We are given the sum and difference of two numbers. We need to find the ratio of the larger number to the smaller number. We can set up a system of two linear equations with two variables and solve for the numbers, then calculate their ratio.
Let the two numbers be \(x\) and \(y\). Assume \(x\) is the larger number and \(y\) is the smaller number.
From the problem statement, the sum of the two numbers is 20. This can be written as an equation:
\[ x + y = 20 \quad \text{(Equation 1)} \]The difference between the two numbers is 16. Since we assumed \(x\) is the larger number, the difference is:
\[ x - y = 16 \quad \text{(Equation 2)} \]Now, we solve this system of linear equations. We can add Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 20 + 16 \]\[ 2x = 36 \]Divide by 2 to find the value of \(x\):
\[ x = \frac{36}{2} \]\[ x = 18 \]Substitute the value of \(x\) into Equation 1 to find \(y\):
\[ 18 + y = 20 \]\[ y = 20 - 18 \]\[ y = 2 \]So, the larger number is 18 and the smaller number is 2.
We need to find the ratio of the larger number to the smaller number, which is \(x : y\):
\[ \text{Ratio} = \frac{x}{y} = \frac{18}{2} \]\[ \text{Ratio} = \frac{9}{1} \]Therefore, the ratio is \(9 : 1\).
A) 9 : 1 is the correct ratio of the larger number to the smaller number, as calculated.
B) 2 : 11 — This would imply numbers like 4 and 22, which do not sum to 20 or have a difference of 16.
C) 1 : 9 — This is the ratio of the smaller number to the larger number, not the larger to the smaller as requested.
D) 11 : 2 — This would imply numbers like 22 and 4, which sum to 26, not 20.
The question asks for the major product when ethanol reacts with potassium dichromate in an acidic medium. This is a classic organic chemistry oxidation reaction. Potassium dichromate (\(\text{K}_2\text{Cr}_2\text{O}_7\)) in an acidic medium is a strong oxidizing agent.
Correct Option: A) Ethanoic acid is the major product because ethanol (a primary alcohol) is fully oxidized to a carboxylic acid by a strong oxidizing agent like acidified potassium dichromate.
To solve this problem, we need to trace Sumit's path step-by-step, noting his direction and distance traveled. We can visualize this by drawing a diagram or by tracking his coordinates. Finally, we calculate the displacement from his final position (Point G) back to his starting position (Point F).
Therefore, to reach Point F from Point G, he should drive 7 km towards the south.
Correct Option: B) 7 km towards the south
This question tests knowledge of the rules of reflection for a concave mirror. Specifically, it asks about the path of a reflected ray when the incident ray passes through the mirror's focus.
A) is parallel to the principal axis
This is a fundamental rule of reflection for concave mirrors. When an incident ray passes through the focus, the reflected ray always emerges parallel to the principal axis.
The question asks about the head of the Council of Ministers in the Government of India. This is a fundamental concept in Indian polity and governance, directly related to the executive branch's structure.
Correct Option: D) The Prime Minister is the head of the Council of Ministers. They are the chief executive authority and the leader of the government in India.
This problem involves analyzing inequalities to determine the logical relationship between different variables. We need to evaluate each conclusion based on the given statement and decide if it necessarily follows.
The given statement is: \(B \le L \le U \le A > D > E\)
Let's break down this statement into individual inequalities:
Now, let's evaluate each conclusion:
From the given statement, we have \(B \le L \le U \le A\). This implies \(B \le A\).
We also have \(A > D\).
Combining \(B \le A\) and \(A > D\), we can say that \(B\) can be less than \(D\), equal to \(D\), or greater than \(D\). For example:
Since \(B \le A\) and \(A > D\), it is not necessarily true that \(B < D\). For instance, if \(B=10\), \(A=10\), and \(D=5\), then \(B \le A\) and \(A > D\) are satisfied, but \(B < D\) is false (as \(10 < 5\) is false). Therefore, conclusion I does not necessarily follow.
From the given statement, we have \(B \le L \le U \le A\).
This chain of inequalities directly implies that \(B\) is less than or equal to \(A\), which can be written as \(B \le A\) or \(A \ge B\).
Therefore, conclusion II necessarily follows from the statement.
Only conclusion II follows.
Correct Option: C) Only conclusion II follows
We need to determine the tallest person among six friends (A, B, C, D, E, F) based on a set of comparative height statements. We will use these statements to establish the relative heights of the friends and then identify the tallest one.
"A is taller than only two other friends."
This implies that there are two friends shorter than A, and three friends taller than A. So, A is the 3rd shortest. We can represent this as:
"C is shorter than only one friend."
This means there is only one friend taller than C, and four friends shorter than C. So, C is the 2nd tallest. We can place C in our sequence:
"E is shorter than A but taller than B."
From this, we know \(B < E < A\). Since A is the 3rd shortest, E and B must be among the two shortest friends. This means E is the 2nd shortest and B is the shortest. Updating the sequence:
At this point, we have placed B, E, A, and C. The remaining friends are D and F. The two empty spots are the 4th shortest and the tallest.
\[ B < E < A < (\text{D or F}) < C < (\text{D or F}) \]"F is shorter than D."
This statement directly tells us the relative heights of D and F: \(F < D\). Since D and F are the only remaining friends to be placed in the 4th shortest and tallest positions, F must be the 4th shortest and D must be the tallest.
Combining all the information, the complete order from shortest to tallest is:
\[ B < E < A < F < C < D \]From this sequence, it is clear that D is the tallest among all the friends.
Correct Option: B) D is the tallest among all the friends.
If ‘+’ means ‘−’, ‘−’ means ‘×’, ‘×’ means ‘÷’, and ‘÷’ means ‘+’, what will come in place of the question mark (?) in the following equation?
This problem requires us to substitute the given symbol transformations into the equation and then evaluate the expression following the order of operations (BODMAS/PEMDAS).
The original equation is: \(21 + 7 - 3 \div 20 \times 5 = ?\)
After substitution, it becomes: \(21 - 7 \times 3 + 20 \div 5 = ?\)
BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. We perform operations in this order from left to right.
A) 4 is the correct answer, as derived from the step-by-step calculation.
The problem presents a series of letter groups, and we need to find the missing group. We will analyze the pattern for each letter's position (first, second, third, fourth, fifth) across the given groups to identify the rule and then apply it to find the missing term.
Combining the letters found: F, K, S, N, I. The missing term is FKSNI.
D) FKSNI is the correct option as it matches the pattern derived for each letter's position in the series.
The question asks to identify the first person believed to have brought Christianity to India. This requires knowledge of the historical origins of Christianity in India.
Correct Option: D) Saint Thomas is traditionally believed to be the first person to bring Christianity to India, arriving in 52 AD.