If x + \( \frac{1}{x} \) = 3, then x⁶ + \( \frac{1}{x^6} \) is:
To find the value of \(x^6 + \frac{1}{x^6}\) given \(x + \frac{1}{x} = 3\), we can use algebraic identities. We will first find \(x^2 + \frac{1}{x^2}\) by squaring the given expression, then find \(x^3 + \frac{1}{x^3}\) using the cubic identity, and finally use these results to find \(x^6 + \frac{1}{x^6}\) by squaring \(x^3 + \frac{1}{x^3}\) or cubing \(x^2 + \frac{1}{x^2}\). A more direct approach is to find \(x^2 + \frac{1}{x^2}\) and then cube this result.
Find \(x^2 + \frac{1}{x^2}\):
Given \(x + \frac{1}{x} = 3\).
Square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 3^2 \]Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\):
\[ x^2 + \frac{1}{x^2} + 2 \cdot x \cdot \frac{1}{x} = 9 \] \[ x^2 + \frac{1}{x^2} + 2 = 9 \] \[ x^2 + \frac{1}{x^2} = 9 - 2 \] \[ x^2 + \frac{1}{x^2} = 7 \]Find \(x^3 + \frac{1}{x^3}\) (Alternative, not strictly needed for \(x^6\) if we cube \(x^2 + \frac{1}{x^2}\)):
Using the identity \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\):
\[ \left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3 \cdot x \cdot \frac{1}{x} \left(x + \frac{1}{x}\right) \]Substitute the given value \(x + \frac{1}{x} = 3\):
\[ 3^3 = x^3 + \frac{1}{x^3} + 3(3) \] \[ 27 = x^3 + \frac{1}{x^3} + 9 \] \[ x^3 + \frac{1}{x^3} = 27 - 9 \] \[ x^3 + \frac{1}{x^3} = 18 \]Find \(x^6 + \frac{1}{x^6}\):
We can find \(x^6 + \frac{1}{x^6}\) by cubing the expression for \(x^2 + \frac{1}{x^2}\) or by squaring the expression for \(x^3 + \frac{1}{x^3}\).
Method 1: Squaring \(x^3 + \frac{1}{x^3}\)
From step 2, we have \(x^3 + \frac{1}{x^3} = 18\).
Square both sides:
\[ \left(x^3 + \frac{1}{x^3}\right)^2 = 18^2 \] \[ (x^3)^2 + \left(\frac{1}{x^3}\right)^2 + 2 \cdot x^3 \cdot \frac{1}{x^3} = 324 \] \[ x^6 + \frac{1}{x^6} + 2 = 324 \] \[ x^6 + \frac{1}{x^6} = 324 - 2 \] \[ x^6 + \frac{1}{x^6} = 322 \]D) 322
The problem presents a series of number pairs where the second number is related to the first number by a specific pattern. We need to identify this pattern and apply it to the last given number to find the missing number.
Find the difference between the second and first number:
\[ 55 - 33 = 22 \]Find the difference between the second and first number:
\[ 68 - 46 = 22 \]In both pairs, the second number is obtained by adding 22 to the first number. This indicates a consistent additive pattern.
To find the missing number, add 22 to 68:
\[ 68 + 22 = 90 \]A) 90 is the correct answer because applying the identified pattern (adding 22) to 68 yields 90.
The question asks to identify the state whose police force introduced "Pink Beat" patrols to enhance the safety of women and children in public places. This is a factual question related to current affairs and specific police initiatives in Indian states.
A) Kerala: The Kerala Police introduced the "Pink Beat" patrols as part of their comprehensive "Pink Protection" project. These patrols, primarily staffed by women police officers, focus on enhancing the safety of women and children in public spaces, including bus stands, commercial centers, and schools.
A certain number of advocates are sitting in a row in a bar council, facing north. Piyush is an immediate neighbour of both Vinod and Taman, while Rudir is an immediate neighbour of both Taman and Biswas. Vinod is sitting at the extreme right end. There are only two advocates to the left of Biswas. If no other advocate is sitting in the row, what is the total number of advocates seated in that row?
This is a linear arrangement puzzle where we need to determine the total number of advocates sitting in a row based on the given positional clues. We will start by placing the individuals whose positions are most clearly defined and then use the relative positions to fill in the rest of the arrangement.
Vinod is sitting at the extreme right end. Since everyone is facing north, the right end is the rightmost position in the row.
Arrangement so far: ... V
Piyush is an immediate neighbour of both Vinod and Taman. This means Piyush (P) must be between Vinod (V) and Taman (T).
Arrangement so far: ... T P V
Rudir is an immediate neighbour of both Taman and Biswas. This means Rudir (R) must be between Taman (T) and Biswas (B).
Arrangement so far: ... B R T P V
There are only two advocates to the left of Biswas. This means Biswas (B) is the third person from the left end.
Let's denote the two advocates to the left of Biswas as X1 and X2.
Final arrangement: X1 X2 B R T P V
Now, count the total number of advocates in the final arrangement.
The advocates are: X1, X2, Biswas, Rudir, Taman, Piyush, Vinod.
Total number of advocates = \(2 + 1 + 1 + 1 + 1 + 1 = 7\).
A) Seven. Based on the step-by-step arrangement, there are a total of seven advocates in the row.
The volume of a cuboid is given by the product of its three dimensions (length, width, and height). We are given the volume as an algebraic expression and need to find which set of dimensions, when multiplied together, yields this expression. We will factorize the given volume expression and then compare it with the product of the dimensions provided in each option.
The volume of the cuboid is given by \(V = 5a^2 - 80b^2\).
First, factor out the common term, which is 5:
\[ V = 5(a^2 - 16b^2) \]The expression inside the parenthesis, \(a^2 - 16b^2\), is in the form of \(x^2 - y^2\), which can be factored as \((x - y)(x + y)\).
Here, \(x = a\) and \(y = \sqrt{16b^2} = 4b\).
So, \(a^2 - 16b^2 = (a - 4b)(a + 4b)\).
Substituting this back into the volume equation:
\[ V = 5(a - 4b)(a + 4b) \]The dimensions of the cuboid must be the factors of this expression. The factors are \(5\), \((a - 4b)\), and \((a + 4b)\).
Correct Option: B) The dimensions are \(5\), \(a - 4b\), \(a + 4b\). When these are multiplied, we get \(5(a - 4b)(a + 4b)\), which simplifies to \(5(a^2 - (4b)^2) = 5(a^2 - 16b^2) = 5a^2 - 80b^2\). This matches the given volume.
The product of these dimensions is \(5(2a - b)(2a + b) = 5((2a)^2 - b^2) = 5(4a^2 - b^2) = 20a^2 - 5b^2\). This does not match the given volume.
The product of these dimensions is \(5(a - b)(a + 4b) = 5(a^2 + 4ab - ab - 4b^2) = 5(a^2 + 3ab - 4b^2) = 5a^2 + 15ab - 20b^2\). This does not match the given volume.
The product of these dimensions is \(5(2a - b)(a + 2b) = 5(2a^2 + 4ab - ab - 2b^2) = 5(2a^2 + 3ab - 2b^2) = 10a^2 + 15ab - 10b^2\). This does not match the given volume.
The question asks for the correct explanation behind silver chloride turning grey in sunlight. This phenomenon is a classic example of a photochemical reaction, specifically a decomposition reaction initiated by light energy.
Correct Option: B) Decomposition of silver chloride into silver and chlorine by light. This option accurately describes the chemical process where silver chloride breaks down into silver and chlorine due to the energy from light (sunlight), causing the grey color change.
The question asks for the establishment year of Coal India Limited (CIL). This is a factual recall question related to the history of public sector undertakings in India. To answer correctly, one needs to know the specific year of CIL's formation.
Therefore, Coal India Limited came into existence in 1975.
A) 1975 — Coal India Limited (CIL) was established in November 1975, following the nationalization of coal mines in India.
The question asks to identify the specific tissue responsible for storing fat. This requires knowledge of different types of animal tissues and their primary functions.
Correct Option: D) adipose tissue is a specialized type of connective tissue that primarily functions in storing fat. It provides insulation, energy reserves, and protection to organs.
An object is placed at a distance of 0.5 m in front of a converging mirror. If the focal length of the mirror is 0.2 m, then the magnification of the image is:
\( \frac{4}{3} \)
\( \frac{1}{3} \)
\( \frac{2}{3} \)
\( \frac{3}{2} \)
To find the magnification of the image formed by a converging mirror, we first need to determine the image distance using the mirror formula. Once the image distance is known, we can calculate the magnification using the magnification formula.
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]Substitute the given values:
\[ \frac{1}{-0.2} = \frac{1}{v} + \frac{1}{-0.5} \] \[ -5 = \frac{1}{v} - 2 \]Rearrange to solve for \( \frac{1}{v} \):
\[ \frac{1}{v} = -5 + 2 \] \[ \frac{1}{v} = -3 \]Therefore, the image distance \(v\) is:
\[ v = -\frac{1}{3} \, \text{m} \]The negative sign indicates that the image is formed on the same side as the object (real image).
The magnification \(m\) is given by the formula:
\[ m = -\frac{v}{u} \]Substitute the values of \(v\) and \(u\):
\[ m = -\frac{\left(-\frac{1}{3}\right)}{(-0.5)} \] \[ m = -\frac{\frac{1}{3}}{0.5} \] \[ m = -\frac{1}{3 \times 0.5} \] \[ m = -\frac{1}{1.5} \]To simplify, multiply the numerator and denominator by 2:
\[ m = -\frac{2}{3} \]The negative sign indicates that the image is inverted.
Correct Option: C) \( \frac{2}{3} \)
The magnitude of the magnification is \( \frac{2}{3} \).
C, E, K, M, O, Q and Z are sitting around a round table facing towards the center (but not necessarily in the same order). C is the immediate neighbour of Z and M. O is neither an immediate neighbour of M nor E. Q, who is second to the left of O, has K and E as immediate neighbours.
Who is sitting to the immediate right of O?