Case study (4 Marks)

Question 14 Marks

Read the following text carefully and answer the questions that follow:
Swimmer in Distress: A lifeguard located $20$ metre from the water spots a swimmer in distress. The swimmer is $30$ metre from shore and $100$ metre east of the lifeguard. Suppose the lifeguard runs and then swims to the swimmer in a direct line, as shown in the figure.

$i$. How far east from his original position will he enter the water? $($Hint: Find the value of $x$ in the sketch$)$.
$ii$. Which similarity criterion of triangle is used?
$iii$. What is the distance of swimmer from the shore?
OR
What is the length of $AD$?

Answer

$\triangle ABC \sim \triangle DEC$
$\frac{20}{30}=\frac{x}{100-x}$
$2000-20 x=30 x$
$2000=50 x$
$x=40 m$
$ii. AA$
$iii. 60$ metres
OR
$A D=A C+C D$
$=\sqrt{20^2+40^2}+{\sqrt{60^2+30^2}}^2$
$=\sqrt{400+1600}+\sqrt{3600+900}$
$=\sqrt{2000}+\sqrt{4500}$
$\Rightarrow 20 \sqrt{5}+30 \sqrt{5}$
$\Rightarrow 50 \sqrt{5} m$

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Question 24 Marks

Read the following text carefully and answer the questions that follow:
Heart Rate : The heart rate is one of the 'vital signs' of health in the human body. It measures the number of times per minute that the heart contracts or beats. While a normal heart rate does not guarantee that a person is free of health problems, it is a useful benchmark for identifying a range of health issues.

Thirty women were examined by doctors of $\text{AIIMS}$ and the number of heart beats per minute were recorded and summarized as follows:

Number of heart beats per minute	Number of Women
$65-68$	$2$
$68-71$	$4$
$71-74$	$3$
$74-77$	$8$
$77-80$	$7$
$80-83$	$4$
$83-86$	$2$

Based on the above information, answer the following questions:
$i$. How many women are having heart beat in the range $68 - 77$ ?
$ii$. What is the median class of heart beats per minute for these women?
$iii$. a. Find the modal value of heart beats per minute for these women.
OR
Find the median value of heart beats per minute for these women.

Answer

$i$. Women having heart beat in range $68 - 77$
$=4+3+8=15$
$ii$. Median class $=74-77$
$iii.$ Mode $=l+\left(\frac{f_1-f_0}{2 f_1-f_0-f_2}\right) \times h$
$ l=74, f_1=8, f_0=3, f_2=7, h=3$
$\therefore$ Modal value $=74+\left(\frac{8-3}{16-3-7}\right) \times 3$
$\quad=76.5$

No. of heart beats	$f$	$cf$
$65-68$	$2$	$2$
$68-71$	$4$	$6$
$71-74$	$3$	$9$
$74-77$	$8$	$17$
$77-80$	$7$	$24$
$80-83$	$4$	$28$
$83-86$	$2$	$30$

Median $=I+\frac{\frac{N}{2}-C f}{f} \times h$
$=74+\frac{(15-9)}{8} \times 3$
$=76.25$

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Question 34 Marks

Read the following text carefully and answer the questions that follow:
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.

$i.$ In the standard form of quadratic polynomial, $a x^2+b x+c$, what are $a, b$ and $c$ ?
$ii.$ If the roots of the quadratic polynomial are equal, what is the discriminant $D$ ?
$iii.$ If $\alpha$ and $\frac{1}{\alpha}$ are the zeroes of the quadratic polynomial are $2 x ^2- x +8 k$, then find the value of $k\ ?$
OR
What is the relation between zeros and coefficient for a quadratic polynomial?

Answer

$i.$ a is a non zero real number and $b$ and $c$ are any real numbers.
$ii. D =0$
$iii. 2 x^2-x+8 k$
$\alpha \times \frac{1}{\alpha}=\frac{8 k}{2}$
$1=4 k$
$k=\frac{1}{4}$
OR
$\alpha+\beta=\frac{-b}{a}$
i.e., $\left(\frac{- \text { coefficient of } x }{\text { coefficient of } x ^2}\right)$
$\alpha \beta=\frac{c}{a}$
i.e., $\left(\frac{\text { constant term }}{\text { coeff of } x^2}\right)$

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Maths Case study (4 Marks)

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