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Nilakantha Somayaji's Work on Calculus and Infinite Series

Description: Nilakantha Somayaji was an Indian mathematician and astronomer who lived in the 15th century. He is best known for his work on calculus and infinite series, which was published in his book Tantrasamgraha. This quiz will test your knowledge of Nilakantha Somayaji's work on calculus and infinite series.
Number of Questions: 15
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Tags: indian mathematics classical indian mathematics texts nilakantha somayaji calculus infinite series
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In which book did Nilakantha Somayaji publish his work on calculus and infinite series?

  1. Tantrasamgraha

  2. Aryabhatiya

  3. Lilavati

  4. Siddhanta Shiromani


Correct Option: A
Explanation:

Nilakantha Somayaji published his work on calculus and infinite series in his book Tantrasamgraha.

What is the name of the series that Nilakantha Somayaji used to calculate the value of pi?

  1. Gregory-Leibniz series

  2. Taylor series

  3. Maclaurin series

  4. Nilakantha series


Correct Option: D
Explanation:

Nilakantha Somayaji used a series that is now known as the Nilakantha series to calculate the value of pi.

What is the formula for the Nilakantha series?

  1. $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$

  2. $$\pi = 4 - \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} - \frac{4}{6 \cdot 7} + \cdots$$

  3. $$\pi = 3 + \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} + \frac{4}{6 \cdot 7} + \cdots$$

  4. $$\pi = 4 - \frac{4}{2 \cdot 3 \cdot 4} + \frac{4}{4 \cdot 5 \cdot 6} - \frac{4}{6 \cdot 7 \cdot 8} + \cdots$$


Correct Option: A
Explanation:

The formula for the Nilakantha series is $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$. This series converges very quickly, and it can be used to calculate the value of pi to a high degree of accuracy.

What is the value of pi that Nilakantha Somayaji calculated using the Nilakantha series?

  1. 3.14159265

  2. 3.141592653589793

  3. 3.1415926535897932384626433832795

  4. 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679


Correct Option: C
Explanation:

Nilakantha Somayaji calculated the value of pi to 33 decimal places using the Nilakantha series. His value of pi is accurate to 33 decimal places.

What is the name of the method that Nilakantha Somayaji used to find the derivative of a function?

  1. Newton's method

  2. Leibniz's method

  3. Rolle's method

  4. Nilakantha's method


Correct Option: D
Explanation:

Nilakantha Somayaji used a method that is now known as Nilakantha's method to find the derivative of a function.

What is the formula for Nilakantha's method for finding the derivative of a function?

  1. $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

  2. $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h}$$

  3. $$f'(x) = \lim_{h \to 0} \frac{f(x + h) + f(x - h)}{2h}$$

  4. $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{2h}$$


Correct Option: B
Explanation:

The formula for Nilakantha's method for finding the derivative of a function is $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h}$$. This formula is very similar to the formula for the derivative of a function that is used in modern calculus.

What is the name of the series that Nilakantha Somayaji used to find the sum of an infinite series?

  1. Gregory-Leibniz series

  2. Taylor series

  3. Maclaurin series

  4. Nilakantha series


Correct Option: D
Explanation:

Nilakantha Somayaji used a series that is now known as the Nilakantha series to find the sum of an infinite series.

What is the formula for the Nilakantha series for finding the sum of an infinite series?

  1. $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$

  2. $$S = \sum_{n=1}^\infty \frac{1}{n}$$

  3. $$S = \sum_{n=1}^\infty \frac{1}{n^2}$$

  4. $$S = \sum_{n=1}^\infty \frac{(-1)^n}{n}$$


Correct Option: A
Explanation:

The formula for the Nilakantha series for finding the sum of an infinite series is $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$. This series converges very quickly, and it can be used to find the sum of many different infinite series.

What is the sum of the infinite series $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$?

  1. 0

  2. 1

  3. $\frac{1}{2}$

  4. $\frac{1}{4}$


Correct Option: C
Explanation:

The sum of the infinite series $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$ is $$\frac{1}{2}$$. This can be shown using the Nilakantha series.

What is the name of the method that Nilakantha Somayaji used to find the area of a circle?

  1. Gregory-Leibniz method

  2. Taylor method

  3. Maclaurin method

  4. Nilakantha method


Correct Option: D
Explanation:

Nilakantha Somayaji used a method that is now known as the Nilakantha method to find the area of a circle.

What is the formula for the Nilakantha method for finding the area of a circle?

  1. $$A = \pi r^2$$

  2. $$A = 2\pi r$$

  3. $$A = \frac{1}{2}\pi r^2$$

  4. $$A = \frac{1}{4}\pi r^2$$


Correct Option: C
Explanation:

The formula for the Nilakantha method for finding the area of a circle is $$A = \frac{1}{2}\pi r^2$$. This formula is very accurate, and it can be used to find the area of a circle to a high degree of accuracy.

What is the area of a circle with radius 10 units?

  1. 100\pi

  2. 200\pi

  3. 300\pi

  4. 400\pi


Correct Option: A
Explanation:

The area of a circle with radius 10 units is $$A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi (10)^2 = 100\pi$$. Therefore, the area of the circle is 100\pi square units.

What is the name of the method that Nilakantha Somayaji used to find the volume of a sphere?

  1. Gregory-Leibniz method

  2. Taylor method

  3. Maclaurin method

  4. Nilakantha method


Correct Option: D
Explanation:

Nilakantha Somayaji used a method that is now known as the Nilakantha method to find the volume of a sphere.

What is the formula for the Nilakantha method for finding the volume of a sphere?

  1. $$V = \frac{4}{3}\pi r^3$$

  2. $$V = \frac{1}{3}\pi r^3$$

  3. $$V = \frac{2}{3}\pi r^3$$

  4. $$V = \frac{3}{4}\pi r^3$$


Correct Option: A
Explanation:

The formula for the Nilakantha method for finding the volume of a sphere is $$V = \frac{4}{3}\pi r^3$$. This formula is very accurate, and it can be used to find the volume of a sphere to a high degree of accuracy.

What is the volume of a sphere with radius 10 units?

  1. 4000\pi

  2. 8000\pi

  3. 12000\pi

  4. 16000\pi


Correct Option: A
Explanation:

The volume of a sphere with radius 10 units is $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (10)^3 = 4000\pi$$. Therefore, the volume of the sphere is 4000\pi cubic units.

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