18
SECTION E
This section comprises of $3$ case-study based questions of $4$ marks each.
Case Study – 1
In an auditorium, seats are arranged in rows and columns.
The number of rows are equal to the number of seats in each row in the beginning. When the number of rows are doubled and the number of seats in each row is reduced by $10$, the total number of seats increases by $300$.
Based on the above, answer the following questions :
(a) Taking $x$ as the number of rows in the beginning, represent the above situation by a quadratic equation.
(b) (i) How many rows are there in the original arrangement ?
OR
(ii) How many seats are there in the auditorium in the beginning?
(c) How many seats are there in the auditorium after re-arrangement ?
This section comprises of $3$ case-study based questions of $4$ marks each.
Case Study – 1
In an auditorium, seats are arranged in rows and columns.
The number of rows are equal to the number of seats in each row in the beginning. When the number of rows are doubled and the number of seats in each row is reduced by $10$, the total number of seats increases by $300$.
Based on the above, answer the following questions :
(a) Taking $x$ as the number of rows in the beginning, represent the above situation by a quadratic equation.
(b) (i) How many rows are there in the original arrangement ?
OR
(ii) How many seats are there in the auditorium in the beginning?
(c) How many seats are there in the auditorium after re-arrangement ?
Show SolutionHide Solution↓
Let no. of rows be $x = $ no. of seats in each row.
Total seats in beginning $= x \times x = x^2$.
New number of rows $= 2x$.
New number of seats in each row $= x-10$.
New total seats $= 2x(x-10)$.
According to the problem, new total seats = original total seats + $300$.
(a) $2x(x - 10) = x^2 + 300$
$2x^2 - 20x = x^2 + 300$
$x^2 - 20x - 300 = 0$
(b) (i) To find $x$ (number of rows in original arrangement):
$x^2 - 20x - 300 = 0$
$(x - 30)(x + 10) = 0$
$x = 30$ or $x = -10$.
Since number of rows cannot be negative, $x = 30$.
So, there are $30$ rows in the original arrangement.
OR
(ii) Number of seats in the auditorium in the beginning $= x^2 = 30^2 = 900$.
(c) Number of seats after re-arrangement $= x^2 + 300 = 900 + 300 = 1200$.
Total seats in beginning $= x \times x = x^2$.
New number of rows $= 2x$.
New number of seats in each row $= x-10$.
New total seats $= 2x(x-10)$.
According to the problem, new total seats = original total seats + $300$.
(a) $2x(x - 10) = x^2 + 300$
$2x^2 - 20x = x^2 + 300$
$x^2 - 20x - 300 = 0$
(b) (i) To find $x$ (number of rows in original arrangement):
$x^2 - 20x - 300 = 0$
$(x - 30)(x + 10) = 0$
$x = 30$ or $x = -10$.
Since number of rows cannot be negative, $x = 30$.
So, there are $30$ rows in the original arrangement.
OR
(ii) Number of seats in the auditorium in the beginning $= x^2 = 30^2 = 900$.
(c) Number of seats after re-arrangement $= x^2 + 300 = 900 + 300 = 1200$.