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Published on: 13/05/2022
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Questions + Answers key
Take MCQ Maths Test1.
On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is transitive
2.
On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is symmetric
3.
On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is symmetric
4.
On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is reflexive
5.
The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day function for exchanging Singapore dollar to Indian Rupee is g(y) = 50.50y, Where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee
6.
Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A\(\rightarrow\)B for each of the following:
one-to-one and onto.
7.
Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A\(\rightarrow\)B for each of the following:
one-to-one but not onto.
8.
If n(A\(\cap\)B) = 3 and n(A\(\cup\)B) = 10 then find n(P(A \(\Delta \) B))
9.
The formula for converting from Fahrenheit to Celsius temperatures is \(y={5x\over 9}-{160\over 9}\). Find the inverse of this function and determine whether the inverse is also a function.
10.
The owner of a small restaurant can prepare a particular meal at a cost of Rupee 100. He estimate that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 - x. Express his day revenue total cost and profit on this meal as a function of x.
1.
The relation is defined by aRb if a + b \(\le\) 6 for all a, b \(\in \)N.
a+b \(\le\)6 \(\Rightarrow\) a \(\le\) 6 - b
| a | 5 | 4 | 3 | 2 | 1 |
| b | 1 | 2 | 3 | 4 | 5 |
\(\therefore\) The list of ordered pairs are (5,1) (4, 2) (3, 3) (2, 4) and (1, 5).
(4, 2) \(\in \) R and (2, 4) \(\in \) R \(\Rightarrow\) (4, 4) \(\notin \) R
\(\therefore\) R is not transitive.
2.
The relation is defined by aRb if a + b \(\le\) 6 for all a, b \(\in \)N.
a+b \(\le\)6 \(\Rightarrow\) a \(\le\) 6 - b
| a | 5 | 4 | 3 | 2 | 1 |
| b | 1 | 2 | 3 | 4 | 5 |
\(\therefore\) The list of ordered pairs are (5,1) (4, 2) (3, 3) (2, 4) and (1, 5).
Symmetric : (5, 1) \(\in \) R \(\Rightarrow\) (1, 5) \(\in \) R
(4, 2) \(\in \) R \(\Rightarrow\) (2, 4) \(\in \) R
\(\therefore\) R is symmetric
3.
Given relation is 2a + 3b = 30 for all a, b \(\in \) N.
2a + 3b = 30 \(\Rightarrow\) 2a =30 - 3b
\(\Rightarrow a=\frac { 30-3b }{ 2 } \)
| a | 12 | 9 | 6 | 3 |
| b | 2 | 4 | 6 | 8 |
\(\therefore\) The list of ordered pairs are (12, 2) (9, 4) (6, 6) (3, 8)
Symmetricity : (9, 4) \(\in \) R \(\Rightarrow\) (4, 9) \(\notin \) R
\(\therefore\) R is not symmetric
4.
2a + 3b = 30
R = {(3,8), (6,6), (9,4). (12,2)}
Not reflexive, Not Symmetric, transitive, hence not an equivalence relation.
5.
Given f(x) = 1.23x where x represents the number of American dollars.
and g(y) = 50.50y where y represents the number of Singapore dollars.
To convert American dollars to Indian rupees, we have to find out go f(x)
∴ go f(x) = g(f(x))
= g(1.23x)
= 50.50[1.23x]
= 62.115x
∴ The function for exchange rate of American dollars in terms of Indian rupee is g o f (x) = 62.115x.
6.
Let f: A ⇾ B defined by
f = {(1, a)(2, b)(3, c)(4, d)}
Here different elements have different images
∴ f is one-to-one.
Also Co-domain = {a,b,c,d} = Range.
∴ f is onto.
∴ f is one-to-one and onto.
7.
The function does not exist for one-toone but not onto.
Since f = A\(\rightarrow\) B, f is one-one \(\Rightarrow\) f must be onto [\(\therefore\) n(A) = n(B)]
8.
We know that n(A\(\cup\) B) = n(A-B)+n(B-A)+n(A\(\cap\)B) if A and B are not disjoint.
\(\Rightarrow\) n(A-B)+n(B-A) = n(A\(\cup\)B) - n(A\(\cap\)B)
\(\Rightarrow\) n(A\(\Delta \)B) = 10-3
\(\Rightarrow\) \(\therefore\) n(A\(\Delta \)B) = 7
\(\therefore\) n[P(A\(\Delta \)B)] = 27= 128
9.
Let \(f(x)={5x-160\over 9}\)
Given \(y={5x\over 9}-{160\over 9}\)
\(⇒\ \ y={5x-160\over 9}\)
Then 9y = 5x - 160 ⇒ 5x = 9y + 160 ⇒ \(x={9y+160\over5}\)
Let \(g(y)={9y+160\over 5}\)
Now gof(x) = \(g[f(x)=g\left(5x-160\over 6\right)\)
\(={9\left(5x-160\over 9\right)+160\over 5}={5x-160+160\over 5}={5x\over 5}=x\)
and fog(y) = f(g(y)) =\(f\left(9y+160\over 5\right)={5\left(9y+160\over5\right)-160\over 5}={9y+160-160\over 9}=y\)
Thus gof = Ix and fog = Iy
This implies that f and g are bijections and inverses to each other
\(f^{-1}(y)={9y+160\over5}\)
Replacing y by x, we get f-1 (x) = \(\frac { 9x+160 }{ 5 } \)
10.
Given cost of one meal = Rs. 100
Number of customers is given by the function D(x) = 200 - x
∴ Day cost function = (200 - x)(100)
= 20000 - 100x
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