The question asks to identify the state through which the Aravalli Range does NOT pass. To answer this, one needs to recall the geographical extent of the Aravalli Range in India.
Correct Option: D) Maharashtra
The problem involves decoding a word based on a given numerical value. We need to find the pattern or rule that converts the word "EDUCATION" into the number 92. Once the rule is identified, we apply the same rule to the word "KNOWLEDGE" to find its corresponding numerical value.
Analyze the given word and its code:
Word: EDUCATION
Code: 92
Assign numerical values to each letter:
We use the standard alphabetical position for each letter (A=1, B=2, ..., Z=26).
Sum the numerical values of the letters in "EDUCATION":
\[ \text{Sum} = 5 + 4 + 21 + 3 + 1 + 20 + 9 + 15 + 14 = 92 \]The sum of the positional values of the letters in "EDUCATION" is 92, which matches the given code. This confirms the rule: the code is the sum of the alphabetical positions of its letters.
Apply the same rule to "KNOWLEDGE":
Assign numerical values to each letter in "KNOWLEDGE":
Sum the numerical values of the letters in "KNOWLEDGE":
\[ \text{Sum} = 11 + 14 + 15 + 23 + 12 + 5 + 4 + 7 + 5 = 96 \]
D) 96 is the correct answer, as the sum of the alphabetical positions of the letters in "KNOWLEDGE" is 96.
To find the speed of the train, we need to use the formula: Speed = Distance / Time. When a train crosses an electric pole, the distance covered is equal to the length of the train itself. We will first calculate the speed in meters per second (m/s) and then convert it to kilometers per hour (km/h).
The distance covered when a train crosses a pole is its own length.
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]Substitute the given values:
\[ \text{Speed} = \frac{300 \text{ m}}{5 \text{ s}} = 60 \text{ m/s} \]To convert speed from m/s to km/h, multiply by \(\frac{18}{5}\) (since \(1 \text{ km} = 1000 \text{ m}\) and \(1 \text{ hour} = 3600 \text{ s}\), so \[1 \text{ m/s} = \frac{1/1000 \text{ km}}{1/3600 \text{ h}} = \frac{3600}{1000} \text{ km/h} = \frac{18}{5} \text{ km/h}\]).
\[ \text{Speed in km/h} = 60 \times \frac{18}{5} \]Calculate the value:
\[ \text{Speed in km/h} = 12 \times 18 = 216 \text{ km/h} \]D) The calculated speed of the train is \(216 \text{ km/h}\).
To find the difference between the two medians, we first need to calculate the median of the original data set. Then, we replace the specified value and calculate the median of the new data set. Finally, we find the absolute difference between these two medians.
Correct Option: B) 10
If \( x^{2} = y + z \), \( y^{2} = z + x \) and \( z^{2} = x + y \), then find the value of \( \frac{1}{x+1} \) + \( \frac{1}{y+1} \) + \( \frac{1}{z+1} \).
The problem provides three equations relating \(x, y, z\). We need to find the value of a sum of three fractions. The key is to manipulate the given equations to simplify the terms in the sum. We can add 1 to both sides of each equation to create terms like \((x+1)\), \((y+1)\), and \((z+1)\) in the denominators.
Given the equations:
Consider the first term we need to evaluate: \( \frac{1}{x+1} \). We can manipulate equation (1) to relate to \(x+1\). Add \(x\) to both sides of equation (1):
\[ x^2 + x = x + y + z \]\[ x(x+1) = x + y + z \]From this, we can express \(x+1\) in terms of \(x, y, z\):
\[ x+1 = \frac{x+y+z}{x} \]Now, substitute this into the first term of the expression we need to find:
\[ \frac{1}{x+1} = \frac{1}{\frac{x+y+z}{x}} = \frac{x}{x+y+z} \]Similarly, for the other two terms, we can follow the same logic:
Now, sum these three simplified terms:
\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} = \frac{x}{x+y+z} + \frac{y}{x+y+z} + \frac{z}{x+y+z} \]Combine the fractions since they have a common denominator:
\[ = \frac{x+y+z}{x+y+z} \]Assuming \(x+y+z \neq 0\), the expression simplifies to 1.
Let's consider the case where \(x+y+z = 0\). If \(x+y+z=0\), then \(y+z = -x\). From the first equation \(x^2 = y+z\), we would have \(x^2 = -x\). This implies \(x^2+x=0\), so \(x(x+1)=0\). Thus \(x=0\) or \(x=-1\). Similarly, \(y=0\) or \(y=-1\), and \(z=0\) or \(z=-1\). If \(x=y=z=0\), then the original equations hold (\(0=0+0\)), but the expression \( \frac{1}{x+1} \) would be \[\frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 3\]. However, if \(x=0\), then \(x+y+z = 0+y+z = 0\), so \(y+z=0\). From \(y^2=z+x\), \(y^2=z\). From \(z^2=x+y\), \(z^2=y\). If \(y+z=0\), then \(z=-y\). Substituting into \(y^2=z\), we get \(y^2=-y\), so \(y(y+1)=0\). Thus \(y=0\) or \(y=-1\). If \(y=0\), then \(z=0\). This leads to \(x=y=z=0\), which results in the sum being 3. However, this case implies \(x+y+z=0\), which would make the denominators \(x+y+z\) zero in our derived terms, making the expression undefined. Thus, \(x+y+z \neq 0\) must be true for the derived expression to be valid.
Consider the case where \(x=y=z\). Then \(x^2 = x+x = 2x\). So \(x^2-2x=0\), \(x(x-2)=0\). This means \(x=0\) or \(x=2\). If \(x=y=z=0\), then the sum is \(1/1 + 1/1 + 1/1 = 3\). If \(x=y=z=2\), then \(2^2 = 2+2\) (which is \(4=4\)), so the equations hold. In this case, the sum is \[\frac{1}{2+1} + \frac{1}{2+1} + \frac{1}{2+1} = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1\]. This confirms our result for a valid non-zero solution.
D) The value of the expression is 1.
We need to find the principal sum. We are given two scenarios involving simple interest on the same sum, and the difference between the interests earned in these two scenarios. We will use the formula for simple interest, set up an equation based on the given difference, and solve for the principal.
Let the principal sum be \(P\).
Recall the formula for simple interest (SI):
\[ \text{SI} = \frac{P \times R \times T}{100} \]Calculate the simple interest for the first scenario:
So, Simple Interest 1 (\(\text{SI}_1\)) is:
\[ \text{SI}_1 = \frac{P \times 8 \times 3}{100} = \frac{24P}{100} \]Calculate the simple interest for the second scenario:
So, Simple Interest 2 (\(\text{SI}_2\)) is:
\[ \text{SI}_2 = \frac{P \times 9 \times 2}{100} = \frac{18P}{100} \]According to the problem, the simple interest in the first case is ₹96 more than in the second case. So, we can write the equation:
\[ \text{SI}_1 - \text{SI}_2 = 96 \]Substitute the expressions for \(\text{SI}_1\) and \(\text{SI}_2\) into the equation:
\[ \frac{24P}{100} - \frac{18P}{100} = 96 \]Simplify the equation:
\[ \frac{(24 - 18)P}{100} = 96 \]\[ \frac{6P}{100} = 96 \]Solve for \(P\):
\[ 6P = 96 \times 100 \]\[ 6P = 9600 \]\[ P = \frac{9600}{6} \]\[ P = 1600 \]Therefore, the sum is ₹1,600.
Correct Option: B) ₹1,600
The question asks to identify a system program that translates and executes instructions simultaneously. This points to a specific type of language processor.
B) Interpreter — An interpreter is a computer program that directly executes instructions written in a programming or scripting language, without requiring them to have been previously compiled into a machine language program. It translates and executes one instruction or statement at a time.
A number is divisible by 6 if it is divisible by both 2 and 3. We will use the divisibility rules for 2 and 3 to find the least natural number A that satisfies the condition.
Correct Option: D) 2
The question asks to identify the specific annual session of the Indian National Congress where Mahatma Gandhi presided for the first and only time. This requires knowledge of the history of the Indian National Congress and Mahatma Gandhi's involvement as its president.
D) Belgaum — Mahatma Gandhi presided over the 39th annual session of the Indian National Congress in Belgaum in 1924. This was the only time he served as the President of the Congress.
The question asks to identify the location that is NOT connected with India's space programmes. We need to evaluate each option to determine its relevance to Indian space activities.
A) Sriperumbudur is not connected with India's space programmes. It is an industrial and religious town with no direct link to space research, development, or launch activities.