The number 0.1235235235 in the \( \frac{p}{q} \) form is equal to:
\(\frac{1}{12352}\)
\(\frac{127}{12352}\)
\(\frac{617}{4995}\)
\(\frac{100}{12525}\)
To convert a repeating decimal into a fraction \( \frac{p}{q} \) form, we can use an algebraic method. We set the given decimal equal to a variable, multiply it by powers of 10 to align the repeating part, and then subtract the equations to eliminate the repeating digits.
Correct Option: C) \( \frac{617}{4995} \) is the simplified fraction form of the given repeating decimal.
The question asks to identify the first metal used by humans. This is a historical and archaeological fact often covered in general knowledge and history sections of competitive exams. The key is to recall the properties of early metals and their availability.
D) Copper — Copper was the first metal to be extensively used by humans. It was found in its native form in many places and could be easily shaped by hammering, making it suitable for tools, weapons, and ornaments even before smelting techniques were developed.
The question asks for the establishment year of the Global Green Growth Institute (GGGI). To answer this, one needs to recall or know the historical information about the organization's founding.
B) 2010 — The Global Green Growth Institute (GGGI) was initially established as a non-profit organization (think tank) in 2010 by the Republic of Korea. It later transitioned into an international organization in 2012.
This problem requires substituting given numerical values for letters into a mathematical expression and then evaluating the expression following the order of operations (BODMAS/PEMDAS).
The final value of the expression \(A \div Z \times H - D\) is 20.
Correct Option: A) 20
The question asks for the location and year of the founding of the Ghadar Party. This requires knowledge of modern Indian history, specifically the revolutionary movements abroad during the early 20th century.
Correct Option: D) America, 1913. The Ghadar Party was founded in San Francisco, America, in 1913 by Indian immigrants, primarily Sikhs, with the objective of liberating India from British rule.
We are given two equations involving variables \(a\) and \(b\), and we need to find the value of an expression involving \(a\) and \(b\). We will use algebraic identities to simplify the given equations and find the required value.
We are given the equation:
\[ a + b = 1 \]Square both sides of the equation \(a + b = 1\):
\[ (a + b)^2 = 1^2 \]\[ a^2 + b^2 + 2ab = 1 \]We are also given the equation:
\[ a^4 + b^4 = 17 \]We know the identity \( (x+y)^2 = x^2 + y^2 + 2xy \). Let \(x = a^2\) and \(y = b^2\). Then,
\[ (a^2 + b^2)^2 = (a^2)^2 + (b^2)^2 + 2a^2b^2 \]\[ (a^2 + b^2)^2 = a^4 + b^4 + 2a^2b^2 \]Substitute the given value \(a^4 + b^4 = 17\) into the equation from step 4:
\[ (a^2 + b^2)^2 = 17 + 2a^2b^2 \]From step 2, we have \(a^2 + b^2 = 1 - 2ab\). Substitute this into the equation from step 5:
\[ (1 - 2ab)^2 = 17 + 2a^2b^2 \]Expand the left side using the identity \( (x-y)^2 = x^2 - 2xy + y^2 \). Here, \(x=1\) and \(y=2ab\):
\[ 1^2 - 2(1)(2ab) + (2ab)^2 = 17 + 2a^2b^2 \]\[ 1 - 4ab + 4a^2b^2 = 17 + 2a^2b^2 \]Rearrange the terms to solve for \(a^2b^2 - 2ab\). Move all terms to one side:
\[ 4a^2b^2 - 2a^2b^2 - 4ab + 1 - 17 = 0 \]\[ 2a^2b^2 - 4ab - 16 = 0 \]Divide the entire equation by 2:
\[ a^2b^2 - 2ab - 8 = 0 \]Isolate the expression \(a^2b^2 - 2ab\):
\[ a^2b^2 - 2ab = 8 \]C) 8
This problem involves the relationship between speed, distance, and time. When the distance traveled is constant, speed and time are inversely proportional. We will use this principle to find the ratio of the times taken by the bikes.
Let the speeds of the three bikes be \(S_1\), \(S_2\), and \(S_3\). The given ratio of their speeds is:
\[ S_1 : S_2 : S_3 = 3 : 4 : 5 \]Let the time taken by these bikes to travel the same distance be \(T_1\), \(T_2\), and \(T_3\). Let the common distance be \(D\).
The fundamental relationship between speed, distance, and time is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]Since the distance \(D\) is constant for all three bikes, we can write:
\[ S = \frac{D}{T} \]From this, we can see that time is inversely proportional to speed when distance is constant:
\[ T = \frac{D}{S} \]Therefore, the ratio of the times taken will be the inverse ratio of their speeds:
\[ T_1 : T_2 : T_3 = \frac{1}{S_1} : \frac{1}{S_2} : \frac{1}{S_3} \]Substitute the given speed ratio:
\[ T_1 : T_2 : T_3 = \frac{1}{3} : \frac{1}{4} : \frac{1}{5} \]To eliminate the fractions and express this ratio in whole numbers, find the least common multiple (LCM) of the denominators (3, 4, 5). The LCM of 3, 4, and 5 is \(3 \times 4 \times 5 = 60\).
Multiply each term in the ratio by the LCM (60):
\[ T_1 : T_2 : T_3 = \left(\frac{1}{3} \times 60\right) : \left(\frac{1}{4} \times 60\right) : \left(\frac{1}{5} \times 60\right) \]\[ T_1 : T_2 : T_3 = 20 : 15 : 12 \]A) 20 : 15 : 12 is the correct ratio of the time taken by the bikes.
The problem involves solving an arithmetic addition puzzle where letters represent single digits. We need to find the highest possible value for X in the given equation by analyzing the column-wise addition, considering carries.
Write down the given equation in a column format:
7 X 5 + 2 Y 3 + 8 Z 7 ------- 18 0 5
Analyze the units column (rightmost column):
The sum of the units digits is \(5 + 3 + 7\).
\[ 5 + 3 + 7 = 15 \]
The units digit of the sum is 5, which matches the given result. This means there is a carry-over of 1 to the tens column.
Analyze the tens column (middle column):
The sum of the tens digits, plus the carry-over from the units column, must result in a number whose units digit is 0 (from 1805). The tens digits are X, Y, and Z.
\[ X + Y + Z + 1 (\text{carry-over}) = \text{_0} \]
Since X, Y, and Z are single digits (0-9), their maximum sum is \(9+9+9 = 27\). Adding the carry-over of 1, the maximum possible sum for this column is \(27+1 = 28\). Therefore, \(X + Y + Z + 1\) must be either 10 or 20.
This implies that \(X + Y + Z\) must be either 9 (if \(10-1=9\)) or 19 (if \(20-1=19\)).
Let's denote the carry-over from this column to the hundreds column as \(C_2\). If \(X+Y+Z+1=10\), then \(C_2=1\). If \(X+Y+Z+1=20\), then \(C_2=2\).
Analyze the hundreds column (leftmost column):
The sum of the hundreds digits, plus the carry-over from the tens column (\(C_2\)), must result in 18 (from 1805). The hundreds digits are 7, 2, and 8.
\[ 7 + 2 + 8 + C_2 = 18 \]
Calculate the sum of the known digits:
\[ 7 + 2 + 8 = 17 \]
Now substitute this into the equation:
\[ 17 + C_2 = 18 \]
Solving for \(C_2\):
\[ C_2 = 18 - 17 = 1 \]
This means the carry-over from the tens column (\(C_2\)) must be 1.
Relate \(C_2\) back to the tens column:
We found that \(C_2 = 1\). This implies that \(X + Y + Z + 1\) must have been 10 (because if it were 20, \(C_2\) would be 2).
\[ X + Y + Z + 1 = 10 \]
Therefore,
\[ X + Y + Z = 9 \]
Find the highest value of X:
We need to maximize X. Since Y and Z are single digits (0-9), their minimum value is 0. To make X as large as possible, Y and Z should be as small as possible.
Let \(Y = 0\) and \(Z = 0\).
Substitute these values into the equation \(X + Y + Z = 9\):
\[ X + 0 + 0 = 9 \]
Thus,
\[ X = 9 \]
This is the highest possible value for X, as Y and Z cannot be negative.
A) 9 is the highest value X can take. This is achieved when Y and Z are both 0.
The question asks to identify the oceanic ridge of which the Telegraph Plateau is a part. This requires knowledge of major submarine topographic features and their geographical locations.
Correct Option: B) North Atlantic Ridge
The Telegraph Plateau is a well-known feature of the Mid-Atlantic Ridge, specifically located in its northern segment, hence it is part of the North Atlantic Ridge.
The problem presents a series of letter groups, and we need to identify the pattern to find the missing term. We will analyze the position of each letter in the alphabet for each term in the series.
The number of letters in each term is decreasing by 1: 4, 3, 2, 1. So, the missing term will be a single letter.
Now let's examine the first letter of each term:
This doesn't seem to follow a simple arithmetic progression. Let's look at the last letter of each term:
This also doesn't show an obvious pattern.
Let's reconsider the sequence as a whole and look for a different type of pattern, perhaps related to the number of letters or their positions in a more complex way.
Let's look at the sum of the positional values of the letters in each term:
It appears the sum of the positional values of the letters in each term is consistently 13.
The next term in the series should have 1 letter, and its positional value must be 13.
The 13th letter of the alphabet is M.
B) M — The pattern identified is that the sum of the positional values of the letters in each term is 13. The number of letters in each term decreases by one (4, 3, 2, 1). Therefore, the next term should be a single letter whose positional value is 13. The 13th letter of the alphabet is M.