Archimedes's Method

Q.

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Q. twenty meters of wire is available for fencing of a flower bed in the form of a circular sector. then the maximum area of the flower bed in square meters is


A) 25 B) 30 C) 12.5 D) 10

Q.

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

1. True

2. False

Q. The rate of convergence to the area of a semicircle is proportional to the number of sides of the polygon inscribed in it.

1. directly
2. indirectly
3. not

Q. If $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k},\overrightarrow{b}=-\hat{i}+\hat{k},\overrightarrow{c}=2\hat{j}-\hat{k}$ are three vectors, find the area of the parallelogram having diagonals $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\left(\overrightarrow{b}+\overrightarrow{c}\right)$. [CBSE 2014]

Q. A square has two of its vertices on a circle and the other two on a tangent to the circle.if the diameter of the circle is 10 cm then the area of the square is

Q. In Archimedes's method for finding the area of a curve(function), we divide the area into intervals of width ‘1’ and construct rectangles of width ‘1’ and height as the value of the function on the rightmost point of the interval.

Q. Which of the following gives the area of y = (/x^{2} /) between x = 0, x = b and x-axis?

1. where k varies from 1 to n and n tends to infinity
2. where k varies from 1 to n and n tends to infinity
3. \sum \frac{b^{3}}{n^{3}} k^{3}where k varies from 1 to n and n tends to infinity
4. where k varies from 1 to n and n tends to infinity

Q. The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
(a) π cm²/sec
(b) 2π cm²/sec
(c) π² cm²/sec
(d) 2π² cm²/sec²

Q. Which of the following gives the area under the curve $y=x^2$, between x = 0 and x = b?

1. $\sum \frac{b^2}{n^2}{k}^3$, where k varies from 1 to n and n tends to infinity
2. $\sum \frac{b^3}{n^3}{k}^2$, where k varies from 1 to n and n tends to infinity
3. $\sum \frac{b^2}{n^2}{k}^2$, where k varies from 1 to n and n tends to infinity
4. $\sum \frac{b^3}{n^3}{k}^3$, where k varies from 1 to n and n tends to infinity

Q. Find the area of the parallelogram determined by the vectors:
(i)

(ii)

(iii)

(iv)

Q. Find the area of the parallelogram determined by the vectors:
(i) $2\hat{i}\mathrm{and}3\hat{j}$

(ii) $2\hat{i}+\hat{j}+3\hat{k}\mathrm{and}\hat{i}-\hat{j}$

(iii) $3\hat{i}+\hat{j}-2\hat{k}\mathrm{and}\hat{i}-3\hat{j}+4\hat{k}$

(iv) $\hat{i}-3\hat{j}+\hat{k}\mathrm{and}\hat{i}+\hat{j}+\hat{k}.$

Q. The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}\mathrm{and}\hat{i}-2\hat{j}-3\hat{k}.$ Find the unit vector parallel to one of its diagonals. Also, find its area.

Q. The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

1. True
2. False

Q. In Archimedes's method for finding the area of a curve(function), we divide the area into intervals of width ‘1 unit’ and construct rectangles of width ‘1 unit’ and height as the value of the function on the rightmost point of the interval.

1. False
2. True

Q.

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

1. True

2. False

Q. If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to
(a) $\frac{2}{\mathrm{\pi }}\mathrm{unit}$

(b) $\frac{1}{\mathrm{\pi }}\mathrm{unit}$

(c) $\frac{\mathrm{\pi }}{2}\mathrm{units}$

(d) π units

Q. Integration can be regarded as a process of summation.

1. False
2. True

Q. Which of the following gives the area under the curve $y=x^2$, between x = 0 and x = b?

1. $\sum \frac{b^2}{n^2}{k}^3$, where k varies from 1 to n and n tends to infinity
2. $\sum \frac{b^3}{n^3}{k}^2$, where k varies from 1 to n and n tends to infinity
3. $\sum \frac{b^2}{n^2}{k}^2$, where k varies from 1 to n and n tends to infinity
4. $\sum \frac{b^3}{n^3}{k}^3$, where k varies from 1 to n and n tends to infinity

Q. Integration can be regarded as a process of summation.

1. False
2. True

Q. In Archimedes's method for finding the area of a curve(function), we divide the area into intervals of width ‘1 unit’ and construct rectangles of width ‘1 unit’ and height as the value of the function on the rightmost point of the interval.

1. True
2. False

Q. In Archimedes's method for finding the area of a curve(function), we divide the area into intervals of width ‘1 unit’ and construct rectangles of width ‘1 unit’ and height as the value of the function on the rightmost point of the interval.

1. True
2. False

Q. lf $x$ is real and $2x^2-4x+5>0$, then $x$ lies in the interval

1. $\varnothing$
2. $(-\infty ,0)$
3. $(0,\infty )$
4. $(-\infty ,\infty )$