To find the simple interest, we use the formula for simple interest: \[\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\], where P is the principal amount, R is the rate of interest per annum, and T is the time in years.
The formula for simple interest (SI) is:
\[ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \]The interest to be paid at the end of two years is ₹3,000.
Correct Option: C) ₹3,000 is the correct interest amount calculated using the simple interest formula.
This problem can be solved using the concept of modular arithmetic or by taking a specific example. We are given a number that leaves a remainder of 4 when divided by 7. We need to find the remainder when the square of this number is divided by 7.
Let the number be \(N\). According to the problem statement, when \(N\) is divided by 7, the remainder is 4. This can be expressed using modular arithmetic as:
\[ N \equiv 4 \pmod{7} \]Alternatively, we can write \(N\) in the form \(7k + 4\), where \(k\) is an integer.
We need to find the remainder when the square of the number, \(N^2\), is divided by 7.
Using modular arithmetic, if \(N \equiv 4 \pmod{7}\), then squaring both sides:
\[ N^2 \equiv 4^2 \pmod{7} \] \[ N^2 \equiv 16 \pmod{7} \]Now, we need to find the remainder when 16 is divided by 7:
\[ 16 = 2 \times 7 + 2 \]So, \(16 \equiv 2 \pmod{7}\).
Therefore, \(N^2 \equiv 2 \pmod{7}\).
Alternatively, using the form \(N = 7k + 4\):
\[ N^2 = (7k + 4)^2 \] \[ N^2 = (7k)^2 + 2(7k)(4) + 4^2 \] \[ N^2 = 49k^2 + 56k + 16 \]We want to find the remainder when \(N^2\) is divided by 7. Notice that \(49k^2\) and \(56k\) are both multiples of 7. So, the remainder will depend only on the remainder of 16 when divided by 7.
\[ N^2 = 7(7k^2 + 8k) + 16 \]Let \(Q = 7k^2 + 8k\). Then \(N^2 = 7Q + 16\). When \(16\) is divided by \(7\), the remainder is \(2\). So, we can write \(16 = 2 \times 7 + 2\).
\[ N^2 = 7Q + (2 \times 7 + 2) \] \[ N^2 = 7Q + 14 + 2 \] \[ N^2 = 7(Q + 2) + 2 \]This shows that when \(N^2\) is divided by 7, the remainder is 2.
D) 2
The question asks for the year in which Article 35A was incorporated into the Indian Constitution. This requires knowledge of the historical context of constitutional amendments and presidential orders related to Jammu and Kashmir.
Therefore, based on the historical record, Article 35A was added in 1954.
Correct Option: D) 1954
To find the 10th term of an Arithmetic Progression (AP), we use the formula for the \(n\)-th term of an AP, which is \(a_n = a + (n-1)d\). We first identify the first term (\(a\)) and the common difference (\(d\)) from the given sequence.
The given Arithmetic Progression is 2, 7, 12, ...
The first term, \(a\), is the first number in the sequence.
\[ a = 2 \]The common difference is the difference between any term and its preceding term. We can calculate it using the first two terms:
\[ d = \text{second term} - \text{first term} \]\[ d = 7 - 2 \]\[ d = 5 \]The formula for the \(n\)-th term of an AP is:
\[ a_n = a + (n-1)d \]We need to find the 10th term, so \(n = 10\).
Substitute \(a=2\), \(d=5\), and \(n=10\) into the formula:
\[ a_{10} = 2 + (10-1) \times 5 \]\[ a_{10} = 2 + (9) \times 5 \]\[ a_{10} = 2 + 45 \]\[ a_{10} = 47 \]The 10th term of the Arithmetic Progression is 47.
B) 47 — This is the correct value calculated using the formula for the \(n\)-th term of an Arithmetic Progression.
To solve this problem, we need to identify the characteristics of each person based on the given information and then find the person who fits the specific criteria: intelligent, but neither honest nor hardworking.
Correct Option: B) Shiva is intelligent, and the information does not state that he is hardworking or honest. Therefore, he fits the description of being intelligent but neither honest nor hardworking.
The question asks to identify the famous scientist who is credited with the quote, "Give me a place to stand and I will move the world." This is a well-known historical quote associated with the principles of levers and mechanics.
D) Archimedes — Archimedes is widely credited with the statement, "Give me a place to stand, and I will move the Earth (or world)." This quote reflects his profound understanding of the principle of the lever and mechanical advantage, which he articulated in his work on mechanics.
The problem involves finding an unknown number given its Highest Common Factor (HCF) and Least Common Multiple (LCM) with another known number. The key property to use here is that for any two positive integers, the product of the numbers is equal to the product of their HCF and LCM.
Let the two numbers be \(N_1\) and \(N_2\).
We are given:
The fundamental relationship between two numbers, their HCF, and their LCM is:
\[ N_1 \times N_2 = \text{HCF} \times \text{LCM} \]Substitute the given values into the formula:
\[ 55 \times N_2 = 11 \times 330 \]Now, solve for \(N_2\):
\[ N_2 = \frac{11 \times 330}{55} \]Simplify the expression:
\[ N_2 = \frac{11 \times 330}{5 \times 11} \]\[ N_2 = \frac{330}{5} \]Perform the division:
\[ N_2 = 66 \]Thus, the other number is 66.
A) 66 — As calculated, the other number is 66.
To find the number of smaller cubes that can be formed by melting a larger cuboid, we need to understand that the volume of the material remains constant. Therefore, we calculate the volume of the cuboid and the volume of one small cube, and then divide the total volume of the cuboid by the volume of one cube.
The dimensions of the cuboid are given as length (\(L\)) = 9 cm, breadth (\(B\)) = 6 cm, and height (\(H\)) = 6 cm.
\[ \text{Volume of cuboid} = L \times B \times H \]\[ \text{Volume of cuboid} = 9 \text{ cm} \times 6 \text{ cm} \times 6 \text{ cm} \]\[ \text{Volume of cuboid} = 324 \text{ cm}^3 \]The side of the small cube (\(s\)) is given as 3 cm.
\[ \text{Volume of cube} = s^3 \]\[ \text{Volume of cube} = (3 \text{ cm})^3 \]\[ \text{Volume of cube} = 3 \text{ cm} \times 3 \text{ cm} \times 3 \text{ cm} \]\[ \text{Volume of cube} = 27 \text{ cm}^3 \]The number of cubes formed is the total volume of the cuboid divided by the volume of one small cube.
\[ \text{Number of cubes} = \frac{\text{Volume of cuboid}}{\text{Volume of cube}} \]\[ \text{Number of cubes} = \frac{324 \text{ cm}^3}{27 \text{ cm}^3} \]\[ \text{Number of cubes} = 12 \]Thus, 12 cubes of side 3 cm can be formed by melting the given cuboid.
B) 12 — As calculated, the number of cubes is 12.
The question asks to identify the first official international step taken to protect the environment. This requires knowledge of key environmental conferences and conventions and their chronological order.
Correct Option: C) Stockholm conference — The United Nations Conference on the Human Environment, held in Stockholm, Sweden, from June 5–16, 1972, was the first major international conference to address environmental issues. It laid the foundation for global environmental governance and led to the establishment of the United Nations Environment Programme (UNEP).
This problem can be solved using the principles of set theory, specifically by understanding the relationship between the number of elements in unions and intersections of sets. We are given the total number of teachers, the number of teachers teaching only one subject, and the number of teachers teaching both subjects. We need to find the number of teachers teaching only the other subject.
D) 4 is the correct answer. There are 4 teachers who teach only physics.