The problem asks us to find the value of an expression involving absolute values, given a linear relationship between two variables, \(a\) and \(b\). We will first use the given equation to find the relationship between \(a-b\) and \(b-a\), and then substitute these values into the absolute value expression.
We are given the equation:
\[ a - 5 = b \]To find \(a-b\), we can subtract \(b\) from both sides of the equation and add \(5\) to both sides:
\[ a - b = 5 \]Since \(a-b = 5\), then \(b-a\) will be the negative of this value:
\[ b - a = -(a - b) = -5 \]The expression we need to evaluate is \(|a - b| + |b - a|\). Substitute the values we found:
\[ |a - b| + |b - a| = |5| + |-5| \]The absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart.
\[ |5| = 5 \]\[ |-5| = 5 \]Add the absolute values:
\[ 5 + 5 = 10 \]A) 10 — The value of \(|a - b| + |b - a|\) is \(|5| + |-5| = 5 + 5 = 10\).
The problem asks for the second term in a sequence of numbers that satisfy specific remainder conditions when divided by 2, 3, and 8. This is a classic problem that can be solved using the Chinese Remainder Theorem concept or by systematically finding numbers that satisfy the conditions. We will find the general form of such numbers and then identify the second smallest positive integer.
Let the number be \(N\). We are given the following conditions:
From the first condition, \(N \equiv 1 \pmod{2}\), we know that \(N\) must be an odd number. So, \(N = 2k_1 + 1\) for some integer \(k_1\).
From the third condition, \(N \equiv 7 \pmod{8}\), we know that \(N\) can be written as \(N = 8k_2 + 7\) for some integer \(k_2\).
Let's check the first condition with \(N = 8k_2 + 7\):
\(N = 8k_2 + 7 = 2(4k_2 + 3) + 1\). This form always leaves a remainder of 1 when divided by 2, so the first condition is satisfied by any number of the form \(8k_2 + 7\).
Now, let's incorporate the second condition: \(N \equiv 2 \pmod{3}\). Substitute \(N = 8k_2 + 7\) into this congruence:
\[ 8k_2 + 7 \equiv 2 \pmod{3} \]Simplify the terms modulo 3:
So, the congruence becomes:
\[ 2k_2 + 1 \equiv 2 \pmod{3} \]Subtract 1 from both sides:
\[ 2k_2 \equiv 1 \pmod{3} \]To solve for \(k_2\), we need to find the multiplicative inverse of 2 modulo 3. Since \(2 \times 2 = 4 \equiv 1 \pmod{3}\), the inverse of 2 modulo 3 is 2.
Multiply both sides by 2:
\[ 2 \times 2k_2 \equiv 2 \times 1 \pmod{3} \]\[ 4k_2 \equiv 2 \pmod{3} \]\[ k_2 \equiv 2 \pmod{3} \]This means \(k_2\) can be written as \(k_2 = 3m + 2\) for some integer \(m\).
Substitute this expression for \(k_2\) back into the equation for \(N\):
\[ N = 8k_2 + 7 \]\[ N = 8(3m + 2) + 7 \]\[ N = 24m + 16 + 7 \]\[ N = 24m + 23 \]This is the general form of numbers that satisfy all three conditions. The numbers in the sequence are obtained by substituting integer values for \(m\).
The question asks for the second term in the sequence, which is 47.
C) 47 is the second term in the sequence. Let's verify:
37 is the first term \(N=24(0)+23=23\), not 37. The first term is 23. If we take \(N=24m+23\), for \(m=0\), \(N=23\). For \(m=1\), \(N=47\). So 37 is not part of the sequence.
The question asks for the original name of the Mahabharata. This requires knowledge of ancient Indian texts and their historical evolution.
Correct Option: A) Jaya Samhita is the original and earliest known name for the core narrative of what later evolved into the Mahabharata.
This problem involves compound interest. We need to divide a total amount between two individuals, A and B, such that their final amounts after different time periods are equal, given a fixed annual compound interest rate. We will use the compound interest formula to set up an equation and solve for the initial amounts.
Let the initial amount received by A be \(P_A\) and the initial amount received by B be \(P_B\).
The total amount to be divided is ₹20,609. So, \(P_A + P_B = 20609\).
The interest rate \(R\) is 3% per annum, compounded yearly. So, \(R = 0.03\).
The time period for A is \(T_A = 7\) years.
The time period for B is \(T_B = 9\) years.
The formula for compound amount \(A\) is \(A = P(1 + R)^T\).
According to the problem, the amount of A after 7 years is equal to the amount of B after 9 years. Therefore,
\[ P_A (1 + 0.03)^7 = P_B (1 + 0.03)^9 \]Simplify the equation:
\[ P_A (1.03)^7 = P_B (1.03)^9 \]Divide both sides by \((1.03)^7\):
\[ P_A = P_B \frac{(1.03)^9}{(1.03)^7} \]\[ P_A = P_B (1.03)^{9-7} \]\[ P_A = P_B (1.03)^2 \]Calculate \((1.03)^2\):
\[ (1.03)^2 = 1.03 \times 1.03 = 1.0609 \]So, we have the relationship:
\[ P_A = 1.0609 P_B \]Now, substitute this into the total amount equation: \(P_A + P_B = 20609\)
\[ 1.0609 P_B + P_B = 20609 \]\[ (1.0609 + 1) P_B = 20609 \]\[ 2.0609 P_B = 20609 \]Solve for \(P_B\):
\[ P_B = \frac{20609}{2.0609} \]\[ P_B = 10000 \]Now, find \(P_A\):
\[ P_A = 20609 - P_B \]\[ P_A = 20609 - 10000 \]\[ P_A = 10609 \]Thus, A receives ₹10,609 and B receives ₹10,000.
Correct Option: A) A = ₹10,609; B = ₹10,000
To find the domain of a trigonometric function like cosecant, we need to identify the values of \(x\) for which the function is defined. The cosecant function is the reciprocal of the sine function, so it is undefined when the sine function is zero.
Correct Option: A) R - {nπ|n ∈ Z}
The question asks to identify the engineer responsible for bringing the ruins of Hampi to light. This requires knowledge of historical figures associated with archaeological discoveries in India.
C) Colonel Colin Mackenzie — He was the first Surveyor General of India and played a pivotal role in surveying and documenting numerous historical sites, including Hampi, in the early 19th century.
To solve this problem, we need to know the basic conversion factor between a nibble and a bit. Once this fundamental relationship is established, we can simply multiply to find the total number of bits for 10 nibbles.
From the definition, we know that:
\[ 1 \text{ nibble} = 4 \text{ bits} \]To find out how many bits are in 10 nibbles, we multiply the number of nibbles by the conversion factor:
\[ 10 \text{ nibbles} = 10 \times (4 \text{ bits}) \] \[ 10 \text{ nibbles} = 40 \text{ bits} \]Therefore, 10 nibbles are equal to 40 bits.
Correct Option: C) 40
For a moderately asymmetrical distribution, the empirical relationship between the mean, median, and mode is given by Karl Pearson's formula. We will use this formula to find the median when the mean and mode are known.
For a moderately asymmetrical distribution, the relationship between the three measures of central tendency (mean, median, and mode) is approximately:
\[ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \]Let the Median be \( M_d \).
\[ 25.2 = 3 \times M_d - 2 \times 27.5 \]Add 55 to both sides of the equation:
\[ 25.2 + 55 = 3 M_d \]\[ 80.2 = 3 M_d \]Divide both sides by 3:
\[ M_d = \frac{80.2}{3} \]\[ M_d \approx 26.7333... \]Rounding to two decimal places, the median is approximately 26.73.
A) 26.73 is the correct value for the median, calculated using Karl Pearson's empirical formula for a moderately asymmetrical distribution.
The problem involves finding the highest possible value of X in a given addition equation where Y and Z are single digits. We need to perform column-wise addition, considering carries, to determine the possible values for X and then identify the highest among them.
Thus, the highest possible value of X is \(9\).
B) 9 — As derived, by minimizing Y and Z to 0, the maximum value for X is 9.
The question asks us to find a relationship between the first two terms and apply the same relationship to the third term to find the fourth term. This is a common type of analogy question where we need to identify the logical connection between the given pair of words.
Correct Option: B) Theatre