The question asks to identify which biotic component of an ecosystem is present in the greatest number. This relates to the concept of ecological pyramids, specifically the pyramid of numbers, which illustrates the number of individual organisms at each trophic level.
B) Producers. Producers, such as plants, form the base of the food chain and are typically the most numerous organisms in an ecosystem to support the energy requirements of all higher trophic levels. This is a fundamental principle of ecological pyramids, particularly the pyramid of numbers.
The problem asks us to identify the possible effects of a given cause. The cause is heavy rain and lightning in City X, with more rain predicted. We need to evaluate each effect (I, II, and III) to determine if it is a logical consequence of the cause.
Correct Option: C) Both II and III are possible effects.
The question asks about the fundamental phenomenon responsible for image formation in mirrors. We need to recall the basic principles of how light interacts with a mirror surface.
Correct Option: C) reflection
To find the greatest number, we first need to determine the value of one part of the ratio. We can do this by summing the ratio parts and then dividing the total sum of the numbers by this sum of ratio parts. Once we have the value of one part, we multiply it by the largest ratio component to find the greatest number.
Let the three numbers be \(5x\), \(6x\), and \(8x\), where \(x\) is a common multiplier.
The sum of these three numbers is given as 3800.
So, we can write the equation:
\[ 5x + 6x + 8x = 3800 \]Combine the terms on the left side of the equation:
\[ (5 + 6 + 8)x = 3800 \]\[ 19x = 3800 \]Solve for \(x\) by dividing both sides by 19:
\[ x = \frac{3800}{19} \]\[ x = 200 \]Now that we have the value of \(x\), we can find the three numbers:
The greatest number among these is 1600.
Correct Option: D) 1600
This is a circular arrangement puzzle. We need to deduce the seating arrangement of six people (A, B, C, D, E, F) around a circular table based on the given clues and then determine E's position relative to B.
Let's visualize it:
E C
/ \
D A
\ /
F B
Starting from B and moving clockwise (to the right):
D) Third to the right
To find the missing number in the series, we need to identify the pattern or relationship between consecutive terms. We will calculate the differences between adjacent numbers to see if there's a consistent arithmetic progression or another identifiable sequence.
C) 178 – This is the correct number that completes the series based on the identified pattern of increasing differences.
A man invested \( \frac{1}{3} \) of his income at 7% simple interest, \( \frac{1}{4} \) of his income at 8% simple interest, and the remainder of his income at 10% simple interest. If the total interest at the end of one year was ₹561, what was his initial income?
To solve this problem, we need to calculate the simple interest earned on each portion of the income and then sum them up to match the total interest given. We will represent the total income as a variable and express each investment portion and its corresponding interest in terms of this variable. Finally, we will solve for the total income.
Thus, his initial income was ₹6,600.
B) ₹6,600 is the correct answer, as calculated by summing the simple interests from each investment portion and equating it to the total interest received.
This is a coding-decoding problem where letters are assigned specific numerical codes. We need to find the code for the letter 'E' by comparing the given coded words and identifying common letters and their corresponding codes.
Correct Option: C) 2
The problem asks us to express a given algebraic expression as a perfect square. We need to recognize the pattern of the square of a trinomial, which is \((x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx\). By comparing the given expression with this general form, we can determine the values of \(x\), \(y\), and \(z\).
Recall the identity for the square of a trinomial:
The general formula for the square of a trinomial is:
\[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \]Analyze the given expression:
The given expression is \(a^2+9b^2+c^2-6ab+6bc-2ac\).
Identify the squared terms:
We can see \(a^2\), \(9b^2\), and \(c^2\). These correspond to \(x^2\), \(y^2\), and \(z^2\).
So, \(x^2 = a^2 \implies x = a\) (assuming positive \(a\)).
\(y^2 = 9b^2 \implies y = \pm 3b\).
\(z^2 = c^2 \implies z = \pm c\).
Determine the signs of \(y\) and \(z\) using the cross-product terms:
The cross-product terms are \(-6ab\), \(+6bc\), and \(-2ac\).
Consider \(-6ab\): This term is \(2xy\). Since \(x=a\), we have \(2(a)y = -6ab\), which means \(2ay = -6ab\). Dividing by \(2a\), we get \(y = -3b\).
Consider \(-2ac\): This term is \(2zx\). Since \(x=a\), we have \(2z(a) = -2ac\), which means \(2az = -2ac\). Dividing by \(2a\), we get \(z = -c\).
Verify with \(+6bc\): This term is \(2yz\). Using \(y=-3b\) and \(z=-c\), we get \(2(-3b)(-c) = 2(3bc) = +6bc\). This matches the given term.
Form the perfect square:
Based on our findings, \(x=a\), \(y=-3b\), and \(z=-c\).
Therefore, the expression can be written as \((a + (-3b) + (-c))^2\), which simplifies to \((a-3b-c)^2\).
Expand to verify (optional but good practice):
\[ (a-3b-c)^2 = a^2 + (-3b)^2 + (-c)^2 + 2(a)(-3b) + 2(-3b)(-c) + 2(-c)(a) \]\[ = a^2 + 9b^2 + c^2 - 6ab + 6bc - 2ac \]This matches the original expression.Correct Option: C) \((a-3b-c)^2\)
We are given the sum of ages of 5 children born at intervals of 2 years. This means their ages form an arithmetic progression. We can set up an equation based on their ages and the given sum to find the age of the youngest child, and then determine the age of the eldest child.
Let the age of the youngest child be \(x\) years.
Since the children are born at intervals of 2 years, their ages will be:
The sum of their ages is given as 40 years. So, we can write the equation:
\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 40 \]Combine the terms:
\[ 5x + (2 + 4 + 6 + 8) = 40 \]\[ 5x + 20 = 40 \]Solve for \(x\):
\[ 5x = 40 - 20 \]\[ 5x = 20 \]\[ x = \frac{20}{5} \]\[ x = 4 \]So, the age of the youngest child is 4 years.
The age of the eldest child is \(x + 8\). Substitute the value of \(x\):
\[ \text{Eldest child's age} = 4 + 8 \]\[ \text{Eldest child's age} = 12 \text{ years} \]Correct Option: D) 12 years