To find the ratio of the bases of two triangles, we will use the formula for the area of a triangle, which is \( \frac{1}{2} \times \text{base} \times \text{height} \). We are given the ratio of their areas and the ratio of their heights. By setting up equations based on these ratios, we can solve for the ratio of their bases.
Let the two triangles be Triangle 1 and Triangle 2.
Let \(A_1\) and \(A_2\) be their respective areas.
Let \(b_1\) and \(b_2\) be their respective bases.
Let \(h_1\) and \(h_2\) be their respective heights.
The formula for the area of a triangle is \(A = \frac{1}{2} \times b \times h\).
So, for Triangle 1: \(A_1 = \frac{1}{2} b_1 h_1\)
And for Triangle 2: \(A_2 = \frac{1}{2} b_2 h_2\)
We are given the ratio of their areas: \[\frac{A_1}{A_2} = \frac{4}{3}\].
We are given the ratio of their heights: \[\frac{h_1}{h_2} = \frac{3}{4}\].
Now, let's find the ratio of their areas using the area formula:
\[ \frac{A_1}{A_2} = \frac{\frac{1}{2} b_1 h_1}{\frac{1}{2} b_2 h_2} \]
\[ \frac{A_1}{A_2} = \frac{b_1 h_1}{b_2 h_2} \]
\[ \frac{A_1}{A_2} = \left(\frac{b_1}{b_2}\right) \times \left(\frac{h_1}{h_2}\right) \]
Substitute the given ratios into this equation:
\[ \frac{4}{3} = \left(\frac{b_1}{b_2}\right) \times \left(\frac{3}{4}\right) \]
Now, solve for the ratio of the bases \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{4}{3} \div \frac{3}{4} \]
\[ \frac{b_1}{b_2} = \frac{4}{3} \times \frac{4}{3} \]
\[ \frac{b_1}{b_2} = \frac{16}{9} \]
Thus, the ratio of their bases is 16 : 9.
Correct Option: C) 16 : 9
To find the measure of the smallest interior angle of a pentagon, we first need to determine the sum of the interior angles of a pentagon. Then, using the given ratio of the angles, we can set up an equation to find the value of a common multiplier and subsequently calculate each angle.
The formula for the sum of interior angles of a polygon with \(n\) sides is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]For a pentagon, \(n = 5\). So, the sum of its interior angles is:\[ (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]The interior angles are in the ratio 1 : 3 : 5 : 7 : 11. Let the common multiplier be \(x\). Then the angles can be represented as \(1x\), \(3x\), \(5x\), \(7x\), and \(11x\).
The sum of these angles must be equal to the total sum of interior angles of a pentagon:
\[ 1x + 3x + 5x + 7x + 11x = 540^\circ \]\[ 27x = 540^\circ \]Now, solve for \(x\):\[ x = \frac{540^\circ}{27} \]\[ x = 20^\circ \]The smallest interior angle is \(1x\).
\[ \text{Smallest angle} = 1 \times 20^\circ = 20^\circ \]Correct Option: B) 20°
The question asks to identify the classical Indian dance form that has been significantly influenced by the Mughal tradition. This requires knowledge of the historical development and cultural influences on various classical dance forms.
Correct Option: A) Kathak is the classical dance form that was significantly influenced by the Mughal tradition. It flourished in the Mughal courts, where it absorbed Persian and Central Asian elements, leading to its present form characterized by intricate footwork, spins, and storytelling.
The question asks to identify which country was NOT among the founding members of the United Nations. To answer this, one needs to recall the historical context of the UN's formation and the status of various nations at that time.
Correct Option: B) Germany was not a founding member of the United Nations. It was a defeated Axis power and was under occupation when the UN was formed. West Germany joined the UN in 1973, and East Germany also joined in 1973. After reunification, Germany's membership continued as a single state.
When did an Indian chief minister hoist the national flag on Independence Day for the first time?
15th August 1977
15th August 1950
15th August 1947
15th August 1974
The question asks for the first time an Indian Chief Minister hoisted the national flag on Independence Day. This is a historical fact related to a significant change in protocol for Independence Day celebrations at the state level.
Correct Option: D) 15th August 1974. This date marks the first time Chief Ministers were permitted to hoist the national flag on Independence Day in their respective states, a significant change in the ceremonial protocol.
The question asks for the full form of SGST in the context of the GST framework in India. Understanding the different components of GST is key to answering this question.
A) State Goods & Service Tax is the correct full form of SGST. This component of GST is levied by the state governments on goods and services consumed within their respective states.
The question asks to identify the scientist who first demonstrated the dispersion of light into its constituent colors (spectrum) using a prism. This is a historical fact related to the study of optics.
Correct Option: D) Isaac Newton was the first person to obtain a spectrum by passing light through a prism. His experiments in the 1660s, detailed in his work "Opticks," conclusively showed that white light is a compound of various colors, each with its own refractive index, which causes them to separate when passing through a prism.
The question asks to identify which of the given options is NOT an instrument of the monetary policy of the Reserve Bank of India (RBI). We need to understand the functions of the RBI and differentiate between monetary policy tools and fiscal policy tools.
Correct Option: A) Goods and Services Tax (GST) is a fiscal policy instrument, not a monetary policy instrument of the RBI. It is a tax levied by the government.
This is an alphanumeric series problem. We need to identify the patterns in both the numerical and alphabetical parts of each term in the series to predict the next term.
The numbers in the series are 2, 4, 7, 12, 19, 30, 43.
Let's find the differences between consecutive numbers:
The differences are 2, 3, 5, 7, 11, 13. This is a sequence of prime numbers.
The next prime number after 13 is 17.
So, the next number in the series will be \(43 + 17 = 60\).
The letters in the series are B, D, G, L, S, D, Q.
Let's find the position of each letter in the English alphabet (A=1, B=2, ...):
Let's find the difference in positions:
The differences are 2, 3, 5, 7. This is the same sequence of prime numbers we found for the numerical part. This pattern continues up to 'S'.
Now consider the transition from 'S' to 'D':
Now consider the transition from 'D' to 'Q':
Following this pattern, the next prime difference after 13 is 17.
The current letter is Q (17th letter).
So, the position of the next letter will be \(17 + 17 = 34\).
Since there are 26 letters in the alphabet, we find the letter corresponding to \(34 - 26 = 8\).
The 8th letter of the alphabet is H.
The next number is 60.
The next letter is H.
Therefore, the next term in the series is 60H.
A) 60H — This matches our derived next term based on the identified patterns for both the numerical and alphabetical components.
The question asks to identify the tallest statue in the world from the given options. This requires knowledge of the heights of prominent statues globally.
Correct Option: D) 'Statue of Unity' in India is the tallest statue in the world, standing at 182 meters (597 feet). It is dedicated to Sardar Vallabhbhai Patel and is located in Gujarat, India.