NCERT Solutions for Maths Class 12 Exercise 3.2

# NCERT Solutions for Maths Class 12 Exercise 3.2

## NCERT Solutions for Maths Class 12 Exercise 3.2

### Maths Class 12 Ex 3.2 Question 1.

Let

Find each of the following:
(i) A + B                         (ii) A – B                          (iii) 3A – C
(iv) AB                           (v) BA

Solution:

### Maths Class 12 Ex 3.2 Question 2.

Compute the following:

Solution:

### Maths Class 12 Ex 3.2 Question 3.

Compute the indicated products.

Solution:

### Maths Class 12 Ex 3.2 Question 4.

If ,

then compute (A + B) and (B – C). Also verify that A + (B – C) = (A + B) – C.

Solution:

### Maths Class 12 Ex 3.2 Question 5.

If ,
then compute 3A – 5B.

Solution:

Simplify:
Solution:

Find X and Y, if

Solution:

Find X, if

Solution:
We have,

### Maths Class 12 Ex 3.2 Question 9.

Find x and y, if

Solution:

We have,

Equating the elements, we have 2 + y = 5 and 2x + 2 = 8
This implies y = 3 and 2x = 6.
Hence, x = 3 and y = 3.

### Maths Class 12 Ex 3.2 Question 10.

Solve the equation for x, y, z and t, if

Solution:
We have,

### Maths Class 12 Ex 3.2 Question 11.

If ,

find the values of x and y.

Solution:
We have,

### Maths Class 12 Ex 3.2 Question 12.

Given
find the values of x, y, z and w.

Solution:
We have,

3x = x + 4 x = 2
And 3y = 6 + x + y
y = 4
Also, 3w = 2w + 3
w = 3
Again, 3z = – 1 + z + w
2z = – 1 + 3
2z = 2
z = 1
Hence, x = 2, y = 4, z = 1, w = 3.

### Maths Class 12 Ex 3.2 Question 13.

If show that F(x).F(y) = F(x + y).

Solution:

### Maths Class 12 Ex 3.2 Question 14.

Show that

Solution:

Hence, L.H.S. ≠ R.H.S.

### Maths Class 12 Ex 3.2 Question 15.

Find A² – 5A + 6I, if A =

Solution:

We have A² = A.A

### Maths Class 12 Ex 3.2 Question 16.

If A = , prove that A³ – 6A² + 7A + 2I = 0

Solution:
We have A² = A.A

### Maths Class 12 Ex 3.2 Question 17.

If  , find k so that A² = kA – 2I

Solution:
We are given that,

We have to find the value of k.
Now, A² = kA – 2I

### Maths Class 12 Ex 3.2 Question 18.

If  and I is the identity matrix of order 2, show that

Solution:
On the L.H.S:

On the R.H.S:

### Maths Class 12 Ex 3.2 Question 19.

A trust fund has  30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and second bond pays 7% interest per year. Using matrix multiplication, determine how to divide  30,000 among the two types of bond if the trust fund must obtain an annual total interest of
(a)  1800              (b)  2000

Solution:
(a)
Let  x be invested in the first bond. Then, the sum of money invested in the second bond will be  (30,000 − x).

It is given that the first bond pays 5% interest per year and the second bond pays 7% interest per year.

Therefore, in order to obtain an annual total interest of Rs 1800, we have:

Thus, in order to obtain an annual total interest of  1800, the trust fund should invest  15,000 in the first bond and the remaining  15,000 in the second bond.

(b) Let  x be invested in the first bond. Then, the sum of money invested in the second bond will be  (30,000 − x).

Therefore, in order to obtain an annual total interest of  2000, we have:

Thus, in order to obtain an annual total interest of  2000, the trust fund should invest  5000 in the first bond and the remaining  25,000 in the second bond.

### Maths Class 12 Ex 3.2 Question 20.

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling price are  80,  60 and  40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Solution:
The bookshop has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen economics books.

The selling prices of a chemistry book, a physics book, and an economics book are given as  80,  60 and  40, respectively.

The total amount of money that will be received from the sale of all these books can be represented in the form of a matrix as:

Thus, the bookshop will receive  20160 from the sale of all these books.

Assuming X, Y, Z, W and P are the matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Choose the correct answer in questions 21 and 22.

### Maths Class 12 Ex 3.2 Question 21.

The restrictions on n, k and p so that PY + WY will be defined are
(A) k = 3, p = n
(B) k is arbitrary, p = 2
(C) p is arbitrary, k = 3
(D) k = 2, p = 3

Solution:
Given: X2 x n, Y3 x k, Z2 x p, Wn x 3, Pp x k
Now PY + WY = Pp x k × Y3 + k + Wn x 3 × Y3 x k
Clearly, k = 3 and p = n
Hence, option (A) is correct.

### Maths Class 12 Ex 3.2 Question 22.

If n = p, then the order of the matrix 7X – 5Z is:
(A) p × 2
(B) 2 × n
(C) n × 3
(D) p × n

Solution:
7X – 5Z = 7X2 x n – 5Z2 x p
We can add two matrices if their order is the same, i.e., n = P.
Therefore, the order of 7X – 5Z is 2 × n.
Hence, option (B) is correct.