Answer:

Shopkeepers	Notebooks In dozen	Pens In dozen	Pencils In dozen
A	12	5	6
B	10	6	7
C	11	3	8

Here,
Cost of notebooks per dozen =

(12 \times 40) paise

= Rs 4.80
Cost of pens per dozen =

Rs . (12 \times 1.25)

= Rs 15
Cost ofpPencils per dozen =

(12 \times 35) paise

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as

$A = [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] \begin{matrix} Telephone \\ House calls \\ Letters \end{matrix}$

The number of contacts of each type made in two cities X and Y is given in the matrix B as

$\begin{matrix} Telephone & House calls & Letters \end{matrix} B = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] \begin{matrix} City X \\ City Y \end{matrix}$

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity of their social activities?

Answer:

According to the question,

Let A be the matrix showing the cost per contact (in paisa).
$A = [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] \begin{matrix} Telephone \\ House calls \\ Letters \end{matrix}$

And, B be a matrix showing the number of contacts of each type made in two cities X and Y.
$\begin{matrix} Telephone & House calls & Letters \end{matrix} B = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] \begin{matrix} City X \\ City Y \end{matrix}$

Now, the total amount spent by the party in the two cities will be shown by BA.
$B A = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] = [\begin{matrix} 140000 + 100000 + 750000 \\ 420000 + 200000 + 1500000 \end{matrix}] = [\begin{matrix} 990000 \\ 2120000 \end{matrix}]$

Hence, the total amount spent by the party in the two cities is
X: ₹9900
Y: ₹21200

One should consider social activities of a party before casting his/her vote.

Page No 4.48:

Question 78:

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Answer:

$If A is a skew - symmetric matrix, then A^{T} = - A . {(A^{n})}^{T} = {(A^{T})}^{n} [For all n \in N] \Rightarrow {(A^{n})}^{T} = {(- A)}^{n} [∵ A^{T} = - A] \Rightarrow {(A^{n})}^{T} = {(- 1)}^{n} A^{n} \Rightarrow {(A^{n})}^{T} = A^{n}, if n is even or - A^{n}, if n is odd .$

Hence, $A^{n}$ is symmetric when n is an even natural number.

Page No 4.70:

Question 29:

If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of the two.

$[\begin{matrix} x + y \\ x - y \end{matrix}] = [\begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix}] [\begin{matrix} 1 \\ - 2 \end{matrix}] \Rightarrow [\begin{matrix} x + y \\ x - y \end{matrix}] = [\begin{matrix} 2 - 2 \\ 4 - 6 \end{matrix}] \Rightarrow [\begin{matrix} x + y \\ x - y \end{matrix}] = [\begin{matrix} 0 \\ - 2 \end{matrix}] Corresponding elements of equal matrices are equal . ∴ x + y = 0 and x - y = - 2 \Rightarrow x = - y and - y - y = - 2 \Rightarrow x = - y and y = 1 \Rightarrow x = - 1 and y = 1 Hence, (x, y) = (- 1, 1) .$

Page No 4.71:

Question 64:

Matrix A = $[\begin{matrix} 0 & 2 b & - 2 \\ 3 & 1 & 3 \\ 3 a & 3 & - 1 \end{matrix}]$ is given to be symmetric, find values of a and b.

Answer:

We have

$A = [\begin{matrix} 0 & 2 b & - 2 \\ 3 & 1 & 3 \\ 3 a & 3 & - 1 \end{matrix}]$

$A' = [\begin{matrix} 0 & 3 & 3 a \\ 2 b & 1 & 3 \\ - 2 & 3 & - 1 \end{matrix}]$

We know that a matrix is symmetric if A = A'.

Thus,

For what values of a and b if A = B, where

$A = [\begin{matrix} a + 4 & 3 b \\ 8 & - 6 \end{matrix}], B = [\begin{matrix} 2 a + 2 & b^{2} + 2 \\ 8 & b^{2} - 5 b \end{matrix}]$

Disclaimer: There is a misprint in the question, b² − 5b should be written instead of b² − 56.

Answer:

As the given matrices A and B are equal, therefore, their corresponding elements must be equal. Comparing the corresponding elements, we get

a + 4 = 2a + 2 3b = b² + 2
8 = 8 −6 = b² − 5b

On simplifying, we get
a = 2 and the common value of b = 2.

Hence, the values of a and b are 2, 2.

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