CDS II 2024 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The function to minimize is f(x) = 4x^2 + x. This is a quadratic expression of the form ax^2 + bx + c. The minimum value occurs at the vertex, where x = -b/(2a). Substituting a = 4 and b = 1, we get x = -1/(2*4) = -1/8 = -0.125.

Explanation

The equation is m^2 - n^2 = 72, which factors as (m-n)(m+n) = 72. Let u = m-n and v = m+n. Then uv = 72, with v > u. For m and n to be natural numbers, u and v must be of the same parity. Since their product 72 is even, both must be even. The possible pairs (u, v) are (2, 36), (4, 18), and (6, 12). This gives 3 pairs of (m, n): (19, 17), (11, 7), and (9, 3).

Explanation

The remainders are 2, 8, 11, 20 when N is divided by 6, 12, 15, 24. The difference between divisor and remainder is constant: 6-2=4, 12-8=4, 15-11=4, 24-20=4. Thus, N+4 is the LCM of 6, 12, 15, and 24. LCM(6, 12, 15, 24) = 120. The general form is N = 120k - 4. For the greatest 5-digit number, 120k - 4 <= 99999 implies 120k <= 100003, so max k = 833. N = 120(833) - 4 = 99960 - 4 = 99956.

Explanation

Modulo 5: 111 is 1, 222 is 2, and 333 is 3 (or -2). 111^222 mod 5 = 1^222 = 1. 222^333 mod 5 = 2^333. Since 2^4 = 16 = 1 mod 5, and 333 mod 4 = 1, 2^333 = 2^1 = 2. 333^444 mod 5 = (-2)^444 = 2^444. Since 444 mod 4 = 0, 2^444 = 2^4 = 1 mod 5. Sum = 1 + 2 + 1 = 4.

Explanation

To find the last three digits, we calculate (4321012345 98766789) mod 1000. This is equivalent to 345 789 mod 1000. 345 * 789 = 272205. The last three digits are 205.

Explanation

The relation is p = k(x^2 + y^2 + z^2). Using the first set of values: 70 = k(1^2 + 2^2 + 3^2) = k(1 + 4 + 9) = 14k. Thus, k = 5. For the second set: p = 5((-1)^2 + 1^2 + 5^2) = 5(1 + 1 + 25) = 5(27) = 135.

Explanation

To find the value of N, we use the property that N leaves a remainder of 5 when divided by 6, 12, 15, and 18. This means N can be expressed in the form N = LCM(6, 12, 15, 18)m + 5, where m is a positive integer. First, we calculate the Least Common Multiple (LCM) of the given numbers: 6 = 2 × 3 12 = 2² × 3 15 = 3 × 5 18 = 2 × 3² LCM(6, 12, 15, 18) = 2² × 3² × 5 = 4 × 9 × 5 = 180.

Thus, N = 180m + 5. We are given that N is a multiple of 11, so 180m + 5 ≡ 0 (mod 11). Simplifying 180 modulo 11: 180 = 11 × 16 + 4, so 180 ≡ 4 (mod 11). The equation becomes 4m + 5 ≡ 0 (mod 11), which simplifies to 4m ≡ -5 (mod 11) or 4m ≡ 6 (mod 11). Dividing by 2, we get 2m ≡ 3 (mod 11). Adding 11 to the right side, 2m ≡ 14 (mod 11), which gives m ≡ 7 (mod 11). The least positive integer m is 7. Substituting m = 7 back into the expression for N: N = 180(7) + 5 = 1260 + 5 = 1265. Comparing this value with the given options, 1265 falls in the range 1200 < N < 1300, which corresponds to option (d).

Explanation

Rationalizing each term 1/(sqrt(n+1) + sqrt(n)) gives (sqrt(n+1) - sqrt(n)). The sum is a telescoping series: (sqrt(10)-sqrt(9)) + (sqrt(11)-sqrt(10)) + ... + (sqrt(196)-sqrt(195)). The sum simplifies to sqrt(196) - sqrt(9) = 14 - 3 = 11.

Explanation

Let the speeds of trains X and Y be sₓ and sᵧ respectively. The length of a train is calculated as the product of its speed and the time it takes to cross a stationary point. Therefore, the length of train X is Lₓ = 24 sₓ and the length of train Y is Lᵧ = 18 sᵧ. When two trains move in opposite directions, their relative speed is the sum of their individual speeds, sₓ + sᵧ. The time taken for them to cross each other is the total length divided by the relative speed: t = (Lₓ + Lᵧ)/(sₓ + sᵧ). Substituting the given values, we have 20 = (24 sₓ + 18 sᵧ)/(sₓ + sᵧ). Cross-multiplying the equation gives 20 sₓ + 20 sᵧ = 24 sₓ + 18 sᵧ. Rearranging the terms to isolate the speeds on opposite sides, we get 20 sᵧ - 18 sᵧ = 24 sₓ - 20 sₓ, which simplifies to 2 sᵧ = 4 sₓ. This results in the ratio sₓ/sᵧ = 2/4 = 1/2. Thus, the ratio of the speed of X to the speed of Y is 1:2, matching option a.

Explanation

Using the relations between the roots and coefficients of a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. For the first equation x² + mx - n = 0, the sum of the roots p + q = -m and the product pq = -n. For the second equation x² + px - q = 0, the sum of the roots m + n = -p and the product mn = -q.

From the sum relations, we have p = -(m + n). Substituting this into the first sum relation gives -(m + n) + q = -m, which simplifies to -m - n + q = -m, implying q = n. Substituting q = n into the second product relation mn = -q gives mn = -n. Since n is a non-zero number, we can divide by n to find m = -1. Substituting m = -1 into the sum relation m + n = -p gives -1 + n = -p, or p + n = 1. From the first product relation pq = -n, and since q = n, we have pn = -n, which simplifies to p = -1 as n is non-zero. Now, using p + n = 1 with p = -1, we find n = 2, and consequently q = 2.

Evaluating the statements with m = -1, n = 2, p = -1, q = 2: Statement I: m(m + n) = -1(-1 + 2) = -1(1) = -1. This is correct. Statement II: p + q = -1 + 2 = 1. This is also correct. Therefore, both statements I and II are correct.

Explanation

Let u = sin(theta), where -1 <= u <= 1. The expression is f(u) = -4u^2 + 8u. This is a downward-opening parabola with a vertex at u = -8/(2*-4) = 1. The maximum value is f(1) = -4(1)^2 + 8(1) = 4.

Explanation

Expanding the terms: (1 + 2 tan alpha tan beta + tan^2 alpha tan^2 beta) + (tan^2 alpha - 2 tan alpha tan beta + tan^2 beta). The 2 tan alpha tan beta terms cancel, leaving: 1 + tan^2 alpha + tan^2 beta + tan^2 alpha tan^2 beta = (1 + tan^2 alpha)(1 + tan^2 beta) = sec^2 alpha sec^2 beta.

Explanation

To determine the signs of these expressions, we use the fundamental property that tan 45° = cot 45° = 1. For angles in the first quadrant (0° < θ < 90°), the tangent function is strictly increasing, while the cotangent function is strictly decreasing.

For Statement I, the angle is 50°, which is greater than 45°. Since tan θ increases as θ increases, tan 50° > tan 45° = 1. Conversely, since cot θ decreases as θ increases, cot 50° < cot 45° = 1. Therefore, the expression tan 50° - cot 50° involves subtracting a value less than 1 from a value greater than 1, resulting in a positive value. Statement I is correct.

For Statement II, the angle is 25°, which is less than 45°. In this range, cot 25° > cot 45° = 1 and tan 25° < tan 45° = 1. Thus, cot 25° - tan 25° involves subtracting a value less than 1 from a value greater than 1, which yields a positive result. Since the statement claims the result is negative, Statement II is incorrect. Only option a is correct.

Explanation

Given the values of the trigonometric functions within the specified range 0 ≤ (α - β) ≤ (α + β) ≤ π/2, we can determine the angles directly using inverse trigonometric values. Since tan (α + β) = √3, it follows that α + β = π/3 or 60°. Similarly, since tan (α - β) = 1/√3, it follows that α - β = π/6 or 30°. Adding these two equations gives 2α = π/3 + π/6 = π/2, which implies α = π/4 or 45°. Subtracting the two equations gives 2β = π/3 - π/6 = π/6 or 30°. We need to evaluate the expression tan α · cot 2β. Substituting the values we found, tan π/4 = 1 and cot π/6 = √3. Therefore, the product is 1 · √3 = √3, which corresponds to option c.

Explanation

sec^2(theta) + cosec^2(theta) = 1/cos^2(theta) + 1/sin^2(theta) = (sin^2(theta) + cos^2(theta))/(sin^2(theta) cos^2(theta)) = 1/(sin^2(theta) cos^2(theta)). Multiplying this by sin^2(theta) cos^2(theta) gives 1.

Explanation

Let 64^sin^2(theta) = y. Then 64^cos^2(theta) = 64^(1-sin^2(theta)) = 64/y. The equation is y + 64/y = 16, which simplifies to y^2 - 16y + 64 = 0, or (y-8)^2 = 0. Thus y = 8. 64^sin^2(theta) = 8 implies (8^2)^sin^2(theta) = 8^1, so 2 sin^2(theta) = 1, meaning sin^2(theta) = 1/2. This occurs at theta = 45 degrees, where tan(theta) + cot(theta) = 1 + 1 = 2.

Explanation

Using the fundamental trigonometric identity csc²θ - cot²θ = 1, we can write this as (cscθ - cotθ)(cscθ + cotθ) = 1. Given that cscθ - cotθ = m, we have m(cscθ + cotθ) = 1, which implies cscθ + cotθ = 1/m. By adding the two equations cscθ - cotθ = m and cscθ + cotθ = 1/m, we get 2cscθ = m + 1/m, which simplifies to cscθ = 1/2(m + 1/m).

Similarly, we use the identity sec²θ - tan²θ = 1, which can be written as (secθ - tanθ)(secθ + tanθ) = 1. Given that secθ - tanθ = n, we have n(secθ + tanθ) = 1, which implies secθ + tanθ = 1/n. By adding the two equations secθ - tanθ = n and secθ + tanθ = 1/n, we get 2secθ = n + 1/n, which simplifies to secθ = 1/2(n + 1/n).

To find the value of cscθ + secθ, we sum the derived expressions: cscθ + secθ = 1/2(m + 1/m) + 1/2(n + 1/n) = 1/2(m + n + 1/m + 1/n). This matches option (a).

Explanation

Let the point X be the observer on the bridge and let the projection of X on the river surface be X'. The width of the river is the sum of the distances from X' to the two points P and Q on opposite banks, which is PQ = PX' + X'Q. Using the definition of the tangent of the angle of depression in right triangles ΔPX'X and ΔQX'X, we have tan α = h/PX' and tan β = h/X'Q. Rearranging these, we get PX' = h cot α and X'Q = h cot β.

The total width of the river is W = h(cot α + cot β). Since α and β are complementary angles, β = 90° - α, which implies cot β = tan α. Substituting this into the width equation gives W = h(cot α + tan α). Applying the trigonometric identity tan α + cot α = sin α/cos α + cos α/sin α = (sin² α + cos² α)/(sin α cos α) = 1/(sin α cos α), we can rewrite the expression. Since 1/cos α = sec α and 1/sin α = csc α, the width becomes W = h sec α csc α. This matches option d.

Explanation

The fastest way to calculate the area of a triangle when two sides and the included angle are given is by using the trigonometric area formula: Area = 1/2 × a × c × sin(B). Here, the sides are AB = 6 cm and BC = 8 cm, and the included angle is ∠ABC = 60°.

Substituting these values into the formula: Area = 1/2 × 6 × 8 × sin(60°) Using the standard value sin(60°) = √3/2, we get: Area = 1/2 × 48 × √3/2 Area = 24 × √3/2 = 12√3 square cm.

Alternatively, if we first find the altitude AD, we use the relation AD = AB sin(60°) = 6 × √3/2 = 3√3 cm. Using the base formula Area = 1/2 × base × height = 1/2 × 8 × 3√3 = 12√3 square cm. This corresponds to option (b).

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. Given the equation x² - sin²θ x - cos²θ = 0, we have p + q = sin²θ and pq = -cos²θ. We need to find the minimum value of the expression p² + q². Using the algebraic identity p² + q² = (p + q)² - 2pq, we substitute the values: p² + q² = (sin²θ)² - 2(-cos²θ) = sin⁴θ + 2cos²θ To find the minimum value, we express the equation in terms of sin²θ by substituting cos²θ = 1 - sin²θ. Let s = sin²θ, where 0 ≤ s ≤ 1. The expression becomes: f(s) = s² + 2(1 - s) = s² - 2s + 2 This is a quadratic function in s. The minimum value of a quadratic as² + bs + c occurs at s = -b/2a. Here, s = -(-2)/2(1) = 1. Since s = 1 is within the valid range for sin²θ, we substitute s = 1 into the expression: f(1) = 1² - 2(1) + 2 = 1 Thus, the minimum value of p² + q² is 1, which corresponds to option b.

Explanation

The arithmetic mean formula is given by: Arithmetic Mean = (Sum of all observations)/(Total number of observations)

According to the problem, the number of observations is n and the arithmetic mean is M. Therefore, the total sum of all n numbers can be calculated as: Total Sum = n × M = nM

We are given that the sum of the first (n - 1) terms is k. Let the nth number be represented by x. The total sum of n numbers is the sum of the first (n - 1) terms plus the nth term: Total Sum = k + x

Equating the two expressions for the total sum, we get: nM = k + x

To find the nth number x, we rearrange the equation by subtracting k from both sides: x = nM - k

Thus, the nth number is nM - k, which corresponds to option (b).

Explanation

The numbers are 3^1, 3^2, 3^3, 3^4, 3^5, 3^6, 3^7. The geometric mean is the 7th root of their product: (3^(1+2+3+4+5+6+7))^(1/7) = (3^28)^(1/7) = 3^4 = 81.

Explanation

The total quantity of tea powder purchased is 1 kg × 4 = 4 kg. According to the given rates, the cost of 1kg from each place is calculated as follows: the cost from place A is 1000 / 1 = 1000, from place B is 1000 / 2 = 500, from place C is 1000 / 4 = 250, and from place D is 1000 / 5 = 200. The total expenditure for 4kg of tea powder is the sum of these costs, which is 1000 + 500 + 250 + 200 = 1950. To find the average quantity x purchased per ₹1000, we apply the formula x = Total Quantity / (Total Expenditure / 1000). Substituting the values, we get x = 4 / (1950 / 1000) = 4 / 1.95. Performing the division, x ≈ 2.05128. Thus, the approximate value of x is 2.05, which corresponds to option (c).

Explanation

The single-digit primes are 2, 3, 5, 7. The largest 4-digit number without repetition is 7532 and the smallest is 2357. Their sum is 7532 + 2357 = 9889.

Explanation

We use modulo 28. 3^3 = 27 = -1 mod 28. Therefore, 3^255 = (3^3)^85 = (-1)^85 = -1 mod 28. -1 mod 28 is 27.

Explanation

Modulo 9: 100 is 1. So 100^97 is 1^97 = 1 and 100^54 is 1^54 = 1. The expression is 1 + 1 + x + 1 = x + 3. For x + 3 to be divisible by 9, x must be 6 (since 0 <= x <= 8).

Explanation

The problem provides the condition AB · DC = AC · BD. By rearranging this equation, we get the ratio BD/DC = AB/AC. This relationship is the fundamental statement of the Angle Bisector Theorem, which states that an internal angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. Since the ratio of the segments BD and DC on side BC is equal to the ratio of the sides AB and AC, it follows that the line segment AD is the internal angle bisector of ∠BAC. Because AD bisects ∠BAC, the two resulting angles ∠BAD and ∠CAD must be equal. Given that ∠BAD = α and ∠CAD = β, we have α = β, which corresponds to option a.

Explanation

A number is divisible by 9 if and only if the sum of its digits is divisible by 9. For the given 10-digit number N = 12345678AB, the sum of the digits is calculated as follows: Sum = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + A + B = 36 + A + B. Since N is divisible by 9, the expression 36 + A + B must be a multiple of 9. Because 36 is already divisible by 9, it follows that A + B must also be divisible by 9. This proves that statement I is correct.

To evaluate statement II, we check if A being odd necessitates B being odd. The possible values for A + B that are multiples of 9 are 0, 9, and 18. If we choose A = 1, which is an odd digit, then A + B = 9 implies B = 8. In this case, B is an even digit. Since we found a counterexample where A is odd and B is even while the number remains divisible by 9, statement II is incorrect. Consequently, only statement I is correct, which corresponds to option (a).

Explanation

For x^3 + 1/x^3 = 65/8, x = 2 (since 8 + 1/8 = 65/8) or x = 1/2. For y^3 + 1/y^3 = 730/27, y = 3 (since 27 + 1/27 = 730/27) or y = 1/3. The possible values for xy are 6, 2/3, 3/2, or 1/6. Option B is 6.

Explanation

We test small natural numbers for x, y. For x+y=2: (1,1) gives 16 (not prime). For x+y=3: (1,2) gives 21, (2,1) gives 27 (neither prime). For x+y=4: (1,3) gives 26, (2,2) gives 32, (3,1) gives 38 (none prime). For x+y=5: (1,4) gives 31, (2,3) gives 37, (3,2) gives 43 (all prime). The minimum value is 5.

Explanation

Number of divisors d(N) = 15. The prime factorization of N could be p^14 or p^2 q^4. If N = p^14, d(N^2) = d(p^28) = 29. If N = p^2 q^4, d(N^2) = d(p^4 * q^8) = (4+1)(8+1) = 45. 45 is in the options.

Explanation

Based on the Triangle Inequality Theorem, the sum of any two sides of a triangle must always be greater than the third side. For a triangle with sides p, q, and r, the following conditions must hold: p + q > r, q + r > p, and r + p > q.

Rearranging these inequalities to match the terms in the expression, we get: p - q - r < 0 (since p < q + r) q - r - p < 0 (since q < r + p) r - p - q < 0 (since r < p + q)

The expression (p - q - r)(q - r - p)(r - p - q) is the product of three negative values. Mathematically, the product of three negative numbers is always negative: (negative) × (negative) × (negative) = negative. Since the sides of a triangle must strictly follow these inequalities, the result cannot be zero or positive. Therefore, the expression is negative only, making option b the correct choice.

Explanation

The expression is ((a^8 + a^4 + 1)/a^4) ((b^8 + b^4 + 1)/b^4) = (a^4 + 1 + 1/a^4)(b^4 + 1 + 1/b^4). By AM-GM, a^4 + 1/a^4 >= 2, so (a^4 + 1 + 1/a^4) >= 3. Similarly, (b^4 + 1 + 1/b^4) >= 3. The minimum value is 3 3 = 9.

Explanation

Let Tea = T and Coffee = C. Given |T intersect C| = n, |C| = 2n, |T| = 3n, and |Neither| = 4n. The number of students who prefer at least one is |T union C| = |T| + |C| - |T intersect C| = 3n + 2n - n = 4n. Total students = |T union C| + |Neither| = 4n + 4n = 8n. Since 8n = 200, n = 25.

Explanation

The area of a triangle given two sides and the included angle is calculated using the formula Area = 1/2 ab sin θ. Substituting the given values, we have 36 = 1/2 × 9 × 12 × sin θ. Simplifying the right side gives 36 = 54 sin θ, which leads to sin θ = 36/54 = 2/3. To find cos θ, we use the fundamental trigonometric identity sin² θ + cos² θ = 1. Substituting the value of sin θ, we get cos² θ = 1 - (2/3)² = 1 - 4/9 = 5/9. Taking the square root of both sides, we find cos θ = √5/3. This matches option a.

Explanation

Given that the HCF(n, n+10) = 10, we know that 10 must divide both n and n+10. This implies n is a multiple of 10. Let n = 10a, where a is a natural number. Then n+10 = 10(a+1). Since any two consecutive integers a and a+1 are always coprime, the HCF will remain 10 for any natural number a.

We use the fundamental relation HCF(A, B) × LCM(A, B) = A × B. Substituting the given values: 10 × x = n × (n+10) 10x = 10a × 10(a+1) 10x = 100a(a+1) x = 10a(a+1)

The problem states that x is a 2-digit number, which means 10 ≤ x ≤ 99. 10 ≤ 10a(a+1) ≤ 99 Dividing by 10: 1 ≤ a(a+1) ≤ 9.9

Now we check possible natural number values for a: If a = 1, a(a+1) = 1 × 2 = 2. This is within range. x = 10 × 2 = 20. If a = 2, a(a+1) = 2 × 3 = 6. This is within range. x = 10 × 6 = 60. If a = 3, a(a+1) = 3 × 4 = 12. This exceeds 9.9, making x = 120, which is a 3-digit number.

The only possible values for x are 20 and 60. Therefore, there are only two such values, making option b correct.

Explanation

To find the HCF, we factorize both polynomial expressions. For the first polynomial, a⁴ + 2a³ + 3a² + 2a + 1, we observe that it follows the pattern of a squared trinomial. Using the identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx, we can express the polynomial as (a²)² + a² + 1² + 2(a²)(a) + 2(a²)(1) + 2(a)(1), which simplifies to (a² + a + 1)².

For the second polynomial, a⁶ - 2a³ + 1, we recognize it as a perfect square of the form x² - 2xy + y² = (x - y)² where x = a³ and y = 1. Thus, a⁶ - 2a³ + 1 = (a³ - 1)². Using the difference of cubes identity a³ - b³ = (a - b)(a² + ab + b²), we can further factorize this as [(a - 1)(a² + a + 1)]² = (a - 1)² (a² + a + 1)².

The HCF is the highest common factor shared by both polynomials. Comparing (a² + a + 1)² and (a - 1)² (a² + a + 1)², the highest common factor is (a² + a + 1)², which corresponds to option c.

Explanation

For equal roots, the discriminant D = 0. D = (k-2)^2 - 4(1)(k+1) = k^2 - 4k + 4 - 4k - 4 = k^2 - 8k. Setting k^2 - 8k = 0 gives k(k-8) = 0, so k = 0 or k = 8.

Explanation

To evaluate the expression E = ((cos θ - sin θ + 1)/(cos θ + sin θ - 1)) (cot θ - csc θ), the fastest method for competitive exams is substituting a standard angle such as θ = 45°. For θ = 45°, we have cos θ = 1/√2 and sin θ = 1/√2. Substituting these values into the first term gives (1/√2 - 1/√2 + 1)/(1/√2 + 1/√2 - 1) = 1/(2/√2 - 1) = 1/(√2 - 1). Rationalizing the denominator yields √2 + 1. The second term becomes cot 45° - csc 45° = 1 - √2. The total product is (√2 + 1)(1 - √2) = 1² - (√2)² = 1 - 2 = -1.

Algebraically, divide the numerator and denominator of the first fraction by sin θ to obtain (cot θ + csc θ - 1)/(cot θ - csc θ + 1). Using the identity 1 = csc² θ - cot² θ = (csc θ - cot θ)(csc θ + cot θ), the numerator becomes (csc θ + cot θ) - (csc θ - cot θ)(csc θ + cot θ) = (csc θ + cot θ)(1 - csc θ + cot θ). Dividing this by the denominator (1 - csc θ + cot θ) simplifies the first part to csc θ + cot θ. Multiplying this result by the second term (cot θ - csc θ) yields cot² θ - csc² θ, which equals -1 according to the identity csc² θ - cot² θ = 1. Thus, option a is correct.

Explanation

To solve this expression, we factorize the numerator and the denominator to identify common trigonometric identities. For the numerator, we take sinθ as a common factor: sinθ - 2sin³θ = sinθ(1 - 2sin²θ). For the denominator, we take cosθ as a common factor: 2cos³θ - cosθ = cosθ(2cos²θ - 1).

Now, we apply the double angle formulas for cos 2θ. We know that cos 2θ = 1 - 2sin²θ and cos 2θ = 2cos²θ - 1. Substituting these values into our factored expression, we get (sinθ cos 2θ)/(cosθ cos 2θ).

The term cos 2θ in the numerator and denominator cancels out, leaving us with sinθ/cosθ. According to the basic trigonometric identity tanθ = sinθ/cosθ, the simplified value of the expression is tanθ, which corresponds to option (d).

Explanation

The fastest way to solve this problem is using the rule of alligation, which is used to find the ratio in which two components are mixed to achieve a specific weighted average. The formula for the ratio of the number of boys to the number of girls is:

Ratio = (Overall Mean - Girls Mean)/(Boys Mean - Overall Mean)

Substituting the given values into the formula: Overall Mean = 60 kg Boys Mean = 70 kg Girls Mean = 55 kg

Ratio of Boys to Girls = (60 - 55)/(70 - 60) Ratio = 5/10 = 1/2

Thus, the ratio of the number of boys to the number of girls is 1:2, which corresponds to option (a). This method avoids the need to calculate the total weight of the class or use simultaneous equations, providing the result directly through the difference between the group means and the collective mean.

Explanation

To find the average speed of the monkey, we first determine the total distance traveled, which is the length of the rod connecting the tops of the two towers. This distance forms the hypotenuse of a right angled triangle where one side is the horizontal distance between the towers and the other side is the difference in the heights of the towers.

The horizontal distance d = 9 m. The difference in height Δh = 23 m - 11 m = 12 m.

Using the Pythagorean theorem, the length of the rod L is calculated as: L = √(d² + (Δh)²) = √(9² + 12²) = √(81 + 144) = √225 = 15 m.

The monkey covers this distance in t = 5 minutes. The average speed in meters per minute is: Speed = Distance/Time = 15 m/5 min = 3 m/min.

To match the options, we convert this speed into centimeters per second: 3 m/min = (3 × 100 cm)/60 sec = 300/60 cm/sec = 5 cm/sec.

Thus, the average speed of the monkey is 5 cm/sec, which corresponds to option d.

Explanation

The total surface area of a spherical ball is given by the formula 4πr². When the sphere is divided into eight identical parts by planes passing through the same diameter, each resulting piece is a spherical wedge. The surface area of each such wedge consists of two components: the curved outer surface and the flat planar surfaces created by the cuts. The curved surface area of each piece is one eighth of the total surface area of the sphere, which is computed as 4πr²/8 = πr²/2. Each wedge is bounded by two flat faces, and since the cutting planes pass through the diameter of the sphere, these faces are semicircles with radius r. The area of a single semicircle is πr²/2, meaning the total area of the two flat faces for one piece is 2 × πr²/2 = πr². Adding the curved surface area and the flat surface area together, the total surface area of each final piece is πr²/2 + πr² = 3πr²/2, which corresponds to option (b).

Explanation

Moving clockwise, the right wheel is on the inner circle (radius 50m) and the left wheel is on the outer circle (radius 50+1 = 51m). The distance traveled by each wheel is proportional to its radius. Thus, the number of turns N_L / N_R = R_L / R_R = 51/50. N_L = (51/50) * 1000 = 1020.

Explanation

The product is 70 71 ... 79. For it to be divisible by 1000 (8 125), we need factors of 2^3 and 5^3. There are multiple even numbers (70, 72, 74, 76, 78), providing more than enough 2s. For 5s, 70 provides one 5 and 75 provides two 5s (5^2), totaling 5^3. Since both 8 and 125 are factors, the product is divisible by 1000, so the remainder is 0.

Explanation

Let the height of the vertical pole be h = 80 m. The base of the pole is at P. The distances from the base to points A and B are PA = 64 m and PB = PA + AB = 64 + 36 = 100 m.

Using the formula for the tangent of the angle of elevation, tan θ = Perpendicular/Base, we calculate the tangents for α and β: tan α = h/PA = 80/64 = 5/4 tan β = h/PB = 80/100 = 4/5

Multiplying these two values, we get tan α · tan β = 5/4 · 4/5 = 1. Since tan β = 1/tan α, we can write tan β = cot α. From the trigonometric complementary angle identity, we know that cot α = tan(90° - α). Therefore, tan β = tan(90° - α), which implies β = 90° - α. Rearranging this gives α + β = 90°. This matches option b.

Explanation

The given dividend is P(x) = x⁸ᵏ⁺³ + x⁸ᵏ⁺⁶ + x⁸ᵏ⁺⁹ + x⁸ᵏ⁺¹². By factoring out the lowest power of x, which is x⁸ᵏ⁺³, we can rewrite the expression as P(x) = x⁸ᵏ⁺³(1 + x³ + x⁶ + x⁹).

Now, consider the divisor D(x) = (1 + x³)(1 + x⁶). Expanding this product using the distributive property: D(x) = 1(1 + x⁶) + x³(1 + x⁶) = 1 + x⁶ + x³ + x⁹. Rearranging the terms, we get D(x) = 1 + x³ + x⁶ + x⁹.

To find the quotient Q(x), we divide the dividend by the divisor: Q(x) = P(x)/D(x) = (x⁸ᵏ⁺³(1 + x³ + x⁶ + x⁹))/(1 + x³ + x⁶ + x⁹).

Since the term (1 + x³ + x⁶ + x⁹) appears in both the numerator and the denominator, they cancel each other out, leaving Q(x) = x⁸ᵏ⁺³. This corresponds to option d.

Explanation

The side of the outer square is s = 14 cm. The area of the outer square is Aₒᵤₜₑᵣ = s² = 14² = 196 sq cm. The inner square is formed by joining the midpoints of the outer square. Each side of the inner square acts as the hypotenuse of a right-angled triangle with legs equal to half the side of the outer square, which is 14/2 = 7 cm.

Using the Pythagorean theorem, the area of the inner square Aᵢₙₙₑᵣ is a² = 7² + 7² = 49 + 49 = 98 sq cm.

The area lying between the two squares is the difference between their areas: Area = Aₒᵤₜₑᵣ - Aᵢₙₙₑᵣ = 196 - 98 = 98 sq cm.

Alternatively, using the geometric property that a square formed by joining the midpoints of another square has exactly half the area of the original square, the area between them is simply half the area of the outer square: Area = 1/2 × 14² = 196/2 = 98 sq cm. This corresponds to option a.

Explanation

Using the formula for the area of a triangle, Area = 1/2 × base × height. Let the height be h and the base be b. According to the problem, b = 4/3h and the area is 54 square cm. Substituting these values into the formula, we get 54 = 1/2 × 4/3h × h, which simplifies to 54 = 2/3h². Solving for h², we find h² = (54 × 3)/2 = 81, which gives h = 9 cm. The base is then b = 4/3 × 9 = 12 cm. To find the perimeter, we first calculate the hypotenuse c using the Pythagorean theorem: c = √(b² + h²) = √(12² + 9²) = √(144 + 81) = √225 = 15 cm. The perimeter of the triangle is the sum of its sides, P = b + h + c = 12 + 9 + 15 = 36 cm. Thus, option c is correct.

Explanation

To calculate the area of a triangle with sides a=4, b=4, and c=6, we use Heron's formula as it is the most direct method for triangles where all three sides are known. First, we find the semi-perimeter s using the formula s = (a+b+c)/2. Substituting the given values, we get s = (4+4+6)/2 = 14/2 = 7 units.

Next, we apply Heron's formula for the area: Area = √s(s-a)(s-b)(s-c). Substituting the values into the formula gives Area = √7(7-4)(7-4)(7-6) = √7 × 3 × 3 × 1. This simplifies to √7 × 9, which equals 3√7 square units.

Alternatively, since this is an isosceles triangle, we can find the height h by dropping a perpendicular from the vertex to the base of 6 units, which bisects the base into two segments of 3 units each. Using the Pythagorean theorem, h = √4² - 3² = √16 - 9 = √7. The area is then 1/2 × base × height = 1/2 × 6 × √7 = 3√7 square units. Thus, option a is the correct result.

Explanation

In the right-angled triangle ABP, we use the tangent ratio for ∠BAP = 45° to find the length of BP. Since tan(∠BAP) = BP/AB, we have tan 45° = BP/10, which gives 1 = BP/10, so BP = 10 cm. It is given that P is the midpoint of BC, meaning BP = PC = 10 cm. Therefore, the total length of the base BC = BP + PC = 10 + 10 = 20 cm.

Now, considering the larger right-angled triangle ABC, the tangent of the angle ∠BAC is tan(∠BAC) = BC/AB = 20/10 = 2. We are given that ∠BAC is the sum of ∠BAP and ∠CAP, so ∠BAC = 45° + θ. Applying the provided identity tan(α + β) = (tan α + tan β)/(1 - tan α tan β), we substitute α = 45° and β = θ:

tan(45° + θ) = (tan 45° + tan θ)/(1 - tan 45° tan θ) = 2

Since tan 45° = 1, the equation becomes (1 + tan θ)/(1 - tan θ) = 2. Cross-multiplying gives 1 + tan θ = 2(1 - tan θ), which simplifies to 1 + tan θ = 2 - 2 tan θ. Combining like terms, we get 3 tan θ = 1, which results in tan θ = 1/3. This matches option b.

Explanation

In triangle ABP, since the angle at B is 90° and ∠BAP = 45°, we use the tangent formula tan(∠BAP) = BP/AB. Substituting the given values, we have tan(45°) = BP/10. Since tan(45°) = 1, it follows that BP = 10 cm.

The problem states that P is the midpoint of BC, meaning BP = PC. Therefore, PC = 10 cm, and the total length of the base BC = BP + PC = 10 + 10 = 20 cm.

Now, consider the larger right-angled triangle ABC. The angle γ is ∠ACP, which is the same as ∠ACB because P lies on the segment BC. Using the tangent formula for triangle ABC, we get tan γ = tan(∠ACB) = AB/BC. Substituting the known lengths, tan γ = 10/20 = 1/2. This corresponds to option a.

Explanation

Statement I is correct because a line segment drawn from a vertex to the midpoint of the opposite side, known as a median, always divides the triangle into two triangles of equal area. In this case, P is the midpoint of the base BC, which makes AP the median of triangle ABC. Statement III is correct as the area of triangle APC is calculated using the formula 1/2 × base × height. Given the dimensions where the base PC = 10 cm and the perpendicular height from A to BC is 10 cm, the area is 1/2 × 10 × 10 = 50 square cm. Statement II is correct because the perimeter of triangle APC is the sum of the lengths of sides AP, PC, and AC. Using the Pythagorean theorem, AP = √(10² + 10²) ≈ 14.14 cm, PC = 10 cm, and AC = √(20² + 10²) ≈ 22.36 cm. Adding these values, the total perimeter is 10 + 14.14 + 22.36 = 46.5 cm, which is greater than 46 cm. Thus, statements I, II, and III are all correct, making option d the right choice.

Explanation

To calculate the median of a grouped frequency distribution with inclusive class intervals, we must first convert them into exclusive boundaries. Since the gap between the upper limit of one class and the lower limit of the next is 1 (e.g., 26 to 27), we subtract 0.5 from the lower limits and add 0.5 to the upper limits. The true class boundaries are 17.5-26.5, 26.5-35.5, 35.5-44.5, 44.5-53.5, 53.5-62.5, 62.5-71.5, and 71.5-80.5.

Next, we calculate the cumulative frequencies (cf): 5, 12, 22, 37, 45, 48, and 50. The total number of observations N is 50. The median position is N/2 = 50/2 = 25. The cumulative frequency just exceeding 25 is 37, which corresponds to the median class 44.5-53.5.

We use the median formula: Median = L + ( (N/2 - cf)/f ) × h

Where: L = 44.5 (lower boundary of the median class) N/2 = 25 cf = 22 (cumulative frequency of the preceding class) f = 15 (frequency of the median class) h = 9 (class width, 53.5 - 44.5)

Substituting the values: Median = 44.5 + ( (25 - 22)/15 ) × 9 Median = 44.5 + ( 3/15 ) × 9 Median = 44.5 + 0.2 × 9 Median = 44.5 + 1.8 = 46.3

Thus, the median of the distribution is 46.3, which corresponds to option d.

Explanation

To find the mode of a grouped frequency distribution, we identify the modal class, which is the class with the highest frequency. In the given data, the highest frequency is 15, corresponding to the class 45-53. Since the class intervals are inclusive (discrete), we must first convert them into continuous boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. Thus, the modal class is 44.5-53.5.

We use the mode formula: Mode = L + ( (f₁ - f₀) / ((f₁ - f₀) + (f₁ - f₂)) ) × h.

Here, L is the lower boundary of the modal class, which is 44.5. The frequency of the modal class f₁ is 15, the frequency of the preceding class f₀ is 10, and the frequency of the succeeding class f₂ is 8. The class width h is 53.5 - 44.5 = 9.

Substituting the values into the formula: Mode = 44.5 + ( (15 - 10) / ((15 - 10) + (15 - 8)) ) × 9 Mode = 44.5 + ( 5 / (5 + 7) ) × 9 Mode = 44.5 + ( 5/12 ) × 9 Mode = 44.5 + 45/12 Mode = 44.5 + 3.75 = 48.25

Therefore, the mode of the distribution is 48.25, which corresponds to option c.

Explanation

In a right-angled triangle ABC right-angled at B, let the length of side AB be x cm. Given that AC - AB = 2 cm, the length of the hypotenuse AC is (x + 2) cm. The length of the perpendicular BC is given as 16 cm. Applying the Pythagoras theorem, AB² + BC² = AC², we substitute the known values: x² + 16² = (x + 2)². Expanding the equation gives x² + 256 = x² + 4x + 4. Subtracting x² from both sides, we get 4x = 252, which simplifies to x = 63. Therefore, AB = 63 cm and AC = 63 + 2 = 65 cm. For ∠BAC = θ, the trigonometric ratios are defined as sin θ = BC/AC and cos θ = AB/AC. Substituting the values, sin θ = 16/65 and cos θ = 63/65. The required value is sin θ + cos θ = 16/65 + 63/65 = (16 + 63)/65 = 79/65, which corresponds to option d.

Explanation

In right-angled triangle ABC with the right angle at B, the Pythagoras theorem states AC² = AB² + BC². Let AC = x cm. According to the problem, AB = x - 2 cm and BC = 16 cm. Substituting these values into the formula, we get x² = (x - 2)² + 16². Expanding this gives x² = x² - 4x + 4 + 256, which simplifies to 4x = 260, resulting in x = 65. Therefore, the hypotenuse AC = 65 cm and the side AB = 63 cm.

The area of triangle ABC can be calculated in two ways. First, using the legs AB and BC, Area = 1/2 × AB × BC = 1/2 × 63 × 16. Second, using the hypotenuse AC and the altitude BD, Area = 1/2 × AC × BD = 1/2 × 65 × BD. Equating these two expressions, we have 65 × BD = 63 × 16. Calculating the product 63 × 16 = 1008, we find BD = 1008/65 cm, which matches option (a).

Explanation

In a circle, a perpendicular drawn from the center to a chord bisects the chord. Given that OP intersects the chord MN perpendicularly at Q, Q is the midpoint of MN. Therefore, the length of MQ is half of MN, which is MQ = 16/2 = 8 cm.

We consider the right-angled triangle ΔOQM, where OM is the radius of the circle and OQ is the distance from the center to the chord. Using the Pythagoras Theorem: OM² = OQ² + MQ² Substituting the given values: 10² = OQ² + 8² 100 = OQ² + 64 OQ² = 100 - 64 OQ² = 36 OQ = √36 = 6 cm.

Thus, the length of OQ is 6 cm, which corresponds to option (b).

Explanation

Given the chord MN = 16 cm and radius OM = 10 cm, the perpendicular from the center O to the chord MN bisects it at Q. Therefore, MQ = 1/2 MN = 16/2 = 8 cm. In the right-angled triangle OQM, we use the Pythagorean theorem to find OQ: OQ = √(OM² - MQ²) = √(10² - 8²) = √(100 - 64) = √36 = 6 cm.

Since MP is a tangent at M, OM ⊥ MP, making ΔOMP a right-angled triangle. MQ is the altitude to the hypotenuse OP. In such a configuration, ΔOQM is similar to ΔMPQ because ∠OMQ = ∠MPQ (both are complementary to ∠MOQ).

Using the property of similar triangles, the ratio of their corresponding sides is equal: MP/OM = MQ/OQ

Substituting the known values: MP = (OM × MQ)/OQ = (10 × 8)/6 = 80/6 = 40/3 cm.

This corresponds to option (c).

Explanation

In triangle OMN, the sides OM and ON are radii of the circle, so OM = ON = 10 cm. The chord MN serves as the base of the triangle with a length of 16 cm. Since OP is the perpendicular bisector of the chord MN at point Q, it serves as the altitude of triangle OMN. To find the height OQ, we use the Pythagorean theorem in the right-angled triangle OQM, where MQ is half of the chord MN, meaning MQ = 16/2 = 8 cm.

The height OQ is calculated as follows: OQ = √(OM² - MQ²) = √(10² - 8²) = √(100 - 64) = √36 = 6 cm.

Now, using the standard formula for the area of a triangle: Area = 1/2 × base × height = 1/2 × MN × OQ Area = 1/2 × 16 × 6 = 8 × 6 = 48 square cm.

Thus, the area of triangle OMN is 48 square cm, which corresponds to option d.

Explanation

For a quadratic expression ax² + bx + c to be positive for all real values of x, two conditions must be satisfied: the coefficient of x² must be positive (a > 0) and the discriminant must be negative (D < 0). In the expression 4x² - kx + 1, we have a = 4, which is already positive. Using the discriminant formula D = b² - 4ac, we get D = (-k)² - 4(4)(1) = k² - 16. Setting D < 0 gives k² - 16 < 0, which simplifies to k² < 16, implying -4 < k < 4. The possible integral values for k are {-3, -2, -1, 0, 1, 2, 3}.

Evaluating Statement-I, the condition k < -2 combined with -4 < k < 4 results in -4 < k < -2. The only integral value in this range is k = -3. Thus, Statement-I alone is sufficient to answer the question. Evaluating Statement-II, the condition k > -4 combined with -4 < k < 4 results in -4 < k < 4. This range contains multiple integral values {-3, -2, -1, 0, 1, 2, 3}, meaning a unique integral value cannot be determined. Since only Statement-I provides a unique answer, option a is the correct choice.

Explanation

To find the number of days A, B, and C together take to finish the work, we need to determine their combined efficiency, which is the sum of their individual work rates per day. Let the work rates of A, B, and C be represented by a, b, and c respectively.

From Statement I, we have the combined rate of A and B: a + b = 1/24

From Statement II, we have the combined rate of B and C: b + c = 1/36

To find the total time taken by A, B, and C together, we need to calculate the sum a + b + c. If we add the two given equations, we get: (a + b) + (b + c) = 1/24 + 1/36 a + 2b + c = (3 + 2)/72 = 5/72

This equation gives the value of a + 2b + c, but it does not provide the value of a + b + c unless the individual work rate of B is known. Since there are three variables and only two independent equations, the system is underdetermined. Therefore, the value of a + b + c cannot be uniquely determined. Thus, the question cannot be answered even by using both the statements together, making option d correct.

Explanation

To determine if a common solution exists and is prime, we must solve both quadratic equations. For Statement-I, the equation is x² - 26x + 133 = 0. We look for two numbers that multiply to 133 and sum to 26. These numbers are 7 and 19, as 7 × 19 = 133 and 7 + 19 = 26. Thus, the solutions for Statement-I are x = 7 and x = 19. Both are prime numbers.

For Statement-II, the equation is x² - 44x + 475 = 0. We look for two numbers that multiply to 475 and sum to 44. These numbers are 19 and 25, as 19 × 25 = 475 and 19 + 25 = 44. Thus, the solutions for Statement-II are x = 19 and x = 25.

Comparing the two sets of solutions, {7, 19} and {19, 25}, the common solution is x = 19. Since 19 is a prime number, the answer to the question is yes. Because a common solution cannot be identified without knowing the roots of both equations, neither statement alone is sufficient. Both statements together are required to identify the common root and verify its primality, making option c the correct choice.

Explanation

To determine the divisibility of the expression 327ⁿ + 173ⁿ by 500, we apply the algebraic identity that aⁿ + bⁿ is divisible by a + b if and only if n is an odd positive integer. Here, a = 327 and b = 173. The sum of the bases is a + b = 327 + 173 = 500.

Using Statement-I, we are told n is an odd natural number. Since n is odd, the identity aⁿ + bⁿ being divisible by a + b holds true. Therefore, 327ⁿ + 173ⁿ is definitely divisible by 500. Statement-I alone is sufficient to answer the question with a definitive Yes.

Using Statement-II, we are told n is a positive integer. This means n can be either even or odd. While the expression is divisible by 500 when n is odd, it is not generally divisible when n is even. For example, if n = 2, the expression is 327² + 173². Modulo 500, 327 ≡ -173 (mod 500), so 327² + 173² ≡ (-173)² + 173² ≡ 2 × 173² (mod 500). Since 173² = 29929, then 2 × 29929 = 59858, which leaves a remainder of 358 when divided by 500. Thus, the expression is not divisible by 500 for even n. Statement-II alone is insufficient.

Consequently, the question can be answered using Statement-I alone but not using Statement-II alone, making option a the correct choice.

Explanation

To find the percentage reduction in consumption when the price increases and the total expenditure remains constant, we use the standard formula: Percentage reduction = r/(100 + r) × 100, where r is the percentage increase in price. Substituting the given value r = 20, we get 20/(100 + 20) × 100 = 20/120 × 100 = 1/6 × 100 = 16.67%. Since this formula depends only on the percentage change in price and not on the absolute initial price or absolute initial consumption, Statement I (Price = Rs. 90) and Statement II (Consumption = 24 litres) are completely irrelevant. The answer can be determined using only the data provided in the question stem. Therefore, option d is the correct choice as the question can be answered even without using any of the statements.

Explanation

To find the expenditure of P, we use the fundamental relation Salary = Expenditure + Savings. From the question, the salaries of P and Q can be represented as 6x and 5x respectively, where x is a constant multiplier.

Statement I provides the ratio of savings as 3:2, which can be represented as 3y and 2y. Statement II provides the ratio of expenditures as 1:1, which means both P and Q have the same expenditure, say E. Combining all the information, we get two equations: 6x = E + 3y and 5x = E + 2y. Subtracting the second equation from the first gives x = y. Substituting this back into the second equation, we find 5x = E + 2x, which simplifies to E = 3x.

While we have established a relationship between expenditure and the salary multiplier x, no absolute numerical value for salary or savings is provided in either statement. Since x remains an unknown variable, the actual numerical value of P's expenditure E cannot be determined. Thus, the question cannot be answered even by using both the statements together, making option d the correct choice.

Explanation

Let the five different integers be x₁, x₂, x₃, x₄, x₅ such that 2 = x₁ < x₂ < x₃ < x₄ < x₅ = 8. The sum of these integers is S = 2 + x₂ + x₃ + x₄ + 8 = 10 + (x₂ + x₃ + x₄). Since the integers are different and lie between 2 and 8, the available values for x₂, x₃, x₄ are {3, 4, 5, 6, 7}.

The minimum possible sum of these three middle integers is 3 + 4 + 5 = 12, and the maximum possible sum is 5 + 6 + 7 = 18. Consequently, the total sum S must fall in the range [10 + 12, 10 + 18], which is [22, 28].

Using Statement I, we know S is a multiple of 5. In the range 22 to 28, the only multiple of 5 is 25. Thus, S = 25 and the average is 25 / 5 = 5. Statement I alone is sufficient.

Using Statement II, we know the number of odd integers is odd. Since 2 and 8 are even, either one or three of the remaining integers must be odd. For example, the set {2, 4, 6, 3, 8} has one odd integer and a sum of 23, while {2, 3, 5, 7, 8} has three odd integers and a sum of 25. Since different sets yield different averages, Statement II alone is insufficient.

Therefore, option a is correct.

Explanation

Let the three distinct integer weights be x, y, z. The given condition is x + y + z = P, where P is a prime number.

Using Statement I alone, we have y = 2x. The sum becomes 3x + z = P. Many sets of integers satisfy this, such as {1, 2, 4} where 1+2+4=7 and {1, 2, 10} where 1+2+10=13. Since no unique set is determined, Statement I is insufficient.

Using Statement II alone, we have y = 3x. The sum becomes 4x + z = P. Multiple sets satisfy this, such as {1, 3, 7} where 1+3+7=11 and {1, 3, 9} where 1+3+9=13. Since no unique set is determined, Statement II is insufficient.

Combining both statements, we have y = 2x and another weight must be three times one of the others. If z = 3x, the sum is x + 2x + 3x = 6x, which is composite for any integer x ≥ 1. If z = 3y = 6x, the sum is x + 2x + 6x = 9x, which is composite. However, if y = 3z, then 2x = 3z. For x and z to be integers, let z = 2k and x = 3k. The weights are 3k, 6k, 2k. Their sum is 3k + 6k + 2k = 11k. For 11k to be a prime number, k must be 1, yielding the unique set {2, 3, 6}. Thus, both statements together are necessary.

Explanation

To calculate the final amount A at the end of a given period, the formula used depends on whether the interest is calculated as Simple Interest (SI) or Compound Interest (CI). The formula for Simple Interest amount is A = P(1 + rt), and the formula for Compound Interest amount is A = P(1 + r)ᵗ, where P is the principal, r is the rate of interest, and t is the time period.

In this problem, the question provides the time t = 10 years. Statement I provides the principal P = 1,00,000 and Statement II provides the rate r = 10%. Even when both statements are used together, the type of interest (Simple or Compound) is not specified. Since the final amount would be 2,00,000 under Simple Interest and approximately 2,59,374 under Compound Interest, a unique answer cannot be determined. Therefore, the question cannot be answered even by using both the statements together, making option d the correct choice.

Explanation

The parity of the expression p² + pq + q² depends on the parities of p and q. We can evaluate the statements based on the properties of even and odd integers.

For Statement I, if p + q is even, then p and q must have the same parity. This leads to two cases: either both p and q are even, or both are odd. If both are even, p² + pq + q² = even + even + even = even. If both are odd, p² + pq + q² = odd + odd + odd = odd. Since we get two different possible results, Statement I alone is not sufficient to determine if the expression is odd.

For Statement II, if pq is odd, it is a mathematical necessity that both p and q must be odd integers. Substituting this into the expression, we get p² + pq + q² = odd + odd + odd = odd. This provides a definitive yes to the question.

Since Statement II alone is sufficient and Statement I alone is not, option a is the correct choice.

Explanation

Total Population = 10,000. Since males and females are equal in number, the Total Males = 5,000 and Total Females = 5,000.

First, calculate the total number of newspaper readers using the percentage provided: Total Newspaper Readers = 30% of 10,000 = 0.30 × 10,000 = 3,000.

Next, find the number of English newspaper readers: English Newspaper Readers = 1/3 × 3,000 = 1,000.

Now, determine the distribution of these English newspaper readers by gender: Female English Newspaper Readers = 20% of 1,000 = 0.20 × 1,000 = 200. Male English Newspaper Readers = Total English Readers - Female English Readers = 1,000 - 200 = 800.

To find the number of males who do not read English newspapers, subtract the male English readers from the total male population: Males who do not read English Newspaper = Total Males - Male English Readers = 5,000 - 800 = 4,200.

Thus, the final answer is 4,200, which corresponds to option (c).

Explanation

For a rectangle with fixed perimeter P = 400 cm, the area is maximised when the rectangle is a square. Side = 400 ÷ 4 = 100 cm. Maximum area = 100 × 100 = 10,000 sq cm.

Explanation

3^1 = 3, 3^2 = 2, 3^3 = 6 = -1 mod 7. Then 3^10 = (3^3)^3 3^1 = (-1)^3 3 = -3 = 4 mod 7.

Explanation

To find the square root of a percentage, first convert the percentage into its decimal or fractional form. Since 64% is equivalent to 64/100 or 0.64, we apply the square root operation to this value. Using the mathematical property √(a/b) = √a/√b, the calculation becomes √(64/100) = √64/√100. Calculating the individual square roots gives 8/10, which simplifies to the decimal 0.8. To express this result as a percentage, multiply the decimal by 100, resulting in 0.8 × 100 = 80%. Therefore, the square root of 64% is 80%, which corresponds to option d.

Explanation

The difference is (10^31 - 5) - (10^30 + p) = 10^31 - 10^30 - p - 5. Modulo 3, 10 is 1. The expression becomes 1^31 - 1^30 - p - 5 = 1 - 1 - p - 5 = -p - 5. For this to be divisible by 3, -p - 5 = 0 mod 3, which means p + 5 = 0 mod 3, or p + 2 = 0 mod 3, which implies p = 1, 4, or 7. There are 3 possible values for p.

Explanation

For Statement I, we use modular arithmetic to check if 107¹⁰⁰ - 76¹⁰⁰ ≡ 0 (mod 61). We observe that 107 can be expressed as 2 × 61 - 15, which implies 107 ≡ -15 (mod 61). Similarly, 76 can be expressed as 1 × 61 + 15, which implies 76 ≡ 15 (mod 61). Substituting these values into the expression, we get (-15)¹⁰⁰ - 15¹⁰⁰ (mod 61). Since the exponent 100 is an even number, (-15)¹⁰⁰ = 15¹⁰⁰, resulting in 15¹⁰⁰ - 15¹⁰⁰ = 0 (mod 61). Thus, 61 divides the expression, and Statement I is correct.

For Statement II, we apply the algebraic identity that aⁿ + bⁿ is divisible by a + b whenever n is an odd positive integer. In the given expression 67⁵ + 33⁵, we have a = 67, b = 33, and n = 5. Since n = 5 is an odd number, the expression is divisible by 67 + 33, which equals 100. Therefore, 100 divides 67⁵ + 33⁵, and Statement II is correct. As both statements are correct, option (c) is the correct answer.

Explanation

To find the maximum possible temperature for one day, we use the formula for the total sum: Total Sum = Average × Number of days. Given the average is 31°C for 7 days (Monday to Sunday), the Total Sum = 31 × 7 = 217°C. To maximize the temperature of one particular day, we must minimize the temperatures of the remaining six days. Since the lowest temperature recorded is 30°C, the minimum possible sum for the other six days is 6 × 30 = 180°C. The maximum temperature for the remaining day is calculated as: Maximum Temperature = Total Sum - Sum of other six days = 217 - 180 = 37°C. Thus, the maximum possible temperature recorded on any one of the days is 37°C, which corresponds to option d.

Explanation

The given equation is in the form A - B = B - C, which simplifies to A + C = 2B. Substituting the expressions provided in the question, we obtain (x + 1/yz) + (z + 1/xy) = 2(y + 1/zx). Rearranging the terms to separate linear variables and fractions, we get x + z + 1/yz + 1/xy = 2y + 2/zx. Combining the fractions on the left side using the common denominator xyz gives x + z + (x + z)/xyz = 2y + 2/zx. Further rearranging the equation leads to (x + z) - 2y = 2/zx - (x + z)/xyz. By simplifying the right side, we get (x + z - 2y) = (2y - (x + z))/xyz, which is equivalent to (x + z - 2y) = -(x + z - 2y)/xyz. Since the problem states x + z ≠ 2y, the term (x + z - 2y) is non-zero and can be divided from both sides. This leaves 1 = -1/xyz, which results in xyz = -1. Thus, option (b) is the correct result.

Explanation

The expression p = n(n+1)(n+2)(n+3) + 1 involves the product of four consecutive natural numbers. For statement I, in any set of four consecutive integers, there are always two even numbers (one of which must be a multiple of 4). The product of an even number and any other integer is always even. Therefore, n(n+1)(n+2)(n+3) is always even. Adding 1 to any even number results in an odd number, making p always odd.

For statement II, we apply the algebraic identity for the product of four consecutive integers plus one. By regrouping the terms, we get p = [n(n+3)][(n+1)(n+2)] + 1, which simplifies to p = (n² + 3n)(n² + 3n + 2) + 1. Let x = n² + 3n. The expression becomes p = x(x + 2) + 1 = x² + 2x + 1. This is a standard quadratic identity equal to (x + 1)². Substituting the value of x back, we find p = (n² + 3n + 1)². Since n is a natural number, n² + 3n + 1 is an integer, confirming that p is always a perfect square. Thus, both statements I and II are correct, making option c the right choice.

Explanation

The average of the first n even natural numbers is given by the formula n+1. For n=50, the average of the first 50 even natural numbers is 50+1 = 51. The average of the first n odd natural numbers is given by the formula n. For n=50, the average of the first 50 odd natural numbers is 50. The difference between these two averages is 51 - 50 = 1. Thus, the difference is 1, which matches option c. This result can be intuitively understood by comparing the two sets: the first 50 even numbers are 2, 4, 6, ..., 100 and the first 50 odd numbers are 1, 3, 5, ..., 99. Since each term in the even sequence is exactly 1 greater than the corresponding term in the odd sequence, the overall average of the even sequence must be exactly 1 greater than the average of the odd sequence.

Explanation

The formula for compound interest (CI) is given by CI = P[(1 + r/100)ⁿ - 1], where P is the principal, r is the rate of interest per annum, and n is the time period in years. Let the constant value [(1 + r/100)ⁿ - 1] be represented by k since the rate and time are the same for both scenarios.

According to the problem, y is the compound interest on x, which gives the equation y = x · k. Similarly, z is the compound interest on y, which gives the equation z = y · k.

From the first equation, we can express k as k = y/x. From the second equation, we can express k as k = z/y. Since the value of k is the same in both cases, we equate the two expressions: y/x = z/y. Cross-multiplying these terms yields y² = zx. This relationship shows that x, y, and z form a geometric progression, and the correct option is (b).

Explanation

For the innermost square, the area is A₁ = 1, so the side length is s₁ = 1 and the diagonal is D₁ = √(1² + 1²) = √2. Let Dₙ be the diagonal of the n-th square. The distance between corresponding corners of consecutive squares is given as 1 unit. Since the squares are concentric, the distance from the center to a corner of the n-th square is Dₙ / 2. The difference between these distances for consecutive squares is (Dₙ / 2) - (Dₙ₋₁ / 2) = 1, which simplifies to Dₙ - Dₙ₋₁ = 2. This forms an arithmetic progression for the diagonals with first term D₁ = √2 and common difference d = 2. Using the formula Dₙ = a + (n-1)d, we get Dₙ = √2 + (n-1)2 = 2n + √2 - 2. Thus, statement I is correct.

To evaluate statement II, we use the area formula for a square in terms of its diagonal, A = D²/2. The area between the n-th and (n-1)-th square is ΔA = Aₙ - Aₙ₋₁ = (Dₙ² - Dₙ₋₁²)/2. Applying the difference of squares formula, ΔA = ((Dₙ - Dₙ₋₁)(Dₙ + Dₙ₋₁))/2. Substituting Dₙ - Dₙ₋₁ = 2, we find ΔA = Dₙ + Dₙ₋₁. Substituting the values for Dₙ and Dₙ₋₁, we get ΔA = (2n + √2 - 2) + (2(n-1) + √2 - 2) = 4n + 2√2 - 6. Since this expression depends on n, the area is not independent of n. Thus, statement II is incorrect. Therefore, only statement I is correct, making option a the right choice.

Explanation

Let the length AB = L, width AD = W, and diagonal AC = D. According to the properties of a rectangle, the relationship between these is given by the Pythagorean theorem: D² = L² + W². From the problem, we have two equations: D + L = 3W and D - W = 4. From the second equation, we can express the diagonal as D = W + 4. Substituting this into the first equation, we get L = 3W - (W + 4) = 2W - 4.

Now, substitute the expressions for D and L into the Pythagorean formula: (W + 4)² = (2W - 4)² + W² W² + 8W + 16 = 4W² - 16W + 16 + W² W² + 8W + 16 = 5W² - 16W + 16 4W² - 24W = 0 4W(W - 6) = 0

Since the width W cannot be zero, we have W = 6 units. Consequently, the length L = 2(6) - 4 = 8 units and the diagonal D = 6 + 4 = 10 units. The area of the triangle formed by the diagonal (either ΔABC or ΔADC) is half the area of the rectangle: Area = 1/2 × L × W = 1/2 × 8 × 6 = 24 square units. Thus, option a is correct.

Explanation

Let the radius of each of the three identical smaller circles be r. When three identical circles touch each other, their centers form an equilateral triangle with side length a = 2r. The radius of the larger circumscribing circle, R, is equal to the distance from the centroid of this equilateral triangle to the center of one of the smaller circles, plus the radius of that smaller circle.

The distance from the centroid to the vertex of an equilateral triangle is given by the formula Rₜᵣᵢ = a / √3. Substituting a = 2r, we get Rₜᵣᵢ = 2r / √3. Therefore, the radius of the circumscribing circle is R = 2r / √3 + r, which can be simplified to R = r(2 + √3) / √3.

The area of the larger circle is given as πR² = π(2 + √3)² / 3. Equating the two expressions for the area: π[r(2 + √3) / √3]² = π(2 + √3)² / 3 r²(2 + √3)² / 3 = (2 + √3)² / 3

Dividing both sides by (2 + √3)² / 3, we find r² = 1, which gives r = 1 cm. Thus, option b is correct.

Explanation

First, find the area of triangle ABC using Heron formula. The semi-perimeter s is calculated as (21 + 20 + 13) / 2 = 27 cm. The area of triangle ABC is √s(s-a)(s-b)(s-c) = √27(27-20)(27-13)(27-21) = √27 × 7 × 14 × 6 = √15876 = 126 square cm.

Since the perpendicular CD is drawn on the longest side AB, the area of triangle ABC can also be expressed as 1/2 × AB × CD. Substituting the known values, 126 = 1/2 × 21 × CD, which gives CD = (126 × 2) / 21 = 12 cm.

Now, consider triangle BCD, which is a right-angled triangle at D. Using the Pythagoras theorem, BD² = BC² - CD² = 20² - 12² = 400 - 144 = 256. Thus, BD = √256 = 16 cm. The area of triangle BCD is 1/2 × BD × CD = 1/2 × 16 × 12 = 96 square cm. Therefore, option a is correct.

Explanation

Using the weighted average formula for concentrations, let Cₐ be the concentration of milk in container A, Cᵦ be the concentration of milk in container B, and C_F be the concentration of milk in the final mixture.

The concentration of milk in A is Cₐ = 1/(1+3) = 1/4. The concentration of milk in the final mixture is C_F = 3/(3+7) = 3/10. The containers are mixed in the ratio nₐ : nᵦ = 2 : 3.

Applying the formula C_F = (nₐ Cₐ + nᵦ Cᵦ)/(nₐ + nᵦ): 3/10 = (2(1/4) + 3(Cᵦ))/(2+3) 3/10 = (1/2 + 3 Cᵦ)/5

Multiplying both sides by 5: 3/2 = 1/2 + 3 Cᵦ 3 Cᵦ = 3/2 - 1/2 = 1 Cᵦ = 1/3

Since Cᵦ represents the ratio of milk to the total volume in container B, we have: m/(m+n) = 1/3 Cross multiplying gives 3m = m + n, which simplifies to 2m = n. Thus, the value of m/n is 1/2, which corresponds to option a.

Explanation

The volumes of the cone, hemisphere, and cylinder with equal radius r and same height h (where h = r for the hemisphere) are V_cone = (1/3)πr^3, V_hemi = (2/3)πr^3, and V_cyl = πr^3. Their sum is V_sum = (1/3 + 2/3 + 1)πr^3 = 2πr^3. For a sphere of radius R, V_sphere = (4/3)πR^3. Setting 2πr^3 = (4/3)πR^3 gives R^3/r^3 = 6/4 = 1.5. However, 1.5 is not an option. If the hemisphere is replaced by a full sphere of radius r, the sum becomes V_sum = (1/3 + 4/3 + 1)πr^3 = (8/3)πr^3. Then (4/3)πR^3 = (8/3)πr^3 implies R^3/r^3 = 2, which matches option C. Given the provided answer key, it is likely the question intended a sphere instead of a hemisphere.

Explanation

Let f(x) = x³ + px² + qx + r. For f(x) to be an integer for all integral values of x, we test specific integers. For x = 0, f(0) = r. Since f(0) must be an integer, r must be an integer, which confirms statement III. For x = 1, f(1) = 1 + p + q + r, and since f(1) and r are integers, p + q must be an integer. For x = -1, f(-1) = -1 + p - q + r, and since f(-1) and r are integers, p - q must be an integer. Adding these two results gives (p + q) + (p - q) = 2p, implying 2p must be an integer, meaning p can be a half-integer. Subtracting them gives (p + q) - (p - q) = 2q, implying 2q must be an integer, meaning q can be a half-integer. For example, if we let p = 1/2 and q = 1/2, the expression becomes f(x) = x³ + 1/2x² + 1/2x + r = x³ + x(x+1)/2 + r. Since the product of two consecutive integers x(x+1) is always even, the term x(x+1)/2 is always an integer. Thus, f(x) is an integer for all x even when p and q are not integers. Therefore, statements I and II are incorrect, and only statement III is correct, making option b the right choice.

Explanation

To determine the expenditure of P, we apply the basic financial formula: Salary = Expenditure + Savings. Let the salaries of P and Q be 6x and 5x respectively. From Statement I, we can denote the savings of P and Q as 3y and 2y. From Statement II, the expenditures of P and Q are equal, which we can denote as z. This gives us two linear equations: 6x = z + 3y and 5x = z + 2y. Subtracting the second equation from the first, we get (6x - 5x) = (z + 3y) - (z + 2y), which simplifies to x = y. Substituting y = x back into the second equation, we find 5x = z + 2x, resulting in z = 3x. Although we have established that P's expenditure is 3x, we lack any absolute numerical value for salary, savings, or expenditure in either statement. Since all given information consists of ratios, we can only determine the relative expenditure, not the actual amount. Therefore, the question cannot be answered even by using both statements together, making option d the correct choice.

Explanation

The fundamental formula used here is Income = Expenditure + Savings. Let the salaries of P and Q be 6x and 5x respectively. According to Statement I, the savings of P and Q are in the ratio 3:2, so let them be 3y and 2y. According to Statement II, the expenditure of P and Q are equal, so let them be z. This gives us two equations: 6x = z + 3y and 5x = z + 2y. Subtracting the second equation from the first, we get x = y. Substituting this back into the first equation, 6x = z + 3x, which simplifies to z = 3x. Although we have established that P's expenditure is 3x, the value of x is unknown because no absolute monetary value for salary, savings, or expenditure is provided in either statement. Since we only have ratios and no actual figures, the absolute amount of P's expenditure cannot be determined. Therefore, the question cannot be answered even by using both the Statements together, which corresponds to option d.

Explanation

The HCF of numbers of the form (a^m - 1) and (a^n - 1) is given by the formula a^gcd(m, n) - 1. Here, a = 2, m = 36, and n = 45. The gcd of 36 and 45 is 9. Therefore, the HCF is 2^9 - 1. Since 2^9 = 512, the HCF is 512 - 1 = 511.

Explanation

The axial cross-section (through vertex, perpendicular to base) is an equilateral triangle of side 14 cm, so the diameter 2r = 14 ⇒ r = 7 cm and slant height l = 14 cm. Cone height: h = √(l² - r²) = √(196 - 49) = √147 = 7√3 cm. Volume = 1/3π r² h = 1/3 × 22/7 × 49 × 7√3 = (22 × 49√3)/3 = 1078√3/3 = 1078/√3 cubic cm.

Explanation

The centers of the bases of the three identical cones form an equilateral triangle because each cone touches the other two. Since the radius of each base is r = 3 cm, the distance between any two centers is equal to the sum of their radii, making the side length of the equilateral triangle s = 3 + 3 = 6 cm. The vertices of the cones are located directly above the centers of their bases. Since the cones are identical and placed on the same plane, their vertices also form an equilateral triangle with the same side length s = 6 cm. The circle that passes through these vertices is the circumcircle of this equilateral triangle. The formula for the circumradius R of an equilateral triangle with side s is R = s/√3. Substituting s = 6 cm, we get R = 6/√3 = 2√3 cm. The area of the circle is calculated using the formula A = πR². Thus, A = π(2√3)² = π(4 × 3) = 12π square cm. This corresponds to option d.

Explanation

For a right-angled triangle, the radius of the inscribed circle (inradius) can be calculated directly using the formula r = (a + b - c)/2, where a and b are the lengths of the two legs and c is the length of the hypotenuse.

First, we find the length of the hypotenuse AC using the Pythagorean theorem: AC = √(AB² + BC²) AC = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.

Now, substituting the values AB = 5 cm, BC = 12 cm, and AC = 13 cm into the inradius formula: r = (5 + 12 - 13)/2 r = (17 - 13)/2 r = 4/2 = 2 cm.

Thus, the radius of the inscribed circle is 2 cm, which corresponds to option (c).

Explanation

Sum of interior angles of a regular polygon with n sides is given by the formula (n-2) × 180°, and the sum of exterior angles for any convex polygon is always 360°. According to the given condition, the ratio is ((n-2) × 180°)/360° = 7/2. Simplifying the left side, we get (n-2)/2 = 7/2, which implies n-2 = 7, hence n = 9. The polygon is a nonagon. The measure of a single interior angle of a regular polygon is calculated using the formula ((n-2) × 180°)/n. Substituting n = 9, the interior angle is ((9-2) × 180°)/9 = (7 × 180°)/9 = 7 × 20° = 140°. Thus, option (d) is correct.

Explanation

The equation is m^2 - n^2 = 199, which can be written as (m - n)(m + n) = 199. Since 199 is a prime number and m, n are natural numbers with m > n, the only possible factors are 1 and 199. Thus, m - n = 1 and m + n = 199. Adding these equations gives 2m = 200, so m = 100. Substituting m = 100 into m - n = 1 gives n = 99. The value of mn is 100 * 99 = 9900.

Explanation

Numbers of the form 2^n - 1 are prime only if n is prime (Mersenne primes). We check n for 2^n - 1 < 2000: For n=2, 2^2-1 = 3 (Prime) For n=3, 2^3-1 = 7 (Prime) For n=5, 2^5-1 = 31 (Prime) For n=7, 2^7-1 = 127 (Prime) For n=11, 2^11-1 = 2047 (Exceeds 2000). Checking non-prime n values: n=1 (1 is not prime), n=4 (15), n=6 (63), n=8 (255), n=9 (511 = 773), n=10 (1023 = 3341). The primes are 3, 7, 31, 127. Total count is 4.

Explanation

Let x be the number of students who opt for all three languages. Let e, h, and s be the students who opt for only English, only Hindi, and only Sanskrit respectively. Total students: e + h + s + x = 160. English: e + x = 130. Hindi: h + x = 120. Sanskrit: s + x = 110. Summing the individual language totals: (e + x) + (h + x) + (s + x) = 130 + 120 + 110 = 360. (e + h + s + x) + 2x = 360. Substituting the total students: 160 + 2x = 360. 2x = 200, so x = 100.

Explanation

S = 5^a + 7^b + 11^c + 13^d. We look at remainders modulo 10: 5^a mod 10 is always 5 for a >= 1. 11^c mod 10 is always 1 for c >= 1. 7^b mod 10 follows the cycle {7, 9, 3, 1}. 13^d mod 10 follows the cycle {3, 9, 7, 1}. S mod 10 = (5 + 1 + 7^b + 13^d) mod 10 = (6 + 7^b + 13^d) mod 10. Possible values for (7^b + 13^d) mod 10: Using sets {7, 9, 3, 1} and {3, 9, 7, 1}, the possible sums are: 1+1=2, 1+3=4, 1+7=8, 1+9=0, 3+3=6, 3+7=0, 3+9=2, 7+7=4, 7+9=6, 9+9=8. The distinct sums are {0, 2, 4, 6, 8}. Adding 6 to these sums mod 10: 6+0=6, 6+2=8, 6+4=0, 6+6=2, 6+8=4. The distinct remainders are {0, 2, 4, 6, 8}, which are 5 in number.

Explanation

In triangle ABC, ∠A = 90° and AD is the altitude drawn from the vertex A to the hypotenuse BC. In the right triangle ADC, we are given that ∠ADC = 90° and ∠CAD = 60°. According to the angle sum property of a triangle, ∠ACD = 180° - (90° + 60°) = 30°. Since ∠ACD is the same as ∠C of the larger triangle ABC, we have ∠C = 30°. To find the length of side AB, we apply the trigonometric ratio for sin C in the right triangle ABC. The formula is sin C = Opposite/Hypotenuse = AB/BC. Substituting the given value BC = 6 cm and the calculated ∠C = 30°, we get sin 30° = AB/6. Using the standard value sin 30° = 1/2, the equation becomes 1/2 = AB/6, which simplifies to AB = 6 × 1/2 = 3 cm. Therefore, option a is the correct answer.