# Chapter 3 ( Miscellaneous)

**class 12 maths chapter 3 Miscellaneous solution**

**Question 1:** Let ,show that , where I is the identity matrix of order 2 and.

**Solution:** It is given that

LHS.

RHS.

Since,

Hence,

We shall prove the result by using the principle of mathematical induction.

For n=1, we have:

P(1):

Therefore, the result is true for n=1 .

Let the result be true for

That is :

P(k):

Now, we have to prove that the result is true for

Consider,

LHS.

Therefore, the result is true for

**Question 2:** If ,prove that

**Solution:** It is given that

We shall prove the result by using the principle of mathematical induction.

Let the result be true for .

Now, we have:

Therefore, the result is true for n=1.

Let the result be true for .

Now, we have to prove that the result is true for .

Since,

Therefore, the result is true for .

**Question 3:** If ,prove that ,where is any positive integer.

**Solution:** It is given that

We shall prove the result by using the principle of mathematical induction.

For n=1 , we have:

Therefore, the result is true for

Let the result be true for n=k .

Now, we have to prove that the result is true for .

Since,

Hence,

Therefore, the result is true for

**Question 4:** If and are symmetric matrices, prove that is a skew symmetric matrix.

**Solution:** It is given that and are symmetric matrices.

Therefore, we have:

and

Now,

Hence,

Thus, is a skew symmetric matrix.

**Question 5:** Show that the matrix is symmetric or skew symmetric according as is symmetric orskew symmetric.

**Solution:** We suppose that is a symmetric matrix, then

Consider,

Therefore,

Thus, if is symmetric matrix, then is a symmetric matrix.

Now, we suppose that is a skew symmetric matrix, then

Consider,

Therefore

Thus, if is a skew symmetric matrix, then is a skew symmetric matrix.

Hence, if is symmetric or skew symmetric matrix, then is symmetric or skew

**symmetric accordingly.**

**Question 6:** Find the values of if the matrix satisfy the equation

**Solution:** It is given that

Therefore,

Now,

Hence,

On comparing the corresponding elements, we have:

Hence,

and

**Question 7:**For what values of x: ?

**Solution:** We have:

Thus, the required value of .

**Question 8:** If , show that

**Solution : ** It is given that

Therefore,

Now,

Thus,

**Question 9: **Find , if

**Solution: **

Hence,

**Question 10:** A manufacturer produces three products which he sells in two markets. Annual sales are

indicated below:

Market Products

product x y z

I 10000 2000 18000

II 6000 20000 8000

(a) If unit sale prices of and are ₹ 2.50 ,₹ 1.50and₹ 1.00, respectively, find the total

revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are ₹2 , ₹1 and 50paise respectively.

Find the gross profit.

**Solution: (a)** The unit sale prices of and are ₹2.50 , ₹1.50 and ₹ 1.00respectively.

Consequently, the total revenue in market I can be represented in the form of a matrix as:

Product matrix:

Sale price matirx:

The total revenue

Total revenue in I market= ₹46000

Total revenue in II market=₹53000

**(b)** The unit costs of and are ₹2.00 , ₹1.00 and 50 paise respectively.

Consequently,

total cost price of each market:

Profit:

Thus, the gross profit in market I is ₹ and in market II is ₹ .

**Question 11:** Find the matrix so that

**Solution:**

The matrix given on the R.H.S. of the equation is a

matrix and the one given on the L.H.S.of the equation is a matrix.

now, Let

Therefore,

Equating the corresponding elements of the two matrices,

we have:

Now,

solving and

we get ,

again solving

and

We get,

Hence the required matrix

**Question 12:** If and are square matrices of the same order such that ,then prove by induction that .Further prove that for all

**Solution:** Given and are square matrices of the same order such that .

To prove:

For ,we have:

.

therefore, the result is true for .

Let the result is true for

Now, we prove that the result is true for

Therefore, the result is true for .

Now, we have to prove that For all

For , we have:

Therefore, the result is true for .

Let the result be true for .

Now, we prove that the result is true for .

**Therefore, the result is true for ** .

**Question 13:** If is such that then,

(A)

(B)

(C)

(D)

**Solution:** It is given that

Therefore,

Now,

Hence,

On comparing the corresponding elements, we have:

Thus, the correct option is C.

**Question 14:** If the matrix is both symmetric and skew symmetric, then

(A) is a diagonal matrix

(B) is a zero matrix

(C) is a square matrix

(D) None of these

**Solution:** If the matrix is both symmetric and skew symmetric, then

and

Hence,

Therefore, is a zero matrix.

Thus, the correct option is B.

**Question 15:** If is a square matrix such that , then is equal to

(A) (B) (C) (D)

**Solution:** It is given that is a square matrix such that .

Now,

Thus, the correct option is .