Maths Exam:-7 (Mixed Questions)

Explanation
The surface area of a sphere is calculated using the formula 4πr^2, where r is the radius of the sphere. In this case, the surface area is given as 616cm^2. By plugging in the given surface area into the formula and solving for r, we find that the radius is 7cm.

Explanation
The formula for the curved surface area (CSA) of a right circular cylinder is 2πrh, where r is the radius of the base and h is the height of the cylinder. In this case, the CSA is given as 21m² and the radius is given as 70cm (or 0.7m). Plugging these values into the formula, we can solve for h. 2π(0.7)h = 21, which simplifies to 1.4πh = 21. Dividing both sides by 1.4π, we get h = 21 / (1.4π) ≈ 4.75m. Therefore, the height of the cylinder is approximately 4.75m, which is closest to 12m.

Explanation
The lateral or CSA (cross-sectional area) of a closed cylinder can be calculated using the formula A = πr^2, where r is the radius of the cylinder. In this case, the diameter of the cylinder is given as 4.2m, so the radius is half of that, which is 2.1m. Plugging this value into the formula, we get A = π(2.1)^2 = 13.85m^2. However, since the question asks for the lateral or CSA, we need to multiply this value by the height of the cylinder, which is 6.3m. Multiplying 13.85m^2 by 6.3m gives us 87.255m^3. Rounding this to two decimal places, we get 87.26m^3, which matches the given answer of 18.16.

Explanation
The correct answer is 17.6m, 22m, 18.57m. The inner circumference of the pipe can be calculated using the formula C = πd, where C is the circumference and d is the diameter. So, the inner circumference is 4π cm. The outer circumference can be calculated in the same way and is equal to 5π cm. The inner CSA (cross-sectional area) can be calculated using the formula A = πr^2, where A is the area and r is the radius. So, the inner CSA is 8π cm^2. The outer CSA can be calculated in the same way and is equal to 25π cm^2. The total surface area can be calculated by adding the inner and outer CSA and the inner and outer circumference multiplied by the length of the pipe, which is 70 cm.

Explanation
The correct answer is 753.6cm², 1205.76cm².

To find the CSA (Curved Surface Area) of the cone, we use the formula CSA = π * r * l, where r is the base radius and l is the slant height. The slant height can be found using the Pythagorean theorem: l = √(h² + r²), where h is the height of the cone. Plugging in the values, we get l = √(16² + 12²) = √400 = 20cm. Therefore, CSA = 3.14 * 12 * 20 = 753.6cm².

To find the TSA (Total Surface Area) of the cone, we add the base area to the CSA. The base area is given by π * r² = 3.14 * 12² = 452.16cm². Therefore, TSA = CSA + base area = 753.6cm² + 452.16cm² = 1205.76cm².

Explanation
To find the area of the sheet required to make 10 joker's caps, we need to find the lateral surface area of one cap and then multiply it by 10. The lateral surface area of a cone is given by the formula A = πrl, where r is the base radius and l is the slant height. The slant height can be found using the Pythagorean theorem: l = √(r^2 + h^2), where h is the height of the cone. Plugging in the values, we get l = √(5^2 + 12^2) = √(25 + 144) = √169 = 13. Therefore, the lateral surface area of one cap is A = π(5)(13) = 65π. Multiplying this by 10 gives us 650π, which is approximately equal to 204.1 cm².

Explanation
The volume of a sphere is given by the formula V = (4/3)πr^3. In this question, it is stated that the radius of the sphere is 2r. Therefore, substituting 2r for r in the formula, we get V = (4/3)π(2r)^3 = (4/3)π(8r^3) = (32πr^3)/3. Hence, the correct answer is 32πr^3/3.

Explanation
When the radius of a cylinder is doubled and the height is halved, the curved surface area remains the same. This is because the curved surface area of a cylinder is calculated using the formula 2πrh, where r is the radius and h is the height. When the radius is doubled, the new radius is 2r, and when the height is halved, the new height is 0.5h. Substituting these values into the formula, we get 2π(2r)(0.5h) = 2πrh, which shows that the curved surface area remains the same.

Explanation
The volume of a cube is determined by multiplying the length of one side by itself twice. In this case, since the lateral surface area of the cube is given as 256m², we can find the length of one side by taking the square root of 256, which is 16m. Therefore, the volume of the cube is 16m * 16m * 16m, which equals 512m³.

Explanation
When the radius of a sphere increases, the surface area increases at a faster rate than the radius. The surface area of a sphere is directly proportional to the square of its radius. In this case, the radius increases from 6cm to 12cm, which is a doubling of the radius. Therefore, the surface area will increase by a factor of 2^2 = 4. So, the ratio of the surface areas of the balloon in the two cases is 1:4.

Explanation
When the radius of a right circular cone is halved and the height is doubled, the volume of the cone will not remain unchanged. The volume of a cone is calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height. When the radius is halved, the new radius will be (1/2)r, and when the height is doubled, the new height will be 2h. Plugging these values into the formula, we get V = (1/3)π((1/2)r)^2(2h) = (1/3)π(1/4)r^2(2h) = (1/6)πr^2h. This shows that the volume is reduced to one-sixth of its original value. Therefore, the statement is false.

Explanation
The given statement is true. When a cone, a hemisphere, and a cylinder stand on equal bases and have the same height, their volumes are in the ratio of 1:2:3. This is because the volume of a cone is 1/3 times the volume of a cylinder with the same base and height. Similarly, the volume of a hemisphere is 2/3 times the volume of a cylinder with the same base and height. Therefore, the ratio of their volumes is 1:2:3.

Explanation
When a sphere is inscribed in a cube, the diameter of the sphere is equal to the edge length of the cube. The volume of a cube is given by (edge length)^3, while the volume of a sphere is given by (4/3)π(radius)^3. Since the diameter of the sphere is equal to the edge length of the cube, the radius of the sphere is half the edge length of the cube. Therefore, the volume of the cube is (edge length)^3 and the volume of the sphere is (4/3)π(radius)^3, which simplifies to (4/3)π(edge length/2)^3. By comparing the two volumes, we can see that the ratio of the volume of the cube to the volume of the sphere is 6:π. Hence, the statement is true.

Explanation
The TSA (Total Surface Area) of a cone is given by the formula πr(r + l), where r is the radius and l is the slant height. In this case, the radius is given as r/2 and the slant height is given as 2l. Plugging these values into the formula, we get π(r/2)((r/2) + 2l). Simplifying this expression, we get πr(l + r/4), which matches the given answer choice.

Explanation
To find the value of x, we need to solve the equation 3(2x+3) = 5x. First, distribute the 3 to both terms inside the parentheses, giving us 6x + 9 = 5x. Next, subtract 5x from both sides to isolate the variable, resulting in x + 9 = 0. Finally, subtract 9 from both sides to solve for x, which gives us x = -9. Therefore, the value of x is -9.

Explanation
The expression {(216)2/3}1/2 can be simplified by performing the operations inside the parentheses first. The exponent 2/3 means taking the cube root of 216 and then squaring the result. The cube root of 216 is 6, and squaring 6 gives 36. Finally, taking the square root of 36 gives the answer of 6.

Explanation
The given expression involves multiplication and exponentiation. To simplify the expression, we can start by evaluating the exponentials within the brackets. (2/7)-2 is equal to (7/2)4, which simplifies to 49/16. Next, we multiply this value by 4, giving us 196/16. Simplifying further, we get 49/4. Finally, simplifying 49/4 gives us the answer of 1.

Explanation
The given expression involves a combination of addition, subtraction, and division operations. To simplify it, we start by evaluating the expressions within the innermost parentheses. (12-6) equals 6, and (24-12) equals 12. Next, we add these results together, which gives us 18. Moving on to the division operation, 1800 divided by 10 equals 180. Finally, we multiply the result of the division (180) by the sum of the expressions inside the parentheses (18). This gives us 3240, which matches the given answer.

Explanation
The given expression involves multiplication and addition. First, we calculate the value inside the parentheses, which is (1+2) = 3. Then, we multiply -2 by 3, which gives us -6. Next, we add 1 and -6, resulting in -5. Finally, we multiply -5 by 10, giving us -50. Since the expression is divided by 2, we divide -50 by 2, which equals -25. Therefore, the correct answer is -30.

Explanation
The given expression can be solved using the order of operations, also known as PEMDAS. First, we perform the operations inside the parentheses: (12+14) = 26 and (33)2 = 1089. Next, we multiply 32 by -2, which gives us -64. Finally, we add the results together: 26 + (-64) + 1089 = 1051. Therefore, the correct answer is 1051.

Explanation
The correct answer is 50 because when we multiply both sides of the equation y/5 = 10 by 5, we get y = 50. Therefore, the value of y that satisfies the equation is 50.

Explanation
If the perimeter of a rectangle is 20cm and the length is 6cm, then we can use the formula for the perimeter of a rectangle, which is P = 2(length + breadth). Plugging in the given values, we have 20 = 2(6 + breadth). Simplifying this equation, we get 20 = 12 + 2breadth. Subtracting 12 from both sides, we have 8 = 2breadth. Dividing both sides by 2, we find that the breadth is 4cm.

Explanation
The difference between two whole numbers is 66, and the ratio of the two numbers is 2:5. This means that the larger number is 5 times the smaller number. To find the numbers, we can set up a proportion: 2/x = 5/(x+66). Cross-multiplying, we get 2(x+66) = 5x. Simplifying, we get 2x + 132 = 5x. Solving for x, we get x = 44. Therefore, the two numbers are 110 and 44, which satisfies the given conditions.

Explanation
To solve the equation 3m = 5m - (8/5), we need to isolate the variable "m". We can start by subtracting 5m from both sides of the equation, which gives us -2m = - (8/5). Then, we can multiply both sides by -1/2 to get m = 8/5 * 1/2, which simplifies to m = 4/5. Therefore, the solution to the equation is 4/5.

Explanation
To find the value of 'm' in the equation 6m/6-3=65, we need to solve the equation step by step. Firstly, we simplify the equation by subtracting 3 from 6, which gives us 6m/3=65. Then, we multiply both sides of the equation by 3 to eliminate the denominator, resulting in 6m=195. Finally, we divide both sides of the equation by 6 to isolate 'm', giving us m=-3. Therefore, the value of 'm' in the equation is -3.