If the difference between the areas of the circumcircle and the incircle of an equilateral triangle is 66 cm², then the area of the triangle is:
2\(\sqrt{3}\) cm²
10\(\sqrt{3}\) cm²
21\(\sqrt{3}\) cm²
20\(\sqrt{3}\) cm²
We need to find the area of an equilateral triangle given the difference between the areas of its circumcircle and incircle. We will use the relationships between the side length of an equilateral triangle and the radii of its circumcircle and incircle. Then, we will set up an equation based on the given area difference and solve for the side length, finally calculating the triangle's area.
Let the side length of the equilateral triangle be \(a\).
The radius of the circumcircle (R) of an equilateral triangle with side \(a\) is given by:
\[ R = \frac{a}{\sqrt{3}} \]The radius of the incircle (r) of an equilateral triangle with side \(a\) is given by:
\[ r = \frac{a}{2\sqrt{3}} \]The area of the circumcircle is \(\text{Area}_{\text{circumcircle}} = \pi R^2\).
\[ \text{Area}_{\text{circumcircle}} = \pi \left(\frac{a}{\sqrt{3}}\right)^2 = \pi \frac{a^2}{3} \]The area of the incircle is \(\text{Area}_{\text{incircle}} = \pi r^2\).
\[ \text{Area}_{\text{incircle}} = \pi \left(\frac{a}{2\sqrt{3}}\right)^2 = \pi \frac{a^2}{12} \]The difference between the areas of the circumcircle and the incircle is given as 66 cm\(\text{}^2\).
\[ \text{Area}_{\text{circumcircle}} - \text{Area}_{\text{incircle}} = 66 \]\[ \pi \frac{a^2}{3} - \pi \frac{a^2}{12} = 66 \]Factor out \(\pi a^2\):
\[ \pi a^2 \left(\frac{1}{3} - \frac{1}{12}\right) = 66 \]\[ \pi a^2 \left(\frac{4-1}{12}\right) = 66 \]\[ \pi a^2 \left(\frac{3}{12}\right) = 66 \]\[ \pi a^2 \left(\frac{1}{4}\right) = 66 \]Substitute \(\pi = \frac{22}{7}\):
\[ \frac{22}{7} \times a^2 \times \frac{1}{4} = 66 \]\[ \frac{11}{14} a^2 = 66 \]\[ a^2 = \frac{66 \times 14}{11} \]\[ a^2 = 6 \times 14 \]\[ a^2 = 84 \]The area of an equilateral triangle with side \(a\) is given by:
\[ \text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} a^2 \]Substitute the value of \(a^2 = 84\):
\[ \text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 84 \]\[ \text{Area}_{\text{triangle}} = 21\sqrt{3} \text{ cm}^2 \]C) \(21\sqrt{3}\) cm\(\text{}^2\)
The question asks to identify the state with the highest percentage of its geographical area covered by forests, according to the India State of Forest Report (IFSR), 2019. This requires knowledge of the findings of the IFSR 2019 report regarding forest cover percentages in different states.
D) Mizoram — As per the India State of Forest Report (IFSR) 2019, Mizoram had the highest percentage of its geographical area (85.41%) covered by forests, making it the correct answer.
The mean proportional (or geometric mean) between two numbers, say \(a\) and \(b\), is a number \(x\) such that the ratio of \(a\) to \(x\) is equal to the ratio of \(x\) to \(b\). This can be expressed as \(a:x = x:b\), or \[\frac{a}{x} = \frac{x}{b}\]. From this relationship, we can derive a formula to find \(x\).
Thus, the mean proportional between 16 and 81 is 36.
D) 36 — As calculated, the mean proportional between 16 and 81 is 36.
Study the given pattern carefully and select the letter that can replace the question mark (?) in it.
To solve this problem, we need to identify the pattern in the arrangement of letters around a central point in the diagram. Each line radiating from the center has a letter, and we need to determine the letter that replaces the question mark.
Correct Option: C) T
The question asks about the materials used for making beads in the Harappan Civilisation. This requires knowledge of the typical crafts and materials associated with the Indus Valley Civilisation.
D) Carnelian stone — Carnelian was a very popular and extensively used material for making beads in the Harappan Civilisation. Its distinctive reddish-orange colour and the craftsmanship involved in shaping and polishing it made carnelian beads highly valued.
The problem provides a relationship between the sides of a triangle and asks to identify the type of triangle. We will manipulate the given algebraic equation to reveal the relationship between the sides \(a\), \(b\), and \(c\).
The given condition is:
\[ a^{2} + b^{2} + c^{2} = ab + bc + ca \]Rearrange the equation by bringing all terms to one side:
\[ a^{2} + b^{2} + c^{2} - ab - bc - ca = 0 \]Multiply the entire equation by 2 to facilitate completing squares:
\[ 2a^{2} + 2b^{2} + 2c^{2} - 2ab - 2bc - 2ca = 0 \]Rearrange the terms to form perfect square expressions:
\[ (a^{2} - 2ab + b^{2}) + (b^{2} - 2bc + c^{2}) + (c^{2} - 2ca + a^{2}) = 0 \]Recognize the perfect square identities \((x-y)^2 = x^2 - 2xy + y^2\):
\[ (a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0 \]The sum of three non-negative squares can only be zero if each individual square term is zero. This is because squares of real numbers are always greater than or equal to zero.
From these conditions, we conclude that \( a = b = c \).
A triangle with all three sides equal is defined as an equilateral triangle.
Correct Option: A) equilateral — The algebraic manipulation of the given condition \( a^{2} + b^{2} + c^{2} = ab + bc + ca \) leads to \( (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \). This implies \( a=b \), \( b=c \), and \( c=a \), meaning all sides of the triangle are equal. Therefore, the triangle is equilateral.
The question asks to identify the nuclear reactor re-commissioned by BARC in 2018 with double its previous capacity. This requires knowledge of India's nuclear research reactors and their recent developments.
Correct Option: C) Apsara. The Apsara reactor, India's first nuclear research reactor, was re-commissioned in 2018 as Apsara-Upgraded (Apsara-U) with a thermal power of 2 MW, which is double its original capacity of 1 MW. This makes it the correct answer.
Select the option that is related to the third term in the same way as the second term is related to the first term.
Tennis : Forehand :: Cricket : ?
The problem asks us to find a term related to 'Cricket' in the same way 'Forehand' is related to 'Tennis'. This is an analogy question where we need to identify the relationship between the first pair of words and apply the same relationship to the second pair.
Correct Option: C) Square Cut is a specific type of shot or stroke in Cricket, similar to how a Forehand is a specific type of shot in Tennis.
This is an analogy question where we need to find the relationship between the first pair of terms (XVI : 32) and then apply the same relationship to the third term (XXIV) to find the missing fourth term.
The first term, XVI, is a Roman numeral. We need to convert it to its decimal equivalent.
So, XVI = \(10 + 5 + 1 = 16\).
Now, we look at the relationship between 16 and 32. It is clear that \(32 = 16 \times 2\).
The third term, XXIV, is also a Roman numeral. Convert it to its decimal equivalent.
In XXIV, 'IV' means \(5 - 1 = 4\). So, XXIV = \(10 + 10 + 4 = 24\).
Now, apply the same rule as found in Step 1: multiply the decimal value by 2.
So, the missing term will be \(24 \times 2 = 48\).
Correct Option: B) 48
The Roman numeral XVI converts to 16. Multiplying 16 by 2 gives 32. Similarly, the Roman numeral XXIV converts to 24. Multiplying 24 by 2 gives 48.
In which city was the 23rd National Conference on e-Governance held?
The question asks for the venue of the 23rd National Conference on e-Governance. This is a factual question requiring knowledge of current affairs or specific event locations.
The 23rd National Conference on e-Governance was indeed held in Mumbai, Maharashtra, on February 7-8, 2020. The theme for this conference was "India 2020: Digital Transformation – Challenges and Way Forward".
A) Mumbai is the correct answer. The 23rd National Conference on e-Governance was held in Mumbai.