CDS II 2017 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

Let the common factor be (x - r). Then r^2 - 5r + alpha = 0 and r^2 - 7r + 2alpha = 0. Subtracting the two equations gives 2r - alpha = 0, so alpha = 2r. Substituting alpha = 2r into the first equation: r^2 - 5r + 2r = 0, which means r^2 - 3r = 0. Since alpha is non-zero, r must be 3. Thus, alpha = 2(3) = 6.

Explanation

A number divisible by 2, 3, 4, and 5 must be divisible by their LCM. LCM(2, 3, 4, 5) = 60. The number of such integers between 1 and 1000 is floor(1000/60) = 16.

Explanation

The fastest way to find the factors of the cubic polynomial f(x) = x³ + 4x² - 11x - 30 is by using Vieta's formulas or the Factor Theorem. According to Vieta's formulas for a cubic equation ax³ + bx² + cx + d = 0, the sum of the roots is given by -b/a and the product of the roots is given by -d/a. In this polynomial, a = 1, b = 4, c = -11, and d = -30. Therefore, the sum of the roots must be -4/1 = -4 and the product of the roots must be -(-30)/1 = 30. For option (c), the roots associated with the factors (x + 2), (x - 3) and (x + 5) are -2, 3, and -5. Checking these values: the sum is -2 + 3 - 5 = -4 and the product is (-2)(3)(-5) = 30. Since both conditions are satisfied, these are the correct factors. Alternatively, applying the Factor Theorem, we can verify that f(-2) = (-2)³ + 4(-2)² - 11(-2) - 30 = -8 + 16 + 22 - 30 = 0, f(3) = (3)³ + 4(3)² - 11(3) - 30 = 27 + 36 - 33 - 30 = 0, and f(-5) = (-5)³ + 4(-5)² - 11(-5) - 30 = -125 + 100 + 55 - 30 = 0. All three values result in zero, confirming that (x + 2), (x - 3) and (x + 5) are the factors.

Explanation

x = (10^20 - 1)/9, y = (10^10 - 1)/3, z = 2(10^10 - 1)/9. y^2 = (10^20 - 2*10^10 + 1)/9. x - y^2 = (10^20 - 1 - 10^20 + 210^10 - 1)/9 = (210^10 - 2)/9 = 2(10^10 - 1)/9. (x - y^2)/z = [2(10^10 - 1)/9] / [2(10^10 - 1)/9] = 1.

Explanation

Let the roots be 3k and 2k. Sum of roots: 5k = -m/12. Product of roots: 6k^2 = 5/12. k^2 = 5/72, so k = sqrt(5/72). m = -60k. For positive m, k = -sqrt(5/72). m = 60 sqrt(5/72) = 60 sqrt(5)/(6sqrt(2)) = 10sqrt(5)/sqrt(2) = 5*sqrt(10).

Explanation

The degree of the product of two polynomials is the sum of their degrees. Degree(f(x)g(x)) = 3 + 4 = 7.

Explanation

According to the Remainder Theorem, when a polynomial P(x) is divided by a linear factor (x - a), the remainder is equal to P(a). In this problem, the polynomial is P(x) = 5x³ + 5x² - 6x + 9 and the divisor is (x + 3). To find the remainder, we set the divisor to zero: x + 3 = 0, which gives x = -3.

Substitute x = -3 into the polynomial P(x): P(-3) = 5(-3)³ + 5(-3)² - 6(-3) + 9 P(-3) = 5(-27) + 5(9) + 18 + 9 P(-3) = -135 + 45 + 18 + 9 P(-3) = -135 + 72 P(-3) = -63

The calculation shows that the remainder is -63, which corresponds to option (d). This method is the fastest way to find the remainder without performing long division of polynomials.

Explanation

Product of two expressions = HCF * LCM. LCM = Product / HCF = [(x + y + z)p^3] / p^2 = (x + y + z)p.

Explanation

The distance between two points x₁ and x₂ on a number line is calculated using the absolute difference formula d = |x₁ - x₂|. In this problem, the points are given as P = 0.83 and Q = 0.62. Substituting these values into the formula, we get d = |0.83 - 0.62|. Subtracting the smaller value from the larger value, we find 0.83 - 0.62 = 0.21. To convert this decimal value into a fraction, we place the number over its corresponding power of ten, which gives 21/100. Comparing this result with the provided options, it matches option (c) exactly. Therefore, the distance between the points P and Q on the number line is 21/100.

Explanation

To find the actual deduction for successive discounts, the fastest method is to use the multiplying factor approach. For discounts of d₁, d₂, and d₃, the single equivalent discount is given by the formula: Total Discount = [1 - (1 - d₁)(1 - d₂)(1 - d₃)] × 100%

Given values are d₁ = 20% = 0.20, d₂ = 12.5% = 0.125, and d₃ = 5% = 0.05. Substituting these into the formula: Remaining Price Factor = (1 - 0.20) × (1 - 0.125) × (1 - 0.05) = 0.8 × 0.875 × 0.95

Calculating the first two terms: 0.8 × 0.875 = 4/5 × 7/8 = 7/10 = 0.7

Now, multiply by the third term: 0.7 × 0.95 = 0.665

The final price is 66.5% of the original price. Therefore, the total actual deduction is: Actual Deduction = (1 - 0.665) × 100% = 0.335 × 100% = 33.5%

This matches option (a).

Explanation

Let the total number of mangoes be X. Remaining after 5% rotten = 0.95X. Remaining after selling 75% of remainder = 25% of 0.95X = 0.25 * 0.95X = 95. 0.2375X = 95, so X = 95 / 0.2375 = 400.

Explanation

To find the time taken to cross 91 km-stones, we first determine the length of the train using the formula Distance = Speed × Time. The speed of the train is 60 km/hr, which is converted to meters per second as 60 × 5/18 = 50/3 m/s. Since the train crosses a single km-stone in 12 seconds, its length L is 50/3 × 12 = 200 meters, or 0.2 km.

To cross 91 km-stones completely, the train must cover the distance between the first and the last stone plus its own length. The distance between 91 km-stones is (91 - 1) × 1 km = 90 km. Therefore, the total distance D to be covered is 90 km + 0.2 km = 90.2 km. Using the formula Time = Distance/Speed, the total time T is 90.2/60 hours. This simplifies to 1 hour + 30.2/60 hours. Converting the fractional part to minutes, 30.2/60 × 60 = 30.2 minutes, which equals 30 minutes and 0.2 × 60 = 12 seconds. Thus, the total time is 1 hr 30 min 12 sec, which corresponds to option (b).

Explanation

First, convert A's speed from km/hr to m/s using the formula Speed in m/s = Speed in km/hr × 5/18. For A, speed = 6 × 5/18 = 5/3 m/s. The time taken by A to complete the 100 m race is Tₐ = Distance/Speed = 100/(5/3) = 300/5 = 60 seconds.

Since A beats B by 9 seconds, B takes 60 + 9 = 69 seconds to finish his race. Because A gave B a start of 8 m, B only needs to cover a distance of 100 - 8 = 92 m. Using the formula Speed = Distance/Time, the speed of B in m/s is Vᵦ = 92/69. Simplifying this fraction by dividing both numerator and denominator by 23, we get Vᵦ = 4/3 m/s.

To convert B's speed back to km/hr, multiply by 18/5. Thus, B's speed = 4/3 × 18/5 = (4 × 6)/5 = 24/5 = 4.8 km/hr. This matches option b.

Explanation

8x^3 - y^3 = (2x - y)((2x)^2 + (2x)(y) + y^2) = (2x - y)(4x^2 + 2xy + y^2). Dividing by (4x^2 + 2xy + y^2), the quotient is (2x - y).

Explanation

Since (x + 2) is a factor, x = -2 is a root for both equations. (-2)^2 + a(-2) + b = 0 => 4 - 2a + b = 0 => 2a - b = 4. (-2)^2 + b(-2) + a = 0 => 4 - 2b + a = 0 => a - 2b = -4. Solving these: 2a - b = 4 and a - 2b = -4. Subtracting: a + b = 8. Adding: 3a = 12 => a = 4. Then b = 4. Ratio a:b = 4:4 = 1.

Explanation

The Remainder Theorem is the most efficient tool to solve this problem. According to this theorem, if a polynomial f(x) is divided by a linear divisor of the form (ax - b), the remainder is equal to the value of the polynomial evaluated at the root of the divisor. To find the root of the divisor, we set the divisor expression to zero: ax - b = 0. Solving for x gives ax = b, which implies x = b/a. Substituting this value back into the polynomial function, the remainder R is given by R = f(b/a). This confirms that option (a) is the correct answer. For any polynomial division where the divisor is a first-degree expression, finding the value of x that nullifies the divisor and substituting it into the dividend always yields the remainder directly without requiring the full long division process.

Explanation

A perfect square number must always end in one of the following digits: 0, 1, 4, 5, 6, or 9. Any integer ending in 2, 3, 7, or 8 can never be a perfect square. This property allows for the rapid elimination of most options in competitive exams.

Applying this unit digit rule to the given numbers: The number 2222 ends in 2, so it cannot be a perfect square. The number 343343 ends in 3, so it cannot be a perfect square. The number 220347 ends in 7, so it cannot be a perfect square.

For 11664, the unit digit is 4, which satisfies the necessary condition. We verify this by calculating the square root: √11664 = 108. Since 108² = (100 + 8)² = 10000 + 1600 + 64 = 11664, it is a perfect square.

Since numbers 1, 3, and 4 fail the unit digit test, they are not perfect squares, which corresponds to option (d).

Explanation

To find the product of the polynomials, we apply algebraic factorization and the identity (a - b)(a + b) = a² - b². First, we observe the two cubic polynomials and factor them by grouping terms. For the polynomial (x³ - 2x² + 4x - 8), we group them as x²(x - 2) + 4(x - 2), which simplifies to (x² + 4)(x - 2). Similarly, for the polynomial (x³ + 2x² + 4x + 8), we group them as x²(x + 2) + 4(x + 2), which simplifies to (x² + 4)(x + 2).

Now, we multiply all four expressions together: Product = (x + 2)(x - 2) × [(x² + 4)(x - 2)] × [(x² + 4)(x + 2)]

Rearranging the factors to group like terms: Product = [(x + 2)(x - 2)] × [(x + 2)(x - 2)] × (x² + 4) × (x² + 4) Product = [(x + 2)(x - 2)]² × (x² + 4)²

Using the identity (a - b)(a + b) = a² - b², we know (x + 2)(x - 2) = x² - 4. Substituting this back: Product = (x² - 4)² × (x² + 4)² Product = [(x² - 4)(x² + 4)]²

Applying the identity (a - b)(a + b) = a² - b² again for the base inside the square, where a = x² and b = 4: Product = (x⁴ - 4²)² = (x⁴ - 16)²

Thus, the product is (x⁴ - 16)², which matches option b.

Explanation

To factorize the expression x(x + 2)(x + 3)(x + 5) - 72, we group the terms such that the sum of the constants in each pair is equal. Pairing x with (x + 5) and (x + 2) with (x + 3) gives 0 + 5 = 5 and 2 + 3 = 5.

The expression becomes [x(x + 5)][(x + 2)(x + 3)] - 72, which simplifies to (x² + 5x)(x² + 5x + 6) - 72.

Let y = x² + 5x. Substituting this into the expression, we get: y(y + 6) - 72 y² + 6y - 72

Now, factorize the quadratic y² + 6y - 72 by finding two numbers that multiply to -72 and add to 6. These numbers are 12 and -6. y² + 12y - 6y - 72 y(y + 12) - 6(y + 12) (y - 6)(y + 12)

Substitute y = x² + 5x back into the factors: (x² + 5x - 6)(x² + 5x + 12)

The quadratic (x² + 5x - 6) can be further factorized into (x + 6)(x - 1) since 6 · (-1) = -6 and 6 + (-1) = 5.

Therefore, the final factors are (x - 1), (x + 6), and (x² + 5x + 12), which matches option (c).

Explanation

The HCF of the polynomials is given as x² + x - 2. Factoring this quadratic expression, we get x² + 2x - x - 2 = x(x + 2) - 1(x + 2) = (x - 1)(x + 2). Since this HCF must divide both f(x) and g(x), both (x - 1) and (x + 2) must be factors of f(x) and g(x).

For f(x) = (x - 1)(x² + 3x + a), the factor (x - 1) is already present. Therefore, (x + 2) must be a factor of the quadratic part x² + 3x + a. Applying the Factor Theorem, substituting x = -2 into the quadratic part must yield zero: (-2)² + 3(-2) + a = 0 4 - 6 + a = 0 -2 + a = 0 ⇒ a = 2.

For g(x) = (x + 2)(x² + 2x + b), the factor (x + 2) is already present. Therefore, (x - 1) must be a factor of the quadratic part x² + 2x + b. Applying the Factor Theorem, substituting x = 1 into the quadratic part must yield zero: (1)² + 2(1) + b = 0 1 + 2 + b = 0 3 + b = 0 ⇒ b = -3.

Thus, the values of a and b are 2 and -3 respectively, which corresponds to option (b).

Explanation

Given that ab divides cd, we can express this relationship using the formula cd = k(ab) for some integer k. This equation implies that a is a divisor of the product cd, written as a ∣ cd. According to Euclid's Lemma, if an integer a divides the product of two integers c and d, and the greatest common divisor of a and c is 1 (meaning they are coprime), then a must divide d. Since the problem explicitly states that a and c are coprime, we have gcd(a, c) = 1. By applying this lemma to the condition a ∣ cd, it follows logically that a ∣ d. Therefore, a is a factor of d, which corresponds to option c.

Explanation

The sum of the coefficients of the equation a(b-c)x^2 + b(c-a)x + c(a-b) = 0 is a(b-c) + b(c-a) + c(a-b) = ab - ac + bc - ba + ca - cb = 0. Therefore, x = 1 is always a root. Since the roots are equal, both roots must be 1. Product of roots = c(a-b) / [a(b-c)] = 1. ca - cb = ab - ac => 2ac = ab + bc. Dividing by abc: 2/b = 1/c + 1/a.

Explanation

Rearrange the given equation by grouping terms with similar denominators on each side. Move the term (b - x²)/cx to the left side and (b - x)/c to the right side:

(a - x²)/bx + (b - x²)/cx = (c - x)/b + (b - x)/c

Now, find a common denominator for both sides. On the left hand side, the common denominator is bcx, and on the right hand side, it is bc:

(c(a - x²) + b(b - x²))/bcx = (c(c - x) + b(b - x))/bc

Simplify the numerators:

(ac - cx² + b² - bx²)/bcx = (c² - cx + b² - bx)/bc

(b² + ac - (b + c)x²)/bcx = (b² + c² - (b + c)x)/bc

Since b ≠ 0 and c ≠ 0, we can cancel the common factor bc from the denominators of both sides:

(b² + ac - (b + c)x²)/x = b² + c² - (b + c)x

Multiply both sides by x to eliminate the fraction:

b² + ac - (b + c)x² = (b² + c²)x - (b + c)x²

The term (b + c)x² appears on both sides of the equation and cancels out:

b² + ac = (b² + c²)x

Solving for x, we get:

x = (b² + ac)/(b² + c²)

This matches option a.

Explanation

For a quadratic equation ax² + bx + c = 0 to have integral roots, the discriminant D = b² - 4ac must be a perfect square of an integer. In the given equation x² + 7x - 14(k² - 7/8) = 0, the coefficients are a = 1, b = 7, and c = -14(k² - 7/8).

Calculating the discriminant D: D = b² - 4ac D = 7² - 4(1)(-14(k² - 7/8)) D = 49 + 56(k² - 7/8) D = 49 + 56k² - (56 × 7/8) D = 49 + 56k² - 49 D = 56k²

For the roots to be integers, D must be a perfect square. We can write D = 4 × 14 × k² = (2k)² × 14. For this expression to be a perfect square, 14 must be a perfect square, which is not possible. The only other possibility is if k = 0, which makes D = 0. If k = 0, the roots are x = -b/2a = -7/2 = -3.5, which is not an integer.

Since the discriminant D = 56k² is never a perfect square for any integer k that would result in an integer root, the roots will always be irrational for k ≠ 0 and non-integral for k = 0. Thus, the equation has no integral root, making option c the correct choice.

Explanation

The first number after 500 divisible by 13 is 507 (13 * 39). The last number before 1000 divisible by 13 is 988 (13 * 76). Number of terms = 76 - 39 + 1 = 38.

Explanation

This problem can be solved efficiently using the Chain Rule formula for work and cost: (M₁ · D₁)/W₁ = (M₂ · D₂)/W₂, where M represents the number of animals, D represents the number of days, and W represents the total cost.

Substituting the given values into the formula: M₁ = 8 cows, D₁ = 60 days, W₁ = 6400 rupees M₂ = 5 cows, D₂ = n days, W₂ = 4800 rupees

(8 · 60)/6400 = (5 · n)/4800

Simplifying the left side: 480/6400 = 5n/4800 3/40 = 5n/4800

Now, solving for n: 5n = (3 · 4800)/40 5n = 3 · 120 5n = 360 n = 360/5 n = 72

Therefore, the value of n is 72 days, which corresponds to option d.

Explanation

Pass marks = 45 + 5 = 50. Since 40% of maximum marks = 50, maximum marks = 50 / 0.4 = 125.

Explanation

To find the value of u, we start by simplifying the given equations. The first equation 3(2u + v) = 7uv expands to 6u + 3v = 7uv. The second equation 3(u + 3v) = 11uv expands to 3u + 9v = 11uv. To eliminate the uv term and find the relationship between u and v, we divide the first equation by the second: (6u + 3v)/(3u + 9v) = 7uv/11uv = 7/11. Cross-multiplying this ratio gives 11(6u + 3v) = 7(3u + 9v), which simplifies to 66u + 33v = 21u + 63v. Rearranging the terms by subtracting 21u and 33v from both sides results in 45u = 30v, which simplifies to v = 45u/30 = 3u/2. Now, we substitute v = 3u/2 back into the first simplified equation 6u + 3v = 7uv to get 6u + 3(3u/2) = 7u(3u/2). Multiplying the entire equation by 2 to remove the fractions gives 12u + 9u = 21u², which simplifies to 21u = 21u². Dividing both sides by 21u for the non-trivial solution yields u = 1, which corresponds to option d.

Explanation

Let the present age of Ram be R and the present age of Shyam be S. Based on the first condition, five years ago, Ram was three times as old as Shyam. This can be written as: R - 5 = 3(S - 5) R - 5 = 3S - 15 R = 3S - 10 (Equation 1)

Based on the second condition, four years from now, Ram will be twice as old as Shyam. This can be written as: R + 4 = 2(S + 4) R + 4 = 2S + 8 R = 2S + 4 (Equation 2)

To find the values, equate the two expressions for R: 3S - 10 = 2S + 4 3S - 2S = 4 + 10 S = 14

Now, substitute the value of S = 14 into Equation 2 to calculate the present age of Ram: R = 2(14) + 4 R = 28 + 4 = 32 years.

Therefore, the present age of Ram is 32 years, which matches option (b).

Explanation

Let the total cost price of 4 chairs be C and the total cost price of 9 stools be S. According to the problem, the total expenditure is C + S = 1340. The total profit is the sum of the individual profits on chairs and stools, given by the equation 0.10C + 0.20S = 188. To solve this efficiently, multiply the profit equation by 10 to eliminate decimals: C + 2S = 1880. Now, subtract the total cost equation from this modified profit equation to find S: (C + 2S) - (C + S) = 1880 - 1340, which simplifies to S = 540. Substitute the value of S back into the total cost equation: C + 540 = 1340. Solving for C gives C = 1340 - 540 = 800. Thus, the amount paid for the chairs is ₹ 800, which matches option c.

Explanation

The given expression {x : x + 5 = 5} is written in set-builder notation, which defines a set by stating the properties that its members must satisfy. To find the elements of this set, we must solve the conditional equation x + 5 = 5. By applying basic algebraic subtraction, we subtract 5 from both sides of the equation: x = 5 - 5, which simplifies to x = 0. Since 0 is the only value that satisfies the condition, the resulting set contains exactly one element, which is 0. In mathematical set notation, a set containing the element 0 is written as {0}. Option (a) is incorrect because the set is not empty. Option (c) is incorrect because a set is a collection of objects and cannot be equal to a scalar numerical value like 0. Option (d) is incorrect because {ϕ} represents a singleton set containing the empty set as its only element, which is fundamentally different from a set containing the number 0. Thus, option (b) is the correct statement.

Explanation

Given ab + bc + ca = 0, then -bc = ab + ac. a^2 - bc = a^2 + ab + ac = a(a + b + c). Similarly, b^2 - ca = b(a + b + c) and c^2 - ab = c(a + b + c). The expression becomes a^2/[a(a+b+c)] + b^2/[b(a+b+c)] + c^2/[c(a+b+c)] = a/(a+b+c) + b/(a+b+c) + c/(a+b+c) = (a+b+c)/(a+b+c) = 1.

Explanation

Percentage of students who failed in at least one subject = 35% + 45% - 20% = 60%. Percentage of students passing both subjects = 100% - 60% = 40%.

Explanation

To solve this expression, we use a standard algebraic identity. Let us substitute the terms as follows: let a = x - y, b = y - z, and c = z - x. First, we check the sum of these variables: a + b + c = (x - y) + (y - z) + (z - x) = 0.

According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc. This is derived from the general identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), where the right side becomes zero when a + b + c = 0.

Substituting our values into the given expression, the numerator is abc and the denominator is a³ + b³ + c³. Since a + b + c = 0, the denominator simplifies to 3abc. Thus, the expression becomes abc/3abc. By cancelling the common term abc, we get 1/3. This matches option (b).

Explanation

Let the given expression be x = √(1 + √(1 + √(1 + ...))). Since the series is infinite, the part under the first square root is also equal to x. Therefore, the equation can be written as x = √(1 + x).

Squaring both sides of the equation, we get x² = 1 + x, which rearranges to the quadratic equation x² - x - 1 = 0. Using the quadratic formula x = (-b ± √(b² - 4ac))/2a where a = 1, b = -1, and c = -1, we have:

x = (-(-1) ± √((-1)² - 4(1)(-1)))/2(1) = (1 ± √(1 + 4))/2 = (1 ± √5)/2

Since the value of the square root expression must be positive, we discard the negative root. Thus, x = (1 + √5)/2. Substituting the approximate value of √5 ≈ 2.236, we get x ≈ (1 + 2.236)/2 = 3.236/2 = 1.618. This value, known as the Golden Ratio, clearly lies between 1 and 2, making option c the correct choice.

Explanation

log 8000 = log(8 * 10^3) = log 8 + 3 = 0.9031 + 3 = 3.9031. log 600 = log(6 * 10^2) = log 6 + 2 = 0.7782 + 2 = 2.7782. Sum = 3.9031 + 2.7782 = 6.6813.

Explanation

Total work = 30 * 40 = 1200 man-days. Work completed in 24 days = 30 * 24 = 720 man-days. Remaining work = 1200 - 720 = 480 man-days. Number of men who stayed = 480 / 40 = 12 men. Number of men who left = 30 - 12 = 18.

Explanation

Since 4 goats or 6 sheep can graze the field in 50 days, their grazing capacities are equivalent. Using the relation 4G = 6S, where G is the grazing rate of one goat and S is the grazing rate of one sheep, we can simplify this to 2G = 3S. This means the work done by 2 goats is equal to the work done by 3 sheep.

To find the time taken by 2 goats and 3 sheep, we calculate their combined grazing rate R = 2G + 3S. Substituting 3S with 2G from our initial relation, we get R = 2G + 2G = 4G.

Since the total work required to graze the field is the amount of work 4 goats can do in 50 days, the total work W = 4G × 50. The time T required for the combined group is T = W/R = (4G × 50)/4G = 50 days. Thus, option d is correct.

Explanation

Let the total capacity of the tub be 1 unit. The filling rate of the tap is R_fill = 1/10 unit per hour. After the outlet was closed, the tap took 7 hours to fill the remaining part of the tub. The amount of water filled in these 7 hours is 7 × 1/10 = 7/10 units. This implies that at the moment the outlet was closed, the tub was filled up to 1 - 7/10 = 3/10 units. This 3/10 units of water was filled during the first 5 hours while both the tap and the outlet were open.

The net rate of filling during the first 5 hours is Rₙₑₜ = Work done/Time = (3/10)/5 = 3/50 unit per hour. Using the relation Rₙₑₜ = R_fill - Rₗₑₐₖ, we can find the rate of the outlet leak: Rₗₑₐₖ = R_fill - Rₙₑₜ = 1/10 - 3/50 = (5 - 3)/50 = 2/50 = 1/25 unit per hour. The time taken by the outlet to empty the full tub is 1/Rₗₑₐₖ = 1/(1/25) = 25 hours. Thus, option b is correct.

Explanation

To find the total distance walked, we first calculate the average speed for the entire journey. Since the distance to school and back is the same, we use the harmonic mean formula for average speed: Average Speed = 2v₁v₂/(v₁ + v₂). Substituting the given speeds v₁ = 12 km/hr and v₂ = 8 km/hr, we get Average Speed = (2 × 12 × 8)/(12 + 8) = 192/20 = 9.6 km/hr.

Next, we convert the total time from minutes to hours to maintain consistency in units. Total time = 50 minutes = 50/60 hours = 5/6 hours.

The total distance walked is the product of the average speed and the total time: Total Distance = Average Speed × Total Time = 9.6 × 5/6. Simplifying this, we get (9.6/6) × 5 = 1.6 × 5 = 8 km. Therefore, the total distance covered by the boy is 8 km, which corresponds to option (b).

Explanation

To find the parts, first simplify the given ratio 1 : 1/3 : 1/6 by multiplying each term by the least common multiple (LCM) of the denominators, which is 6. This transforms the ratio into 6 : 2 : 1. The sum of these ratio parts is 6 + 2 + 1 = 9. Using the formula for calculating a specific part of a total, the middle part is determined by (ratio of middle part/sum of ratios) × total value. Substituting the given values into the formula, we get (2/9) × 78. To simplify this calculation, divide both 78 and 9 by their greatest common divisor, which is 3, resulting in (2 × 26)/3, which equals 52/3. This matches option (c).

Explanation

Ratio 1st:2nd = 2:3 = 8:12. Ratio 2nd:3rd = 4:5 = 12:15. Ratio 1st:2nd:3rd = 8:12:15. Total units = 8 + 12 + 15 = 35. 35 units = 350, so 1 unit = 10. Boys in 1st and 3rd = (8 + 15) 10 = 23 10 = 230.

Explanation

For a period of 2 years, the difference between compound interest (compounded annually) and simple interest can be calculated using the direct formula D = P(R/100)², where D is the difference, P is the principal sum, and R is the annual rate of interest.

Substituting the given values into the formula, we have D = 15 and R = 5. 15 = P(5/100)² 15 = P(1/20)² 15 = P/400

To find the principal P, we rearrange the equation: P = 15 × 400 P = 6000

Therefore, the sum of money deposited is ₹ 6,000, which corresponds to option a. Using this formula is the fastest way to solve such problems in competitive exams as it avoids calculating SI and CI separately.

Explanation

To find the percentage reduction in consumption when the price increases and the expenditure remains constant, we use the standard formula for consumption reduction: Percentage Reduction = r/(100 + r) × 100, where r is the percentage increase in price.

Given that the price rises by r = 12%, we substitute this value into the formula: Percentage Reduction = 12/(100 + 12) × 100 Percentage Reduction = 12/112 × 100

First, simplify the fraction 12/112 by dividing both the numerator and denominator by 4: (12 ÷ 4)/(112 ÷ 4) = 3/28

Now, multiply by 100: 3/28 × 100 = 300/28

Further simplify by dividing both by 4: (300 ÷ 4)/(28 ÷ 4) = 75/7

Converting the improper fraction 75/7 into a mixed fraction: 75 ÷ 7 = 10 with a remainder of 5. Therefore, the reduction is 10 5/7%, which corresponds to option d.

Explanation

Downstream speed = 18/4 = 4.5 km/hr. Upstream speed = 18/10 = 1.8 km/hr. Still water speed = (4.5 + 1.8)/2 = 3.15 km/hr. Stream speed = (4.5 - 1.8)/2 = 1.35 km/hr. All three statements are correct.

Explanation

Sides 5, 12, 13 form a right triangle. For an acute angle theta, sin theta = 5/13 and cos theta = 12/13 (or vice versa). sin theta + cos theta = 5/13 + 12/13 = 17/13.

Explanation

To find the range of the expression sin x + cos x for 0 < x < π/2, we use the auxiliary angle formula a sin θ + b cos θ = √(a² + b²) sin(θ + α), where tan α = b/a. Here, a = 1 and b = 1, so √(a² + b²) = √(1² + 1²) = √2 and tan α = 1, which gives α = π/4. The expression can be rewritten as √2 sin(x + π/4). Given the interval 0 < x < π/2, the argument of the sine function ranges from π/4 < x + π/4 < 3π/4. In this interval, the maximum value of sin(x + π/4) is 1, occurring at x = π/4. Therefore, the maximum value of the expression is √2 × 1 = √2. Since √2 ≈ 1.414, which is strictly less than 2, the value of (sin x + cos x) is always less than 2. This confirms that option (b) is correct.

Explanation

cos^2 theta = 1 - sin^2 theta = 1 - [(m^2 - n^2)/(m^2 + n^2)]^2 = [(m^2 + n^2)^2 - (m^2 - n^2)^2] / (m^2 + n^2)^2 = [4m^2n^2] / (m^2 + n^2)^2. cos theta = 2mn / (m^2 + n^2).

Explanation

According to the Sine Rule in trigonometry, the ratio of a side of a triangle to the sine of its opposite angle is equal to twice the circum-radius of the triangle. The formula is given by a/sin(A) = 2R, where a is the length of the side opposite to angle A, and R is the circum-radius.

In this problem, the side opposite to angle A is BC, the angle A = 30°, and the circum-radius R = 10 cm. Substituting these values into the formula, we get:

BC = 2R sin(A) BC = 2 × 10 × sin(30°)

Since the value of sin(30°) = 1/2, the calculation becomes:

BC = 20 × (1/2) BC = 10 cm

Thus, the length of side BC is 10 cm, which corresponds to option b.

Explanation

A = (1/sqrt(2) - 1/2) / (1/sqrt(2) + 1/2) = (2 - sqrt(2)) / (2 + sqrt(2)) = (2 - sqrt(2))^2 / 2 = (6 - 4sqrt(2))/2 = 3 - 2sqrt(2). B = (sqrt(2) - 1) / (sqrt(2) + 1) = (sqrt(2) - 1)^2 / 1 = 2 + 1 - 2sqrt(2) = 3 - 2sqrt(2). A = B.

Explanation

To find the angle θ between the hour hand and the minute hand of a clock, we use the standard formula: θ = |30H - 5.5M|, where H is the hour and M is the minutes. Given the time 4:36 pm, we substitute H = 4 and M = 36 into the formula.

θ = |30(4) - 5.5(36)| θ = |120 - 198| θ = |-78| = 78°

To convert the angle from degrees to radians, we multiply by π/180: θ = 78 × π/180 = 13π/30 radians.

Now, we compare 13π/30 with the ranges given in the options. Converting the boundaries of option B to a common denominator of 30: 2π/5 = 12π/30 and 3π/5 = 18π/30.

Since 12π/30 < 13π/30 < 18π/30, it is evident that 2π/5 < θ < 3π/5. Therefore, option B is the correct range.

Explanation

For statement 1, we use the identity sec²θ + csc²θ = 1/cos²θ + 1/sin²θ = (sin²θ + cos²θ)/(sin²θ cos²θ) = 1/(sin²θ cos²θ). Using the double angle formula sin(2θ) = 2sinθ cosθ, this becomes 4/sin²(2θ). Given 45° < θ < 60°, the angle 2θ lies between 90° and 120°. In this interval, sin(2θ) decreases from 1 to √3/2, so sin²(2θ) ranges from 1 down to 3/4. Consequently, α² = 4/sin²(2θ) ranges from 4 to 16/3, which implies 2 ≤ α < 4/√3. Thus, α > 1 is true.

For statement 2, we apply the half angle identity (1 + cosθ)/(1 - cosθ) = cot²(θ/2). Let x² = cot²(θ/2), so x = cot(θ/2). For 0° < θ < 45°, the value of θ/2 is between 0° and 22.5°. Since the cotangent function is strictly decreasing in this range, x must be greater than cot(22.5°). Using the formula cot(22.5°) = √2 + 1 ≈ 2.414, we find that x > 2.414. Therefore, the condition x > 2 is true.

For statement 3, the expression simplifies as follows: cosθ/(1 - tanθ) + sinθ/(1 - cotθ) = cosθ/(1 - sinθ/cosθ) + sinθ/(1 - cosθ/sinθ) = cos²θ/(cosθ - sinθ) + sin²θ/(sinθ - cosθ) = (cos²θ - sin²θ)/(cosθ - sinθ) = cosθ + sinθ. The maximum value of cosθ + sinθ = √2 sin(θ + 45°) on the interval 0° < θ < 45° is √2 ≈ 1.414 at θ = 45°, and it approaches 1 as θ approaches 0°. Since the maximum value 1.414 is less than 2, the inequality cosθ + sinθ ≥ 2 is false.

Since only statements 1 and 2 are true, the number of true statements is two.

Explanation

Let the horizontal distance between the building and the tower be d, given by BB' = d. Let the height of the building AB = h and the height of the tower A'B' = h+x. Draw a horizontal line from the top of the building A to a point P on the tower A'B'. The distance AP = BB' = d, and the height PA' = (h+x) - h = x. Using the angle of elevation of A' from A: tan(45°) = PA'/AP ⇒ 1 = x/d ⇒ x = d.

For Statement I, we substitute x = d into the expression h + x. Since h + x = h + d and the height of the building h must be positive (h > 0), it follows that h + d > d. Thus, Statement I is true.

For Statement II, let θ be the angle of depression of B as seen from A'. This angle is the same as the angle of elevation of A' from B. In triangle A'B'B: tan(θ) = A'B'/BB' = (h+x)/d. Substituting x = d: tan(θ) = (h+d)/d = h/d + 1. Since h and d are positive, h/d > 0, which means tan(θ) > 1. Since tan(45°) = 1, it follows that θ > 45°. Therefore, Statement II is false.

Since Statement I is true and Statement II is false, option c is correct.

Explanation

Let the breadth of the river be w and let P be the point on the bank XY such that TP is perpendicular to the bank, where T is the tower. This distance TP = w represents the breadth of the river.

At point X, the bearing of the tower is N α° E, meaning the angle α is measured from the North direction toward the East. In the right-angled triangle XPT, we use the formula tan θ = Opposite/Adjacent. Thus, tan α = XP/TP = XP/w. Given tan α = 2, we find XP = 2w.

At point Y, the bearing of the tower is N β° W, meaning the angle β is measured from the North direction toward the West. In the right-angled triangle YPT, we have tan β = PY/TP = PY/w. Given tan β = 0.5, we find PY = 0.5w.

Since the man walks from X to Y along the bank, the total distance XY is the sum of XP and PY. XY = XP + PY 200 = 2w + 0.5w 200 = 2.5w w = 200/2.5 = 80 m.

The breadth of the river is 80m, which matches option (c).

Explanation

Assuming the question intends to compare sin(1 degree) and sin(1 radian). 1 radian is approximately 57.3 degrees. sin(1 degree) is approximately 0.0175, and sin(1 radian) is approximately 0.8415. The ratio sin(1 degree)/sin(1 radian) is less than 1.

Explanation

Let the radii of the two circles be r₁ and r₂. Since the ratio of their diameters is 12:5, the ratio of their radii is also r₁:r₂ = 12:5. We can let r₁ = 12x and r₂ = 5x.

The formula for the area of a circle is A = πr². The problem states that the sum of the areas of these two circles equals the area of a third circle with a diameter of 65 cm, meaning its radius r₃ = 65/2 = 32.5 cm.

Substituting the values into the area equation: π(12x)² + π(5x)² = π(32.5)²

Dividing both sides by π: 144x² + 25x² = (32.5)² 169x² = 1056.25 x² = 1056.25 / 169 x² = 6.25 x = √6.25 = 2.5

Now, we find the actual radii: r₁ = 12 × 2.5 = 30 cm r₂ = 5 × 2.5 = 12.5 cm

Thus, the radii are 30 cm and 12.5 cm, which corresponds to option d.

Explanation

A cube is a three dimensional solid figure that consists of 6 faces and 12 edges. To form a hollow cube by joining six identical squares, every edge of the cube must be taped to join the two adjacent square faces. Since a cube has exactly 12 edges, a total of 12 pieces of cello tape are required.

The area of a single rectangular piece of cello tape is calculated using the formula Area = Length × Breadth. Substituting the given values: Area of one tape = 4 cm × 0.5 cm = 2 sq cm.

To find the total area of the cello tape used for the entire cube, we multiply the area of one piece by the total number of edges: Total Area = 12 × 2 sq cm = 24 sq cm. Thus, the total area of the tape used is 24 square cm, which corresponds to option B.

Explanation

Let the radius of the smaller circle be r and the radius of the bigger circle be R. Given that the area of the bigger circle is 9 times the area of the smaller circle, we have πR² = 9πr², which simplifies to R = 3r.

The centers of both circles lie on the angle bisector of angle A. Let d₁ and d₂ be the distances from vertex A to the centers of the smaller and bigger circles respectively. Using the properties of tangents to a circle, we have sin(A/2) = r/d₁ and sin(A/2) = R/d₂. Since the two circles touch each other externally, the distance between their centers is d₂ - d₁ = R + r.

Substituting the expressions for d₁ and d₂, we get R/sin(A/2) - r/sin(A/2) = R + r. This simplifies to the general formula for two circles touching each other and two lines: sin(A/2) = (R - r)/(R + r). Substituting R = 3r into this formula:

sin(A/2) = (3r - r)/(3r + r) = 2r/4r = 1/2.

Since sin(30°) = 1/2, we have A/2 = 30°, which leads to A = 60°. Thus, option b is correct.

Explanation

Let the side of the square be s. The perimeter of the original square is 4s. Let the two equal sides of the isosceles triangle be a. The perimeter of the complete figure consists of three sides of the square and the two equal sides of the triangle, which is 3s + 2a. According to the given condition, 3s + 2a = 7/6 × 4s. Simplifying this, we get 3s + 2a = 14s/3. Solving for a, we have 2a = 14s/3 - 3s = 5s/3, which gives a = 5s/6.

To find the area of the triangle, we first determine its height h using the Pythagorean theorem: h = √(a² - (s/2)²). Substituting the value of a, we get h = √((5s/6)² - (s/2)²) = √(25s²/36 - s²/4) = √(25s² - 9s² / 36) = √(16s²/36) = 4s/6 = 2s/3.

The area of the triangle is 1/2 × base × height = 1/2 × s × 2s/3 = s²/3. The area of the square is s². Thus, the ratio of the area of the triangle to the area of the square is (s²/3)/s² = 1/3, which is 1:3. This matches option d.

Explanation

To find the area of a triangle when the lengths of all three sides are given, we use Heron's formula. Let the sides of the triangle be a = 51 cm, b = 37 cm, and c = 20 cm. First, we calculate the semi-perimeter s using the formula s = (a + b + c)/2. Substituting the values, we get s = (51 + 37 + 20)/2 = 108/2 = 54 cm.

Now, we apply Heron's formula for the area: Area = √s(s-a)(s-b)(s-c). Substituting the values of s, a, b, and c, we have Area = √54(54 - 51)(54 - 37)(54 - 20) = √54 × 3 × 17 × 34. To simplify the calculation, we break the numbers into their prime factors: 54 = 2 × 3 × 3 × 3, 3 = 3, 17 = 17, and 34 = 2 × 17. Thus, the expression becomes √(2 × 3³) × 3 × 17 × (2 × 17) = √2² × 3⁴ × 17². Taking the square root, we get 2 × 3² × 17 = 2 × 9 × 17 = 18 × 17 = 306 square cm. This corresponds to option c.

Explanation

The area of the part of the triangle PQR lying outside the circular region is found by subtracting the area of the circular sector inside the triangle from the total area of triangle PQR. Since QR is a tangent at Q, the radius PQ is perpendicular to QR, making triangle PQR a right-angled triangle at Q. Given that the radius PQ = r and the tangent length QR = r, the area of triangle PQR is calculated using the formula Area = 1/2 × base × height = 1/2 × r × r = r²/2.

Because PQ = QR, triangle PQR is an isosceles right triangle, which means the angle ∠QPR is 45°. The part of the triangle inside the circular region is a sector with a central angle of 45°. The area of this sector is given by the formula 45°/360° × πr² = 45°/360° × πr² = 1/8 πr². Therefore, the area outside the circular region is r²/2 - πr²/8, which corresponds to option (b).

Explanation

Area of triangle ABC = 1/2 AB AC = 1/2 c b. Also, Area = 1/2 BC AD = 1/2 a p. Therefore, bc = ap, which means p = bc/a.

Explanation

Let r be the radius of the circle and x be the perpendicular distance from the center of the circle to the longer chord CD. The length of the half-chord CD is 10/2 = 5 cm. According to the Pythagorean theorem, the relation between the radius, the distance from the center, and the half-chord is r² = x² + 5². The shorter chord AB = 4 cm is parallel to CD and situated 3 cm away. The length of the half-chord AB is 4/2 = 2 cm. If the chords are on the same side of the center, the distance from the center to AB is x + 3 cm. Thus, we have r² = (x+3)² + 2². Equating the two expressions for r²: x² + 25 = (x+3)² + 4 x² + 25 = x² + 6x + 9 + 4 25 = 6x + 13 12 = 6x x = 2 cm. Substituting the value of x back into the radius formula: r² = 2² + 25 = 4 + 25 = 29 r = √29 cm. Thus, the radius of the circle is √29 cm, matching option (c).

Explanation

In a cyclic quadrilateral ABCD where diagonals AC and BD intersect at P, the triangles ΔAPB and ΔDPC are similar. This similarity arises because ∠BAP = ∠CDP and ∠ABP = ∠DCP, as these angles are subtended by the same arcs BC and AD respectively. Additionally, ∠APB = ∠DPC due to vertically opposite angles.

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Applying this formula to ΔAPB and ΔDPC:

Area(ΔCPD)/Area(ΔAPB) = (CD/AB)²

Substituting the given values Area(ΔAPB) = 24 cm², AB = 8 cm, and CD = 5 cm into the equation:

Area(ΔCPD)/24 = (5/8)²

Area(ΔCPD) = 24 × 25/64

By simplifying the fraction (dividing both 24 and 64 by 8):

Area(ΔCPD) = 3 × 25/8 = 75/8 = 9.375 square cm

Thus, the area of triangle CPD is 9.375 square cm, which corresponds to option d.

Explanation

In an equilateral triangle, the altitude drawn from a vertex to the opposite side bisects that side. Let the side of the equilateral triangle ABC be a. Since BD is perpendicular to AC, D is the midpoint of AC. Therefore, the length of AD is given by the formula AD = a/2.

The altitude BD of an equilateral triangle is given by the formula BD = (√3/2)a. To find the value of BD², we square the expression for the altitude: BD² = ((√3/2)a)² = 3/4 a².

Now, consider the expression for AD² by squaring the length of AD: AD² = (a/2)² = a²/4. By comparing the two results, we can see that BD² = 3 × (a²/4) = 3 AD². Thus, the value of BD² is equal to 3AD², which corresponds to option c.

Explanation

To find the length of the direct common tangent of two circles, we use the standard formula L = √(d² - (r₁ - r₂)²), where d is the distance between the centers of the two circles, and r₁ and r₂ are the radii of the larger and smaller circles respectively. In this problem, the distance between the centers d is given as 13 cm, the radius of the larger circle r₁ is 9 cm, and the radius of the smaller circle r₂ is 4 cm. Substituting these values into the formula, we first calculate the difference between the radii as r₁ - r₂ = 9 - 4 = 5 cm. Then, the length of the tangent is L = √(13² - 5²). This simplifies to L = √(169 - 25), which equals √144. Taking the square root of 144, we get L = 12 cm. Therefore, option a) is the correct result.

Explanation

In a triangle PQR, if O is the orthocenter, then the points P, Q, R, and O form an orthocentric system. In such a system, any one of the four points is the orthocenter of the triangle formed by the other three. Thus, Q is the orthocenter of triangle OPR.

Explanation

In a right triangle ABC with angle C = 90 and altitude CD to the hypotenuse AB: Area = 1/2 BC CA = 1/2 AB CD. Therefore, BC CA = AB CD.

Explanation

This problem can be solved using the British Flag Theorem, which states that for any point O inside a rectangle ABCD, the sum of the squares of the distances from the point to opposite vertices is equal. The mathematical expression for this theorem is OA² + OC² = OB² + OD².

To derive this using coordinate geometry, let the vertices of the rectangle be D(0, 0), C(w, 0), B(w, h), and A(0, h), where w is the width and h is the height of the rectangle. Let point O be at coordinates (x, y). Using the distance formula d² = (x₂ - x₁)² + (y₂ - y₁)², we calculate the squares of the distances: OA² = (x - 0)² + (y - h)² = x² + (y - h)² OC² = (x - w)² + (y - 0)² = (x - w)² + y² OB² = (x - w)² + (y - h)² OD² = (x - 0)² + (y - 0)² = x² + y²

Adding the pairs of opposite vertices: OA² + OC² = x² + (y - h)² + (x - w)² + y² OB² + OD² = (x - w)² + (y - h)² + x² + y²

Since both expressions are identical, we have OB² + OD² = OA² + OC², which corresponds to option c.

Explanation

Sphere radius R = 2x. Cylinder height h = 2x. Using the relation between sphere radius and cylinder dimensions, R^2 = r^2 + (h/2)^2, where r is the cylinder radius. (2x)^2 = r^2 + (2x/2)^2 implies 4x^2 = r^2 + x^2, so r^2 = 3x^2 and r = xsqrt(3). Curved surface area of cylinder = 2pirh = 2pi(xsqrt(3))(2x) = 4pix^2sqrt(3). Surface area of sphere = 4piR^2 = 4pi(2x)^2 = 16pix^2. Ratio = (4pix^2sqrt(3)) / (16pix^2) = sqrt(3)/4.

Explanation

Volume of sphere = (4/3)pi(15^3) = 4500pi cubic cm. Volume of water displaced in cylinder = piR^2delta_h, where R = 30 cm. 4500pi = pi(30^2)delta_h implies 4500 = 900*delta_h, so delta_h = 5 cm.

Explanation

Given vertical angle is pi/2 (90 degrees), the angle between the axis and slant height is 45 degrees. Slant height l = sqrt(2)r. Height h = lcos(45) = sqrt(2)r (1/sqrt(2)) = r. Volume of cone = (1/3)pir^2h = (1/3)pir^2r = (pi*r^3)/3.

Explanation

The volume of a cone is proportional to the cube of its linear dimensions when the cones are similar. Let R be the radius of the base of the whole cone and r be the radius of the top of the frustum, which also serves as the base of the smaller cone removed from the top. Given the ratio R:r = 2:1, the ratio of the radii is r/R = 1/2. Since the smaller cone and the whole cone are similar solids, the ratio of their volumes is the cube of the ratio of their corresponding radii: Vₛₘₐₗₗ/Vᵩₕₒₗₑ = (r/R)³ = (1/2)³ = 1/8. The volume of the frustum is the difference between the volume of the whole cone and the volume of the removed smaller cone, represented as V_frustum = Vᵩₕₒₗₑ - Vₛₘₐₗₗ. To find the required ratio, we divide by the volume of the whole cone: V_frustum/Vᵩₕₒₗₑ = 1 - Vₛₘₐₗₗ/Vᵩₕₒₗₑ = 1 - 1/8 = 7/8. Therefore, the ratio of the volume of the frustum to that of the whole cone is 7:8, matching option (d).

Explanation

The inner surface area of the conical cavity corresponds to the curved surface area of the cone that has been hollowed out from the cylinder. To find this area, we first need to determine the slant height l of the cone using the Pythagorean theorem, as the radius r and height h are given as 6 cm and 8 cm respectively.

The formula for slant height is l = √(r² + h²). Substituting the values, we get l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

The curved surface area (CSA) of a cone is given by the formula CSA = πrl. Substituting the known values of r = 6 cm and l = 10 cm, we calculate CSA = π × 6 × 10 = 60π square cm. Therefore, the inner surface area of the cavity is 60π square cm, which corresponds to option (d).

Explanation

Volume of cylinder = pir^2h = pi(50^2)10 = 25000pi. Height of cone = Total height - Cylinder height = 15 - 10 = 5 m. Volume of cone = (1/3)pir^2h = (1/3)pi(50^2)5 = 12500pi/3. Total volume = 25000pi + 12500pi/3 = (75000pi + 12500pi)/3 = 87500*pi/3.

Explanation

The volume of a right circular cylinder is given by the formula V = πr²h, where r is the radius and h is the height. When a rectangular sheet of dimensions L × W is rolled to form a cylinder, one side becomes the height h and the other becomes the circumference C = 2πr. Thus, r = C / (2π).

For the first sheet of size 2π × 4π: Case 1: Let h = 2π and C = 4π. Then r = 4π / (2π) = 2. The volume is V = π(2²)(2π) = 8π². Case 2: Let h = 4π and C = 2π. Then r = 2π / (2π) = 1. The volume is V = π(1²)(4π) = 4π².

For the second sheet of size π × 5π: Case 3: Let h = π and C = 5π. Then r = 5π / (2π) = 2.5. The volume is V = π(2.5²)(π) = 6.25π². Case 4: Let h = 5π and C = π. Then r = π / (2π) = 0.5. The volume is V = π(0.5²)(5π) = 1.25π².

Comparing the volumes 8π², 4π², 6.25π², and 1.25π², the maximum possible volume is 8π², which corresponds to option b.

Explanation

In triangle ABC, BE is a median, which implies that point E is the midpoint of side AC. Given that AC = 9 cm, the length of EC = AC / 2 = 9 / 2 = 4.5 cm. Since AD is also a median, D is the midpoint of side BC. Now, consider triangle CBE. In this triangle, D is the midpoint of side BC and it is given that the line DF is drawn parallel to the side BE, with point F located on the line segment AC (which contains CE). According to the Converse of the Midpoint Theorem, a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. Therefore, since D is the midpoint of BC and DF ∥ BE, the point F must be the midpoint of CE. The length of CF is then calculated as CF = CE / 2 = 4.5 / 2 = 2.25 cm. This corresponds to option a.

Explanation

According to the Converse of the Basic Proportionality Theorem, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. In triangle PQR, the point X lies on PR and the point Y lies on QR. We compare the ratios of the segments created on sides PR and QR starting from the common vertex R.

The ratio on side PR is given by RX/RP = 4/10 = 2/5. The ratio on side QR is given by RY/RQ = 2/5.

Since RX/RP = RY/RQ = 2/5, it is mathematically proven that XY is parallel to PQ, which makes option (a) correct. Furthermore, because XY || PQ, triangle RXY is similar to triangle RPQ by AA similarity. The ratio of their corresponding sides is XY/PQ = RX/RP = 2/5, which implies PQ = 5/2XY = 2.5XY. This confirms that options (b) and (d) are incorrect. For option (c), PX = PR - RX = 10 - 4 = 6cm and QY = QR - YR = 5 - 2 = 3cm, so PX ≠ QY.

Explanation

Statement 1 is based on the transitive property of parallel lines in a plane. According to Euclidean geometry, if two lines are each parallel to a third line, they are parallel to each other. Mathematically, if A ∥ C and B ∥ C, then A ∥ B. This ensures that lines A and B will never intersect and maintain a constant distance from each other.

Statement 2 is a property of perpendicularity in two-dimensional space. If line A is perpendicular to line C (A ⊥ C) and line B is perpendicular to line C (B ⊥ C), then both A and B are perpendicular to the same transversal C. Since the corresponding angles formed with the transversal are both 90°, the sum of the consecutive interior angles is 90° + 90° = 180°. This is a sufficient condition for lines A and B to be parallel.

Because both statements 1 and 2 are logically and geometrically correct, option (b) is the correct choice.

Explanation

In a rhombus, the diagonals bisect each other at right angles, dividing the figure into four congruent right-angled triangles. In each of these triangles, the two legs are half the lengths of the diagonals, and the hypotenuse corresponds to the side of the rhombus. Using the Pythagorean theorem, the side a can be calculated using the formula a = √((d₁/2)² + (d₂/2)²), where d₁ and d₂ are the lengths of the diagonals. Substituting the given values d₁ = 20 cm and d₂ = 48 cm, we find that the half-lengths are 20/2 = 10 cm and 48/2 = 24 cm. Applying these to the formula, a = √(10² + 24²) = √(100 + 576) = √676 = 26 cm. Thus, the length of the side of the rhombus is 26 cm, which matches option B.

Explanation

The length of an arc of a circle can be calculated using the standard formula l = rθ, where l represents the length of the arc, r is the radius of the circle, and θ is the angle subtended by the arc at the centre expressed in radians. In this problem, we are given that the arc length l = 22 cm and the angle θ = π radians. Substituting these values into the formula, we get 22 = r × π. Given that π = 22/7, the equation becomes 22 = r × 22/7. To solve for r, we multiply both sides by 7/22, which gives r = 22 × 7/22. This simplifies to r = 7 cm. Therefore, the radius of the circle is 7 cm, which corresponds to option b.

Explanation

Since DE is parallel to BC, the triangle ADE is similar to triangle ABC by the AA similarity criterion. For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions. Let the area of triangle ADE be Aₐ₀ₑ and the area of triangle ABC be Aₐᵦ꜀. According to the problem, Aₐ₀ₑ = 1/5 Aₐᵦ꜀.

Using the similarity property: Aₐ₀ₑ/Aₐᵦ꜀ = (DE/BC)²

Substituting the given values Aₐ₀ₑ/Aₐᵦ꜀ = 1/5 and BC = 10 cm: 1/5 = (DE/10)²

Solving for DE: 1/5 = DE²/100 DE² = 100/5 DE² = 20 DE = √20 = √(4 × 5) = 2√5 cm

Thus, the length of DE is 2√5 cm, which corresponds to option b.

Explanation

The maximum number of intersection points for n lines, where no two are parallel and no three are concurrent, is given by nC2 = n(n-1)/2. For n=8, intersections = 8*7/2 = 28.

Explanation

The sum of the interior angles of a closed polygon with n sides is given by the formula (n - 2) × 180°. For a hexagon where n = 6, the total sum of interior angles is (6 - 2) × 180° = 4 × 180° = 720°.

Let the value of each of the five equal angles be x. According to the problem, the sixth angle is 30° greater than these, meaning it is x + 30°. Summing these angles, we get the equation 5x + (x + 30°) = 720°. Simplifying this, we have 6x + 30° = 720°, which leads to 6x = 690°. Dividing by 6, we find x = 115°. Therefore, the value of one of the equal angles is 115°, which corresponds to option (b).

Explanation

Statement 1 describes the circumcenter, which is the point where the perpendicular bisectors of the sides of a triangle intersect. The position of the circumcenter depends on the type of triangle: it lies inside an acute-angled triangle, at the midpoint of the hypotenuse in a right-angled triangle, and outside the triangle in an obtuse-angled triangle. Therefore, statement 1 is correct.

Statement 2 describes the orthocenter, which is the point where the altitudes (perpendiculars drawn from the vertices to the opposite sides) intersect. In an acute-angled triangle, the orthocenter is inside, and in an obtuse-angled triangle, it is outside. However, in a right-angled triangle, the altitudes from the two acute angles are the legs of the triangle themselves. These altitudes, along with the altitude from the right angle, intersect precisely at the vertex of the right angle. Since a vertex is the common endpoint of two sides, the orthocenter in a right-angled triangle lies on two sides of the triangle. Therefore, statement 2 is also correct.

Since both statements are mathematically accurate, option (c) is correct.

Explanation

Total students = 1200. Mathematics = 20% of 1200 = 240. Physics = 1/4 of 1200 = 300. Statistics = 320. Chemistry = 1200 - (240+300+320) = 340. Physics boys = 60% of 300 = 180. Statistics boys = 320 - 250 girls = 70. Total boys in Statistics and Physics = 180 + 70 = 250.

Explanation

Total students = 1200. Number of Mathematics students = 20% of 1200 = 20/100 × 1200 = 240. Number of Physics students = 1/4 of 1200 = 1200/4 = 300. Number of Statistics students = 320. The number of students studying Chemistry is the remainder of the total: Chemistry students = 1200 - (240 + 300 + 320) = 1200 - 860 = 340. The number of girls studying Statistics is given as 250. To find the required percentage, we use the standard percentage formula: Percentage = (Number of girls in Statistics / Total Chemistry students) × 100 Substituting the values: Percentage = (250 / 340) × 100 = 25/34 × 100 = 2500/34 ≈ 73.529% Rounding to one decimal place, we get 73.5%, which corresponds to option b.

Explanation

Total students = 1200. Mathematics students = 20% of 1200 = 240. Physics students = 1/4 of 1200 = 300. Statistics students = 320. Chemistry students = 1200 - (240 + 300 + 320) = 340.

Now we calculate the number of boys and girls for each subject to find the difference: For Mathematics, boys = 150, so girls = 240 - 150 = 90. The difference is 150 - 90 = 60. For Physics, boys = 60% of 300 = 180, so girls = 300 - 180 = 120. The difference is 180 - 120 = 60. For Statistics, girls = 250, so boys = 320 - 250 = 70. The difference is |70 - 250| = 180. For Chemistry, girls = 3/5 of 340 = 204, so boys = 340 - 204 = 136. The difference is |136 - 204| = 68.

Comparing the results, the difference between the number of boys and girls is exactly 60 for both Mathematics and Physics. This matches option c.

Explanation

Mathematics boys = 150. Physics total = 300, Physics boys = 60% of 300 = 180, so Physics girls = 300 - 180 = 120. Difference = 150 - 120 = 30.

Explanation

First, calculate the number of students in each subject. Mathematics has 20% of 1200 = 240 students. Physics has 1/4 × 1200 = 300 students. Statistics is given as 320 students. Chemistry students are the remainder: 1200 - (240 + 300 + 320) = 1200 - 860 = 340 students.

Next, determine the distribution of boys and girls for each subject. For Mathematics, boys are 150, so girls are 240 - 150 = 90. For Physics, boys are 60% of 300 = 180, so girls are 300 - 180 = 120. For Statistics, girls are 250, so boys are 320 - 250 = 70. For Chemistry, girls are 3/5 × 340 = 204, so boys are 340 - 204 = 136.

Now, sum the totals for boys and girls. Total Boys = 150 + 180 + 70 + 136 = 536. Total Girls = 90 + 120 + 250 + 204 = 664.

The ratio of the total number of boys to the total number of girls is 536 : 664. Dividing both sides by the highest common factor, 8, we get 536 ÷ 8 : 664 ÷ 8 = 67 : 83. This corresponds to option a.

Explanation

Frequency density for a class is defined as the class frequency divided by the class width.

Explanation

Total salary = (5 20,000) + (3 25,000) + (1 * 65,000) = 100,000 + 75,000 + 65,000 = 240,000. Mean salary = 240,000 / 9 = 26,666.67. Workers earning less than the mean are the 5 category C workers (20,000) and 3 category B workers (25,000), totaling 8 workers.

Explanation

The average speed for a journey involving multiple segments is calculated using the formula: Average Speed = Total Distance / Total Time.

In this problem, the total distance travelled is the sum of the two segments: Total Distance = 12 km + 10 km = 22 km

The total time is the sum of the time taken for each segment. Time is calculated as Distance divided by Speed: Time for first segment = 12 km / 4 km/hr = 3 hours Time for second segment = 10 km / 5 km/hr = 2 hours Total Time = 3 hours + 2 hours = 5 hours

Substituting these values into the average speed formula: Average Speed = 22 km / 5 hours = 4.4 km/hr

Thus, the average speed of the man is 4.4 km/hr, which corresponds to option a.

Explanation

In a pie chart, the area of the circle represents the total value of the data being visualized. Since the area of a circle is given by the formula Area = πr², the ratio of the expenditures of the two families is proportional to the ratio of the areas of their respective pie diagrams.

Given the ratio of the radii as rₐ : rᵦ = 16 : 9, we can determine the ratio of their expenditures Eₐ : Eᵦ using the relation Eₐ / Eᵦ = (πrₐ²) / (πrᵦ²) = (rₐ / rᵦ)². Substituting the given values, we get Eₐ / Eᵦ = (16 / 9)² = 256 / 81.

To find the matching data from the options, we check which pair of values simplifies to the ratio 256 : 81. In option c, the values are 25,600 and 8,100. Dividing these gives 25600 / 8100 = 256 / 81, which perfectly matches the calculated ratio. Thus, option c provides the appropriate expenditure data.

Explanation

Statement I is correct because the mode is defined as the value that appears most frequently in a given set of data or a probability distribution. In a frequency distribution, the mode corresponds to the value of the random variable associated with the highest frequency or the peak of the probability density function.

Statement II is incorrect because the mode is not necessarily unique. While many distributions are unimodal, meaning they have a single peak, others can be bimodal (having two modes) or multimodal (having multiple modes). A bimodal distribution occurs when two different values share the same highest frequency. Furthermore, if all values in a data set occur with the same frequency, the set is said to have no mode. Therefore, the claim that the mode is always unique is mathematically false. Consequently, Statement I is true but Statement II is false, making option c the correct choice.

Explanation

In a pie chart, the proportion p of a particular item relative to the total is represented by several geometric properties. The central angle θ of a slice is given by θ = p × 360°, making it directly proportional to p. Similarly, the area of the slice is A = p × πr², and the length of the curved arc is L = p × 2πr. Both A and L are directly proportional to the proportion p because they are products of p and a constant.

However, the perimeter of a slice consists of the curved arc length plus the two straight radii that form the slice. The formula for the perimeter P is P = 2πrp + 2r. Due to the addition of the constant term 2r (the two radii), the perimeter does not vary linearly with the proportion p. If the proportion p doubles, the perimeter P does not double. Therefore, the perimeter of the slices is not proportional to the proportion of the items, making option (d) the incorrect statement.

Explanation

GM(x, y) = sqrt(xy) = 6 implies xy = 36. GM(x, y, z) = cube_root(xyz) = 6 implies xyz = 216. Substituting xy = 36 into xyz = 216 gives 36z = 216, so z = 6.

Explanation

Data is classified into two main categories based on the source of collection: primary and secondary. Primary data is the original information collected firsthand by a researcher through methods such as surveys, interviews, experiments, or direct observations. In contrast, secondary data is information that has already been collected, processed, and recorded by some other person, agency, or organization for their own purposes.

In this scenario, the researcher is obtaining the number of live births from the office of the Birth Registrar. Since the data was originally collected and maintained by the government registry office and not by the researcher themselves, it constitutes secondary data. The researcher is simply utilizing existing records for their analysis, which is the defining characteristic of secondary data collection. Option B is correct because the registrar's office acts as an external source of pre-existing information.

Explanation

Mean is calculated using the formula Mean = Σx/n. In this case, the sum of the heights of the 5 students is 150 + 165 + 161 + 144 + 155 = 775. Dividing this sum by the number of observations n = 5, we get Mean = 775/5 = 155 cm. To find the median, the observations must first be arranged in ascending order: 144, 150, 155, 161, 165. For an odd number of data points, the median is the value at the (n + 1)/2-th position. Here, the median is the (5 + 1)/2 = 3rd term in the sorted list, which is 155 cm. Since both the mean and median are 155 cm, option b is the correct choice.

Explanation

Total height = (22 140) + (28 152) = 3080 + 4256 = 7336 cm. Total students = 22 + 28 = 50. Average height = 7336 / 50 = 146.72 cm.