We are given the sum of two natural numbers and their HCF. We need to find the number of possible pairs of these numbers, given an additional condition that one number is greater than the other. We will use the property that if HCF of two numbers is \(h\), then the numbers can be expressed as \(hx\) and \(hy\), where \(x\) and \(y\) are coprime.
Let the two natural numbers be \(x\) and \(y\). We are given:
Since the HCF of \(x\) and \(y\) is 20, we can write \(x = 20a\) and \(y = 20b\), where \(a\) and \(b\) are coprime natural numbers (i.e., \(\text{HCF}(a, b) = 1\)).
Substitute these expressions into the sum equation:
\[ 20a + 20b = 320 \]Divide by 20:
\[ 20(a + b) = 320 \]\[ a + b = \frac{320}{20} \]\[ a + b = 16 \]Now we need to find pairs of natural numbers \((a, b)\) such that their sum is 16 and they are coprime. Also, we are given \(x > y\), which implies \(20a > 20b\), so \(a > b\).
Let's list all pairs \((a, b)\) such that \(a + b = 16\) and \(a > b\):
The valid pairs \((a, b)\) are \((15, 1)\), \((13, 3)\), \((11, 5)\), and \((9, 7)\).
Each of these pairs corresponds to a unique pair of \((x, y)\):
There are 4 such possible pairs of \(x\) and \(y\).
B) 4
The problem requires us to code a given sequence of numbers and symbols based on a provided table and a set of conditions. We need to first find the direct codes for each element from the table and then apply the conditions in the given order to modify the codes if necessary.
D) A T A Y D X — This matches our derived code after applying Condition (iii).
Study the given diagram carefully and answer the question. The numbers in different sections indicate the number of persons living in a village.
How many surgeons are living in the village?
To find the number of surgeons living in the village, we need to analyze the Venn diagram provided, which shows the distribution of people among different professions: painters, musicians, and surgeons.
Thus, the total number of surgeons in the village is 185.
Correct Option: A) 185
The problem asks us to modify a given set of 3-letter words by changing each letter to the next letter in the English alphabetical order. Then, we need to count how many of the newly formed words will contain no vowels. The vowels are A, E, I, O, U.
From the analysis, only one word (DSZ) has no vowels.
B) One is the correct answer because only one of the newly formed words (DSZ) contains no vowels.
This question deals with electromagnetic induction, specifically Faraday's Law of Induction and Lenz's Law. Faraday's Law states that the magnitude of the induced electromotive force (EMF) is proportional to the rate of change of magnetic flux. Lenz's Law determines the direction (sign) of the induced EMF, stating that the induced current will flow in a direction that opposes the change in magnetic flux that produced it.
Correct Option: C) The induced potential differences are of the same magnitude(s) and have opposite sign(s) in the above two cases.
The question asks to identify the digestive enzyme present in saliva. This requires knowledge of the initial stages of digestion and the enzymes involved in breaking down different food components.
Correct Option: A) Amylase is the digestive enzyme present in saliva. It initiates the breakdown of starches into simpler sugars.
The question asks to identify the company booked by the CBI in February 2022 for India's largest bank fraud involving ₹22,842 crore and 28 banks. This is a factual question related to current affairs and major financial frauds in India.
A) ABG Shipyard — In February 2022, the CBI booked ABG Shipyard Ltd. and its directors for allegedly defrauding a consortium of 28 banks of ₹22,842 crore. This was reported as the biggest bank fraud in India's history at that time.