To find the mean deviation of a set of observations, we first need to calculate the mean of the observations. Then, we find the absolute difference between each observation and the mean. Finally, the mean deviation is the average of these absolute differences.
List the observations:
\(x = \{6, 7, 10, 12, 13, 4, 8, 12\}\)
Calculate the number of observations (n):
\(n = 8\)
Calculate the mean (\(\bar{x}\)) of the observations:
The sum of observations (\(\sum x\)) is:
\[ \sum x = 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72 \]The mean is:
\[ \bar{x} = \frac{\sum x}{n} = \frac{72}{8} = 9 \]Calculate the absolute deviations from the mean (\(|x_i - \bar{x}|\)):
Calculate the sum of the absolute deviations (\(\sum |x_i - \bar{x}|\)):
\[ \sum |x_i - \bar{x}| = 3 + 2 + 1 + 3 + 4 + 5 + 1 + 3 = 22 \]Calculate the Mean Deviation (MD):
\[ \text{MD} = \frac{\sum |x_i - \bar{x}|}{n} = \frac{22}{8} \] \[ \text{MD} = 2.75 \]The question asks what is essential for booting a computer. Booting is the process of starting up a computer. We need to identify the core software component that manages this process and allows the user to interact with the hardware and other software.
Correct Option: C) Operating system
An operating system (OS) is the most crucial software that manages computer hardware and software resources and provides common services for computer programs. It is absolutely essential for a computer to boot up and function. Without an OS, the computer cannot start up beyond the basic BIOS/UEFI level.
The question asks to identify the scientist credited with discovering the laws of planetary motion. This is a direct knowledge-based question related to the history of astronomy and physics.
Correct Option: B) Kepler
Johannes Kepler (1571–1630) formulated the three laws of planetary motion based on the precise observational data collected by Tycho Brahe. These laws describe how planets orbit the Sun:
The question asks for the term that describes how the brain interprets the frequency of an emitted sound. This is a fundamental concept in the physics of sound and human perception.
C) pitch — Pitch is the perceptual property of sounds that allows their ordering on a frequency-related scale. It is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies.
The question asks to identify the portal launched in February 2021 for providing municipal services to citizens in Cantonment boards across India. This requires knowledge of recent government initiatives related to digital services and cantonment areas.
Correct Option: C) e-Chhawani is the correct answer. It is a digital portal launched in February 2021 to provide online municipal services to residents of Cantonment Boards across India, enhancing ease of living and transparency.
This problem can be solved using the division algorithm (Euclidean algorithm) which states that for any integers \(a\) (dividend) and \(b\) (divisor) with \(b > 0\), there exist unique integers \(q\) (quotient) and \(r\) (remainder) such that \(a = bq + r\), where \(0 \le r < b\). We will apply this principle twice.
According to the problem, when the positive integer K is divided by 18, the remainder is 9. We can write this relationship using the division algorithm:
\[ K = 18q + 9 \]where \(q\) is some non-negative integer quotient.
We need to find the remainder when K is divided by 6. Substitute the expression for K into the division by 6:
\[ K = 6m + r' \]where \(m\) is the new quotient and \(r'\) is the remainder we need to find, with \(0 \le r' < 6\).
Substitute the first equation into the second:
\[ 18q + 9 = 6m + r' \]We can rewrite the terms on the left side to explicitly show multiples of 6:
\[ (18q) + 9 = (6 \times 3q) + (6 \times 1 + 3) \]Group the terms that are multiples of 6:
\[ 6(3q) + 6 + 3 \]\[ 6(3q + 1) + 3 \]Comparing this with \(6m + r'\), we can see that \(m = 3q + 1\) and \(r' = 3\).
Since \(0 \le 3 < 6\), the remainder \(r' = 3\) is valid.
B) 3 is the correct remainder when K is divided by 6.
Which of the following options has the fractions in ascending order?
\( \frac{3}{8} \), \( \frac{2}{5} \), \( \frac{9}{11} \),\( \frac{5}{6} \), \( \frac{6}{7} \)
\( \frac{2}{5} \), \( \frac{6}{7} \), \( \frac{9}{11} \), \( \frac{3}{8} \), \( \frac{5}{6} \)
\( \frac{9}{11} \), \( \frac{6}{7} \), \( \frac{5}{6} \), \( \frac{2}{5} \),\( \frac{3}{8} \)
\( \frac{6}{7} \), \( \frac{5}{6} \), \( \frac{9}{11} \), \( \frac{2}{5} \), \( \frac{3}{8} \)
To arrange fractions in ascending order, we need to compare their values. This can be done by converting them to decimal form or by finding a common denominator. Converting to decimal form is often quicker for competitive exams.
This sequence matches option A.
Correct Option: A) \( \frac{3}{8} \), \( \frac{2}{5} \), \( \frac{9}{11} \),\( \frac{5}{6} \), \( \frac{6}{7} \)
We need to find the height of the electric pole using trigonometry. We will use the angles of depression and the height of the tower to find the horizontal distance between the tower and the pole, and then use this distance to find the height of the pole.
Let the tower be AB and its height be \(h_T = 12 \, \text{m}\). Let the electric pole be CD and its height be \(h_P\).
The angle of depression from the top of the tower (A) to the bottom of the pole (D) is 60°.
In the right-angled triangle ABD:
\[ \tan(60^\circ) = \frac{AB}{BD} \] \[ \sqrt{3} = \frac{12}{BD} \] \[ BD = \frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3} \, \text{m} \]So, the horizontal distance between the tower and the pole is \(4\sqrt{3} \, \text{m}\).
The angle of depression from the top of the tower (A) to the top of the pole (C) is 30°.
Draw a line AE parallel to BD, such that E is on BC. Then AE = BD = \(4\sqrt{3} \, \text{m}\).
In the right-angled triangle AEC:
\[ \tan(30^\circ) = \frac{AC'}{AE} \]Where AC' is the vertical distance from A to C, which is AB - EB. Since CDEB forms a rectangle, EB = CD = \(h_P\).
So, AC' = \(12 - h_P\).
\[ \frac{1}{\sqrt{3}} = \frac{12 - h_P}{4\sqrt{3}} \]Now, solve for \(h_P\):
\[ 4\sqrt{3} = \sqrt{3}(12 - h_P) \]Divide both sides by \(\sqrt{3}\):
\[ 4 = 12 - h_P \] \[ h_P = 12 - 4 \] \[ h_P = 8 \, \text{m} \]Therefore, the height of the electric pole is 8 m.
Correct Option: C) 8 m
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we perform operations in the following order: Brackets (Parentheses), Orders (Exponents/Powers/Roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
So, the final answer is 33.
A) 33 is the correct answer, as derived by following the BODMAS/PEMDAS rule.
The two sides of a triangle measure 10 cm and 8 cm respectively. The area of the triangle is equal to 16\(\sqrt{6}\) cm². Find the length of the third side, given that it is a rational number.
We are given two sides of a triangle and its area. We need to find the length of the third side, which is stated to be a rational number. We can use Heron's formula for the area of a triangle, which relates the area to the lengths of its sides. Let the sides be \(a\), \(b\), and \(c\). We are given \(a = 10\) cm, \(b = 8\) cm, and the area \(A = 16\sqrt{6}\) cm\(^2\). We need to find \(c\).
Let the sides of the triangle be \(a = 10\) cm, \(b = 8\) cm, and the third side be \(c\).
Calculate the semi-perimeter \(s\) of the triangle:
\[ s = \frac{a + b + c}{2} = \frac{10 + 8 + c}{2} = \frac{18 + c}{2} \]Apply Heron's formula for the area of a triangle:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the given values into Heron's formula:
\[ 16\sqrt{6} = \sqrt{\frac{18+c}{2} \left(\frac{18+c}{2} - 10\right) \left(\frac{18+c}{2} - 8\right) \left(\frac{18+c}{2} - c\right)} \]Simplify the terms inside the square root:
\[ 16\sqrt{6} = \sqrt{\frac{18+c}{2} \left(\frac{18+c-20}{2}\right) \left(\frac{18+c-16}{2}\right) \left(\frac{18+c-2c}{2}\right)} \]\[ 16\sqrt{6} = \sqrt{\frac{18+c}{2} \cdot \frac{c-2}{2} \cdot \frac{c+2}{2} \cdot \frac{18-c}{2}} \]\[ 16\sqrt{6} = \sqrt{\frac{(18+c)(c-2)(c+2)(18-c)}{16}} \]Square both sides of the equation:
\[ (16\sqrt{6})^2 = \frac{(18+c)(18-c)(c+2)(c-2)}{16} \]\[ 256 \cdot 6 = \frac{(18^2 - c^2)(c^2 - 2^2)}{16} \]\[ 1536 = \frac{(324 - c^2)(c^2 - 4)}{16} \]Multiply both sides by 16:
\[ 1536 \cdot 16 = (324 - c^2)(c^2 - 4) \]\[ 24576 = (324 - c^2)(c^2 - 4) \]Let \(x = c^2\). The equation becomes a quadratic in \(x\):
\[ 24576 = (324 - x)(x - 4) \]\[ 24576 = 324x - 1296 - x^2 + 4x \]\[ 24576 = -x^2 + 328x - 1296 \]\[ x^2 - 328x + 1296 + 24576 = 0 \]\[ x^2 - 328x + 25872 = 0 \]Solve the quadratic equation for \(x\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) or factorize. Let's try to factorize by checking the options. Since \(c\) is a rational number, \(c^2\) must be a perfect square of a rational number. The options for \(c\) are 16, 10, 14, 12. So \(c^2\) could be \(16^2=256\), \(10^2=100\), \(14^2=196\), \(12^2=144\).
Test the options for \(c^2\):
Since \(x = c^2 = 196\), then \(c = \sqrt{196} = 14\) cm (length must be positive).
Check triangle inequality: \(10 + 8 > 14\) (18 > 14, true), \(10 + 14 > 8\) (24 > 8, true), \(8 + 14 > 10\) (22 > 10, true). All conditions are met.
C) 14 cm is the correct length for the third side.