Find the variance of the following data points:
6, 7, 5, 9, 12, 15
\(\frac{37}{6}\)
\(\frac{67}{6}\)
\(\frac{81}{3}\)
\(\frac{37}{3}\)
To find the variance of a set of data points, we first need to calculate the mean (average) of the data. Then, we find the squared difference of each data point from the mean, sum these squared differences, and finally divide by the total number of data points.
The given data points are \(X = \{6, 7, 5, 9, 12, 15\}\).
There are 6 data points, so \(n = 6\).
The mean is the sum of all data points divided by the number of data points.
\[ \bar{X} = \frac{\sum X}{n} = \frac{6 + 7 + 5 + 9 + 12 + 15}{6} \]\[ \bar{X} = \frac{54}{6} = 9 \]The variance is the sum of the squared deviations divided by the number of data points (for population variance, which is typically assumed in such problems unless specified otherwise for sample variance).
\[ \sigma^2 = \frac{\sum (X - \bar{X})^2}{n} = \frac{74}{6} \]\[ \sigma^2 = \frac{37}{3} \]Correct Option: D) \(\frac{37}{3}\)
The question asks to identify the outermost layer of the Earth. To answer this, one needs to recall the basic structure of the Earth's interior, which is typically divided into several concentric layers.
Correct Option: C) Crust is the outermost solid layer of the Earth, where all life exists.
The question asks for another common name for a computer monitor. We need to identify the acronym that refers to the display unit of a computer system.
Correct Option: D) VDU
VDU stands for Visual Display Unit. This term is synonymous with a computer monitor, as both refer to the output device that displays visual information from a computer.
To answer this question, one needs to recall the list of the world's largest countries by area and identify the sixth one in that ranking.
Correct Option: C) Australia is the sixth largest country in the world by area.
The question asks to identify a cartilaginous fish from the given options. This requires knowledge of fish classification, specifically distinguishing between cartilaginous (Chondrichthyes) and bony (Osteichthyes) fish.
Correct Option: A) Shark is a cartilaginous fish, belonging to the class Chondrichthyes, characterized by a skeleton made of cartilage.
The question asks about the specific phase of the moon during which a lunar eclipse occurs. Understanding the geometry of the Earth, Moon, and Sun during different lunar phases is key to answering this question.
Correct Option: C) A lunar eclipse occurs during the Full Moon phase. This is because, during a full moon, the Earth is positioned directly between the Sun and the Moon. If the alignment is perfect, the Earth's shadow falls upon the Moon, causing an eclipse.
The question asks about the ministry responsible for publishing the Goods and Service Tax (GST) Act, 2017. This requires knowledge of the legislative process and the roles of different government ministries in India.
Correct Option: A) Ministry of Law and Justice is responsible for the formal publication of all Acts passed by the Parliament, including the Goods and Service Tax (GST) Act, 2017, in the Gazette of India. This is part of its core function related to legislative drafting and notification.
The problem asks us to find a number based on a relationship between its fractions. We can set up an algebraic equation to represent the given condition and then solve for the unknown number.
Correct Option: B) 80
The number is 80. Let's verify: Four-fifth of 80 is \(\frac{4}{5} \times 80 = 4 \times 16 = 64\). Three-fourth of 80 is \(\frac{3}{4} \times 80 = 3 \times 20 = 60\). The difference is \(64 - 60 = 4\), which matches the condition given in the problem.
The question asks to identify the organization among the given options of which India is NOT a member. This requires knowledge of India's membership status in various international organizations.
Based on this analysis, NATO is the organization of which India is not a member.
D) North Atlantic Treaty Organisation (NATO) — India is not a member of NATO. NATO is a collective defense alliance primarily consisting of North American and European countries. India maintains a policy of strategic autonomy and is not part of such military alliances.
Find the angle of elevation of a 200\(\sqrt{3}\) m tower's top from a point 200 m away from its base.
This problem involves trigonometry, specifically finding the angle of elevation. We can model the situation as a right-angled triangle where the tower is the perpendicular side, the distance from the base is the base side, and the angle of elevation is the angle between the base and the hypotenuse. We will use the tangent function, which relates the opposite side (height of the tower) to the adjacent side (distance from the base).
Identify the given values:
Choose the appropriate trigonometric ratio:
The tangent function relates the opposite side to the adjacent side:
Substitute the given values into the formula:
\[ \tan(\theta) = \frac{h}{d} = \frac{200\sqrt{3}}{200} \]Simplify the expression:
\[ \tan(\theta) = \sqrt{3} \]Find the angle \(\theta\):
We know that \(\tan(60^\circ) = \sqrt{3}\). Therefore,
D) 60° is the correct angle of elevation.