Case study (4 Marks)

Question 14 Marks

Read the text carefully and answer the questions:
An observer on the top of a $40m$ tall light house $($including height of the observer$)$ observes a ship at an angle of depression $30^{\circ}$ coming towards the base of the light house along straight line joining the ship and the base of the light house.
The angle of depression of ship changes to $45^{\circ}$ after 6 seconds.

$(a)$ Find the distance of ship from the base of the light house after $6$ seconds from the initial position when angle of depression is $45^{\circ}$.
$(b)$ Find the distance between two positions of ship after $6$ seconds?
OR
Find the distance of ship from the base of the light house when angle of depression is $30^{\circ}$.
$(c)$ Find the speed of the ship?

Answer

Read the text carefully and answer the questions:
An observer on the top of a $40 m$ tall light house $($including height of the observer$)$ observes a ship at an angle of depression $30^{\circ}$ coming towards the base of the light house along straight line joining the ship and the base of the light house.
The angle of depression of ship changes to $45^{\circ}$ after 6 seconds.

The distance of ship from the base of the light house after $6$ seconds from the initial position when angle of depression is
$45^{\circ} .$
$\text { In } \triangle ABC$
$\tan 45^{\circ}=\frac{A B}{B C}$
$\Rightarrow 1=\frac{40}{B C}$
$\Rightarrow BC=40 m$
$(ii)$

The distance between two positions of ship after $6$ seconds
$CD=BD-BC$
$\Rightarrow CD=40 \sqrt{3}-40=40(\sqrt{3}-1)$
$\Rightarrow CD=29.28 m$
OR

The distance of ship from the base of the light house when angle of depression is $30^{\circ}$.
In $\triangle ABD$
$\tan 30^{\circ}=\frac{A B}{B D}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{40}{B D}$
$\Rightarrow BD=40 \sqrt{3} m$

Speed of ship $=\frac{\text { Distance }}{\text { Time }}=\frac{29.28}{6}=4.88 m / \sec$

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Question 24 Marks

Read the text carefully and answer the questions:
Karan went to the Lab near to his home for $\text{COVID} \ 19$ test along with his family members.
The seats in the waiting area were as per the norms of distancing during this pandemic $($as shown in the figure$)$.
His family member took their seats surrounded by red circular area.

$(a)$ What is the distance between Neena and Karan?
$(b)$ What are the coordinates of seat of Akash?
OR
Find distance between Binu and Karan.
$(c)$ What will be the coordinates of a point exactly between Akash and Binu where a person can be?

Answer

Read the text carefully and answer the questions:
Karan went to the Lab near to his home for $\text{COVID} \ 19$ test along with his family members.
The seats in the waiting area were as per the norms of distancing during this pandemic $($as shown in the figure$)$. His family member took their seats surrounded by red circular area.

$(i)$ Position of Neena $=(3,6)$
Position of Karan $=(6,5)$
Distance between Neena and Karan $=\sqrt{(6-3)^2+(5-6)^2}$
$=\sqrt{9+(-1)^2}$
$=\sqrt{10}$
$(ii)$ Co $-$ ordinate of seat of Akash $=2,3$
OR
$\text { Binu }=(5,5)$
$ \text {; Karan }=(6,5)$
$\text { Distance }=\sqrt{(6-5)^2+(5-2)^2}$
$=\sqrt{1+9}$
$=\sqrt{10}$
$(iii)$

Co $-$ ordinate of middle point $=\left(\frac{2+5}{2}, \frac{3+2}{2}\right)$
$=3.5,2.5$

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Question 34 Marks

Read the text carefully and answer the questions:
Kamla and her husband were working in a factory in Seelampur, New Delhi. During the pandemic, they were asked to leave the job.As they have very limited resources to survive in a metro city, they decided to go back to their hometown in Himachal Pradesh.After a few months of struggle, they thought to grow roses in their fields and sell them to local vendors as roses have been always in demand.Their business started growing up and they hired many workers to manage their garden and do packaging of the flowers.

In their garden bed, there are $23$ rose plants in the first row $, 21$ are in the $2^{\text {nd }}, 19$ in $3^{\text {rd }}$ row and so on. There are $5$ plants in the last row.
$(a)$ How many rows are there of rose plants?
$(b)$ Also, find the total number of rose plants in the garden.
OR
If total number of plants are $80$ in the garden, then find number of rows?
$(c)$ How many plants are there in 6th row.

Answer

Read the text carefully and answer the questions:
Kamla and her husband were working in a factory in Seelampur, New Delhi.
During the pandemic, they were asked to leave the job. As they have very limited resources to survive in a metro city, they decided to go back to their hometown in Himachal Pradesh.
After a few months of struggle, they thought to grow roses in their fields and sell them to local vendors as roses have been always in demand.
Their business started growing up and they hired many workers to manage their garden and do packaging of the flowers.

In their garden bed, there are $23$ rose plants in the first row, $21$ are in the $2^{\text {nd }}, 19$ in $3^{\text {rd }}$ row and so on.
There are $5$ plants in the last row.
$(i)$ The number of rose plants in the $1^{ st }, 2^{\text {nd }}, \ldots$ are $23,21,19, \ldots 5$
$a=23, d=21-23=-2, a_n=5$
$\therefore a_n=a+(n-1) d$
$\text { or, } 5=23+(n-1)(-2)$
$\text { or, } 5=23-2 n+2$
$\text { or, } 5=25-2 n$
$\text { or, } 2 n=20$
$\text { or, } n=10$
$(ii)$ Total number of rose plants in the flower bed,
$S_n=\frac{n}{2}[2 a+(n-1) d]$
$S_{10}=\frac{10}{2}[2(23)+(10-1)(-2)]$
$S_{10}=5[46-20+2]$
$S_{10}=5(46-18)$
$S_{10}=5(28)$
$S_{10}=140$
OR
$S_{n}=80$
$S_{n}=\frac{n}{2}[2 a+(n-1) d]$
$\Rightarrow 80=\frac{n}{2}[2 \times 23+(n-1) \times-2]$
$\Rightarrow 80=23 n-n^2+n$
$\Rightarrow n^2-24 n+80=0$
$\Rightarrow(n-4)(n-20)=0$
$\Rightarrow n=4 \text { or } n=20$
$n=20 \text { not possible }$
$a_{20}=23+19 \times(-2)=-15$
Number of plants cannot be negative.
$n=4$
$\text { (iii) } a_n =a+(n-1) d$
$\Rightarrow a_6=23+5 \times(-2)$
$\Rightarrow a_6=13$

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Maths Case study (4 Marks)

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