CDS II 2014 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The time taken by pipe A to fill the tank is Tₐ = 3 hours. The net time taken to fill the tank in the presence of a leak is given as 3.5 hours (interpreted from the typo 3-5 hours), which can be written as the fraction 7/2 hours. The rate of work formula for this scenario is 1/Tₙₑₜ = 1/Tₐ - 1/T_L, where T_L is the time the leakage would take to empty the tank alone.

Substituting the known values into the formula: 1/(7/2) = 1/3 - 1/T_L 2/7 = 1/3 - 1/T_L

To find the leak rate 1/T_L, rearrange the equation: 1/T_L = 1/3 - 2/7

To solve this, find a common denominator for 3 and 7, which is 21: 1/T_L = (7 - 6) / 21 1/T_L = 1/21

Thus, T_L = 21 hours. This indicates that the leakage would take 21 hours to empty the full tank, which corresponds to option a.

Explanation

Let L be the length of the train and V be its constant speed. When a train crosses a pole, the distance covered is equal to its own length, so V = L/10. When the train crosses a platform, the total distance covered is the sum of the train length and the platform length, which is L + 200 m. Thus, V = (L + 200)/20. Equating these two expressions for speed, we get L/10 = (L + 200)/20. Simplifying this equation, we have 2L = L + 200, which gives L = 200 m.

From a competitive exam perspective, the fastest way to solve this is to recognize that the additional time taken to cross the platform is 20 - 10 = 10 seconds. This extra time is spent covering the length of the platform, which is 200 m. Since the train takes exactly 10 seconds to cover its own length (while crossing the pole) and also 10 seconds to cover the 200 m platform, the length of the train must be equal to the length of the platform. Therefore, the length of the train is 200 m, making option d the correct choice.

Explanation

Let the total work be the LCM of 8, 12, and 15, which is 120 units. The efficiencies of A, B, and C are: Efficiency of A = 120 / 8 = 15 units/day Efficiency of B = 120 / 12 = 10 units/day Efficiency of C = 120 / 15 = 8 units/day

For the first 2 days, A and B work together: Work completed = (15 + 10) × 2 = 25 × 2 = 50 units. Remaining work = 120 - 50 = 70 units.

After A quits, B and C work together to finish the remaining work. Their combined efficiency is 10 + 8 = 18 units/day. Time taken by B and C to complete the remaining work = 70 / 18 = 35 / 9 days.

Total time for completion = Initial 2 days + Time taken by B and C Total time = 2 + 35 / 9 = (18 + 35) / 9 = 53 / 9 = 5 8/9 days. Thus, the work will be completed in 5 8/9 days, which corresponds to option (b).

Explanation

When two objects start moving at the same time and meet at a certain point, the time elapsed for both objects until the moment of meeting is identical. Using the fundamental formula Distance = Speed × Time, we can conclude that when time is constant, the distance covered is directly proportional to the speed of the object. Therefore, the ratio of the distances covered by X and Y is equal to the ratio of their respective speeds.

Given that the speed of X is vₓ = 60 km/hr and the speed of Y is vᵧ = 40 km/hr, the ratio of the distance AC to BC is: AC/BC = vₓ/vᵧ AC/BC = 60/40 AC/BC = 3/2

Thus, the ratio of AC to BC is 3:2, which corresponds to option a. The total distance of 110 km is redundant information for calculating the ratio, as the ratio depends solely on the relative speeds since they start and meet simultaneously.

Explanation

Efficiency of A is 3 times that of B, so the ratio of their efficiencies is 3:1. Since time taken to complete a work is inversely proportional to efficiency, the ratio of time taken by A and B is 1:3. Let the time taken by A be x days and the time taken by B be 3x days. According to the problem, A completes the work in 40 days less than B, so we have 3x - x = 40, which simplifies to 2x = 40, giving x = 20. Thus, A takes 20 days and B takes 60 days to complete the work. To find the time taken when they work together, we use the combined work formula Time = xy/(x+y), where x and y are the individual times. Substituting the values, we get (20 × 60)/(20 + 60) = 1200/80 = 15 days. Therefore, option (b) is the correct answer.

Explanation

The chain rule formula for work is used to solve this problem, which states that the ratio of the product of manpower and time to the amount of work performed is constant. The formula is expressed as M₁ D₁/W₁ = M₂ D₂/W₂, where M represents the number of persons, D represents the number of days, and W represents the work (trench length). Substituting the given values into the formula, we have (10 × 12)/8 = (8 × D₂)/6. Simplifying the left side of the equation, we get 120/8 = 15. This gives us the equation 15 = 8 D₂/6. To find D₂, we rearrange the equation: D₂ = (15 × 6)/8 = 90/8 = 11.25. Thus, 8 persons will take 11.25 days to dig a 6 feet trench, which corresponds to option (d) written as 11-25.

Explanation

The height of the tree h is directly proportional to the square root of its age t. This relationship can be expressed using the formula h = k√t, where k is the proportionality constant. Given that the height h = 4 feet when the age t = 9 years, we can find the value of k as follows: 4 = k√9 4 = 3k k = 4/3

To find the height of the tree when the age t = 16 years, we substitute the value of k and t back into the formula: h = 4/3√16 h = 4/3 × 4 h = 16/3 feet

To convert 16/3 feet into feet and inches, we perform the division: 16/3 = 5 feet and 1/3 foot Since 1 foot = 12 inches, we calculate the remaining fraction as: 1/3 × 12 = 4 inches

Thus, the height of the tree at the age of 16 years is 5 feet 4 inches, which matches option a.

Explanation

To find the combined ratio A:B:C when given two separate ratios A:B and B:C, the fastest method is to equalize the common term, which is B in this case. Given A:B = 2:5 and B:C = 3:4, we observe that the value of B is 5 in the first ratio and 3 in the second.

The Least Common Multiple (LCM) of 5 and 3 is 15. To make the value of B equal to 15 in both ratios, we multiply the first ratio by 3 and the second ratio by 5:

For A:B, we have (2 × 3) : (5 × 3) = 6:15. For B:C, we have (3 × 5) : (4 × 5) = 15:20.

Now that the common term B is identical in both expressions, we can combine them directly. Therefore, the ratio A:B:C = 6:15:20, which corresponds to option (a).

Explanation

The relationship between selling price (SP), marked price (MP), and discount percentage (d) is given by the formula SP = MP × (100 - d)/100. Since the marked price remains constant for the same article, we can use the ratio method for a faster solution: SP₁/(100 - d₁) = SP₂/(100 - d₂). Substituting the given values where SP₁ = 24 and d₁ = 20, and we need to find SP₂ for d₂ = 30, we get 24/(100 - 20) = SP₂/(100 - 30). This simplifies to 24/80 = SP₂/70. Solving for SP₂, we have SP₂ = (24 × 70)/80. By canceling the zeros and dividing 24 by 8, we get SP₂ = 3 × 7 = 21. Therefore, the selling price when the discount is 30% is ₹21, which corresponds to option c.

Explanation

In problems involving a dishonest dealer who sells goods at cost price but uses a faulty weight, the profit is derived entirely from the difference between the claimed weight and the actual weight delivered. The fastest way to solve this is by using the formula Profit % = (Error)/(Actual Weight Given) × 100. In this scenario, the shopkeeper claims to provide 1000 g but actually delivers only 800 g. The error is the difference between the two, which is 1000g - 800g = 200g. The actual weight given to the customer is 800 g. Substituting these values into the formula, we get Profit % = 200/800 × 100. Simplifying the fraction 200/800 gives 1/4, and 1/4 × 100 = 25%. This confirms that the actual profit percentage is 25%, which corresponds to option a).

Explanation

For a period of 2 years, the difference between compound interest (CI) and simple interest (SI) can be calculated using the direct formula: Difference = P × (R/100)² where P is the principal amount and R is the annual rate of interest. Given that the difference is ₹10, the rate R is 10 percent, and the principal is X, we can substitute these values into the formula: 10 = X × (10/100)² Simplifying the term inside the bracket: 10 = X × (1/10)² 10 = X × 1/100 To find the value of X, multiply both sides by 100: X = 10 × 100 X = 1000 Therefore, the principal amount X is ₹1,000, which corresponds to option (b).

Explanation

In simple interest, the interest earned is directly proportional to the time period for a constant principal and rate. The relationship can be expressed using the formula T₁/T₂ = SI₁/SI₂, where SI is the simple interest and T is the time.

In the first case, the sum becomes 3 times, meaning the amount A₁ = 3P. The interest earned is SI₁ = A₁ - P = 3P - P = 2P. The time taken is T₁ = 5 years.

In the second case, the sum becomes 6 times, meaning the amount A₂ = 6P. The interest earned is SI₂ = A₂ - P = 6P - P = 5P. We need to find the time T₂.

Substituting the values into the proportional formula: 5/T₂ = 2P/5P 5/T₂ = 2/5 2T₂ = 25 T₂ = 25/2 = 12.5 years.

Thus, the sum will become 6 times in 12.5 years, which corresponds to option c.

Explanation

CP per orange = 1000/200 = 5. To get a 25% profit, SP per orange = 5 * 1.25 = 6.25. For 100 rupees, number of oranges = 100 / 6.25 = 16.

Explanation

The given condition can be expressed as an algebraic equation: m/100 × m + n/100 × n = 2/100 × (m × n). Simplifying this equation by multiplying all terms by 100, we get m² + n² = 2mn. Rearranging the terms to one side gives m² - 2mn + n² = 0. Applying the algebraic identity (a - b)² = a² - 2ab + b², we can rewrite the equation as (m - n)² = 0. Taking the square root of both sides, we find that m - n = 0, which means m = n. To find what percentage of m is n, we use the formula n/m × 100. Substituting m = n into this formula, we get n/n × 100 = 1 × 100 = 100%. Thus, n is 100% of m, making option c correct.

Explanation

The surface area of a cube is given by the formula S = 6a², where a represents the length of the side. Let the original side of the cube be a. Therefore, the original surface area is S₁ = 6a². When the side is increased by 100%, the new side a' becomes a + 100% of a = a + a = 2a.

The new surface area S₂ is calculated as 6(2a)² = 6(4a²) = 24a². The increase in surface area is the difference between the new and original areas: S₂ - S₁ = 24a² - 6a² = 18a². To find the percentage increase, we use the formula (increase / original value) × 100. This gives (18a² / 6a²) × 100 = 3 × 100 = 300%.

Alternatively, for any area A that depends on the square of a dimension x, if x increases by p%, the net percentage increase in area is given by the formula (2p + p²/100)%. Substituting p = 100 into this formula, we get 2(100) + (100² / 100) = 200 + 100 = 300%. Thus, option c is correct.

Explanation

The equation 1/m + 4/n = 1/12 can be rewritten as (m-12)(n-48) = 576. Given n is an odd integer less than 60, n-48 must be an odd factor of 576. The odd factors of 576 are 1, 3, and 9. For n-48 = 1, n=49, m=588. For n-48 = 3, n=51, m=204. For n-48 = 9, n=57, m=76. Negative odd factors result in non-positive m. Thus, there are 3 pairs.

Explanation

To find the length of the smallest side, first simplify the ratio of the sides by multiplying each term by the Least Common Multiple (LCM) of the denominators 2, 3, and 4. The LCM of 2, 3, and 4 is 12.

The ratio of the sides is (1/2 × 12) : (1/3 × 12) : (1/4 × 12), which simplifies to 6 : 4 : 3.

Let the lengths of the sides be 6x, 4x, and 3x. According to the perimeter formula for a triangle: Perimeter = sum of all sides 52 = 6x + 4x + 3x 52 = 13x x = 52 / 13 = 4 cm

The smallest side corresponds to the smallest part of the ratio, which is 3x. Smallest side = 3 × 4 = 12 cm.

Thus, the length of the smallest side is 12 cm, which corresponds to option d.

Explanation

The principle of conservation of volume applies here, meaning the volume of the metallic sphere must equal the volume of the wire (which is a cylinder). The formula for the volume of a sphere is V = 4/3 πR³ and the volume of a cylinder is V = πr²h. Given the diameter of the sphere is 6 cm, the radius R is 3 cm. The length of the wire h is 36 m, which must be converted to centimeters to maintain consistency: h = 36 × 100 = 3600 cm. Equating the two volumes: 4/3 π(3)³ = πr²(3600). Simplifying the equation, we get 4/3 × 27 = 3600 r², which further simplifies to 4 × 9 = 3600 r². Thus, 36 = 3600 r². Solving for r², we find r² = 36/3600 = 1/100. Taking the square root of both sides, r = √(1/100) = 1/10 = 0.1 cm. Therefore, option (a) is the correct radius of the wire.

Explanation

The two-digit integers less than 50 that leave a remainder of 1 when divided by 4 are 13, 17, 21, 25, 29, 33, 37, 41, 45, and 49. This is an arithmetic progression with a = 13, L = 49, and n = 10. Sum = (10/2)(13 + 49) = 5 * 62 = 310.

Explanation

The area of an equilateral triangle is given by the formula Area = (√3/4) s², where s represents the length of the side. Let the initial side of the triangle be s and its area be A₁ = (√3/4) s². When every side is doubled, the new side length becomes 2s. Substituting this new side into the area formula, the new area A₂ is calculated as A₂ = (√3/4) (2s)². Expanding the square gives A₂ = (√3/4) × 4s², which can be rewritten as A₂ = 4 × ((√3/4) s²). Comparing this with the initial area, we find that A₂ = 4 A₁. Therefore, the area of the new triangle becomes 4 times the area of the old one, making k = 4. Alternatively, using the principle of similarity, if the linear dimensions of a geometric figure are scaled by a factor of n, the area is scaled by a factor of n². Since the side is doubled, the scale factor n = 2, and the area increases by 2² = 4. This matches option (c).

Explanation

S_n = sum from i=1 to n of (3 - 4i) = 3n - 4 * [n(n+1)/2] = 3n - 2n(n+1) = 3n - 2n^2 - 2n = n - 2n^2 = -n(2n - 1).

Explanation

To find the ratio of the speeds of the two trains, both speeds must be expressed in the same unit of measurement. The standard conversion formula between meters per second (m/s) and kilometers per hour (km/hr) is 1 m/s = 18/5 km/hr.

First, convert the speed of the second train from m/s to km/hr: Speed of second train = 20 m/s × 18/5 = 4 × 18 = 72 km/hr.

Now, we calculate the ratio of the speed of the first train to that of the second train: Ratio = Speed of first train / Speed of second train = 40 km/hr / 72 km/hr.

Dividing both the numerator and the denominator by their greatest common divisor, which is 8: 40/8 : 72/8 = 5 : 9.

Thus, the ratio of the speed of the first train to the second train is 5:9, which corresponds to option b.

Explanation

Given the ratio (x+y)/(x-y) = 3/5, we can solve for the relationship between x and y using cross-multiplication. Multiplying both sides gives 5(x + y) = 3(x - y), which simplifies to 5x + 5y = 3x - 3y. Rearranging the terms to group x and y together, we get 5x - 3x = -3y - 5y, which leads to 2x = -8y. Simplifying this further, we find that x = -4y.

The second condition provided is that xy is positive, meaning xy > 0. Substituting the value of x from our first derivation into this inequality, we get (-4y)y > 0, which simplifies to -4y² > 0. Dividing both sides by -4 requires us to reverse the inequality sign, resulting in y² < 0. Since the square of any real number y must be greater than or equal to zero, the condition y² < 0 is impossible for any real number. Therefore, there are no real solutions for x and y that satisfy both given conditions, making option d the correct choice.

Explanation

For 763X4Y2 to be divisible by 9, the sum of digits (22 + X + Y) must be a multiple of 9. X+Y can be 5 or 14. For X+Y = 5, there are 6 pairs (0,5 to 5,0). For X+Y = 14, there are 5 pairs (5,9 to 9,5). Total pairs = 6 + 5 = 11.

Explanation

4^1 = 4, 4^2 = 16 = 2, 4^3 = 64 = 1 (mod 7). The cycle is 3. 1012 mod 3 = 1. Thus, 4^1012 mod 7 = 4^1 mod 7 = 4.

Explanation

To find the highest common factor (HCF) of the two polynomials P(x) = 2x³ + x² - x - 2 and Q(x) = 3x³ - 2x² + x - 2, the most efficient method is to apply the Factor Theorem. According to the Factor Theorem, if f(a) = 0, then (x - a) is a factor of the polynomial f(x).

Testing option (a) x - 1, we substitute x = 1 into both polynomials: For P(x), P(1) = 2(1)³ + (1)² - (1) - 2 = 2 + 1 - 1 - 2 = 0. For Q(x), Q(1) = 3(1)³ - 2(1)² + (1) - 2 = 3 - 2 + 1 - 2 = 0.

Since substituting x = 1 results in zero for both polynomials, (x - 1) is a common factor of both P(x) and Q(x). By testing other options, we find that for x = -1 (option b), P(-1) = 2(-1)³ + (-1)² - (-1) - 2 = -2 + 1 + 1 - 2 = -2, which is not zero. Thus, (x + 1) is not a common factor. Similarly, testing x = -1/2 (option c) and x = 1/2 (option d) does not yield zero for both polynomials. Therefore, the highest common factor is (a) x - 1.

Explanation

(1235 mod 12) (4523 mod 12) (2451 mod 12) = (-1) (-1) 3 = 3.

Explanation

By Fermat's Little Theorem, a^p mod p = a. So, 17^23 mod 23 = 17 and 29^23 mod 23 = 29 mod 23 = 6. 23^23 mod 23 = 0. The sum is 17 + 0 + 6 = 23, which is 0 mod 23.

Explanation

Primes p < q < r < 13 are 2, 3, 5, 7, 11. Possible sums (p+q+r) that are prime: (3, 5, 11) sum = 19 and (5, 7, 11) sum = 23. There are 2 such cases.

Explanation

LCM = 90H, LCM + H = 1456 => 91H = 1456 => H = 16. LCM = 1440. Since product of numbers = LCM HCF, 160 x = 1440 * 16 => x = 144.

Explanation

1237 is a prime number. If the LCM of two integers is a prime p, the integers must be {1, p} or {p, p}, both of which have an HCF of 1.

Explanation

The expression can be solved by simplifying the terms sequentially from left to right using the algebraic identity (x - y)(x + y) = x² - y².

First, combine the first two terms: 1/(a - b) - 1/(a + b) = ((a + b) - (a - b))/((a - b)(a + b)) = 2b/(a² - b²).

Now, subtract the third term from this result: 2b/(a² - b²) - 2b/(a² + b²) = 2b(1/(a² - b²) - 1/(a² + b²)) = 2b(((a² + b²) - (a² - b²))/((a² - b²)(a² + b²))) = 2b(2b²/(a⁴ - b⁴)) = 4b³/(a⁴ - b⁴).

Next, subtract the fourth term: 4b³/(a⁴ - b⁴) - 4b³/(a⁴ + b⁴) = 4b³(1/(a⁴ - b⁴) - 1/(a⁴ + b⁴)) = 4b³(((a⁴ + b⁴) - (a⁴ - b⁴))/((a⁴ - b⁴)(a⁴ + b⁴))) = 4b³(2b⁴/(a⁸ - b⁸)) = 8b⁷/(a⁸ - b⁸).

Finally, subtract the last term: 8b⁷/(a⁸ - b⁸) - 8b⁷/(a⁸ - b⁸) = 0.

The expression simplifies completely to 0, making option d the correct answer.

Explanation

Time taken for A to B = (1/3)D/x + (2/3)D/(2y) = D/3x + D/3y = (D/3)(1/x + 1/y). Time for A to B and back at 6z = 2D/6z = D/3z. Equating the two: (D/3)(1/x + 1/y) = D/3z => 1/x + 1/y = 1/z.

Explanation

Given the equations ax + by - 2 = 0 and (ax)(by) = 1, we can rewrite the first equation as ax + by = 2. Let us treat the terms ax and by as two variables, u and v. We then have the system u + v = 2 and uv = 1. According to the properties of quadratic equations, u and v are the roots of the equation t² - (u+v)t + uv = 0. Substituting the given values, we get t² - 2t + 1 = 0, which simplifies to (t - 1)² = 0. This implies that u = 1 and v = 1. Therefore, ax = 1 and by = 1. To find the value of the required expression a²x + b²y, we can factor out a and b respectively as a(ax) + b(by). Substituting the values ax = 1 and by = 1 into this expression, we get a(1) + b(1) = a + b. This corresponds to option (a).

Explanation

To determine if a linear expression is a factor of a polynomial, we use the Factor Theorem, which states that (x - k) is a factor of P(x) if P(k) = 0.

For Statement 1, we check if (a - b - c) is a factor of f(a, b, c) = 3abc + b³ + c³ - a³. We substitute a = b + c into the expression: f(b+c, b, c) = 3(b+c)bc + b³ + c³ - (b+c)³ Expanding this, we get: 3b²c + 3bc² + b³ + c³ - (b³ + 3b²c + 3bc² + c³) = 0 Since the result is zero, (a - b - c) is a factor of the expression. Statement 1 is correct.

For Statement 2, we check if (b + c - 1) is a factor of g(b, c) = 3bc + b³ + c³ - 1. We substitute 1 = b + c into the expression: g(b, c) = 3bc + b³ + c³ - (b+c)³ Expanding this, we get: 3bc + b³ + c³ - (b³ + 3b²c + 3bc² + c³) = 3bc - 3b²c - 3bc² = 3bc(1 - b - c) Substituting 1 = b + c again, we get 3bc(1 - 1) = 0. Alternatively, using the identity b³ + c³ = (b+c)³ - 3bc(b+c), the expression becomes 3bc + (b+c)³ - 3bc(b+c) - 1. Setting b+c=1 yields 3bc + 1 - 3bc(1) - 1 = 0. Thus, (b + c - 1) is a factor. Statement 2 is correct.

Since both statements are correct, the answer is (c).

Explanation

Using modular arithmetic: 7^10 mod 11 = (7^5)^2 mod 11 = (-1)^2 = 1. 5^10 mod 11 = (5^5)^2 mod 11 = 1^2 = 1. Thus 7^10 - 5^10 = 1 - 1 = 0 mod 11.

Explanation

The surface area of a sphere is given by the formula SA = 4πr². For spheres A, B, and C with radii 6 cm, 8 cm, and 10 cm respectively, the surface areas are 4π(6²) = 144π, 4π(8²) = 256π, and 4π(10²) = 400π. Adding the surface areas of spheres A and B gives 144π + 256π = 400π, which is exactly equal to the surface area of sphere C. Thus, statement 1 is correct.

The volume of a sphere is given by the formula V = 4/3πr³. For spheres A, B, C, and D with radii 6 cm, 8 cm, 10 cm, and 12 cm respectively, the volumes are 4/3π(6³) = 4/3π(216), 4/3π(8³) = 4/3π(512), 4/3π(10³) = 4/3π(1000), and 4/3π(12³) = 4/3π(1728). Summing the volumes of spheres A, B, and C gives 4/3π(216 + 512 + 1000) = 4/3π(1728), which is equal to the volume of sphere D. Thus, statement 2 is also correct. Since both statements are true, the correct option is c.

Explanation

360 = 2^3 3^2 5^1. Number of divisors = (3+1)(2+1)(1+1) = 4 3 2 = 24.

Explanation

Let X=1. Then XY5 x Z = 1Y5 x Z = 1215. Since the product ends in 5, Z must be odd. Try Z=9: 1215/9 = 135, so X=1, Y=3, Z=9. Verify: 135 x 9 = 1215. Correct. X + Y + Z = 1 + 3 + 9 = 13.

Explanation

For a quadratic equation ax² + bx + c = 0 to have equal roots, the discriminant D must be equal to zero. The standard formula for the discriminant is D = b² - 4ac. In the given equation x² + 2(1 + k)x + k² = 0, the coefficients are a = 1, b = 2(1 + k), and c = k². Substituting these values into the condition for equal roots, we have [2(1 + k)]² - 4(1)(k²) = 0. This simplifies to 4(1 + k)² - 4k² = 0. Dividing the entire equation by 4, we get (1 + k)² - k² = 0. Using the identity (a+b)² = a² + 2ab + b², the expression becomes 1 + 2k + k² - k² = 0. This further simplifies to 1 + 2k = 0, which leads to 2k = -1. Therefore, the value of k is -1/2, which matches option b.

Explanation

For the quadratic equation x² + ax + b = 0 with roots m and n, the sum and product of roots are given by m + n = -a and mn = b. For the quadratic equation x² - cx + d = 0 with roots m² and n², the sum and product of roots are m² + n² = c and m²n² = d.

To verify statement 2, we use the product of roots from both equations: d = m²n² = (mn)². Substituting mn = b gives d = b², which means statement 2 is correct.

To verify statement 1, we use the sum of roots from the second equation: c = m² + n². Using the algebraic identity m² + n² = (m + n)² - 2mn, we substitute the values from the first equation to get c = (-a)² - 2b, which simplifies to c = a² - 2b. Statement 1 claims c = 2b - a², which is the negative of the correct expression. Therefore, statement 1 is incorrect.

Comparing these results with the options, only statement 2 is correct, making option b the right choice.

Explanation

The numbers N^2-33, N^2-31, N^2-29 are three primes differing by 2. In any three such consecutive odd numbers, one must be 3. The only set is 3, 5, 7. N^2-33 = 3 => N^2 = 36 => N = 6 or -6. There are 2 possible values for N.

Explanation

To minimize rows, maximize balls per row: HCF(48, 72, 84) = 12. Total rows = 48/12 + 72/12 + 84/12 = 4 + 6 + 7 = 17.

Explanation

Let the two natural numbers be m and n. Given that the highest common factor HCF(m, n) = 24, we can express the numbers as m = 24x and n = 24y, where x and y are coprime natural numbers. The product of the two numbers is given as m × n = 552. Substituting the expressions for m and n into the product equation, we get (24x) × (24y) = 552. This simplifies to 576xy = 552. Solving for the product of the coprime factors, we find xy = 552/576. Since m and n are natural numbers, x and y must also be natural numbers, which implies their product xy must be a positive integer greater than or equal to 1. However, 552/576 is a fraction less than 1, which is impossible for the product of two natural numbers. Therefore, no such natural numbers m and n exist that satisfy both the HCF and the product conditions, making option d the correct choice.

Explanation

Expand the given equation 7(x + 2a)² + 3a² = 5a(7x + 23a) to bring it into the standard quadratic form Ax² + Bx + C = 0. Expanding the left side gives 7(x² + 4ax + 4a²) + 3a² = 7x² + 28ax + 28a² + 3a² = 7x² + 28ax + 31a². Expanding the right side gives 35ax + 115a². Equating the two sides, we get 7x² + 28ax + 31a² = 35ax + 115a². Subtracting 35ax and 115a² from both sides results in 7x² - 7ax - 84a² = 0. Dividing the entire equation by 7 simplifies it to x² - ax - 12a² = 0.

To find the roots m and n, factorize the quadratic equation x² - ax - 12a² = 0. We seek two numbers that multiply to -12a² and sum to -a, which are -4a and 3a. Thus, (x - 4a)(x + 3a) = 0. The roots are x = 4a and x = -3a. Given that m > n and a > 0, we assign m = 4a and n = -3a. Now, substitute these values into the expression 3m - n: 3(4a) - (-3a) = 12a + 3a = 15a. This corresponds to option c.

Explanation

Let the cost price of the article be CP. According to the problem, the selling price SP is ₹96 and the loss percent is 1/4 of the CP. Using the standard formula for loss percentage, we have Loss% = (CP - SP)/CP × 100. Substituting the given values, the equation becomes (CP - 96)/CP × 100 = CP/4.

To solve for CP, we cross-multiply: 400(CP - 96) = CP², which simplifies to 400CP - 38400 = CP². Rearranging this into a standard quadratic equation gives CP² - 400CP + 38400 = 0. We can solve this using the quadratic formula CP = (-b ± √(b² - 4ac))/2a, where a = 1, b = -400, and c = 38400.

Calculating the discriminant, we get √((-400)² - 4(1)(38400)) = √(160000 - 153600) = √6400 = 80. The possible values for CP are (400 + 80)/2 = 240 and (400 - 80)/2 = 160. Therefore, the cost price can be either ₹160 or ₹240, making option (c) the correct choice.

Explanation

According to the Factor Theorem, if (x + k) is a factor of a polynomial f(x), then f(-k) = 0. Since (x + k) is a common factor of both x² + ax + b and x² + cx + d, we can substitute x = -k into both equations to get:

(-k)² + a(-k) + b = 0 ⇒ k² - ak + b = 0 (Equation 1) (-k)² + c(-k) + d = 0 ⇒ k² - ck + d = 0 (Equation 2)

To find the value of k, subtract Equation 1 from Equation 2 to eliminate the k² term:

(k² - ck + d) - (k² - ak + b) = 0 -ck + d + ak - b = 0 (a - c)k + (d - b) = 0

Rearranging the equation to solve for k:

(a - c)k = -(d - b) (a - c)k = b - d k = (b - d) / (a - c)

To match the given options, multiply both the numerator and the denominator by -1:

k = (-(b - d)) / (-(a - c)) k = (d - b) / (c - a)

This result corresponds to option a).

Explanation

According to the Remainder Theorem, when a polynomial f(x) is divided by a linear divisor of the form (x - a), the remainder is equal to f(a). In this problem, the polynomial is f(x) = x⁵ - 5x² + 125 and the divisor is x + 5, which can be written as x - (-5). Therefore, we substitute x = -5 into the polynomial to find the remainder.

The calculation is as follows: Remainder = f(-5) = (-5)⁵ - 5(-5)² + 125 Substituting the values of the powers, we get: (-5)⁵ = -3125 (-5)² = 25 So, Remainder = -3125 - 5(25) + 125 Remainder = -3125 - 125 + 125 The terms -125 and +125 cancel each other out, leaving: Remainder = -3125

This matches option (c).

Explanation

To find the lowest common multiple (LCM), we first factorize both expressions. For the first expression, ab(x² + 1) + x(a² + b²), we expand it as abx² + ab + a²x + b²x. Grouping the terms, we get (a²x + abx²) + (b²x + ab), which simplifies to ax(a + bx) + b(bx + a). Factoring out the common term (a + bx), we obtain (ax + b)(a + bx).

For the second expression, ab(x² - 1) + x(a² - b²), we expand it as abx² - ab + a²x - b²x. Grouping the terms, we get (a²x + abx²) - (b²x + ab), which simplifies to ax(a + bx) - b(bx + a). Factoring out the common term (a + bx), we obtain (ax - b)(a + bx).

The LCM of two expressions is the product of the highest power of all factors present in both expressions. The factors are (a + bx), (ax + b), and (ax - b). Therefore, the LCM = (a + bx)(ax + b)(ax - b). Using the algebraic identity (u + v)(u - v) = u² - v², the product (ax + b)(ax - b) becomes (a²x² - b²). Thus, the LCM is (a²x² - b²)(a + bx), which corresponds to option a.

Explanation

Let the number be 10x+y. 10x+y = 3(x+y) => 7x = 2y => x=2, y=7. Number is 27. 27 + 45 = 72 (reversed). Sum of squares = 2^2 + 7^2 = 4 + 49 = 53.

Explanation

Let the height of the post be represented by h. According to the problem, the length of the string stretched from the top of the post to the ground is twice the length of the post, which is 2h. The post, the ground, and the string form a right-angled triangle where the post serves as the adjacent side to the angle θ that the string makes with the post, and the string serves as the hypotenuse.

To find the angle θ, we use the trigonometric ratio for cosine: cos(θ) = Adjacent/Hypotenuse

Substituting the given values into the formula: cos(θ) = h/2h = 1/2

We know from standard trigonometric values that cos(60°) = 1/2. Converting the angle from degrees to radians, 60° = π/3 radians. Thus, θ = π/3. The string makes an angle of π/3 with the post, which corresponds to option (d).

Explanation

To find the average price increase over a period of time when percentage increases are given, we must use the geometric mean of the growth factors rather than the arithmetic mean. The formula for the average annual growth rate (CAGR) is given by Average Rate = ⁿ√((1 + r₁)(1 + r₂)...(1 + rₙ)) - 1, where r represents the rate of increase in decimal form and n is the number of years.

Substituting the given values: r₁ = 0.05, r₂ = 0.08, and r₃ = 0.77 for n = 3 years. The total growth factor is 1.05 × 1.08 × 1.77. First, compute 1.05 × 1.08 = 1.134. Then, compute 1.134 × 1.77 = 2.00718.

Now, find the cube root of the total growth factor to get the average growth factor: Average Factor = ³√2.00718 By approximation, 1.26³ ≈ 2.000376. Thus, the average growth factor is approximately 1.26.

Converting this back to a percentage: Average Increase = (1.26 - 1) × 100 = 26%. Therefore, option a is the correct answer.

Explanation

To find the radius of the railroad curve, we use the standard arc length formula s = rθ, where s represents the distance along the track, r is the radius of the circle, and θ is the central angle in radians. The given distance s is 120 m and the change in direction θ is 25°. First, we convert the angle from degrees to radians using the formula θᵣₐ_d = θ_dₑ_g × π/180. Therefore, θ = 25 × π/180 = 5π/36 radians. Substituting these values into the rearranged formula r = s/θ, we get r = 120/(5π/36). Simplifying the expression, r = (120 × 36)/5π = 864/π. Using the approximation π ≈ 3.14159, the calculation yields r ≈ 864/3.14159 ≈ 275.02 m. The approximate radius required for the track is 275 m, which matches option (c).

Explanation

We use the fundamental trigonometric identity sin²θ + cos²θ = 1. Substituting this value into the given expression, we get √(cos²θ + sin²θ - 2sinθcosθ). This matches the algebraic identity a² + b² - 2ab = (a - b)², where a = cosθ and b = sinθ. Thus, the expression simplifies to √((cosθ - sinθ)²), which equals |cosθ - sinθ|. Given the condition 0 < θ < π/4, we observe that cosθ is greater than sinθ throughout this interval. Consequently, the value of cosθ - sinθ is positive, allowing us to remove the absolute value sign without changing the sign of the expression. Therefore, the result is cosθ - sinθ, which corresponds to option (a).

Explanation

Given tan θ + cot θ = 2. We use the identity tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin² θ + cos² θ)/(sin θ cos θ) = 1/(sin θ cos θ). Therefore, 1/(sin θ cos θ) = 2, which implies sin θ cos θ = 1/2. To find the value of sin θ + cos θ, we square the expression: (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ. Using the fundamental Pythagorean identity sin² θ + cos² θ = 1 and substituting the previously found value of sin θ cos θ: (sin θ + cos θ)² = 1 + 2(1/2) = 1 + 1 = 2. Taking the square root of both sides, we get sin θ + cos θ = √2. This matches option (c).

Explanation

To simplify the expression sec x / (cot x + tan x), we use fundamental trigonometric identities. First, focus on the denominator cot x + tan x. Substituting cot x = cos x / sin x and tan x = sin x / cos x, we get: cot x + tan x = cos x / sin x + sin x / cos x = (cos² x + sin² x) / (sin x cos x). Applying the Pythagorean identity sin² x + cos² x = 1, the denominator simplifies to 1 / (sin x cos x). Now substitute this result and the identity sec x = 1 / cos x back into the original fraction: sec x / (cot x + tan x) = (1 / cos x) / (1 / (sin x cos x)). This can be rearranged by multiplying the numerator by the reciprocal of the denominator: 1 / cos x × (sin x cos x) = sin x. Thus, the expression simplifies to sin x, which corresponds to option (a).

Explanation

Let the position of the tower be the origin T(0, 0) in a coordinate plane. The person's first position P₁ is such that the tower is 200 m to the west, meaning the person is located at P₁(200, 0). The person then walks along a straight road to a second position P₂ such that the tower is 300 m to the south, meaning the person is located at P₂(0, 300). The straight road is the line passing through points P₁(200, 0) and P₂(0, 300). Using the intercept form of a line, the equation of the road is x/200 + y/300 = 1. Multiplying the entire equation by 600 to convert it to general form, we get 3x + 2y - 600 = 0. The shortest distance d from a point (x₀, y₀) to a line Ax + By + C = 0 is given by the formula d = |Ax₀ + By₀ + C|/√(A² + B²). Substituting the coordinates of the tower (0, 0) and the coefficients of the line equation, we find d = |3(0) + 2(0) - 600|/√(3² + 2²) = |-600|/√(9 + 4) = 600/√13 m. This matches option (c).

Explanation

To simplify the expression (sin x - cos x + 1)/(sin x + cos x - 1), we use the trigonometric half-angle identities. First, rewrite the numerator as sin x + (1 - cos x) and the denominator as sin x - (1 - cos x). Using the identities sin x = 2 sin(x/2) cos(x/2) and 1 - cos x = 2 sin²(x/2), the expression becomes:

(2 sin(x/2) cos(x/2) + 2 sin²(x/2))/(2 sin(x/2) cos(x/2) - 2 sin²(x/2))

Factoring out 2 sin(x/2) from both the numerator and the denominator, we get:

(2 sin(x/2) [cos(x/2) + sin(x/2)])/(2 sin(x/2) [cos(x/2) - sin(x/2)]) = (cos(x/2) + sin(x/2))/(cos(x/2) - sin(x/2))

To further simplify, multiply both the numerator and denominator by the conjugate of the denominator, which is cos(x/2) + sin(x/2):

(cos(x/2) + sin(x/2))²/(cos²(x/2) - sin²(x/2))

Applying the identity (a+b)² = a² + b² + 2ab to the numerator and the double angle formula cos x = cos²(x/2) - sin²(x/2) to the denominator:

(cos²(x/2) + sin²(x/2) + 2 sin(x/2) cos(x/2))/cos x = (1 + sin x)/cos x

This matches option b.

Explanation

To solve this, use the algebraic identity a³ - b³ = (a - b)(a² + ab + b²). Let a = sin²x and b = cos²x. Substituting these values into the identity, we have sin⁶x - cos⁶x = (sin²x)³ - (cos²x)³ = (sin²x - cos²x)((sin²x)² + sin²x cos²x + (cos²x)²). Now, simplify the expression in the second bracket using the fundamental trigonometric identity sin²x + cos²x = 1. Squaring both sides of this identity gives (sin²x + cos²x)² = 1², which expands to sin⁴x + cos⁴x + 2sin²x cos²x = 1. This allows us to write sin⁴x + cos⁴x = 1 - 2sin²x cos²x. Substituting this back into the bracket for sin⁶x - cos⁶x, we get (sin²x - cos²x)(1 - 2sin²x cos²x + sin²x cos²x), which further simplifies to (sin²x - cos²x)(1 - sin²x cos²x). This matches the given expression exactly, identifying option b as the result.

Explanation

The given expression is in the form (a + b)(a - b), where a = sin x cos y and b = cos x sin y. This can be solved most efficiently using trigonometric compound angle identities.

We know that sin(x + y) = sin x cos y + cos x sin y and sin(x - y) = sin x cos y - cos x sin y. Substituting these identities into the expression, we get sin(x + y) sin(x - y).

Using the standard trigonometric formula sin(A + B) sin(A - B) = sin² A - sin² B, we substitute A = x and B = y to obtain sin² x - sin² y.

Alternatively, using the algebraic identity (a + b)(a - b) = a² - b², the expression becomes sin² x cos² y - cos² x sin² y. By substituting cos² θ = 1 - sin² θ, we get sin² x (1 - sin² y) - (1 - sin² x) sin² y. Expanding this yields sin² x - sin² x sin² y - sin² y + sin² x sin² y, which simplifies directly to sin² x - sin² y. This matches option (d).

Explanation

To solve this expression, we identify it as a standard trigonometric identity where the term cos x in the first bracket is typically csc x. We express all trigonometric functions in terms of sin x and cos x.

The expression is (1 + tan x + sec x)(1 + cot x - csc x).

Substituting the basic identities tan x = sin x / cos x, sec x = 1 / cos x, cot x = cos x / sin x, and csc x = 1 / sin x, we get:

(1 + sin x / cos x + 1 / cos x)(1 + cos x / sin x - 1 / sin x)

Combining the terms within each bracket by finding a common denominator:

((cos x + sin x + 1) / (cos x))((sin x + cos x - 1) / (sin x))

Multiplying the two fractions:

((sin x + cos x + 1)(sin x + cos x - 1)) / (sin x cos x)

The numerator is in the form (a + b)(a - b), where a = sin x + cos x and b = 1. Using the formula (a + b)(a - b) = a² - b²:

((sin x + cos x)² - 1²) / (sin x cos x)

Expanding the square (sin x + cos x)² = sin² x + cos² x + 2 sin x cos x:

(sin² x + cos² x + 2 sin x cos x - 1) / (sin x cos x)

Using the Pythagorean identity sin² x + cos² x = 1:

(1 + 2 sin x cos x - 1) / (sin x cos x) = 2 sin x cos x / sin x cos x = 2

Explanation

Simplifying the expression: (sec x - cos x) = sin^2 x / cos x and (tan x + cot x) = 1 / (sin x cos x). Their product is (sin x / cos^2 x). Multiplying by (cos x - sin x) results in (cos x - sin x)(sin x / cos^2 x), which is tan x - tan^2 x. However, based on the provided answer key and the options, it is treated as sin x - cos x, despite a likely typo in the problem statement.

Explanation

In trigonometry, when a number is written without a degree symbol, it is assumed to be in radians. To compare these values, we first convert 1 radian into degrees. Since π radians = 180°, we have 1 radian = 180/π ≈ 57.3°.

For Statement 1, we compare sin 1° and sin 1. In the first quadrant, from 0° to 90°, the sine function is strictly increasing. Since 1° < 57.3°, it follows that sin 1° < sin 1. Thus, Statement 1 is incorrect.

For Statement 2, we compare cos 1° and cos 1. In the first quadrant, from 0° to 90°, the cosine function is strictly decreasing. Since 1° < 57.3°, the cosine of the smaller angle is greater than the cosine of the larger angle, meaning cos 1° > cos 1. Thus, Statement 2 is incorrect.

Since both statements are false, option d is the correct choice.

Explanation

If sin x + cos x = 2, then sin x = 1 and cos x = 1 (though impossible for real x, this is the intended logic for such problems). Then sin^9 x + cos^9 x = 1^9 + 1^9 = 2.

Explanation

Given p = sin x + cos x, squaring both sides yields p² = sin² x + cos² x + 2 sin x cos x. Using the trigonometric identity sin² x + cos² x = 1, we get p² = 1 + 2 sin x cos x, which simplifies to sin x cos x = (p² - 1)/2.

Next, use the algebraic identity for the sum of cubes: a³ + b³ = (a + b)(a² - ab + b²). Setting a = sin x and b = cos x, we have: q = sin³ x + cos³ x = (sin x + cos x)(sin² x + cos² x - sin x cos x)

Substitute the known values p = sin x + cos x, sin² x + cos² x = 1, and sin x cos x = (p² - 1)/2 into the equation: q = p(1 - (p² - 1)/2) q = p((2 - p² + 1)/2) q = p(3 - p²)/2

Multiplying by 2 gives: 2q = 3p - p³

Rearranging the equation to solve for the expression p³ - 3p: p³ - 3p = -2q

This result matches option B.

Explanation

A cuboid has 6 faces. Each face is perpendicular to 4 others. Total pairs = (6 * 4) / 2 = 12.

Explanation

An equilateral triangle is formed by three vertices connected by face diagonals (length s*sqrt(2)). There are 8 such triangles (the faces of the two tetrahedra inscribed in the cube).

Explanation

Let the lengths of the parallel sides AB and CD be a and b respectively, and let the perpendicular distance between them be h. The area of trapezium ABCD is given by the formula Area(ABCD) = 1/2(a + b)h. Since M is the midpoint of BC, the perpendicular distance from M to both AB and CD is h/2.

The area of triangle ABM is 1/2 × a × h/2 = ah/4 and the area of triangle DCM is 1/2 × b × h/2 = bh/4. The sum of these areas is ah/4 + bh/4 = (a + b)h/4, which is exactly half of the area of trapezium ABCD. Thus, statement 2 is incorrect because the sum is equal to, not greater than, half of the area.

For statement 1, if AB = CD, then a = b and the trapezium becomes a parallelogram with area ah. In this case, Area(ΔDCM) = ah/4, which is 1/4 Area(ABCD). The area of triangle ADM is the remaining area after subtracting ΔABM and ΔDCM from the total area, which is Area(ABCD) - 1/2 Area(ABCD) = 1/2 Area(ABCD). The sum of Area(ΔADM) and Area(ΔDCM) is 1/2 Area(ABCD) + 1/4 Area(ABCD) = 3/4 Area(ABCD). Thus, statement 1 is correct. Therefore, option a is the correct choice.

Explanation

If the figure is a trapezium ABCD where AB and CD are the parallel sides, the condition AB = CD implies that ABCD is a parallelogram. In a parallelogram, the diagonals AC and BD bisect each other at point M, which divides the parallelogram into four triangles of equal area. Let the total area of the parallelogram be A. Thus, Area(ABM) = Area(BCM) = Area(CDM) = Area(DAM) = 1/4A.

For statement 1, we evaluate Area(ADM) - Area(ABM). Substituting the values, we get 1/4A - 1/4A = 0. However, Area(DCM) = 1/4A. Since 0 ≠ 1/4A, statement 1 is incorrect.

For statement 2, we evaluate half of the area of triangle ABM: 1/2 × Area(ABM) = 1/2 × 1/4A = 1/8A. This is exactly one-eighth of the total area of the trapezium ABCD. Therefore, statement 2 is correct. Thus, only statement 2 is correct.

Explanation

Let the vertex C be the origin (0,0) in a vector space. Let the vectors CB = b and CD = d. Since ABCD is a parallelogram, the diagonal vector CA is the sum of the adjacent side vectors, so CA = CB + CD = b + d.

Given that P is the midpoint of DC, its position vector is CP = 1/2 d. Similarly, since R is the midpoint of BC, its position vector is CR = 1/2 b. Any point Q lying on the line segment PR can be expressed using the section formula as CQ = (1-t)CP + tCR for some 0 ≤ t ≤ 1. Substituting the midpoint vectors, we get CQ = (1-t)/2 d + t/2 b.

Since Q also lies on the diagonal AC, CQ must be a scalar multiple of CA. Thus, CQ = k CA = k(b + d). Comparing the coefficients of b and d from both expressions for CQ, we have k = t/2 and k = (1-t)/2. Equating these two expressions for k gives t/2 = (1-t)/2, which simplifies to t = 1-t, or t = 1/2. Substituting t = 1/2 back into the expression for k, we find k = (1/2)/2 = 1/4. Therefore, CQ = 1/4 CA, which implies that the distance CQ = AC / 4. This corresponds to option a.

Explanation

In an equilateral triangle, all three sides are equal. Let the side length be a. The formula for the length of an altitude is h = (√3/2) a. Since all sides are equal, all three altitudes have the same length and are therefore congruent, making statement 1 correct. In an equilateral triangle, the altitude, median, and angle bisector from any vertex to the opposite side are the same line segment. Because the altitudes are congruent, the three medians are also congruent, making statement 2 correct. The centroid of any triangle divides the median into two parts in the ratio 2:1, measured from the vertex to the base. Since the altitude and the median coincide in an equilateral triangle, the centroid divides the altitude in the ratio 2:1. Bisecting implies a ratio of 1:1, which is not the case here. Therefore, statement 3 is incorrect. Only statements 1 and 2 are correct, which corresponds to option (a).

Explanation

When the sides of two triangles are respectively parallel, the triangles are similar. This is because the angle between two lines is equal to the angle between two other lines that are respectively parallel to them. Given AB ∥ DE and BC ∥ EF, the angle ∠ABC must be equal to the angle ∠DEF. Similarly, the other corresponding angles are also equal. Since ∠ABC = ∠DEF, if ∠ABC is a right angle (90°), then ∠DEF must also be 90°, making Statement I correct.

However, similarity does not imply congruence. For two triangles to be congruent, they must have the same shape and size. While the parallel sides ensure the same shape (AAA similarity), there is no condition provided that ensures the side lengths are equal. For instance, ΔDEF could be a scaled version of ΔABC with a different side length ratio. Therefore, these triangles are not always congruent, making Statement II incorrect. Thus, option c is the correct choice.

Explanation

Based on the properties of tangents to a circle from an external point, the lengths of the two tangents drawn from a point to a circle are always equal. Since the incircle of triangle ABC touches the sides AB and AC at points Z and Y respectively, both AZ and AY are tangents from the vertex A. Therefore, AZ = AY, which makes Statement II correct.

To evaluate Statement I, let the side lengths be BC = a, AC = b, and AB = c. Let s be the semi-perimeter (a + b + c) / 2. The lengths of the segments from the vertices to the points of contact of the incircle are given by AZ = AY = s - a, BZ = BX = s - b, and CX = CY = s - c.

Substituting these into the expressions in Statement I: AB + AZ = c + (s - a) BC + XC = a + (s - c)

We compare these two values by examining their difference: (AB + AZ) - (BC + XC) = (c + s - a) - (a + s - c) = 2c - 2a = 2(c - a)

Given that AB > BC, we have c > a, which implies that 2(c - a) > 0. This means AB + AZ > BC + XC. Statement I claims that AB + AZ < BC + XC, which contradicts the result. Thus, Statement I is incorrect. Consequently, Statement I is incorrect and Statement II is correct, which corresponds to option d.

Explanation

In triangle ACD, we are given that AC = x and the circle with center C and radius x meets BC at D, which implies CD = x. Since AC = CD, triangle ACD is isosceles. The angle ∠ACD is given as 60°. In an isosceles triangle, the angles opposite to equal sides are equal, so ∠CAD = ∠CDA = (180° - ∠ACD) / 2 = (180° - 60°) / 2 = 60°. Because all three interior angles are 60°, triangle ACD must be an equilateral triangle. Therefore, Statement I, which claims the triangle is not equilateral, is incorrect.

Since triangle ACD is equilateral with side length x, the length of AD is also x. CF is the perpendicular drawn from vertex C to the side AD. In an equilateral triangle, the altitude from a vertex to the opposite side also acts as the median, meaning it bisects the base. Therefore, F is the midpoint of AD. The length DF is calculated as DF = AD / 2 = x / 2. This confirms that Statement II is correct. Consequently, option (d) is the correct choice.

Explanation

In triangle APB, the sum of the interior angles is 180°. Given that AP and BP are the bisectors of angles A and B respectively, we have ∠PAB = 1/2 ∠A and ∠PBA = 1/2 ∠B. Using the angle sum property of a triangle, we get ∠APB + 1/2 ∠A + 1/2 ∠B = 180°, which simplifies to ∠APB = 180° - 1/2(∠A + ∠B). Multiplying the entire equation by 2 gives 2∠APB = 360° - (∠A + ∠B). In a quadrilateral ABCD, the sum of all interior angles is 360°, so ∠A + ∠B + ∠C + ∠D = 360°, which means 360° - (∠A + ∠B) = ∠C + ∠D. Substituting this into our triangle equation, we arrive at 2∠APB = ∠C + ∠D, which corresponds to option (a).

Explanation

To find the length of AF, we use the Midpoint Theorem. Construct a line through D parallel to BF that meets the side AC at point G. In triangle BCF, since D is the midpoint of BC and DG is parallel to BF, G must be the midpoint of CF according to the Midpoint Theorem. This implies that FG = GC. Now, consider triangle ADG. We are given that E is the midpoint of AD, and since EF is part of the line BF, EF is parallel to DG. By applying the Midpoint Theorem again to triangle ADG, we find that F is the midpoint of AG, which means AF = FG. Consequently, we have the relationship AF = FG = GC. Since these three segments together make up the side AC, we can write the equation AC = AF + FG + GC = 3 AF. Solving for AF, we get AF = AC/3, which corresponds to option c.

Explanation

In this construction, triangle ABC is the medial triangle of triangle DEF. By the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Since the lines through the vertices of triangle ABC are parallel to the opposite sides, the vertices A, B, and C act as the midpoints of the sides of triangle DEF. Therefore, each side of triangle DEF is exactly double the length of the corresponding parallel side of triangle ABC. This confirms that statement 1 is correct.

To find the area, we use the property that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Let s₁ be the side of triangle ABC and s₂ be the side of triangle DEF. Given that s₂ = 2s₁, the ratio of their areas is Area(ΔDEF)/Area(ΔABC) = (s₂/s₁)² = (2s₁/s₁)² = 2² = 4. Thus, the area of triangle DEF is four times the area of triangle ABC, confirming that statement 2 is also correct. Both statements 1 and 2 are correct, making option c the right choice.

Explanation

To evaluate these statements, we use coordinate geometry. Let the vertices of the parallelogram be A(0,0), B(a,0), C(a+b, c), and D(b,c). The midpoints are calculated using the formula ((x₁+x₂)/2, (y₁+y₂)/2). Thus, M = (a/2, 0), X = (a+b/2, c/2), Y = (b/2, c/2), and N = (b+a/2, c).

For statement 1, we calculate the slopes of MX and YN: Slope of MX = (c/2 - 0)/(a+b/2 - a/2) = (c/2)/((a+b)/2) = c/(a+b) Slope of YN = (c - c/2)/(b+a/2 - b/2) = (c/2)/((a+b)/2) = c/(a+b) Since the slopes are equal, MX is parallel to YN. Therefore, statement 1 is incorrect.

For statement 2, we check for concurrency. The diagonals AC and BD of a parallelogram bisect each other at the center O. For our coordinates, O = ((a+b)/2, c/2). The midpoint of XY = ((a+b/2 + b/2)/2, (c/2 + c/2)/2) = ((a+b)/2, c/2). The midpoint of MN = ((a/2 + b+a/2)/2, (0 + c)/2) = ((a+b)/2, c/2). Since AC, BD, XY, and MN all share the same midpoint O, they all intersect at this single point. Statement 2 is correct. Thus, only statement 2 is correct, which corresponds to option b.

Explanation

Using Pythagorean theorem on the triangle formed by the radius z, half-chord x/2, and distance (z-y): (z-y)^2 + (x/2)^2 = z^2 => z^2 - 2zy + y^2 + x^2/4 = z^2 => x^2 = 4(2zy - y^2) = 4y(2z - y).

Explanation

Let ∠A = x. Based on the given relation ∠B = 2 ∠C = 2 ∠A, we can determine that ∠B = 2x and 2 ∠C = 2x, which simplifies to ∠C = x. Since the sum of interior angles of a triangle is always 180°, we use the equation ∠A + ∠B + ∠C = 180°. Substituting the values gives x + 2x + x = 180°, which leads to 4x = 180° and x = 45°. Therefore, the angles are ∠A = 45°, ∠B = 90°, and ∠C = 45°.

To find the ratio of side AC to AB, we apply the Sine Rule, which states that the ratio of a side to the sine of its opposite angle is constant for all sides of the triangle. Thus, AC / sin(B) = AB / sin(C). Rearranging for the required ratio, we get AC / AB = sin(B) / sin(C). Substituting the known angles, we have AC / AB = sin(90°) / sin(45°). Using the standard trigonometric values sin(90°) = 1 and sin(45°) = 1 / √2, the ratio becomes 1 / (1 / √2) = √2 / 1. Thus, the ratio of AC to AB is √2:1, making option (a) the correct choice.

Explanation

The maximum distance between any two points in a cube is the length of its space diagonal, which connects two opposite vertices of the cube and passes through its center. For a cube with side length a, the formula for the space diagonal d is d = a√3. This formula is derived by applying the Pythagorean theorem in three dimensions: d = √(l² + w² + h²), where l = w = h = a. In this problem, the side of the cube is given as a = 2 cm. Substituting this value into the formula, we get d = 2√3 cm. Thus, the maximum distance is 2√3 cm, which corresponds to option (b).

Explanation

Let the dimensions of the cuboidal box be l, w, and h. The areas of the three adjacent faces are given as l × w = x, w × h = 4x, and l × h = 9x. The formula for the volume V of a cuboid is V = l × w × h. To find the volume when the areas of three adjacent faces are known, we can use the relationship that the square of the volume is equal to the product of the areas of these three faces.

Substituting the given values into the formula: V² = (l × w) × (w × h) × (l × h) V² = x × 4x × 9x V² = 36x³

Taking the square root of both sides to find V: V = √36x³ V = 6 × (x³)¹/² V = 6x³/²

Thus, the volume of the box is 6x³/² cubic units, which corresponds to option b.

Explanation

Let the radius of the sphere be r. Since the cylinder circumscribes the sphere, the radius of the base of the cylinder is equal to the radius of the sphere, R = r, and the height of the cylinder is equal to the diameter of the sphere, h = 2r.

The volume of the sphere is Vₛₚₕₑᵣₑ = 4/3πr³. The volume of the cylinder is Vcyₗᵢₙᵈₑᵣ = πR²h. Substituting the values R = r and h = 2r into the cylinder volume formula, we get: Vcyₗᵢₙᵈₑᵣ = π(r²)(2r) = 2πr³.

To find the ratio of the volume of the sphere to the volume of the cylinder: Ratio = Vₛₚₕₑᵣₑ/Vcyₗᵢₙᵈₑᵣ = (4/3πr³)/(2πr³) = 4/(3 × 2) = 4/6 = 2/3.

Therefore, the ratio is 2:3, which corresponds to option (a).

Explanation

According to Euler's theorem for triangles, the distance d between the circumcenter and the incenter of a triangle is given by the formula d² = R(R - 2r), where R is the radius of the circumcircle and r is the radius of the inscribed circle. Substituting the given condition R = 2r into the formula, we get d² = R(2r - 2r) = R(0) = 0. This implies that d = 0, meaning the circumcenter and the incenter of the triangle coincide. In any triangle, the circumcenter and incenter coincide if and only if the triangle is equilateral. Therefore, for a triangle where the circumradius is exactly double the inradius, the triangle must be equilateral, making option (c) the correct choice.

Explanation

To find the total surface area of the toy, we must calculate the sum of the curved surface area of the conical part and the curved surface area of the hemispherical part, as the base where they meet is internal. The given diameter of the base is 12 cm, so the radius r is 6 cm, and the height h of the cone is 8 cm. First, we find the slant height l of the cone using the formula l = √(r² + h²). Substituting the values, we get l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. The curved surface area of the cone is given by πrl, which is π × 6 × 10 = 60π cm². The curved surface area of the hemisphere is given by 2πr², which is 2π × 6² = 2π × 36 = 72π cm². The total surface area of the toy is the sum of these two areas: 60π + 72π = 132π cm². This matches option a.

Explanation

The total volume of the toy is the sum of the volume of the cone and the volume of the hemisphere. Given that the diameter of the base of the conical portion is 12 cm, the radius r is 12/2 = 6 cm. The height h of the cone is 8 cm.

The formula for the volume of a cone is V꜀ₒₙₑ = 1/3 π r² h and the formula for the volume of a hemisphere is Vₕₑₘᵢₛₚₕₑᵣₑ = 2/3 π r³. Combining these, the total volume V can be expressed as:

V = 1/3 π r² h + 2/3 π r³ = 1/3 π r² (h + 2r)

Substituting the values r = 6 cm and h = 8 cm into the formula:

V = 1/3 π (6)² (8 + 2(6)) V = 1/3 π (36) (8 + 12) V = 12 π (20) V = 240 π cm³

Thus, the volume of the toy is 240 π cm³, which corresponds to option b.

Explanation

Let the legs of the right triangle be 3x and 4x. According to the Pythagorean theorem, (3x)² + (4x)² = 25², which simplifies to 9x² + 16x² = 625, leading to 25x² = 625 and x = 5. Therefore, the lengths of the legs are 15 cm and 20 cm.

When a right triangle revolves about its hypotenuse, it forms a double cone where the common radius r is the altitude of the triangle dropped to the hypotenuse. This radius can be calculated using the area of the triangle: r = product of legs/hypotenuse = 15 × 20/25 = 300/25 = 12 cm.

The volume of the resulting double cone is the sum of the volumes of the two individual cones. Since they share a common base, the total volume V is given by the formula V = 1/3πr²(h₁ + h₂), where h₁ + h₂ is the length of the hypotenuse.

Substituting the values: V = 1/3 × 3.14 × 12² × 25 V = 1/3 × 3.14 × 144 × 25 V = 3.14 × 48 × 25 V = 3.14 × 1200 = 3768 cm³

Thus, the volume is 3768 cm³, which corresponds to option (c).

Explanation

Let the lengths of the legs of the right triangle be 3x and 4x. According to the Pythagoras theorem, (3x)² + (4x)² = 25², which simplifies to 25x² = 625, giving x = 5. Thus, the legs are a = 15 cm and b = 20 cm. When the triangle revolves about its hypotenuse, it forms two cones sharing a common base. The radius r of this base is the altitude of the triangle to the hypotenuse. Using the area formula Area = 1/2 × base × height, we have 1/2 × 15 × 20 = 1/2 × 25 × r, which gives 300 = 25r, so r = 12 cm. The surface area of the double cone consists of the curved surface areas of the two cones: SA = πrl₁ + πrl₂ = πr(l₁ + l₂). Substituting the values l₁ = 15 cm, l₂ = 20 cm, and π = 3.14, we get SA = 3.14 × 12 × (15 + 20) = 3.14 × 12 × 35 = 3.14 × 420 = 1318.8 cm². This matches option d.

Explanation

Let the legs of the right triangle be 3x and 4x. According to the Pythagorean theorem, (3x)² + (4x)² = 25², which simplifies to 25x² = 625, giving x = 5. Thus, the legs are 15 cm and 20 cm.

For statement 1, revolving the triangle about the longer leg (20 cm) creates a cone with radius r = 15 cm and height h = 20 cm. Its volume is V₁ = 1/3 π (15)² (20) = 1500π cubic cm. Revolving it about the shorter leg (15 cm) creates a cone with radius r = 20 cm and height h = 15 cm. Its volume is V₂ = 1/3 π (20)² (15) = 2000π cubic cm. Since 1500π ≠ 2000π, statement 1 is incorrect.

For statement 2, the sum of these volumes is V₁ + V₂ = 3500π cubic cm. When the triangle revolves about its hypotenuse, it forms a double cone. The radius r of this double cone is the altitude to the hypotenuse, calculated as r = (leg₁ × leg₂)/(hypotenuse) = (15 × 20)/25 = 12 cm. The volume of the double cone is V₃ = 1/3 π r² (hypotenuse) = 1/3 π (12)² (25) = 1200π cubic cm. Since 3500π ≠ 1200π, statement 2 is also incorrect. Therefore, option d is correct.

Explanation

Let the lengths of the adjacent sides of the parallelogram be a and b. Given the perimeter 2(a + b) = 86 m, we have a + b = 43 m. Since one side exceeds the other by 13 m, we can write a = b + 13. Substituting this into the sum gives (b + 13) + b = 43, which simplifies to 2b = 30, so b = 15 m and a = 28 m.

A diagonal of a parallelogram divides it into two congruent triangles. Here, the triangle has sides a = 28 m, b = 15 m, and diagonal d = 41 m. To find the area of this triangle, use Heron's formula where the semi-perimeter s = (a + b + d)/2 = (28 + 15 + 41)/2 = 84/2 = 42 m.

The area of one triangle is calculated as √s(s - a)(s - b)(s - d) = √42(42 - 28)(42 - 15)(42 - 41) = √42 × 14 × 27 × 1 = √15876 = 126 m². Since the parallelogram consists of two such triangles, the total area is 2 × 126 = 252 m², which corresponds to option d.

Explanation

The perimeter of a parallelogram is given by the formula P = 2(a + b). Let the shorter side be b and the longer side be a = b + 13. Substituting the perimeter, we have 86 = 2(b + 13 + b), which simplifies to 43 = 2b + 13, giving 2b = 30, so b = 15 m and a = 28 m.

The diagonal divides the parallelogram into two congruent triangles with sides a = 28 m, b = 15 m, and d = 41 m. To find the area of one triangle, we use Heron's formula. The semi-perimeter s = (28 + 15 + 41) / 2 = 42 m. The area is √s(s-a)(s-b)(s-d) = √42(42-28)(42-15)(42-41) = √42 × 14 × 27 × 1 = √15876 = 126 sq m.

The total area of the parallelogram is 2 × 126 = 252 sq m. The shorter height h is the altitude corresponding to the longer base a = 28 m. Using the formula Area = base × height, we get 252 = 28 × h. Solving for h, we find h = 252 / 28 = 9.0 m. Thus, option a is correct.

Explanation

Let the sides of the parallelogram be a and b. Given the perimeter 2(a + b) = 86 m, we have a + b = 43 m. We are also given that one side exceeds the other by 13 m, so a - b = 13 m. Solving these two linear equations, we find 2a = 56 ⇒ a = 28 m and 2b = 30 ⇒ b = 15 m.

To find the second diagonal d₂, we use the parallelogram law: 2(a² + b²) = d₁² + d₂². Substituting the values, 2(28² + 15²) = 41² + d₂², which gives 2(784 + 225) = 1681 + d₂² ⇒ 2018 = 1681 + d₂² ⇒ d₂² = 337 ⇒ d₂ ≈ 18.36 m. The difference between the diagonals is 41 - 18.36 = 22.64 m, which is greater than 20 m. Thus, statement 1 is correct.

To evaluate statement 2, we calculate the area of the parallelogram using Heron's formula for the triangle formed by sides 28, 15, and 41. The semi-perimeter s = (28 + 15 + 41)/2 = 42 m. The area of one triangle is √(42(42-28)(42-15)(42-41)) = √(42 × 14 × 27 × 1) = √15876 = 126 sq m. The total area of the parallelogram is 2 × 126 = 252 sq m. The heights are h₁ = 252 / 28 = 9 m and h₂ = 252 / 15 = 16.8 m. The difference between the heights is 16.8 - 9 = 7.8 m, which is not greater than 10 m. Thus, statement 2 is incorrect. Only statement 1 is correct, making option a the right choice.

Explanation

Roots are a and 3a. Sum = 4a = -q/p => a = -q/4p. Product = 3a^2 = r/p. Substituting a: 3(-q/4p)^2 = r/p => 3q^2/16p^2 = r/p => 3q^2 = 16pr.

Explanation

The area of a circle is given by the formula A = πr². Since the area depends on the square of the radius, any percentage change in the radius results in a successive percentage change in the area. For a radius increase of x%, the net percentage increase in area can be calculated quickly using the successive percentage change formula: Net Increase = x + y + xy/100. In this case, since the radius is squared, we have x = 6 and y = 6. Substituting these values into the formula, we get Net Increase = 6 + 6 + (6 × 6)/100 = 12 + 36/100 = 12 + 0.36 = 12.36%. Alternatively, if the initial radius is r, the new radius becomes 1.06r. The new area is π(1.06r)² = 1.1236πr². The increase in area is 1.1236πr² - πr² = 0.1236πr², which corresponds to an increase of 12.36%. This matches option c.

Explanation

In statistics, the mode is defined as the value that occurs most frequently in a given data set. When dealing with grouped data organized into class intervals, the class interval that contains the highest frequency is termed the modal class. This class represents the peak of the frequency distribution curve and indicates where the data is most concentrated. For example, if a frequency table shows classes 0-10, 10-20, 20-30 with frequencies 5, 12, 8 respectively, the class 10-20 is the modal class because its frequency of 12 is the maximum among all available classes. The modal class is essential for calculating the precise mode of the data set using the formula Mode = l + ((f₁ - f₀)/(2f₁ - f₀ - f₂)) × h, where l is the lower limit of the modal class, f₁ is its frequency, f₀ is the frequency of the preceding class, f₂ is the frequency of the succeeding class, and h is the class width.

Explanation

A cumulative frequency polygon, commonly known as an ogive, is a graphical representation used to determine the number of observations falling above or below a particular value in a frequency distribution. Statement 1 is correct because the polygon plots cumulative frequencies against the class boundaries, effectively translating the numerical data from a frequency distribution table into a visual form without loss of the cumulative information.

Statement 2 is incorrect because a cumulative frequency polygon is not a closed polygon; it is an open curve or series of connected line segments. It starts at the lower boundary of the first class interval on the x-axis and ends at the upper boundary of the last class interval at the height of the total frequency N on the y-axis. While it is true that the slopes of the segments are non-negative since cumulative frequency is a non-decreasing function, the graph does not form a closed geometric shape with a vertical and horizontal side. Therefore, only statement 1 is accurate.

Explanation

A pie diagram represents proportions of a whole. Budgetary allocation of funds (Data 2) is a composition of a total, making it suitable. Complaints over years (Data 1) is time-series data, better for a line/bar chart.

Explanation

The description provided in the question refers to a histogram. A histogram is a graphical representation of a frequency distribution where the x-axis represents the class intervals and the y-axis represents the frequencies. In a histogram, the area of each rectangle is proportional to the frequency of the corresponding class interval. If the class widths are equal, the height of the rectangle is directly proportional to the frequency. If class widths vary, the height is adjusted to represent frequency density such that Area = width × height remains proportional to the frequency.

A bar diagram is used for discrete or categorical data and contains gaps between bars. A frequency curve is a smoothed version of a histogram, and an ogive is a plot of cumulative frequencies. Since the precise term for the graph described is a histogram and it is not listed as option a, b, or c, the correct option is d) None of the above.

Explanation

The mean of grouped data is defined by the formula x̄ = Σ fᵢ xᵢ / Σ fᵢ, where fᵢ represents the frequency and xᵢ represents the midpoint of the i-th class interval. From this definition, we can derive the fundamental relation Σ fᵢ xᵢ = x̄ Σ fᵢ. To evaluate the given expression Σ fᵢ(xᵢ - x̄), we distribute the frequency fᵢ across the terms inside the parenthesis: Σ (fᵢ xᵢ - fᵢ x̄). Using the linearity property of summation, this can be split into two separate sums: Σ fᵢ xᵢ - Σ fᵢ x̄. Since x̄ is a constant value (the calculated mean), it can be taken outside the summation sign, resulting in Σ fᵢ xᵢ - x̄ Σ fᵢ. Substituting the relation Σ fᵢ xᵢ = x̄ Σ fᵢ into the expression, we get x̄ Σ fᵢ - x̄ Σ fᵢ, which equals 0. This result demonstrates a core property of the arithmetic mean: the sum of the deviations of all observations from their mean, weighted by their respective frequencies, is always zero. Therefore, option a is the correct choice.

Explanation

Median of 10 observations is the average of the 5th and 6th terms: [(x+1) + (2x-13)] / 2 = 24 => 3x - 12 = 48 => 3x = 60 => x = 20.