If the radius of a sphere is increased by 2 cm, then its surface area increases by 704 cm². Using π = \( \frac{22}{7} \), find the radius of the sphere before the increase.
We are given that the radius of a sphere increases by 2 cm, and its surface area increases by 704 cm². We need to find the original radius of the sphere. We will use the formula for the surface area of a sphere and set up an equation based on the given information.
Let the original radius of the sphere be \(r\) cm.
The original surface area of the sphere, \(S_1\), is given by the formula:
\[ S_1 = 4 \pi r^2 \]When the radius is increased by 2 cm, the new radius becomes \((r+2)\) cm.
The new surface area of the sphere, \(S_2\), is:
\[ S_2 = 4 \pi (r+2)^2 \]We are given that the surface area increases by 704 cm². This means:
\[ S_2 - S_1 = 704 \]Substitute the expressions for \(S_1\) and \(S_2\) into the equation:
\[ 4 \pi (r+2)^2 - 4 \pi r^2 = 704 \]Factor out \(4 \pi\):
\[ 4 \pi [(r+2)^2 - r^2] = 704 \]Expand \((r+2)^2\) and simplify the term inside the brackets:
\[ (r+2)^2 - r^2 = (r^2 + 4r + 4) - r^2 = 4r + 4 \]Substitute this back into the equation:
\[ 4 \pi (4r + 4) = 704 \]We are given \( \pi = \frac{22}{7} \). Substitute this value:
\[ 4 \times \frac{22}{7} \times (4r + 4) = 704 \]Simplify the left side:
\[ \frac{88}{7} \times 4(r + 1) = 704 \]\[ \frac{352}{7} (r + 1) = 704 \]Solve for \((r+1)\):
\[ r + 1 = \frac{704 \times 7}{352} \]\[ r + 1 = 2 \times 7 \]\[ r + 1 = 14 \]Solve for \(r\):
\[ r = 14 - 1 \]\[ r = 13 \text{ cm} \]D) 13 cm is the correct radius of the sphere before the increase, as calculated from the given information.
The question asks to identify the state where Monazite ore is primarily found in its sands. This requires knowledge of the geographical distribution of important mineral resources in India, specifically Monazite.
D) Kerala — Kerala's coastal sands are well-known for their rich deposits of heavy minerals, including Monazite, which is a primary source of thorium.
To solve this number series problem, we need to identify the pattern or rule governing the sequence of numbers. We will calculate the differences between consecutive terms to find a hidden sequence, which often reveals the underlying pattern.
Thus, the missing number is 114.
D) 114 is the correct answer because it follows the established pattern of increasing differences, where the differences themselves follow a sequence of prime numbers.
The problem asks us to identify a set of three numbers that follows the same relationship as the given sets: (11, 64, 3) and (17, 36, 11). We need to find a mathematical operation or pattern connecting the three numbers in each set.
A) (24, 169, 11)
Here, \(A=24\), \(B=169\), \(C=11\).
Let's check the pattern: \( (A - C)^2 = (24 - 11)^2 = 13^2 = 169 \).
Since \(169\) is equal to \(B\), this option follows the established relationship.
This problem involves the relationship between the product of two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). A fundamental property states that the product of two numbers is equal to the product of their HCF and LCM. We will use this property to find the LCM.
Product of two numbers \( = 1472 \)
HCF of the two numbers \( = 8 \)
For any two positive integers, say \(a\) and \(b\), the following relationship holds:
\[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \]Let the two numbers be \(a\) and \(b\). We are given \(a \times b = 1472\) and \(\text{HCF}(a, b) = 8\). Let \(\text{LCM}(a, b) = X\).
\[ 1472 = 8 \times X \]To find \(X\), divide the product of the numbers by their HCF:
\[ X = \frac{1472}{8} \]The LCM of the two numbers is 184.
Correct Option: C) 184
To find the missing number in the series, we need to identify the pattern or rule governing the sequence of numbers. We will look for differences between consecutive terms, or relationships involving squares, cubes, or other mathematical operations.
It appears the pattern for the differences is \(n^3 + 1\), where \(n\) is the position of the difference in the sequence (starting from \(n=1\)).
Thus, the missing number in the series is 246.
Correct Option: B) 246
The question asks to identify the part of the Indian Constitution that contains Articles 12 to 35, which deal with Fundamental Rights. This requires knowledge of the structure and parts of the Indian Constitution.
Correct Option: B) Part III of the Constitution contains Articles 12 to 35, which enumerate the Fundamental Rights of the citizens.
The question asks about the direct consequence of ozone depletion in the stratosphere. We need to understand the primary role of the stratospheric ozone layer to determine what happens when it is depleted.
Correct Option: C) Increase of UV radiations reaching Earth’s surface
Ozone depletion directly leads to a reduction in the absorption of UV radiation in the stratosphere, allowing more of it to reach the Earth's surface. This is the primary and most direct consequence.
Harjeet walked to his school at a speed of 4 km/h and returned on a scooter at 20 km/h. His average speed during the two-way journey is:
To find the average speed for a two-way journey where the distance is the same for both parts but the speeds are different, we use the formula for average speed, which is total distance divided by total time. Since the distance is not given, we can assume a convenient distance or use a general formula for average speed when distances are equal.
Correct Option: A) \( \frac{20}{3} \) kmph
The question asks us to evaluate two statements regarding the properties of ionic compounds. We need to determine the truthfulness of each statement based on the known characteristics of ionic bonds and structures.
Correct Option: C) Statement-I is true, and Statement-II is false.