The problem asks us to find the next number in the given series: 5, 7, 14, 36, ?. We need to identify the pattern or rule governing the sequence to determine the missing term. We will look for relationships between consecutive terms, such as addition, subtraction, multiplication, division, or powers.
Let's summarize the pattern:
We observed that the 2nd differences form a geometric progression with a common ratio of 3:
\[ 5 \times 3 = 15 \]\[ 15 \times 3 = 45 \]Therefore, \(\text{Y} = 45\).Now, we can find \(\text{X}\):
\[ \text{X} - 22 = 45 \]\[ \text{X} = 45 + 22 \]\[ \text{X} = 67 \]Finally, we can find the missing term:\[ ? = 36 + \text{X} \]\[ ? = 36 + 67 \]\[ ? = 103 \]To find the mean (average) of a given set of data, we need to sum all the values in the data set and then divide by the total number of values. This is a fundamental concept in statistics.
The mean of the given data is 8.38.
B) 8.38 is the correct mean calculated by summing the data points and dividing by their count.
This problem involves solving an arithmetic addition puzzle where letters represent digits. We need to find the highest possible value for X by analyzing the addition column by column, starting from the units place and considering carry-overs.
Analyze the units column: The units column is \(1 + 7 + 3\). The sum is \(11\). In the result \(1471\), the units digit is \(1\). This means there is a carry-over of \(1\) to the tens column.
\[ 1 + 7 + 3 = 11 \]So, we write down \(1\) in the units place of the sum and carry over \(1\) to the tens place.
Analyze the tens column: The tens column is \(X + Y + Z\) plus the carry-over from the units column, which is \(1\). The sum of this column should result in a units digit of \(7\) (from \(1471\)).
\[ X + Y + Z + 1 = \text{_}7 \]This implies that \(X + Y + Z\) must end in \(6\). So, \(X + Y + Z = 6\) or \(X + Y + Z = 16\) or \(X + Y + Z = 26\), and so on. Since X, Y, Z are single digits (0-9), the maximum sum \(9+9+9 = 27\). So, \(X + Y + Z\) can be \(6\) or \(16\) or \(26\).
Let's consider the carry-over to the hundreds column. If \(X + Y + Z + 1\) is \(7\), the carry-over to the hundreds column is \(0\). If \(X + Y + Z + 1\) is \(17\), the carry-over is \(1\). If \(X + Y + Z + 1\) is \(27\), the carry-over is \(2\).
Analyze the hundreds column: The hundreds column is \(5 + 6 + 3\) plus the carry-over from the tens column. The sum of this column should result in \(14\) (from \(1471\)).
\[ 5 + 6 + 3 + \text{carry-over from tens} = 14 \] \[ 14 + \text{carry-over from tens} = 14 \]This means the carry-over from the tens column must be \(0\).
Deduce the value of \(X + Y + Z\): Since the carry-over from the tens column is \(0\), it means \(X + Y + Z + 1\) must be \(7\).
\[ X + Y + Z + 1 = 7 \] \[ X + Y + Z = 6 \]Find the highest value of X: We need to find the highest possible value for \(X\). Since \(X, Y, Z\) are distinct digits (implied by typical puzzles of this type, though not explicitly stated, we assume they can be different digits) and they are single digits from \(0\) to \(9\).
To maximize \(X\), we must minimize \(Y\) and \(Z\). The smallest possible distinct digits for \(Y\) and \(Z\) are \(0\) and \(1\).
If \(Y=0\) and \(Z=1\), then:
\[ X + 0 + 1 = 6 \] \[ X = 5 \]However, the problem does not state that X, Y, Z must be distinct. If they can be the same, to maximize X, we minimize Y and Z. The smallest possible value for Y and Z is \(0\).
If \(Y=0\) and \(Z=0\), then:
\[ X + 0 + 0 = 6 \] \[ X = 6 \]This is the highest possible value for X when \(X+Y+Z=6\) and Y, Z are minimized.
Correct Option: C) 6
Solve the given equation.
\(\sqrt{12321}\) × \(\sqrt{?}\) = 13431
To solve the given equation, we first need to find the square root of 12321. Then, we can rearrange the equation to isolate the unknown term, which is under a square root. Finally, we will square both sides of the equation to find the value of the unknown.
We can find the square root of 12321 by estimation or by prime factorization. Let's try estimation:
The given equation is \(\sqrt{12321} \times \sqrt{?} = 13431\).
Substitute \(\sqrt{12321} = 111\):
\[ 111 \times \sqrt{?} = 13431 \]Divide both sides by 111:
\[ \sqrt{?} = \frac{13431}{111} \]Let's divide 13431 by 111:
\[ \frac{13431}{111} = 121 \]So,\[ \sqrt{?} = 121 \]To find the unknown, we need to square both sides of the equation:
\[ ? = 121^2 \]\[ ? = 121 \times 121 \]\[ ? = 14641 \]C) 14641 is the correct answer.
Which two signs need to be interchanged to make the following equation correct?
9 ÷ 3 - 15 × 5 ÷ 6 = 2
The problem requires us to find which two mathematical signs, when interchanged in the given equation, will make the equation correct. We will test each option by substituting the interchanged signs into the original equation and then evaluating the expression using the BODMAS/PEMDAS rule.
The given equation is:
\(9 \div 3 - 15 \times 5 \div 6 = 2\)
Let's test each option:
Option A: Interchanging - and \(\div\)
The equation becomes:
\[ 9 - 3 \div 15 \times 5 \div 6 \]First, perform divisions and multiplications from left to right:
\[ 9 - \frac{3}{15} \times 5 \div 6 \] \[ 9 - \frac{1}{5} \times 5 \div 6 \] \[ 9 - 1 \div 6 \] \[ 9 - \frac{1}{6} \]Now, perform subtraction:
\[ \frac{54 - 1}{6} = \frac{53}{6} \]This does not equal 2. Let's re-check the problem statement. The original equation is \(9 \div 3 - 15 \times 5 \div 6 = 2\). The options are signs to be interchanged. Let's assume the last operation in the original equation is not division, but the options are for the signs given. The question has a typo in the original equation. Let's assume the equation is \(9 \div 3 - 15 \times 5 + 6 = 2\) or similar, or the last sign is not a division.
Let's re-evaluate the question as it is written: \(9 \div 3 - 15 \times 5 \div 6 = 2\). We need to interchange two signs.
Let's assume the last operation is not a division, but a plus sign, as is common in such problems. If the original equation was \(9 \div 3 - 15 \times 5 + 6 = 2\), then:
Original equation evaluation: \(3 - 75 + 6 = -66\). This is not 2.
Let's stick to the given equation: \(9 \div 3 - 15 \times 5 \div 6 = 2\)
Option A: Interchanging - and \(\div\)
The equation becomes:
\[ 9 - 3 \div 15 \times 5 - 6 \]Perform divisions and multiplications:
\[ 9 - \frac{3}{15} \times 5 - 6 \] \[ 9 - \frac{1}{5} \times 5 - 6 \] \[ 9 - 1 - 6 \]Perform subtractions:
\[ 8 - 6 = 2 \]This matches the right-hand side of the equation. So, Option A is correct.
A) When '-' and '\(\div\)' are interchanged, the equation becomes:
\[ 9 - 3 \div 15 \times 5 - 6 \]Following the order of operations (BODMAS/PEMDAS):
First, division:
\[ 9 - \frac{3}{15} \times 5 - 6 \] \[ 9 - \frac{1}{5} \times 5 - 6 \]Next, multiplication:
\[ 9 - 1 - 6 \]Finally, subtraction:
\[ 8 - 6 = 2 \]Since \(2 = 2\), the equation is correct with this interchange.
B) +, - (Assuming there is a '+' sign somewhere in the original equation, or if the last '\(\div\)' was meant to be a '+')
If we consider the original equation as given: \(9 \div 3 - 15 \times 5 \div 6 = 2\). There is no '+' sign to interchange with '-'. So this option is not directly applicable to the given equation.
However, if we assume the question implies interchanging '-' with a hypothetical '+' sign, or if the last '\(\div\)' was meant to be a '+', let's test a common variation: \(9 \div 3 + 15 \times 5 - 6 = 2\)
Original equation (with assumed +): \(3 + 75 - 6 = 78 - 6 = 72 \neq 2\)
Interchanging '+' and '-': \(9 \div 3 - 15 \times 5 + 6\)
\[ 3 - 75 + 6 = -72 + 6 = -66 \neq 2 \]Thus, this option is incorrect.
C) \(\times\), + (Again, assuming a '+' sign)
If we consider the original equation as given: \(9 \div 3 - 15 \times 5 \div 6 = 2\). There is no '+' sign to interchange with '\(\times\)'. So this option is not directly applicable to the given equation.
If we assume the last '\(\div\)' was meant to be a '+', then equation is \(9 \div 3 - 15 \times 5 + 6 = 2\)
Interchanging '\(\times\)' and '+': \(9 \div 3 - 15 + 5 \times 6\)
\[ 3 - 15 + 30 = -12 + 30 = 18 \neq 2 \]Thus, this option is incorrect.
D) \(\times\), \(\div\)
The equation becomes:
\[ 9 \times 3 - 15 \div 5 \times 6 \]Perform multiplications and divisions:
\[ 27 - 3 \times 6 \] \[ 27 - 18 \]Perform subtraction:
\[ 9 \neq 2 \]Thus, this option is incorrect.
The question asks for the name of the political party founded by Subhash Chandra Bose after his departure from the Indian National Congress. This requires knowledge of key events in Indian independence history, specifically related to Subhash Chandra Bose's political career.
C) All India Forward Bloc — Subhash Chandra Bose formed the Forward Bloc on May 3, 1939, after resigning from the presidency of the Indian National Congress. This party aimed to consolidate the left-wing elements within the Congress and pursue a more radical approach to achieving independence.
The question asks to identify the first European president of the Indian National Congress (INC). This requires knowledge of the early history of the INC and its presidents.
A) George Yule was the first European (and non-Indian) to become the president of the Indian National Congress. He presided over the INC session in Allahabad in 1888.
The question asks to identify the primary ore of aluminium. This requires knowledge of common metallic ores and their corresponding metals. We will evaluate each option to determine which one is the ore of aluminium.
Correct Option: C) Bauxite is the chief ore of aluminium. It is a sedimentary rock with a relatively high aluminium content, formed from the laterization of aluminous rocks.
For a moderately asymmetrical distribution, there is an empirical relationship between the mean, median, and mode. This relationship is given by the formula: \( \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \). We can use this formula to find the median when the mode and mean are known.
Identify the given values:
Recall the empirical formula for a moderately asymmetrical distribution:
\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]Substitute the given values into the formula:
\[ 7.52 = 3 \times \text{Median} - 2 \times 9.83 \]Calculate the product of \(2 \times \text{Mean}\):
\[ 2 \times 9.83 = 19.66 \]Substitute this value back into the equation:
\[ 7.52 = 3 \times \text{Median} - 19.66 \]Rearrange the equation to solve for the Median:
\[ 3 \times \text{Median} = 7.52 + 19.66 \]\[ 3 \times \text{Median} = 27.18 \]Divide by 3 to find the Median:
\[ \text{Median} = \frac{27.18}{3} \]\[ \text{Median} = 9.06 \]D) 9.06 is the correct answer. The median of the distribution, calculated using the empirical formula for moderately asymmetrical distributions, is 9.06.
The question asks for the city where the National Museum of Indian Cinema was inaugurated in January 2019. This is a factual question related to current affairs and cultural institutions in India.
Correct Option: C) Mumbai
The National Museum of Indian Cinema was inaugurated in Mumbai in January 2019. Mumbai is the hub of the Indian film industry, making it a fitting location for such a museum.