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 $\bar{z} \bar{z}$ ? (1)
iii. Find the value of $| Z |$. (2)
OR
Find the real part of Z. (2)

Answer

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

View full question & answer→

Question 24 Marks

There are 4 red, 5 blue and 3 green marbles in a basket.
i. If two marbles are picked at randomly, find the probability that both red marbles. (1)
ii. If three marbles are picked at randomly, find the probability that all green marbles. (1)
iii. If two marbles are picked at randomly then find the probability that both are not blue marbles. (2)
OR
If three marbles are picked at randomly, then find the probability that atleast one of them is blue. (2)

Answer

i. Total marbles = 4 + 5 + 3 = 12
Required probability $=\frac{{ }^4 C_2}{{ }^{12} C_2}=\frac{\frac{4 \times 3}{2 \times 1}}{\frac{12 \times 11}{2 \times 1}}=\frac{1}{11}$
ii. Total marbles = 4 + 5 + 3 = 12
Required probability $=\frac{{ }^3 C_3}{{ }^{12} C_3}=\frac{1}{\frac{12 \times 11 \times 10}{3 \times 2}}=\frac{1}{220}$
iii. Total marbles = 4 + 5 + 3 = 12
Required probability $=\frac{{ }^7 C_2}{{ }^{12} C_2}=\frac{\frac{7 \times 6}{2 \times 1}}{\frac{12 \times 11}{2 \times 1}}=\frac{21}{66}=\frac{7}{22}$
OR
Total marbles = 4 + 5 + 3 = 12
Required probability = 1 - P (None is blue)
$\begin{array}{l}=1-\frac{{ }^7 C_3}{{ }^{12} C_3} \\ =1-\frac{\frac{7 \times 6 \times 5}{3 \times 2}}{\frac{13 \times 11 \times 10}{3 \times 2}} \\ =1-\frac{7}{44}=\frac{37}{44}\end{array}$

View full question & answer→

Question 34 Marks

A Relation R from A to B can be depicted pictorially using arrow diagram. In arrow diagram, we write down the elements of two sets A and B in two disjoint circles. Then we draw arrow from set A to set B whenever $( A , B ) \in$ R. An example of information depicted through an arrow diagram is shown below. For example:
A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (E). The company provides ₹ 10,000 , ₹ 25,000 , ₹ 50,000 and ₹ 1,00,000 to the people who work in the categories A, C, $M$ and $E$ respectively. Here $A_1, A_2, A_3, A_4$ and $A_5$ are Assistants; $C_1, C_2, C_3, C_4$ are Clerks; $M_1, M_2, M_3$ are Managers and $E_1, E_2$ are Executive Officers then the relation $R$ is defined by xRy,
where x is the salary given to person y.

i. If the number of elements in set A and set B are p and q then find the number of functions from A to B. (1)
ii. If the number of elements in set A and set B are p and q, then find the number of relations from A to B. (1)
iii. Which figures shows a relation between the two non-empty sets? (2)

OR
Show the relation defined in the below arrow diagram from set A to set B. (2)

Answer

i. Number of functions from $A$ to $B$ are $n(B)^{n(A)}=q^p$
ii. Number of relations from A to B is $2^{ n (A) n(B)}=2^{\text {pA }}$.
iii. Figures A and B show relations. Figure C shows a function but not a relation.
OR
x is a factor of y .
$1,2,4$ and 8 are factors of 8 .

View full question & answer→

MATHS Case study (4 Marks)

Test yourself on this topic