There is a rectangular sheet of dimension (2m | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) rectangle is formed by choosing two horizontal lines and two vertical lines. To have odd side lengths,the distance between the chosen horizontal l

Vedclass · Accepted Answer

$m^2n^2$

Vedclass · Answer

(D) rectangle is formed by choosing two horizontal lines and two vertical lines. To have odd side lengths,the distance between the chosen horizontal lines must be odd,and the distance between the chosen vertical lines must be odd.Let the horizontal side length be $L_h$ and the vertical side length be $L_v$. We require $L_h \in \{1, 3, 5, \dots, 2m-1\}$ and $L_v \in \{1, 3, 5, \dots, 2n-1\}$.For a horizontal length $k$,the number of ways to choose two lines such that the distance is $k$ is $(2m-1) - k + 1 = 2m - k$. Summing this for all odd $k \in \{1, 3, \dots, 2m-1\}$:Number of ways $= \sum_{j=1}^{m} (2m - (2j-1)) = \sum_{j=1}^{m} (2m - 2j + 1) = 2m^2 - 2\frac{m(m+1)}{2} + m = 2m^2 - m^2 - m + m = m^2$.Similarly,the number of ways to choose vertical lines with odd distance is $n^2$.Therefore,the total number of rectangles with odd side lengths is $m^2 	imes n^2 = m^2n^2$.

There is a rectangular sheet of dimension $(2m-1) \times (2n-1)$,where $m > 0, n > 0$. It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

Similar Questions

The value of $\lambda$ such that $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines is:

The distance between the parallel lines given by $(x+7y)^2 + 4\sqrt{2}(x+7y) - 42 = 0$ is

Which of the following compounds is optically active?

Cataract and skin cancer are caused by

Tetrahalides of group $14$ elements (except that of carbon) act as:

Vedclass Test Series

Exam Paper Generator

Online Exam Module