Section - B
1.
The population of a village is 3500. 5/7 of the population are males and rest the
females. 40% of the males are married. Find the percentage of married fe-
males.
2.
If 5% of the price of an article is equal to 6% of its cost and 8% of the sale price
exceeds 9% of cost by Rs. 3. Find the sale and cost price.
3.
A cycle is sold for Rs. 150 at a profit of 25%. At what price it should be sold in
order to have 50% profit.
4.
State and prove the Demorgan's law.
5.
Explain the multiplication of two matrices.
6.
Find 43rd term of the progression
16, 13, 10, 7, 4, -1, -2, -5...... Is 302 any term of this A.P.
1+X
dy
Find
dy
If y =
Answers
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Answer:
Solution 1
population of village =3500
5/7 are male = 3500×5/7
= 2500
no of male in the village =2500
No of female population= 3500-2500= 1000
40% male are married = 2500×40%
= 1000
1000 male population are married.
it means 1000 female are also married
% of female population is 100%
Solution 2
Selling price = 600
Cost price = 500
(Attachment for solution)
Solution 3
To get 50% of profit on the same cycle it should be sold for 180Rs.
{attachment for solution}
Solution 4
Definition of De Morgan’s law:
The complement of the union of two sets is equal to the intersection of their complements and the complement of the intersection of two sets is equal to the union of their complements. These are called De Morgan’s laws.
For any two finite sets A and B;
(i) (A U B)' = A' ∩ B' (which is a De Morgan's law of union).
(ii) (A ∩ B)' = A' U B' (which is a De Morgan's law of intersection).
Proof of De Morgan’s law: (A U B)' = A' ∩ B'
Let P = (A U B)' and Q = A' ∩ B'
Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)'
⇒ x ∉ (A U B)
⇒ x ∉ A and x ∉ B
⇒ x ∈ A' and x ∈ B'
⇒ x ∈ A' ∩ B'
⇒ x ∈ Q
Therefore, P ⊂ Q …………….. (i)
Again, let y be an arbitrary element of Q then y ∈ Q ⇒ y ∈ A' ∩ B'
⇒ y ∈ A' and y ∈ B'
⇒ y ∉ A and y ∉ B
⇒ y ∉ (A U B)
⇒ y ∈ (A U B)'
⇒ y ∈ P
Therefore, Q ⊂ P …………….. (ii)
Now combine (i) and (ii) we get; P = Q i.e. (A U B)' = A' ∩ B'
Proof of De Morgan’s law: (A ∩ B)' = A' U B'
Let M = (A ∩ B)' and N = A' U B'
Let x be an arbitrary element of M then x ∈ M ⇒ x ∈ (A ∩ B)'
⇒ x ∉ (A ∩ B)
⇒ x ∉ A or x ∉ B
⇒ x ∈ A' or x ∈ B'
⇒ x ∈ A' U B'
⇒ x ∈ N
Therefore, M ⊂ N …………….. (i)
Again, let y be an arbitrary element of N then y ∈ N ⇒ y ∈ A' U B'
⇒ y ∈ A' or y ∈ B'
⇒ y ∉ A or y ∉ B
⇒ y ∉ (A ∩ B)
⇒ y ∈ (A ∩ B)'
⇒ y ∈ M
Therefore, N ⊂ M …………….. (ii)
Now combine (i) and (ii) we get; M = N i.e. (A ∩ B)' = A' U B'
Solution 5
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix.
Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812,to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra.
