The mean proportional between (13 + 8\(\sqrt{5}\)) and (10 − 2\(\sqrt{5}\)) is:
10\(\sqrt{2}\)
5\(\sqrt{2}\)
\(\sqrt{54+50\sqrt{5}}\)
\(\sqrt{50+54\sqrt{5}}\)
To find the mean proportional between two numbers, we take the square root of their product. We will multiply the given expressions and then simplify the result.
Let the two given numbers be \(a = (13 + 8\sqrt{5})\) and \(b = (10 - 2\sqrt{5})\).
The mean proportional (MP) between \(a\) and \(b\) is given by the formula:
\[ \text{MP} = \sqrt{a \times b} \]Substitute the given expressions for \(a\) and \(b\):
\[ \text{MP} = \sqrt{(13 + 8\sqrt{5}) \times (10 - 2\sqrt{5})} \]Now, multiply the two expressions inside the square root:
\[ \begin{aligned} (13 + 8\sqrt{5})(10 - 2\sqrt{5}) &= 13 \times 10 - 13 \times 2\sqrt{5} + 8\sqrt{5} \times 10 - 8\sqrt{5} \times 2\sqrt{5} \ &= 130 - 26\sqrt{5} + 80\sqrt{5} - 16 \times (\sqrt{5} \times \sqrt{5}) \ &= 130 - 26\sqrt{5} + 80\sqrt{5} - 16 \times 5 \ &= 130 + (80 - 26)\sqrt{5} - 80 \ &= 130 + 54\sqrt{5} - 80 \ &= (130 - 80) + 54\sqrt{5} \ &= 50 + 54\sqrt{5} \end{aligned} \]Therefore, the mean proportional is:
\[ \text{MP} = \sqrt{50 + 54\sqrt{5}} \]D) \(\sqrt{50+54\sqrt{5}}\) is the correct answer, as derived from the calculation.
If both roots of the quadratic equation, \( x^{2} + 3x + 2 = 0 \), are also roots of another quadratic equation, \( ax^{2} + bx + c = 0 \), then which of the following must be true?
a + 3b + 2c = 0
a = b = c
\( \frac{a}{1} \) = \( \frac{b}{2} \) = \( \frac{c}{3} \)
\( ax^{2} + bx + c \) must be a multiple of \( x^{2} + 3x + 2 \)
This problem deals with the relationship between quadratic equations when they share roots. If two quadratic equations share both roots, it implies a direct proportionality between their coefficients, or more fundamentally, that one polynomial is a multiple of the other.
Find the roots of the first quadratic equation:
The given equation is \( x^{2} + 3x + 2 = 0 \).
We can factor this equation:
\[ x^{2} + 2x + x + 2 = 0 \] \[ x(x + 2) + 1(x + 2) = 0 \] \[ (x + 1)(x + 2) = 0 \]So, the roots are \( x = -1 \) and \( x = -2 \).
Relate the roots to the second quadratic equation:
It is given that both roots of \( x^{2} + 3x + 2 = 0 \) are also roots of \( ax^{2} + bx + c = 0 \).
If \( -1 \) and \( -2 \) are the roots of \( ax^{2} + bx + c = 0 \), then we can write the second quadratic equation in factored form as:
\[ a(x - (-1))(x - (-2)) = 0 \] \[ a(x + 1)(x + 2) = 0 \] \[ a(x^{2} + 3x + 2) = 0 \] \[ ax^{2} + 3ax + 2a = 0 \]Compare coefficients:
Comparing this with \( ax^{2} + bx + c = 0 \), we get:
From these relationships, we can see that \( b/a = 3 \) and \( c/a = 2 \). This implies that the coefficients are proportional, i.e., \[\frac{a}{1} = \frac{b}{3} = \frac{c}{2}\].
Evaluate the options:
Since \( ax^{2} + bx + c = 0 \) has the same roots as \( x^{2} + 3x + 2 = 0 \), it means that the polynomial \( ax^{2} + bx + c \) must be a scalar multiple of the polynomial \( x^{2} + 3x + 2 \). That is, \( ax^{2} + bx + c = k(x^{2} + 3x + 2) \) for some non-zero constant \( k \). In our derivation, \( k \) is represented by \( a \).
Correct Option: D) \( ax^{2} + bx + c \) must be a multiple of \( x^{2} + 3x + 2 \)
This is directly derived from the fact that if two quadratic equations have the same roots, their corresponding polynomials must be proportional to each other. Here, \( ax^{2} + bx + c = a(x^{2} + 3x + 2) \), which means \( ax^{2} + bx + c \) is a multiple of \( x^{2} + 3x + 2 \).
To solve this problem, we need to track Raghav's movements step-by-step, considering each turn and distance covered. We can visualize his path on a 2D coordinate system or simply by keeping track of his net displacement in the East-West and North-South directions from his starting point A.
C) 6 km to the east
Raghav ends up 6 km to the west of his starting point A. Therefore, to return to A, he must drive 6 km towards the east.
The question asks about the number of LVM-3 rocket launches planned for the Chandrayaan-4 mission. This requires knowledge of ISRO's future lunar mission plans, specifically the architecture of Chandrayaan-4.
A) Two. The Chandrayaan-4 mission is planned to utilize two separate launches of the LVM-3 rocket to deploy its various components, including the lander, ascent module, and Earth Return Vehicle.
To determine the maximum value of 'p' for which the given eight-digit number is divisible by 9, we will use the divisibility rule for 9. The rule states that a number is divisible by 9 if the sum of its digits is divisible by 9.
Identify the given number: The given eight-digit number is \(31425p99\).
Apply the divisibility rule for 9: Sum of the digits must be divisible by 9.
Sum of the digits \(S = 3 + 1 + 4 + 2 + 5 + p + 9 + 9\)
Calculate the sum of the known digits:
\(S = 3 + 1 + 4 + 2 + 5 + 9 + 9 + p\)
\(S = 33 + p\)
Find the possible values of 'p': Since 'p' is a digit, its value can range from 0 to 9 (i.e., \(0 \le p \le 9\)). We need to find the value of 'p' such that \(33 + p\) is divisible by 9.
The multiples of 9 are 9, 18, 27, 36, 45, etc.
Determine the maximum value of 'p': From the valid possibilities, the only single-digit value for 'p' that makes \(33+p\) divisible by 9 is \(p=3\).
Therefore, the maximum value of 'p' is 3.
Correct Option: A) 3. When \(p=3\), the sum of the digits is \(33+3 = 36\), which is divisible by 9. This is the only single-digit value for 'p' that satisfies the condition.
The question asks for the location where the 2024 International Conference of NGOs, organized by UNESCO, was held. This is a factual question related to current affairs and international organizations.
Correct Option: A) Paris, France
The 2024 International Conference of NGOs was organized by UNESCO and took place at its headquarters in Paris, France. This conference is a significant event for NGOs maintaining official relations with UNESCO, fostering dialogue and cooperation on global issues.
The problem asks us to identify numbers in a given series that are immediately preceded by a symbol and immediately followed by a symbol. We need to scan the series from left to right and apply this condition to each number.
(Left) * 1 * $ 2 3 7 £ 1 7 @ 4 9 9 * 6 * 3 $ 1 \( @ (Right)
B) 4
The question asks to identify the Hindustani music style that originated from the folk songs of camel-riders in North-West India. This requires knowledge of the origins and characteristics of various Hindustani classical music forms.
B) Tappa is a semi-classical vocal form in Hindustani music that originated from the folk songs of camel-riders in the Punjab and Sindh regions of North-West India. It is characterized by its fast, intricate, and undulating melodic movements and short, quick taans (melodic phrases).
To find the volume of a cube, we need to know the length of one of its edges. The volume of a cube is calculated by cubing the length of its edge.
1764
x 42
------
3528 (1764 x 2)
70560 (1764 x 40)
------
74088
So, \(V = 74088 \text{ m}^3\).
D) The calculated volume of the cube is \(74,088 \text{ m}^3\), which matches option D.
This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams to visualize the relationships between the categories mentioned in the statements and then check if the conclusions logically follow.
C) Neither conclusion (I) nor (II) follows.