Case study (4 Marks)

Question 14 Marks

Consider the complex number $Z = 2 - 2i.$
Complex Number in Polar Form

$i.$ Find the principal argument of $Z. (1)$
$ii.$ Find the value of $z \overline { Z }$ ? $(1)$
$iii.$ Find the value of $| Z |. (2)$
OR
Find the real part of $Z. (2)$

Answer

$\text { i. } r =|Z|=2 \sqrt{2}$
$x =2, y =-2$
$\cos \theta=\frac{x}{r}=\frac{2}{2 \sqrt{2}}=\frac{1}{\sqrt{2}}$
$\sin \theta=\frac{y}{r}=\frac{-2}{2 \sqrt{2}}=\frac{-1}{\sqrt{2}}$
$\operatorname{Arg}( Z )=\frac{-\pi}{4}$
ii. $z \bar{z}=|z|^2=(2 \sqrt{2})^2=8$
$\text { iii. }|Z|=\sqrt{2^2+(-2)^2}$
$\quad=\sqrt{8}=2 \sqrt{2}$
$OR$
Real part of $2 - 2i = 2$

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Question 24 Marks

Four friends Dinesh, Yuvraj, Sonu, and Rajeev are playing cards. Dinesh, shuffling a cards and told to Rajeev choose any four cards.

$i$. What is the probability that Rajeev getting all face card. $(1)$
$ii$. What is the probability that Rajeev getting two red cards and two black card. $(1)$
$iii$. What is the probability that Rajeev getting one card from each suit. $(2)$
OR
What is the probability that Rajeev getting two king and two Jack cards. $(2)$

Answer

$i$. Total number of possible outcomes $={ }^{52} C_4$
We know that there are $12$ face cards
$\therefore$ Number of favourable outcomes $={ }^{12} C_4$
$\therefore$ Required probability $=\frac{{ }^{12} C_4}{{ }^{52} C_4}$
$ii.$ Total number of possible outcomes $={ }^{52} C_4$
We know that there are $26$ red and $26$ black cards.
$\therefore$ Number of favourable outcomes $={ }^{26} C_2 \times{ }^{26} C_2$
$\therefore$ Required probability $=\frac{\left({ }^{26} C_2\right)^2}{{ }^{52} C_4}$
$iii.$ Total number of possible outcomes $={ }^{52} C_4$
$\therefore $ Number of favourable outcomes $=\left({ }^{13} C_1\right)^4$
$\therefore $ Required probability $=\frac{(13)^4}{{ }^{52} C_4}$
OR
Total number of possible outcomes $={ }^{52} C_4$
In playing cards there are $4$ king and $4$ jack cards.
$\because $ Number of favourable outcomes $=\left({ }^4 C_2 \times{ }^4 C_2\right)$
$=6 \times 6=36$
$\therefore $ Required probability $=\frac{36}{{ }^{52} C_4}$

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Question 34 Marks

Consider the graphs of the functions f(x), h(x) and g(x).

i. Find the range of h(x). (1)
ii. Find the domain of f(x). (1)
iii. Find the value of f(10). (2)
OR
Find the range of g(x). (2)

Answer

i. $h(x)=[x]$ is the greatest integer function. Its range is $Z$ (set of integers)
ii. $f(x)=|x|$. The domain of $f(x)$ is $R$.
iii. Since 10 > 0, f(10) = 1.
OR
g(x) is the signum function. Its range is {-1, 0, 1}.

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Maths Case study (4 Marks)

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