If (a – \( \frac{1}{a} \)) = 6, then (a⁴ + \( \frac{1}{a^4} \)) =?
The problem asks us to find the value of \(a^4 + \frac{1}{a^4}\) given the value of \(a - \frac{1}{a}\). We can solve this by successively squaring the given expression. First, we find \(a^2 + \frac{1}{a^2}\) by squaring \(a - \frac{1}{a}\). Then, we find \(a^4 + \frac{1}{a^4}\) by squaring the result from the previous step.
We are given \(a - \frac{1}{a} = 6\).
Squaring both sides:
\[ \left(a - \frac{1}{a}\right)^2 = 6^2 \]\[ a^2 + \left(\frac{1}{a}\right)^2 - 2 \cdot a \cdot \frac{1}{a} = 36 \]\[ a^2 + \frac{1}{a^2} - 2 = 36 \]From the previous step, we have:
\[ a^2 + \frac{1}{a^2} = 36 + 2 \]\[ a^2 + \frac{1}{a^2} = 38 \]Now, we need to find \(a^4 + \frac{1}{a^4}\). We can do this by squaring the expression \(a^2 + \frac{1}{a^2}\).
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 38^2 \]\[ \left(a^2\right)^2 + \left(\frac{1}{a^2}\right)^2 + 2 \cdot a^2 \cdot \frac{1}{a^2} = 1444 \]\[ a^4 + \frac{1}{a^4} + 2 = 1444 \]From the previous step, we have:
\[ a^4 + \frac{1}{a^4} = 1444 - 2 \]\[ a^4 + \frac{1}{a^4} = 1442 \]Correct Option: D) 1442
The question asks to identify the Tamil folk singer, composer, and exponent of Tamil folk art who has received the Padma Shri award. This requires knowledge of prominent personalities in Tamil folk art and their accolades.
B) Vijayalakshmi Navaneethakrishnan is a renowned Tamil folk singer, composer, and exponent of Tamil folk art. She, along with her husband M. Navaneethakrishnan, has dedicated her life to researching, collecting, and preserving ancient Tamil folk songs and dances. For her significant contributions to the field of folk arts, she was awarded the Padma Shri by the Government of India in 2018.
The phenomenon of an object floating or sinking in a fluid is governed by its density relative to the density of the fluid. An object floats if its density is less than the fluid's density, and it sinks if its density is greater than the fluid's density.
Correct Option: B) The density of the iron nail is more than the density of water.
This statement correctly explains why the iron nail sinks. Since the density of iron is greater than the density of water (approximately \(7.8 \text{ g/cm}^3\) for iron vs. \(1 \text{ g/cm}^3\) for water), the nail experiences a buoyant force less than its weight, causing it to sink.
\( \frac{189}{135} \) when written in the simplest form is:
\( \frac{14}{10} \)
\( \frac{28}{20} \)
\( \frac{91}{65} \)
\( \frac{7}{5} \)
To simplify a fraction, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). This process is also known as reducing the fraction to its lowest terms.
The given fraction is \( \frac{189}{135} \).
Numerator = \(189\)
Denominator = \(135\)
We can find the GCD by listing factors or using prime factorization.
Let's use prime factorization:
The common prime factors are \(3 \times 3 \times 3 = 27\).
So, the GCD of \(189\) and \(135\) is \(27\).
Divide \(189\) by \(27\):
\[ \frac{189}{27} = 7 \]Divide \(135\) by \(27\):
\[ \frac{135}{27} = 5 \]The simplified form of \( \frac{189}{135} \) is \( \frac{7}{5} \).
D) \( \frac{7}{5} \) is the simplest form of \( \frac{189}{135} \) as found by dividing both the numerator and denominator by their GCD, which is \(27\).
Which of the following awards is conferred on Indian sportspersons for their lifetime contribution to sports?
Dronacharya Award
Sardar Award
Dhyan Chand Award
Indira Gandhi Award
The question asks to identify the award conferred on Indian sportspersons for their lifetime contribution to sports. We need to recall the purpose of various sports awards in India.
Correct Option: C) The Dhyan Chand Award is India's highest award for lifetime achievement in sports and games, given to sportspersons who have contributed to sports both during their active career and after retirement.
Find the height (in metres) of a building if the sun’s shadow increases by 10 m when the angle of elevation of the sun decreases from 60° to 45°.
\(\frac{10\sqrt{3}}{\sqrt{3}+1}\)
\(\frac{10}{\sqrt{3}-1}\)
\(10\sqrt{3}\)
\(\frac{10\sqrt{3}}{\sqrt{3}-1}\)
This problem involves trigonometry, specifically the tangent function, to relate the height of the building to the length of its shadow at different angles of elevation of the sun. We will set up two equations based on the given angles and the change in shadow length, then solve for the height of the building.
Let \(h\) be the height of the building (in metres).
Let \(x\) be the length of the shadow when the angle of elevation is \(60^\circ\).
When the angle of elevation decreases to \(45^\circ\), the shadow increases by \(10 \text{ m}\). So, the new shadow length is \(x + 10 \text{ m}\).
Using the tangent function for the first scenario (angle \(60^\circ\)):
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x} \] \[ \sqrt{3} = \frac{h}{x} \] \[ h = x\sqrt{3} \quad \text{(Equation 1)} \]Using the tangent function for the second scenario (angle \(45^\circ\)):
\[ \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x+10} \] \[ 1 = \frac{h}{x+10} \] \[ h = x+10 \quad \text{(Equation 2)} \]Now we have a system of two equations. Substitute \(x\) from Equation 1 into Equation 2.
From Equation 1, \(x = \frac{h}{\sqrt{3}}\).
Substitute this into Equation 2:
\[ h = \frac{h}{\sqrt{3}} + 10 \]Solve for \(h\):
\[ h - \frac{h}{\sqrt{3}} = 10 \] \[ h \left(1 - \frac{1}{\sqrt{3}}\right) = 10 \] \[ h \left(\frac{\sqrt{3}-1}{\sqrt{3}}\right) = 10 \] \[ h = \frac{10\sqrt{3}}{\sqrt{3}-1} \]Rationalize the denominator (optional, but often done for final answers):
\[ h = \frac{10\sqrt{3}}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} \] \[ h = \frac{10\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3})^2 - 1^2} \] \[ h = \frac{10(3+\sqrt{3})}{3-1} \] \[ h = \frac{10(3+\sqrt{3})}{2} \] \[ h = 5(3+\sqrt{3}) \text{ metres} \]However, the options are not rationalized in this form, so the expression \(\frac{10\sqrt{3}}{\sqrt{3}-1}\) is the desired format.
Correct Option: D) The height of the building is \(\frac{10\sqrt{3}}{\sqrt{3}-1}\) metres.
The question asks to identify the winner of the Best Male Playback Singer Award at the Filmfare Awards 2018. This requires knowledge of the specific awards given out in that particular year.
For the 63rd Filmfare Awards, held in 2018, the Best Male Playback Singer award was presented to Arijit Singh for the song "Roke Na Ruke Naina" from the film Badrinath Ki Dulhania.
C) Arijit Singh — Arijit Singh won the Best Male Playback Singer award at the Filmfare Awards 2018 for his song "Roke Na Ruke Naina" from the movie Badrinath Ki Dulhania.
The question asks to identify the Indian weightlifter who won a gold medal at the Weightlifting World Championships in 2017, marking the first time in over two decades for an Indian athlete. This requires knowledge of significant achievements in Indian sports, specifically weightlifting.
In 2017, Saikhom Mirabai Chanu won a gold medal in the 48 kg category at the World Weightlifting Championships held in Anaheim, USA. This was a historic achievement as it was the first gold medal for India at the World Championships since Karnam Malleswari's gold in 1995, thus ending a 22-year drought.
B) Saikhom Mirabai Chanu — She won the gold medal in the 48 kg category at the 2017 World Weightlifting Championships, becoming the first Indian in over two decades to do so.
The pH value of a salt solution indicates its acidity or alkalinity. A pH of 14 signifies a strongly basic solution. The nature of the salt (acidic, basic, or neutral) depends on the strength of the acid and base from which it is formed. We need to identify the combination of acid and base that results in a salt solution with a very high pH (strongly basic).
C) weak acid and strong base — A salt formed from a weak acid and a strong base undergoes hydrolysis to produce hydroxide ions (OH\(^-\)), making the solution basic. A pH of 14 indicates a very strongly basic solution, which is consistent with a salt derived from a weak acid and a strong base.
Find the odd one out:
The task is to identify the odd one out from a set of symbols. The symbols are presented in a diagram, each labeled with a letter corresponding to an option.
Correct Option: A) \( N \)