formula, the time required for Y to catch up with X is Time = Distance Gap / Relative Speed = 0.8/1.2 = 2/3 hour. The distance from place P where they meet is the total distance travelled by Y during this time, which is Distance = Speed of Y × Time = 6 × 2/3 = 4 km. Therefore, option (c) is the correct answer.

Explanation

Let the initial salary of X be Sₓ and the initial salary of Y be Sᵧ. According to the problem, the final salaries of X and Y are equal. We use the formula for successive percentage changes: Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100).

For employee X, the salary is first increased by 12% and then decreased by 10%, so the final salary is Sₓ × (1 + 12/100) × (1 - 10/100) = Sₓ × 1.12 × 0.90.

For employee Y, the salary is first increased by 10% and then decreased by 12%, so the final salary is Sᵧ × (1 + 10/100) × (1 - 12/100) = Sᵧ × 1.10 × 0.88.

Since the final salaries are equal, we have: Sₓ × 1.12 × 0.90 = Sᵧ × 1.10 × 0.88

To find the ratio of initial salaries, we rearrange the equation: Sₓ/Sᵧ = (1.10 × 0.88)/(1.12 × 0.90) = (110 × 88)/(112 × 90)

Simplifying the fractions: 110/90 = 11/9 and 88/112 = 11/14 (by dividing both by 8)

Thus, Sₓ/Sᵧ = 11/9 × 11/14 = 121/126

The ratio of the initial salary of X to the initial salary of Y is 121:126, which corresponds to option c.

Explanation

Let the number of males in the village be M and the number of females be F. We are given that the total population is p, so M + F = p, which implies F = p - M. The total number of literate persons is the sum of literate males and literate females. Using the given percentages, the equation is x/100 p = y/100 M + z/100 (p - M). To simplify, multiply the entire equation by 100 to remove the denominators: xp = yM + z(p - M). Expanding the brackets gives xp = yM + zp - zM. Rearranging the terms to isolate M, we get xp - zp = yM - zM, which can be factored as p(x - z) = M(y - z). Solving for M, we find M = p(x - z)/(y - z). This corresponds to option a. Alternatively, using the rule of alligation, the ratio of males to females is (x - z) : (y - x). The proportion of males in the population is (x - z)/((x - z) + (y - x)) = (x - z)/(y - z). Multiplying this fraction by the total population p yields the number of males as p(x - z)/(y - z).

Explanation

To find the ratio of copper (Cu) to zinc (Zn) in alloy Z, we use the weighted average method. In alloy X, the ratio is 5:4, making the total parts 5 + 4 = 9. In alloy Y, the ratio is 5:13, making the total parts 5 + 13 = 18. Since equal quantities of X and Y are melted, we assume a common quantity for both, preferably the least common multiple (LCM) of 9 and 18, which is 18 units.

For alloy X (18 units): Amount of Cu = 5/9 × 18 = 10 units Amount of Zn = 4/9 × 18 = 8 units

For alloy Y (18 units): Amount of Cu = 5/18 × 18 = 5 units Amount of Zn = 13/18 × 18 = 13 units

In alloy Z, the total copper is 10 + 5 = 15 units and the total zinc is 8 + 13 = 21 units. Therefore, the ratio of Cu to Zn in alloy Z is 15:21. Simplifying this ratio by dividing both sides by 3, we get 5:7, which corresponds to option b.

Explanation

Let the scores of students X and Y be denoted by x and y respectively. According to the problem, the score of X is 75% of the sum of the scores of X and Y. This can be represented by the equation x = 75% × (x + y). Converting the percentage to a fraction, we have x = 3/4(x + y). Multiplying both sides by 4, we get 4x = 3x + 3y. Subtracting 3x from both sides simplifies the equation to x = 3y. To find the ratio of the score of X to the score of Y, we divide both sides by y, resulting in x/y = 3/1. Thus, the ratio is 3:1, which corresponds to option c. It is important to note that while the problem states X is 20 more than Y, this specific value is redundant for determining the ratio, as the percentage relationship alone defines the proportional link between the two scores.

Explanation

The equation is 2x^2 - 5px + 2p^2 = 0. Factoring: (2x - p)(x - 2p) = 0. Roots are x1 = p/2 and x2 = 2p. The difference is 2p - p/2 = 3p/2. Given 3p/2 = 4, then 3p = 8, so p = 8/3.

Explanation

Installment A = P * [r(1+r)^n] / [(1+r)^n - 1] A = 10000 [0.1 (1.1)^3] / [(1.1)^3 - 1] A = 10000 [0.1 1.331] / [1.331 - 1] = 10000 * 0.1331 / 0.331 = 1331 / 0.331 approx 4021.15.

Explanation

To find the solution of the given system of inequalities, solve each inequality separately and then find the intersection of their solution sets.

For the first inequality, 5x + 3 < 8x - 9, rearrange the terms to isolate x by subtracting 5x from both sides and adding 9 to both sides: 3 + 9 < 8x - 5x 12 < 3x Dividing both sides by 3, we get x > 4.

For the second inequality, 2x + 20 > 5x + 2, rearrange the terms by subtracting 2x from both sides and subtracting 2 from both sides: 20 - 2 > 5x - 2x 18 > 3x Dividing both sides by 3, we get x < 6.

Since the question asks for the solution of both inequalities (the and condition), we must find the values of x that satisfy both x > 4 and x < 6. Combining these two results, the solution is 4 < x < 6, which matches option a.

Explanation

Let the marked price be M and the cost price be CP. The final selling price SP_final after three consecutive discounts of 10%, 20%, and 25% is given by the formula SP_final = M(1 - d₁)(1 - d₂)(1 - d₃). Substituting the values, we get SP_final = M(0.90 × 0.80 × 0.75) = M(0.54). Since the shopkeeper makes a profit of 8% on the cost price, we have SP_final = CP(1 + 0.08) = 1.08 CP. Equating the two expressions for SP_final, we get 0.54 M = 1.08 CP, which simplifies to M = 1.08/0.54 CP = 2 CP. If the article had been sold after only the first discount of 10%, the new selling price SPₙₑᵥᵥ would be M(1 - 0.10) = 0.9 M. Substituting M = 2 CP, we find SPₙₑᵥᵥ = 0.9(2 CP) = 1.8 CP. The profit percentage is calculated as (SPₙₑᵥᵥ - CP)/CP × 100 = (1.8 CP - CP)/CP × 100 = 0.8 × 100 = 80%. Since 80% is not among options a, b, or c, the correct option is d.

Explanation

In triangle ABC, let the altitude AD divide the base BC into two segments BD and CD. By the Pythagorean theorem in right triangles ABD and ACD, we have AD² = AB² - BD² and AD² = AC² - CD². Equating these two expressions for the height squared gives AB² - BD² = AC² - CD². Rearranging the terms to group the sides and the base segments, we get AB² - AC² = BD² - CD². Applying the algebraic identity a² - b² = (a+b)(a-b) to both sides, the equation becomes (AB + AC)(AB - AC) = (BD + CD)(BD - CD). Substituting the given values AB + AC = p, AB - AC = q, and BD - CD = r into this relation, we get p ⋅ q = (BD + CD) ⋅ r. Since the total length of the base BC is the sum of segments BD + CD, we have pq = BC ⋅ r. Solving for BC yields BC = pq/r, which corresponds to option (c).

Explanation

Given the sum of the height h and the radius r of the cylinder as r + h = 21 cm and the curved surface area (CSA) as 616 cm², we use the formula for CSA of a right circular cylinder: CSA = 2πrh. Substituting the given values, we have 2 × 22/7 × rh = 616. Solving for the product rh, we get rh = (616 × 7)/44 = 14 × 7 = 98.

We now have two equations: r + h = 21 and rh = 98. These are the characteristics of a quadratic equation x² - (r + h)x + rh = 0, which is x² - 21x + 98 = 0. Factorizing this, we get (x - 14)(x - 7) = 0. Thus, the possible values for r and h are 7 and 14. Since the condition r < h is given, we must have r = 7 cm and h = 14 cm.

To find the volume V, we use the formula V = πr²h. Substituting the values, V = 22/7 × 7² × 14 = 22/7 × 49 × 14 = 22 × 7 × 14 = 154 × 14 = 2156 cm³. Therefore, option d is correct.

Explanation

Let the length of side AB = c and side BC = a. Since the triangle is right angled at B, its area is given by the formula Area = 1/2 × a × c. To find the area, we need to determine the values of a and c.

From statement I, AQ = √73. Since Q is the midpoint of BC, BQ = a/2. Applying the Pythagorean theorem in ΔABQ: c² + (a/2)² = (√73)² c² + a²/4 = 73 ⇒ 4c² + a² = 292

From statement II, CP = √52. Since P is the midpoint of AB, BP = c/2. Applying the Pythagorean theorem in ΔCBP: a² + (c/2)² = (√52)² a² + c²/4 = 52 ⇒ 4a² + c² = 208

We now have a system of two linear equations in terms of a² and c². Let X = a² and Y = c². The equations are X + 4Y = 292 and 4X + Y = 208. Solving these simultaneous equations: Subtracting the second from four times the first: 4(X + 4Y) - (4X + Y) = 4(292) - 208 15Y = 1168 - 208 = 960 ⇒ Y = 64 ⇒ c = 8 Substituting Y = 64 into X + 4(64) = 292: X + 256 = 292 ⇒ X = 36 ⇒ a = 6

With a = 6 and c = 8, the area is 1/2 × 6 × 8 = 24 square units. Since neither statement alone provides enough information to solve for two variables, both I and II are required. Thus, option c is correct.

Explanation

Let the three consecutive integers representing the length, breadth, and height be n-1, n, and n+1. The formula for the volume of a cuboid is V = l × b × h. Substituting the values, we get (n-1) × n × (n+1) = 336, which simplifies to n(n² - 1) = 336 or n³ - n = 336. In a competitive exam, the fastest way to find n is to identify the perfect cube closest to 336. Since 7³ = 343, we check n = 7, and 343 - 7 = 336. Thus, the dimensions of the cuboid are 6, 7, and 8 units. The formula for the Total Surface Area (TSA) of a cuboid is TSA = 2(lb + bh + hl). Substituting the dimensions l=6, b=7, h=8, we calculate TSA = 2(6 × 7 + 7 × 8 + 8 × 6) = 2(42 + 56 + 48) = 2(146) = 292 square units. This corresponds to option b.

Explanation

Let the center of the circle be O and the radius be R = 14 cm. Since P is the midpoint of the shorter arc APB and C is the midpoint of the chord AB, the line passing through P and C must also pass through the center O. Therefore, P, C, and O are collinear, and the line PO is the perpendicular bisector of the chord AB. Given that the radius OP = 14 cm and PC = 7 cm, the distance OC = OP - PC = 14 - 7 = 7 cm.

In the right-angled triangle OCA, we apply the Pythagorean theorem to find the length of AC: AC² = OA² - OC² AC² = 14² - 7² = 196 - 49 = 147

Now, consider the right-angled triangle PCA. We use the Pythagorean theorem again to find the length of the chord AP: AP² = PC² + AC² AP² = 7² + 147 = 49 + 147 = 196 AP = √196 = 14 cm

Thus, the length of the chord AP is 14 cm, which corresponds to option (d).

Explanation

For problems involving the intersection of lines joining the top of one pole to the foot of another, the height of the point of intersection is independent of the distance between the poles. The height h can be calculated using the harmonic mean formula for heights a and b:

1/h = 1/a + 1/b or h = (a × b)/(a + b)

Given the heights of the two poles are a = 10 m and b = 15 m, we substitute these values into the formula:

h = (10 × 15)/(10 + 15) h = 150/25 h = 6 m

The distance of 25 m given in the question is extra information and does not affect the vertical height of the intersection point. Thus, the height of the point of intersection is 6 m, which corresponds to option c.

Explanation

Let the lengths of the sides be AB = c, BC = a, and the hypotenuse AC = b. According to the problem, the perimeter is given as a + b + c = 60 and the difference between the sum of the legs and the hypotenuse is (a + c) - b = 10. Subtracting the second equation from the first gives (a + b + c) - (a + c - b) = 60 - 10, which simplifies to 2b = 50, so b = 25. Substituting this value back into the perimeter equation, we find a + c = 60 - 25 = 35.

To find the area, we use the Pythagorean identity a² + c² = b² and the algebraic identity (a + c)² = a² + c² + 2ac. Substituting the known values, we get 35² = 25² + 2ac, which leads to 1225 = 625 + 2ac. This simplifies to 2ac = 600, and thus ac = 300. The area of a right-angled triangle is given by the formula Area = 1/2 × base × height = 1/2 ac. Therefore, Area = 1/2 × 300 = 150 square units. This corresponds to option d.

Explanation

The distance between the tips of two vertical poles can be calculated using the Pythagorean theorem, as the horizontal distance between the poles and the difference in their heights form the two legs of a right-angled triangle. Let the horizontal distance between the poles be d = 24 m and the difference in their heights be h = 10 m. The distance between the tips L represents the hypotenuse of this triangle.

Applying the formula L = √(d² + h²), we substitute the given values: L = √(24² + 10²) L = √(576 + 100) L = √676 L = 26 m.

For a faster competition approach, recognize that the values 10 and 24 are multiples of the standard Pythagorean triple (5, 12, 13). Since 10 = 5 × 2 and 24 = 12 × 2, the hypotenuse must be 13 × 2 = 26 m. Thus, option b is the correct answer.

Explanation

According to the Midpoint Theorem, a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. In triangle ABC, since Z is the midpoint of AB and X is the midpoint of BC, the segment ZX is parallel to AC. Since Y lies on AC, ZX ∥ AY. Similarly, since X is the midpoint of BC and Y is the midpoint of AC, the segment XY is parallel to AB. Since Z lies on AB, XY ∥ AZ. A quadrilateral with both pairs of opposite sides parallel is a parallelogram; therefore, AZXY is a parallelogram, making statement I correct.

For the area, connecting the midpoints X, Y, Z divides triangle ABC into four congruent triangles: ΔAZY, ΔZBX, ΔYCX, and ΔXYZ. Each of these smaller triangles has an area equal to 1/4 of the area of ΔABC. The quadrilateral AZXY is composed of two of these triangles, namely ΔAZY and ΔXYZ. Thus, the area of quadrilateral AZXY = 1/4 Area(ΔABC) + 1/4 Area(ΔABC) = 1/2 Area(ΔABC), making statement II correct. Both statements are correct, which corresponds to option c.

Explanation

For a regular polygon with n sides, the sum of all exterior angles is always 360°. Since all exterior angles in a regular polygon are equal, the measure of each exterior angle θ is given by the formula θ = 360°/n. For θ to be the exterior angle of a regular polygon, n must be a positive integer greater than or equal to 3. This implies that θ must be a divisor of 360.

Checking the given options: For θ = 4°, n = 360/4 = 90, which is an integer. For θ = 5°, n = 360/5 = 72, which is an integer. For θ = 6°, n = 360/6 = 60, which is an integer. For θ = 8°, n = 360/8 = 45, which is an integer.

Since all four values are divisors of 360 and result in an integer number of sides n ≥ 3, all four angles can be the exterior angle of a regular polygon. Thus, option d is correct.

Explanation

The area of a triangle can be calculated using the formula Area = 1/2 bc sin A, where b and c are the lengths of two sides and A is the included angle between them. In this problem, the given values are AB = c = 7 cm, AC = b = 12 cm, and ∠A = 30°. Substituting these values into the formula, we get Area = 1/2 × 12 × 7 × sin 30°. Since the value of sin 30° = 1/2, the calculation becomes Area = 1/2 × 12 × 7 × 1/2. Simplifying the expression, we find Area = 6 × 7 × 1/2 = 42 × 1/2 = 21 cm². Therefore, the area of triangle ABC is 21 cm², which matches option (a).

Explanation

In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse can be calculated using the area formula or the relation BD = (AB × BC)/AC. First, calculate the length of the hypotenuse AC using the Pythagorean theorem: AC = √(AB² + BC²) = √(p² + (√3p)²) = √(p² + 3p²) = √(4p²) = 2p. Now, applying the formula for the altitude BD to the hypotenuse, we have BD = (p × √3p)/2p. Simplifying the expression, the p in the denominator cancels one p in the numerator, resulting in BD = √3p²/2p = √3p/2. This matches option c.

Explanation

Interior angle I = (n-2) * 180 / n. Exterior angle E = 360 / n. I - E = 120 => (180n - 360) / n - 360 / n = 120 (180n - 720) / n = 120 180n - 720 = 120n 60n = 720 => n = 12.

Explanation

Complementary angles sum to 90 degrees. theta = 1/4 * (90 - theta) 4theta = 90 - theta 5theta = 90 => theta = 18 degrees.

Explanation

First, check if the given triangle is a right-angled triangle using the Pythagorean theorem a² + b² = c². For the given sides 11 cm, 60 cm, and 61 cm, we have 11² + 60² = 121 + 3600 = 3721, and 61² = 3721. Since the sum of the squares of the two shorter sides equals the square of the longest side, it is a right-angled triangle.

The area of the original triangle is calculated using the formula Area = 1/2 × base × height. Substituting the perpendicular sides, we get Area = 1/2 × 11 × 60 = 330 cm².

According to the properties of triangles, the triangle formed by joining the midpoints of the sides is known as the medial triangle. The area of the medial triangle is always exactly one-fourth of the area of the original triangle. Therefore, the area of the triangle formed by the midpoints is 330/4 = 82.5 cm², which corresponds to option c.

Explanation

The length of the pendulum corresponds to the radius r of the sector formed by the swing. Using the arc length formula for a circle, we have s = θ/360 × 2πr, where s is the arc length and θ is the central angle in degrees.

Substituting the given values s = 55 cm, θ = 30°, and π = 22/7 into the formula:

55 = 30/360 × 2 × 22/7 × r

Simplifying the fraction 30/360 to 1/12, the equation becomes:

55 = 1/12 × 44/7 × r

55 = 11/(3 × 7) × r

55 = 11/21 × r

To find r, rearrange the equation:

r = (55 × 21)/11

r = 5 × 21 = 105 cm

Thus, the length of the pendulum is 105 cm, which corresponds to option c.

Explanation

For a conical tent with a vertex angle of 60°, the semi-vertical angle θ is 30°. Using the geometric properties of a cone, the relationship between the radius r, slant height l, and height h is determined by sin(30°) = r/l, which implies l = 2r. Consequently, the height h is h = √(l² - r²) = √((2r)² - r²) = r√3.

The formula for the curved surface area (CSA) is CSA = πrl. Substituting the given value of 100 m² and l = 2r, we get: 100 = πr(2r) 2πr² = 100 r² = 50/π r = √(50/π) = 5√(2/π)

The volume V of a cone is calculated using the formula V = 1/3 πr²h. Substituting h = r√3 into the formula: V = 1/3 πr²(r√3) = (πr³√3)/3

Now, substituting the value of r: V = (π√3)/3 (50/π)³/² = (π√3)/3 · (50√50)/(π√π) = (50√3 · 5√2)/(3√π) = (250√6)/(3√π)

To match the options, we simplify (250√6)/(3√π) as follows: V = (250√2√3)/(3√π) = (250√2)/((3/√3)√π) = (250√2)/(√3√π) = (250√2)/√(3π)

Thus, the volume is (250√2)/√(3π) m³, which corresponds to option a.

Explanation

Let the radius of the base of the cone be r₁ and the radius of the hemisphere be r₂. Since the base of the cone and the hemisphere are equal, we have r₁ = r₂ = r. Let h be the height of the cone.

According to the formula for volumes, the volume of a right circular cone is V꜀ₒₙₑ = 1/3 π r² h and the volume of a hemisphere is Vₕₑₘᵢₛₚₕₑᵣₑ = 2/3 π r³.

Given that the volumes are equal, we set up the equation: 1/3 π r² h = 2/3 π r³

Dividing both sides by 1/3 π r²: h = 2r

To find the ratio of the height of the cone to the radius of the hemisphere, we calculate: h/r = 2/1

Thus, the ratio is 2:1, which corresponds to option c.

Explanation

Area of an equilateral triangle is given by the formula A = (√3/4) a², where a is the length of the side. Substituting the given area, 36√3 = (√3/4) a², which simplifies to a² = 144, giving a = 12 cm. The total length of the wire is the perimeter of the triangle, L = 3a = 3 × 12 = 36 cm. When this wire is reshaped into a semicircle, its perimeter is the sum of the arc length and the diameter, given by P = πr + 2r = r(π + 2). Equating the length of the wire to the perimeter of the semicircle, we get r(π + 2) = 36, which results in r = 36/(π + 2) cm. Comparing this result with the given options, we see that option (a) 9/π cm, option (b) 18/(π + 2) cm, and option (c) 18/π cm are all incorrect. Thus, the correct choice is (d) None of the above.

Explanation

For the first part, the area of the largest square inscribed in a circle is calculated using the formula x = 2r², where r is the radius of the circle. Since the radius r = 1, the area x = 2(1)² = 2. This is because the diagonal of the inscribed square is equal to the diameter of the circle, d = 2r = 2, and the area of a square is half the square of its diagonal, so x = d²/2 = 2²/2 = 2.

For the second part, the area of the largest circle inscribed in a square is calculated using the formula y = π(a/2)², where a is the side length of the square. Given the side length a = 1, the area y = π(1/2)² = π/4. This is because the diameter of the inscribed circle is equal to the side of the square.

To find the relation between x and y, we substitute the values: πx = π × 2 = 2π 8y = 8 × π/4 = 2π Comparing the two results, we get πx = 8y, which corresponds to option (d).

Explanation

The expression given is x²ⁿ - y²ⁿ + 1. We can analyze the first two terms using the algebraic identity a² - b² = (a - b)(a + b). By substituting a = xⁿ and b = yⁿ, we can rewrite the expression as (xⁿ - yⁿ)(xⁿ + yⁿ) + 1. When this expression is divided by the divisor xⁿ + yⁿ, the term (xⁿ - yⁿ)(xⁿ + yⁿ) is exactly divisible by xⁿ + yⁿ because it contains the divisor as a factor. This results in a remainder of 0 for that specific part of the expression. The only remaining term is the constant 1, which is the final remainder of the division. Since the identity a² - b² = (a - b)(a + b) is valid for all natural numbers n regardless of whether n is odd or even, the remainder will always be 1. Thus, neither Statement-I nor Statement-II is required to find the answer, making option (d) the correct choice.

Explanation

Given the product of a natural number N and its reverse M is 252, we look for pairs of factors of 252 that are reverses of each other. The factors of 252 are (1, 252), (2, 126), (3, 84), (4, 63), (6, 42), (7, 36), (9, 28), (12, 21), and (14, 18). Among these, only the pair (12, 21) satisfies the condition that one is the reverse of the other. Therefore, N must be either 12 or 21.

Evaluating Statement-I: N + M = 33. Since 12 + 21 = 33, both possible values for N (12 and 21) satisfy this equation. Thus, Statement-I alone is not sufficient to determine if N is 12 or 21.

Evaluating Statement-II: N > M. Since the possible values for N and M are 12 and 21, the condition N > M uniquely identifies N = 21 and M = 12. Thus, Statement-II alone is sufficient to answer the question.

Since the question can be answered using Statement-II alone but not using Statement-I alone, option (a) is the correct choice.

Explanation

The last digit of any power depends only on the last digit of the base and the cyclicity of that digit. For a number ending in D raised to the power of 100, the last digit will be 1 if D is 1, 3, 7, or 9. This is because the cyclicity of 1 is 1 (1¹=1), the cyclicity of 3 is 4 (3¹=3, 3²=9, 3³=7, 3⁴=1), the cyclicity of 7 is 4 (7¹=7, 7²=9, 7³=3, 7⁴=1), and the cyclicity of 9 is 2 (9¹=9, 9²=1). Since the exponent 100 is a multiple of 4, any of these digits D ∈ {1, 3, 7, 9} would result in a last digit of 1.

Statement I tells us D > 5, which narrows the possibilities to D = 7 or D = 9. This alone is not sufficient to find a unique value for D. Statement II tells us D is a multiple of 3, which narrows the possibilities to D = 3 or D = 9. This alone is also not sufficient. However, by combining both statements, the only digit that is both greater than 5 and a multiple of 3 from the set {1, 3, 7, 9} is D = 9. Therefore, both statements together are necessary to answer the question, making option c correct.

Explanation

Using the sum of angles property of a triangle, we know that ∠A + ∠B + ∠C = 180°. The given condition states that ∠A = ∠B - ∠C, which can be rearranged as ∠B = ∠A + ∠C. Substituting this value of ∠B into the sum of angles formula, we get ∠A + (∠A + ∠C) + ∠C = 180°, which simplifies to 2∠A + 2∠C = 180°. Dividing by 2, we find that ∠A + ∠C = 90°. Since ∠B = ∠A + ∠C, it follows that ∠B = 90°, making ABC a right-angled triangle. In any triangle, all angles must be greater than 0°. Therefore, if ∠A + ∠C = 90°, then ∠A must be strictly less than 90° and greater than 0°, which by definition makes ∠A an acute angle. Since this result is derived solely from the given equation and the fundamental properties of a triangle, no additional information from Statement I or Statement II is required. Thus, the question can be answered even without using any of the statements, making option d correct.

Explanation

In a right angled triangle, the hypotenuse serves as the diameter of the circumcircle. This is a fundamental property derived from the fact that the angle subtended by a diameter at any point on the circumference is 90°. Therefore, the circumradius R of a right angled triangle is always equal to half the length of its hypotenuse.

The formula to calculate the circumradius R is R = Hypotenuse/2. In the given question, the triangle ABC is right angled at B, which makes AC the hypotenuse. The length of AC is explicitly given as 20 cm. By substituting this value into the formula, we get R = 20/2 = 10 cm.

Since the length of the hypotenuse is already provided in the main question, the circumradius can be determined directly. Statement I providing AB = 12 cm and Statement II providing BC = 16 cm are redundant pieces of information. While these values are consistent with the Pythagorean theorem 12² + 16² = 144 + 256 = 400 = 20², they are not necessary to find the circumradius. Thus, the question can be answered even without using any of the statements, making option d correct.

Explanation

The area of a parallelogram with sides a and b and included angle θ is given by the formula Area = a · b · sin(θ). Substituting the given values, we have 7√3 = a · b · sin(60°). Since sin(60°) = √3/2, the equation becomes 7√3 = a · b · √3/2, which simplifies to a · b = 14. The perimeter is P = 2(a + b).

Using Statement-I, we are told that a and b are prime numbers. The only two prime numbers whose product is 14 are 2 and 7. Thus, a+b = 2+7 = 9, and the perimeter P = 2(9) = 18. Statement-I alone is sufficient.

Using Statement-II, we are told that a and b are natural numbers greater than 1. The pairs of natural numbers that multiply to 14 are (1, 14) and (2, 7). Since both sides must be greater than 1, the only possible pair is (2, 7). Thus, a+b = 2+7 = 9, and the perimeter P = 2(9) = 18. Statement-II alone is sufficient.

Since either statement independently provides the necessary information to find the perimeter, option b is correct.

Explanation

According to the Intersecting Chords Theorem in geometry, if two chords AB and CD of a circle intersect at a point P inside the circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. The mathematical expression for this theorem is AP × PB = CP × PD.

In the given problem, it is already provided that AP × PB = 48 square units. Applying the formula directly, we have CP × PD = AP × PB = 48 square units. Since the value of CP × PD is determined entirely by the information provided in the question itself, neither Statement I (which gives the length of AP) nor Statement II (which gives the length of CP) is necessary to find the answer. Therefore, the question can be answered even without using any of the statements, making option d correct.

Explanation

The solution relies on the Triangle Inequality Theorem, which states that for any triangle with sides a, b, and c, the length of side c must satisfy |a - b| < c < a + b. In quadrilateral ABCD, the diagonal BD divides the figure into two triangles: ΔABD and ΔBCD.

For ΔABD with sides AB = 6 and AD = 9, the range for BD is 9 - 6 < BD < 9 + 6, which simplifies to 3 < BD < 15. For ΔBCD with sides BC = 18 and CD = 6, the range for BD is 18 - 6 < BD < 18 + 6, which simplifies to 12 < BD < 24. For BD to exist, it must satisfy both inequalities, meaning 12 < BD < 15. If BD is an integer, the only possible values are 13 and 14.

Statement I states that BD is an integer greater than 13, which uniquely determines BD = 14. Statement II states that BD is an even integer, which also uniquely determines BD = 14 since 13 is odd. Therefore, either statement alone is sufficient to answer the question, making option (b) the correct choice.

Explanation

Let the base BC be b and the height of the triangle from vertex A to BC be h. In an isosceles triangle, the height bisects the base, creating two right triangles. By the Pythagorean theorem, h² + (b/2)² = 10², which simplifies to h² = 100 - b²/4.

The area of the triangle is given by Area = 1/2 × b × h = 48, so bh = 96, or h = 96/b. Substituting this into the Pythagorean equation:

(96/b)² = 100 - b²/4 9216/b² = (400 - b²)/4 36864 = 400b² - b⁴ b⁴ - 400b² + 36864 = 0

Using the quadratic formula for b²: b² = (400 ± √(400² - 4(36864)))/2 b² = (400 ± √(160000 - 147456))/2 b² = (400 ± √12544)/2 b² = (400 ± 112)/2

This gives two possible values for b²: b² = 512/2 = 256 ⇒ b = 16 b² = 288/2 = 144 ⇒ b = 12

Now evaluate the statements: Statement-I states BC is an even integer. Since both 12 and 16 are even integers, this statement alone is not sufficient to find a unique value for BC. Statement-II states the height h is greater than half of the base b/2. If b = 16, then h = 96/16 = 6. Here, b/2 = 8. Since 6 is not greater than 8, this case is invalid. If b = 12, then h = 96/12 = 8. Here, b/2 = 6. Since 8 > 6, this case is valid. Thus, Statement-II alone is sufficient to determine BC = 12.

Since Statement-II alone works but Statement-I alone does not, option a is correct.

Explanation

Let the diagonals of the rhombus be d₁ and d₂. Given the ratio d₁ : d₂ = 5 : 12, we can represent them as d₁ = 5x and d₂ = 12x for some positive constant x. In a rhombus, diagonals bisect each other at right angles, so the side s can be calculated using the Pythagorean theorem: s = √((d₁/2)² + (d₂/2)²). Substituting the values, we get s = √((5x/2)² + (12x/2)²) = √((25x² + 144x²)/4) = √(169x²/4) = 13x/2 = 6.5x. To determine if one of the diagonals is equal to the side, we compare s = 6.5x with d₁ = 5x and d₂ = 12x. Clearly, 6.5x ≠ 5x and 6.5x ≠ 12x for any x > 0. Since the answer is a definitive No based solely on the provided ratio, no additional information from Statement I or Statement II is required. Therefore, the question can be answered even without using any of the statements, making option d the correct choice.

Explanation

The Median is a positional average, meaning its value is determined by the middle position of the dataset once the observations are arranged in ascending or descending order. Because the Median depends on the rank of the observations rather than their actual numerical magnitude, an extremely high or low value at the fringes of the distribution does not shift the middle point. This makes it a robust measure of central tendency.

In contrast, the Arithmetic Mean is calculated using the formula Mean = Σ xᵢ/n. Since every observation xᵢ is summed, a single extreme outlier can significantly inflate or deflate the result, pulling the mean away from the center of the majority of the data. Similarly, the Geometric Mean and Harmonic Mean are sensitive to extreme values, especially very small values or zeros, which can disproportionately affect the product or the sum of reciprocals. Therefore, the Median is the most reliable measure when dealing with skewed data or datasets containing extreme observations.

Explanation

The Harmonic Mean is specifically designed to calculate the average of rates, ratios, or fractions. In statistics, when we deal with quantities expressed as a ratio of two different units, such as speed (kilometres per hour), price per unit, or work done per hour, the arithmetic mean often provides a biased result. The Harmonic Mean provides the correct average when the numerator of the rate is constant across the observations. For example, if a vehicle travels a fixed distance at different speeds, the average speed for the entire journey is the harmonic mean of those speeds. The formula for the Harmonic Mean of n observations x₁, x₂, ..., xₙ is given by HM = n/(Σᵢ₌₁ⁿ 1/xᵢ). This measure effectively weights the values by their reciprocals, making it the most suitable central tendency for averaging rates and ratios where the unit of measure is in the denominator.

Explanation

To find the average marks, we first convert the given cumulative frequency distribution into a grouped frequency distribution. The frequency for each class interval is calculated by subtracting the subsequent cumulative frequency from the preceding one.

For the class 10 to 20, the frequency f₁ is 100 - 75 = 25. For the class 20 to 30, the frequency f₂ is 75 - 60 = 15. For the class 30 to 40, the frequency f₃ is 60 - 40 = 20. For the class 40 to 50 (assuming the same interval width), the frequency f₄ is the remaining 40 candidates.

Next, we determine the mid-value x for each class interval: x₁ = (10+20)/2 = 15 x₂ = (20+30)/2 = 25 x₃ = (30+40)/2 = 35 x₄ = (40+50)/2 = 45

Using the formula for the mean x̄ = Σfᵢxᵢ/N, where N = 100: Σfᵢxᵢ = (25 × 15) + (15 × 25) + (20 × 35) + (40 × 45) Σfᵢxᵢ = 375 + 375 + 700 + 1800 = 3250

Therefore, the average marks are: x̄ = 3250/100 = 32.5

This corresponds to option d).

Explanation

If every observation in a set is multiplied by a constant k, the arithmetic mean is also multiplied by k. New mean = 5 * 60 = 300.

Explanation

To calculate the mean of a combined distribution, we use the weighted average formula for the combined mean: x̄ = (n₁x̄₁ + n₂x̄₂ + n₃x̄₃)/(n₁ + n₂ + n₃). Here, the frequencies are given as n₁ = 45, n₂ = 40, and n₃ = 55, and their respective means are x̄₁ = 2, x̄₂ = 2.5, and x̄₃ = 2.

Substituting these values into the formula, we first find the sum of the products of frequencies and means: Sum = (45 × 2) + (40 × 2.5) + (55 × 2) Sum = 90 + 100 + 110 = 300

Next, we calculate the total frequency of the combined distribution: Total frequency N = 45 + 40 + 55 = 140

The combined mean is then: x̄ = 300/140 = 30/14 = 15/7 ≈ 2.1428

Rounding to two decimal places, we get 2.14, which corresponds to option (a).

Explanation

A positional average is a measure of central tendency that is determined by the position of values in a data set when they are arranged in ascending or descending order, rather than by a mathematical calculation involving the sum or product of all observations. The Median is the primary example of a positional average because it identifies the middle-most value of a sorted data set. For a set of n observations, the position of the median is calculated using the formula (n+1)/2.

Unlike the arithmetic mean or geometric mean, which are calculated averages that incorporate the specific numerical value of every single data point, the median only depends on the rank or position of the data. This characteristic makes the median more robust than the mean when dealing with skewed distributions or datasets containing extreme outliers. While the mode represents the most frequent value, it does not partition the data based on rank in the same way the median, quartiles, deciles, and percentiles do.

Explanation

First Group: Median = 20.8, L = 20, h = 5, f = 75. Cumulative frequency cf = u + v + 11 + 52 = u + v + 63. Total frequency N = u + v + 160. Median = 20 + [( (u+v+160)/2 - (u+v+63) ) / 75] * 5 0.8 = [ (u+v+160 - 2u-2v-126) / 2 ] / 15 12 = (-u-v+34) / 2 => 24 = -u-v+34 => u + v = 10. Second Group: Mean = 17.3. Midpoints: 2.5, 7.5, 12.5, 17.5, 22.5, 27.5. Sum fx = 2.5(3u) + 7.5(2v) + 12.5(40) + 17.5(50) + 22.5(30) + 27.5(28) = 7.5u + 15v + 2820. N = 3u + 2v + 148. Mean = (7.5u + 15v + 2820) / (3u + 2v + 148) = 17.3. 7.5u + 15v + 2820 = 51.9u + 34.6v + 2560.4 => 44.4u + 19.6v = 259.6. Substitute v = 10 - u: 44.4u + 19.6(10 - u) = 259.6 24.8u = 63.6 => u = 2.56. Checking options: for u=3, v=7, Mean = (22.5+105+2820)/171 = 17.236. For u=2, v=8, Mean = (15+120+2820)/170 = 17.38. Among the options, u=3 is the closest fit for the system when rounding is considered for the mean.

Explanation

Based on the system derived in Question 24, u + v = 10. If u = 3, then v = 7. Checking these values in the First Group median calculation: N = 170, cf = 73. Median = 20 + [(85 - 73)/75] * 5 = 20 + (12/15) = 20.8. This matches the given median exactly. Thus, v = 7.

Explanation

The median value is given as 24, which falls in the class interval 20-30. The cumulative frequency (cf) before the median class is 30. The frequency (f) of the median class is (30+k) - 30 = k. The total number of observations (N) is 55+k. Using the median formula: Median = L + ((N/2 - cf)/f) h, where L=20 and h=10. Substituting the values: 24 = 20 + (((55+k)/2 - 30)/k) 10. This simplifies to 4 = ((k-5)/2k) * 10, which further simplifies to 4k = 5(k-5). Solving for k gives 4k = 5k - 25, so k = 25. Thus, option C is correct.

Explanation

The given data is in a cumulative frequency distribution. First, we convert it into class intervals and their respective frequencies: 0-10 (5), 10-20 (30 - 5 = 25), 20-30 (30 + k - 30 = k), 30-40 (48 + k - (30 + k) = 18), and 40-50 (55 + k - (48 + k) = 7). The total frequency is N = 55 + k.

Since the median is 24, it lies in the class interval 20-30. Using the median formula Median = L + ((N/2 - cf)/f) × h, where L = 20, cf = 30, f = k, and h = 10, we have: 24 = 20 + (((55 + k)/2 - 30)/k) × 10 4 = ((55 + k - 60)/2k) × 10 0.4 = (k - 5)/2k 0.8k = k - 5 0.2k = 5 ⇒ k = 25

With k = 25, the total number of students is N = 55 + 25 = 80. We calculate the mean using the formula Mean = (Σ fᵢ xᵢ)/N, where xᵢ is the class midpoint: Σ fᵢ xᵢ = (5 × 5) + (25 × 15) + (25 × 25) + (18 × 35) + (7 × 45) Σ fᵢ xᵢ = 25 + 375 + 625 + 630 + 315 = 1970 Mean = 1970/80 = 24.625

Thus, the mean of the distribution is 24.625, which corresponds to option d.