The problem provides information about three co-prime numbers \(x, y, z\) in non-decreasing order (\(x \le y \le z\)). We are given the product of the first two numbers (\(x \cdot y = 143\)) and the product of the last two numbers (\(y \cdot z = 195\)). We need to find the sum of these three numbers. The key to solving this is to find the common factor \(y\) from the two products and then determine \(x\) and \(z\).
Identify the given equations:
We are also given that \(x, y, z\) are co-prime to each other, and \(x \le y \le z\).
Find the common factor \(y\):
From the two equations, \(y\) is a common factor of 143 and 195. Therefore, \(y\) must be the Greatest Common Divisor (GCD) of 143 and 195, or a factor of the GCD.
Let's find the prime factors of 143:
\[ 143 = 11 \times 13 \]Let's find the prime factors of 195:
\[ 195 = 3 \times 5 \times 13 \]The common factor between 143 and 195 is 13. So, \(y = 13\).
Determine \(x\) and \(z\):
Verify the conditions:
Calculate the sum of the three numbers:
\[ x + y + z = 11 + 13 + 15 = 39 \]C) The sum of the three numbers \(11 + 13 + 15\) is \(39\).
The question asks for the geometric term that describes a line connecting two specific end points. We need to recall the definitions of basic geometric figures like lines, line segments, rays, parallel lines, and intersecting lines.
Correct Option: B) line segment
The area of a square is 900 cm². Its perimeter is equal to the perimeter of a regular hexagon. What is the area (in cm²) of the hexagon?
The problem requires us to find the area of a regular hexagon given the area of a square and the condition that their perimeters are equal. We will first find the side length of the square from its area, then calculate its perimeter. This perimeter will be used to find the side length of the regular hexagon. Finally, we will calculate the area of the hexagon using its side length.
Find the side length of the square:
Let the side length of the square be \(s\). The area of a square is given by \(s^2\).
Find the perimeter of the square:
The perimeter of a square is given by \(4s\).
Find the side length of the regular hexagon:
The problem states that the perimeter of the square is equal to the perimeter of a regular hexagon. A regular hexagon has 6 equal sides. Let the side length of the regular hexagon be \(a\).
Find the area of the regular hexagon:
The area of a regular hexagon can be calculated using the formula:
Substitute the value of \(a = 20 \text{ cm}\):
\[ \text{Area of hexagon} = \frac{3\sqrt{3}}{2} (20)^2 \] \[ \text{Area of hexagon} = \frac{3\sqrt{3}}{2} \times 400 \] \[ \text{Area of hexagon} = 3\sqrt{3} \times 200 \] \[ \text{Area of hexagon} = 600\sqrt{3} \text{ cm}^2 \]D) \(600\sqrt{3}\)
As calculated, the area of the hexagon is \(600\sqrt{3} \text{ cm}^2\).
The question asks to identify the part of a flower that develops into a fruit after fertilisation. This is a fundamental concept in plant reproduction, specifically focusing on the post-fertilisation changes in angiosperms.
Correct Option: B) Ovary — After fertilisation, the ovary, which contains the ovules, matures and develops into the fruit. The fruit encloses the seeds that develop from the ovules.
This problem involves a property of points inside a rectangle. For any point P inside a rectangle ABCD, the sum of the squares of the distances from P to opposite vertices are equal. That is, \(PA^2 + PC^2 = PB^2 + PD^2\). We can use this property to find the unknown length PD.
D) 18 — The length PD is 18 cm, as calculated using the property \(PA^2 + PC^2 = PB^2 + PD^2\).
This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. The best way to solve such problems is by drawing Venn diagrams for each statement and then checking if the conclusions logically follow from these diagrams. Remember to consider all possible diagrams for ambiguous statements.
This means there is an overlap between the set of 'glasses' and the set of 'windows'.
Diagram 1: Intersecting circles for Glasses and Windows.
This means there is an overlap between the set of 'windows' and the set of 'white'.
Diagram 2: Add a circle for White, overlapping with the Windows circle. The overlap might or might not include the 'glasses' part of 'windows'.
This means the entire set of 'white' is contained within the set of 'brittle'.
Diagram 3: Draw a larger circle for Brittle that completely encloses the 'white' circle.
Let's draw a combined diagram or consider the relationships:
From these, we can infer:
From the statements, we know Some Glasses are Windows, Some Windows are White. It is possible that some white are glasses (if the 'white' part of 'windows' overlaps with the 'glasses' part of 'windows'), but it is not a definite conclusion. We cannot be 100% sure. For example, the 'white' windows could be entirely different from the 'glass' windows.
We know All White are Brittle. If some white are glasses (which is not definite), then some brittle would be glasses. But since 'Some white are glasses' is not definite, 'Some brittle are glasses' is also not definite.
We know 'Some windows are white' and 'All white are brittle'. If some windows are white, and all white things are brittle, then it logically follows that those specific windows that are white must also be brittle. Therefore, 'Some windows are brittle' or 'Some brittle are windows' is a definite conclusion.
C) Only conclusion III is true.
Conclusion III directly follows from the statements: Since some windows are white, and all white things are brittle, it must be true that some windows are brittle. Therefore, some brittle things are windows.
The question asks to identify the term in the Preamble of the Indian Constitution that signifies an 'elected Head of the State'. We need to understand the meaning of each option in the context of a state's governance.
Correct Option: D) republic — The term 'republic' in the Preamble signifies that India has an elected Head of State (the President), who is chosen for a fixed term, rather than a hereditary monarch. This is a fundamental characteristic of a republic.
In this question, a statement is followed by four conclusions. Find out which conclusion is true based on the given statement.
Statement:
B ≤ T = N > O = Q ≥ M = R
Conclusions:
1. N≥B
2. N<R
3. O<R
4. T<M
This problem involves analyzing inequalities to determine the truthfulness of given conclusions based on a statement. We need to combine the parts of the statement and then check each conclusion against the derived relationships.
The given statement is: \(B \le T = N > O = Q \ge M = R\)
Let's break down the statement into individual relationships:
Now let's evaluate each conclusion:
Conclusion 1: \(N \ge B\)
Conclusion 2: \(N < R\)
Conclusion 3: \(O < R\)
Conclusion 4: \(T < M\)
A) 1
Conclusion 1 (\(N \ge B\)) is the only true conclusion based on the given statement.
Study the following number arrangement and answer the question given below. All numbers given below are 2 digit numbers.
13 56 46 23 16 34 25 19 78 85 86 24 29 73 94 52 64 63
What will be the difference between the sum of the first half and second half of the above series?
To find the difference between the sum of the first half and the second half of the given series, we first need to count the total number of elements. Then, we divide the series into two equal halves. We calculate the sum of the numbers in each half and finally find the absolute difference between these two sums.
The given series is: 13 56 46 23 16 34 25 19 78 85 86 24 29 73 94 52 64 63.
Counting the numbers, we find there are 18 numbers in the series.
Since there are 18 numbers, each half will contain \(18 / 2 = 9\) numbers.
First half: 13 56 46 23 16 34 25 19 78
Second half: 85 86 24 29 73 94 52 64 63
Correct Option: B) 260
The question asks for the biological reason behind voice change during puberty. This is a common physiological change associated with adolescence, particularly in males, and is directly linked to the development of specific organs involved in sound production.
D) the development of larynx. During puberty, especially in boys, the larynx (voice box) grows significantly. This growth causes the vocal cords to lengthen and thicken, leading to a deeper voice. This is the primary reason for voice change during this developmental stage.