Correct Answer: A. friends

Correct Answer: B. -3i + 2j

Explanation

\(\overrightarrow{BC} = \overrightarrow{BA} + \overrightarrow{AC}\)

\(\overrightarrow{BA} = - \overrightarrow{AB} = -(5i + 3j)\)

= \(-5i - 3j\)

\(\overrightarrow{BC} = (-5i - 3j) + (2i + 5j)\)

= \(-3i + 2j\)

Correct Answer: C. A plague called Uhud

Explanation

(a) In the set of numbers from 75, 76, 77, ... , 497, 498, 499, 500, you will find that the first term which is the first number divisible by 7 = 77 and the last term = 497.

Given the first and last terms of an A.P, you can use the formula

\(n = \frac{l - a}{d} + 1\) to find the number of terms in the sequence.

First term = a = 77

Last term = l = 497

Common difference = d = 7

\(n = \frac{497 - 77}{7} + 1\)

\(n = \frac{420}{7} + 1\)

= \(60 + 1\)

= 61.

There are 61 numbers between 75 and 500 which are divisible by 7.

(b) \(T_{n} = a + (n - 1)d\) (for an A.P)

\(T_{8} = a + 7d\)

\(T_{3} = a + 2d\)

\(\implies a + 7d = 5(a + 2d)\)

\(T_{7} = a + 6d\)

\(T_{4} = a + 3d\)

\(\implies a + 6d = a + 3d + 9\)

\(a + 7d = 5a + 10d \implies 5a - a + 10d - 7d = 0\)

\(4a + 3d = 0 ... (1)\)

\(a + 6d = a + 3d + 9 \implies a - a + 6d - 3d = 9\)

\(3d = 9 \implies d = 3\)

Putting d = 3 in (1),

\(4a + 3(3) = 0 \implies 4a + 9 = 0\)

\(4a = -9 \implies a = \frac{-9}{4}\)

\(\therefore\) The second term of the sequence = \(\frac{-9}{4} + 3 = \frac{3}{4}\)

Third term = \(\frac{3}{4} + 3 = \frac{15}{4}\)

Fourth term = \(\frac{15}{4} + 3 = \frac{27}{4}\)

Fifth term = \(\frac{27}{4} + 3 = \frac{39}{4}\)

The first 5 terms of the sequence = \(\frac{-9}{4}, \frac{3}{4}, \frac{15}{4}, \frac{27}{4}, \frac{39}{4}\)

Correct Answer: D) cytoplasmic membrane

Explanation

This question is asking which of the given options is essential for a living cell. The options are flagella, capsule, cell wall, cytoplasmic membrane, and capsule and cell wall. Flagella are long whip-like structures that some cells use for movement. While flagella are important for some cells, they are not essential for all living cells. Therefore, option A is not the correct answer. A capsule is a protective layer that surrounds some bacterial cells. It helps the cell to avoid being engulfed by white blood cells and to adhere to surfaces. However, not all living cells have a capsule, so option B is also incorrect. The cell wall is a structure that surrounds the cell membrane and provides support and protection to the cell. While the cell wall is important for many cells, not all living cells have a cell wall. Therefore, option C is not the correct answer. The cytoplasmic membrane, also known as the cell membrane, is a thin layer that surrounds the cell and separates the interior from the exterior. It controls the movement of substances in and out of the cell and is essential for all living cells. Therefore, option D is the correct answer. Option E states that both capsule and cell wall are essential for a living cell. However, this is not true as not all cells have a capsule or a cell wall. Therefore, option E is incorrect. In conclusion, the essential structure for a living cell is the cytoplasmic membrane or the cell membrane. This structure is present in all living cells and is responsible for controlling the movement of substances in and out of the cell.

Correct Answer: B. 4

Correct Answer: D) Researchers have used statistical methods to address questions of authorship within the field of music.

Explanation

The choice that most effectively uses relevant information from the notes to make a generalization about the study conducted by Glickman, Brown, and Song is: D) Researchers have used statistical methods to address questions of authorship within the field of music. This choice accurately reflects the information that Glickman, Brown, and Song used statistical methods to analyze the musical content of Beatles songs and make conclusions about authorship. It acknowledges the broader application of statistical methods in addressing questions of authorship within the field of music, without making specific claims about the authorship of the verse in In My Life.

Explanation

\((x - a)^{2} + (y - b)^{2} = r^{2}\)

\(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)

Using A(3, 2) :

\(9 - 6a + a^{2} + 4 - 4b + b^{2} = r^{2}\)

\(13 - 6a - 4b + a^{2} + b^{2} = r^{2} .... (1)\)

Using B(-1, -2) :

\(1 + 2a + a^{2} + 4 + 4b + b^{2} = r^{2}\)

\(5 + 2a + 4b + a^{2} + b^{2} = r^{2} .... (2)\)

Using C(5, -4) :

\(25 - 10a + a^{2} + 16 + 8b + b^{2} = r^{2}\)

\(41 - 10a + 8b + a^{2} + b^{2} = r^{2} ... (3)\)

(1) - (2) :

\(8 - 8a - 8b = 0 \implies 1 - a - b = 0 .... (4)\)

(3) - (2) :

\(24 - 12a + 12 + 4b = 0 \implies 36 - 12a + 4b = 0 \)

\(\equiv 9 - 3a + b = 0... (5)\)

Solving (4) and (5), we have :

\(a = 2\frac{1}{2} ; b = -1\frac{1}{2}\)

Coordinates of the centre : \(C(2\frac{1}{2}, -1\frac{1}{2})\).

(b) To get the radius, we can make use of any point on the circle as well as the centre.

Using point \(A(3, 2) & C(2\frac{1}{2}, -1\frac{1}{2})\)

Radius = \(\sqrt{(3 - 2\frac{1}{2})^{2} + (2 + 1\frac{1}{2})^{2}}\)

= \(\sqrt{\frac{1}{4} + \frac{49}{4}}\)

= \(\sqrt{\frac{50}{4}}\)

= \(\frac{5\sqrt{2}}{2} cm\)

(c) Equation of the circle :

\((x - 2\frac{1}{2})^{2} + (y + 1\frac{1}{2})^{2} = (\frac{5\sqrt{2}}{2})^{2}\)

\(x^{2} - 11x + \frac{25}{4} + y^{2} + 3y + \frac{9}{4} = \frac{50}{4}\)

\(x^{2} + y^{2} - 11x + 3y + \frac{34}{4} - \frac{50}{4} = 0\)

\(x^{2} + y^{2} - 11x + 3y - 4 = 0\)

Correct Answer: B. Sub-soil

Correct Answer: C. values