Ordered Pairs The ordered pair of two elements a and 3 is denoted… - Vidyadip

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Ordered Pairs The ordered pair of two elements a and 3 is denoted by (a, b) : a is first element (or first component) and d is second element (or second component). Two ordered pairs are equal if their corresponding elements are equal. ie. (a, b) = (c, d)

⇒ a = c and b = d

Cartesian Product of Two Sets For two non-empty sets A and B, the cartesian product A . B is the set of all ordered pairs of elements from sets Aand B. In symbolic form, it can be written as

$\text{A}\cdot\text{B}=\{(\text{a},\text{b}):\text{a}\in\text{A},\text{b}\in\text{B}\}$

Based on the above topics, answer the following questions.

If (a - 3, 6 + 7) = (3, 7), then the value of aand d are:

6, 0

3, 7

7, 0

3, -7

If (x + 6, y - 2) = (0, 6), then the value of x and y are:

6, 8

-6, -8

-6, 8

6, -8

If (x + 2, 4) = (5, 2x + y), then the value of x and y are:

-3, 2

3, 2

-3, -2

Let A and B be two sets such that A . B consists of 6 elements. If three elements of A . B are (1, 4), (2, 6) and (3, 6), then

(A . B) = (B . A)

$(\text{A}\cdot\text{B})\neq(\text{B}\cdot\text{A})$

A . B = {(1, 4), (1, 6), (2, 4)}

None of the above

If m(A . B) = 45, then n(A) cannot be

15

17

5

9

✓

Answer

1

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Method to Find the Sets When Cartesian Product is Given For finding these two sets, we write first element of each ordered pair in first set say A and corresponding second element in second set B (say). Number of Elements in Cartesian Product of Two Sets If there are p elements in set A and g elements in set B, then there will be pq elements in A . B i.e. if n(A) = p and n(B) = q, then n(A . B) = pq.
Based on the above two topic, answer the following questions.

If A . B = {(a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2)}. Then, A and B are:

{1, 3, 2}, {a, b}
{a, b}, {1, 3}
{a, b}, {1, 3, 2}
None of these

If the set A has 3 elements and set B has 4 elements, then the number of elements in A . B is:

3
4
7
12

A and B are two sets given in such a way that A . B contains 6 elements. If three elements of A . B are (1, 3), (2, 5) and (3, 3), then A, B are:

{1, 2, 3}, {3, 5}
{3, 5,}, {1, 2, 3}
{1, 2}, {3, 5}
{1, 2, 3}, {5}

The remaining elements of A . B in (iii) is:

(5, 1), (3, 2), (3, 5)
(1, 5), (2, 3), (3, 5)
(1, 5), (3, 2), (5, 3)
None of the above

The cartesian product P . P has 16 elements among which are found (a, 1) and (b, 2). Then, the set P is:

{a, b}
{1, 2}
{a, b,1, 2}
{0, b, 1, 2, 4}

For a group of 200 candidates, the mean and the standard deviation of scores were found to be 40 and 15 , respectively. Later on it was discovered that the scores of 43 and 35 were misread as 34 and 53, respectively.

Student	English	Hindi	S.st	Science	Maths
Ramu	39	59	84	80	41
Rajitha	79	92	68	38	75
Komala	41	60	38	71	82
Patil	77	77	87	75	42
Pursi	72	65	69	83	67
Gayathri	46	96	53	71	39

i. Find the correct variance. (1)
ii. What is the formula of variance. (1)
iii. Find the correct mean. (2)
OR
Find the sum of correct scores. (2)

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

i
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)

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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)

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)

A manufacturing company produces certain goods. The company manager used to make a data record on daily basis about the cost and revenue of these goods separately. The cost and revenue function of a product are given by $C(x)=20 x+4000$ and $R(x)=60 x+2000$, respectively, where $x$ is the number of goods produced and sold.

Based on above information, answer the following questions.

(i) How many goods must be sold to realise some profit?
(a) $\mathrm{x}<\mathbf{5 0}$
(b) $x>50$
(c) $x \geq 50$
(d) $\mathbf{x} \leq \mathbf{5 0}$

(ii) If the cost and revenue functions of a product are given by $C(x)=3 x+400$ and $R(x)=$ $5 x+20$ respectively, where $x$ is the number of items produced by the manufacturer, then how many items must be sold to realise some profit?
(a) $x \leq 190$
(b) $x \geq 190$
(c) $x<190$
(d) $x>190$

(iii) Let $\mathbf{x}$ and $\mathbf{b}$ are real numbers. If $\mathbf{b} > \mathbf{0}$ and $\mathbf{x}< \mathbf{b}$, then
(a) $x$ is always positive
(b) $\mathbf{X}$ is always negative
(c) $\mathrm{x}$ is real number
(d) None of these

(iv) The solution set of $\mathbf{3}-\mathbf{5}<\mathrm{x}+\mathbf{7}$, when $\mathrm{x}$ is a whole number is given by
(a) $\{0,1,2,3,4,5\}$
(b) $(-\infty, 6)$
(c) $[0,5]$
(d) None of these

(v) Graph of inequality $x>2$ on the number line is represented by
(a)

(d) None of the above

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)

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Rajiv constructs two right angled triangles in the fourth quadrant in such a way that the measure of triangle gives $\cos A=\frac{4}{5}$ and $\cos B=\frac{12}{13}$, where $\frac{3 \pi}{2} < A$ and $B > 2 \pi$.

Based on the above information, answer the following questions.
(i) Find the value of $\cos (A+B)$
(ii) Find the value of $\sin (A-B)$
(iii) Find the value of $\tan (\mathbf{A}+\mathbf{B})$

Consider the data

$x_i$	4	8	11	17	20	24	32
$f _{ i }$	3	5	9	5	4	3	1

i. Find the standard deviation. (1)
ii. Find the variance. (1)
iii. Find the mean. (2)
OR
Write the formula of variance? (2)

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Two students Ankit and Vinod appeared in an examination. The probability that Ankit will qualify the examination is 0.05 and that Vinod will qualify is 0.10. The probability that both will qualify is 0.02.
i. Find the probability that atleast one of them will qualify the exam. (1)
ii. Find the probability that atleast one of them will not qualify the exam. (1)
iii. Find the probability that both Ankit and Vinod will not qualify the exam. (2)
OR
Find the probability that only one of them will qualify the exam. (2)

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