To determine the last digit of \(6^n - 5^n\), we need to analyze the pattern of the last digits of powers of 6 and powers of 5 for any natural number \(n\).
Let's look at the last digits of the first few powers of 6:
It is clear that any positive integer power of 6 will always have 6 as its last digit.
Let's look at the last digits of the first few powers of 5:
It is clear that any positive integer power of 5 will always have 5 as its last digit.
Since the last digit of \(6^n\) is always 6 and the last digit of \(5^n\) is always 5, the last digit of their difference \(6^n - 5^n\) will be the last digit of \(6 - 5\).
\[ \text{Last digit of } (6^n - 5^n) = \text{Last digit of } (6 - 5) = \text{Last digit of } (1) \]Thus, the expression \(6^n - 5^n\) always ends with 1.
Correct Option: D) 1
This problem involves work and time. We need to calculate the individual work rates of A and B, determine the amount of work B completed alone, and then find the time A and B worked together to complete the remaining work. Finally, sum up the time periods to get the total time.
Calculate individual work rates:
A can complete the work in 20 days. So, A's one day's work is:
\[ \text{A's 1 day work} = \frac{1}{20} \text{ of the total work} \]B can complete the work in 30 days. So, B's one day's work is:
\[ \text{B's 1 day work} = \frac{1}{30} \text{ of the total work} \]Calculate work done by B alone:
B alone did the work for the last 5 days. Work done by B in 5 days is:
\[ \text{Work done by B in 5 days} = 5 \times \frac{1}{30} = \frac{5}{30} = \frac{1}{6} \text{ of the total work} \]Calculate remaining work:
The total work is 1 unit. The remaining work after B worked alone is:
\[ \text{Remaining work} = 1 - \frac{1}{6} = \frac{5}{6} \text{ of the total work} \]Calculate combined work rate of A and B:
When A and B work together, their combined one day's work is:
\[ \text{Combined 1 day work} = \frac{1}{20} + \frac{1}{30} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ of the total work} \]Calculate time taken by A and B together to complete the remaining work:
Let \(x\) be the number of days A and B worked together. They completed \(\frac{5}{6}\) of the work.
\[ x \times \frac{1}{12} = \frac{5}{6} \]Solving for \(x\):
\[ x = \frac{5}{6} \times 12 = 5 \times 2 = 10 \text{ days} \]Calculate total time taken:
Total time = (Time A and B worked together) + (Time B worked alone)
\[ \text{Total time} = 10 \text{ days} + 5 \text{ days} = 15 \text{ days} \]C) 15 days
The numerator of a fraction is 2 less than the denominator. If the numerator is multiplied by 2 and the denominator is multiplied by 3, then the fraction becomes \( \frac{2}{9} \). The fraction is:
\( \frac{5}{7} \)
\( \frac{3}{5} \)
\( \frac{7}{9} \)
\( \frac{1}{3} \)
We need to find a fraction based on two given conditions. We can represent the fraction using variables for the numerator and denominator, form equations based on the given conditions, and then solve these equations simultaneously to find the values of the numerator and denominator.
Let the numerator of the fraction be \(x\) and the denominator be \(y\). So the fraction is \( \frac{x}{y} \).
The first condition states that the numerator of a fraction is 2 less than the denominator. This can be written as:
\[ x = y - 2 \quad \text{(Equation 1)} \]The second condition states that if the numerator is multiplied by 2 and the denominator is multiplied by 3, the new fraction becomes \( \frac{2}{9} \). This can be written as:
\[ \frac{2x}{3y} = \frac{2}{9} \quad \text{(Equation 2)} \]Now, we need to solve these two equations simultaneously. From Equation 2, we can simplify it:
\[ 9(2x) = 2(3y) \]\[ 18x = 6y \]\[ 3x = y \quad \text{(Equation 3)} \]Substitute the value of \(y\) from Equation 3 into Equation 1:
\[ x = (3x) - 2 \]\[ x - 3x = -2 \]\[ -2x = -2 \]\[ x = 1 \]Now that we have the value of \(x\), substitute it back into Equation 3 to find \(y\):
\[ y = 3x \]\[ y = 3(1) \]\[ y = 3 \]So, the original fraction is \[\frac{x}{y} = \frac{1}{3}\].
Let's check if this fraction satisfies both conditions:
Condition 1: Numerator is 2 less than the denominator. Here, \(1 = 3 - 2\), which is true.
Condition 2: If the numerator is multiplied by 2 and the denominator by 3, the fraction becomes \( \frac{2}{9} \). Here, \[\frac{2 \times 1}{3 \times 3} = \frac{2}{9}\], which is true.
Correct Option: D) \( \frac{1}{3} \)
We are given the sum of two numbers and their Least Common Multiple (LCM). We need to find the two numbers from the given options. A key property to remember is that if two numbers are \(a\) and \(b\), and their LCM is \(L\), then \(L\) must be divisible by both \(a\) and \(b\). We will check this property for each option.
A) 1489, 11 — The sum of 1489 and 11 is 1500. Also, 16379 is divisible by both 1489 (16379 / 1489 = 11) and 11 (16379 / 11 = 1489). This satisfies all the given conditions.
The question asks to identify which of the given options is NOT a component of the circulatory system. To answer this, we need to recall the main parts and functions of the circulatory system and then evaluate each option.
C) Large intestine — The large intestine is an organ of the digestive system, responsible for water absorption and waste elimination. It is not involved in the circulation of blood, oxygen, or nutrients throughout the body.
The question asks to identify the initiator of the Global Climate Change Alliance (GCCA). This requires knowledge of international environmental initiatives and the organizations behind them.
Correct Option: A) The Global Climate Change Alliance (GCCA) is indeed an initiative launched by the European Union (EU) in 2007. Its primary goal is to foster dialogue and cooperation on climate change between the European Union and developing countries most vulnerable to climate change, especially Least Developed Countries (LDCs) and Small Island Developing States (SIDS).
This problem involves successive percentage changes. When a quantity is increased by a certain percentage and then decreased by the same percentage, the net change is always a decrease. We can solve this by assuming an initial number or by using a direct formula for successive percentage changes.
Alternative Method (Formula for successive percentage change):
If a number is first changed by \(x\%\) and then by \(y\%\), the net percentage change is given by the formula:
\[ \left( x + y + \frac{xy}{100} \right)\% \]Here, an increase is positive, and a decrease is negative.
First change: \(x = +40\%\) (increase)
Second change: \(y = -40\%\) (decrease)
Net percentage change \[= \left( 40 + (-40) + \frac{40 \times (-40)}{100} \right)\%\]
\[ = \left( 0 + \frac{-1600}{100} \right)\% \]\[ = (-16)\% \]This means a 16% decrease.
C) 16% decrease — As calculated, the net effect of increasing a number by 40% and then decreasing the new number by 40% results in a 16% decrease from the original number.
This question tests general knowledge about the currencies of different countries. To answer correctly, one needs to identify the country that uses Yen as its official currency.
Correct Option: A) Japan
The Yen (¥; JPY) is the official currency of Japan. It is the third most traded currency in the foreign exchange market after the US Dollar and the Euro.
The question asks to identify the individual who first proposed the idea of a Constituent Assembly for India. This is a factual question related to the historical background of the Indian Constitution.
Correct Option: B) M N Roy
M.N. Roy, a pioneer of the communist movement in India and an advocate of radical democracy, first proposed the idea of a Constituent Assembly for India in 1934. This idea was later officially adopted by the Indian National Congress in 1935.
This problem involves the concept of work and time. The capacity of working (or efficiency) is inversely proportional to the time taken to complete a work. If the ratio of capacities of two individuals is given, we can find the ratio of the time taken by them to complete the same work.
Let the capacity of working of A be \(C_A\) and that of B be \(C_B\).
Given the ratio of their capacities: \[ C_A : C_B = 3 : 4 \]
The time taken to complete a work is inversely proportional to the capacity (efficiency) of working. This means if someone has higher capacity, they take less time, and vice-versa.
So, the ratio of time taken by A (\(T_A\)) and B (\(T_B\)) will be the inverse of their capacity ratio:
\[ T_A : T_B = \frac{1}{C_A} : \frac{1}{C_B} = \frac{1}{3} : \frac{1}{4} \]To simplify the ratio \[\frac{1}{3} : \frac{1}{4}\], we multiply both sides by the least common multiple (LCM) of 3 and 4, which is 12.
\[ T_A : T_B = \left(\frac{1}{3} \times 12\right) : \left(\frac{1}{4} \times 12\right) \] \[ T_A : T_B = 4 : 3 \]We are given that A takes 12 days to finish the work. So, \(T_A = 12\) days.
From the ratio, we can write:
\[ \frac{T_A}{T_B} = \frac{4}{3} \]Substitute the value of \(T_A\):
\[ \frac{12}{T_B} = \frac{4}{3} \]Now, solve for \(T_B\):
\[ 4 \times T_B = 12 \times 3 \] \[ 4 \times T_B = 36 \] \[ T_B = \frac{36}{4} \] \[ T_B = 9 \text{ days} \]A) 9 days is the correct answer. Based on the inverse relationship between capacity and time, B, being more efficient than A, will take less time to complete the same work.