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#### 2.   Draw a rough sketch of a quadrilateral KLMN. State,             (a) two pairs of opposite sides,            (b) two pairs of opposite angles,            (c) two pairs of adjacent sides,            (d) two pairs of adjacent angles.

The following is the sketch of a quadrilateral KLMN

(a) Two pairs of opposite sides are

(i) KL and MN.

(ii) LM and NK

(b) Two pairs of opposite angles are

(i)  and

(ii)  and

(c) Two pairs of adjacent sides are

(i) KL and LM

(ii) LM and MN

(d) Two pairs of adjacent angles are

(i)  and

(ii)  and

#### 1.   Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is the meeting point of the diagonals in the interior or exterior of the quadrilateral?

Diagonal PR and diagonal  SQ meet  at O which is inside the quadrilaterl

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#### Q : 18       If A is an invertible matrix of order 2, then det   is equal to                (A)        (B)         (C)         (D)

Given that the matrix is invertible hence  exists and

Let us assume a matrix of the order of 2;

.

Then .

and

Now,

Taking determinant both sides;

Therefore we get;

Hence the correct answer is B.

#### Q : 17        Let A be a nonsingular square matrix of order . Then  is equal to                 (A)       (B)       (C)       (D)

We know the identity

Hence we can determine the value of .

Taking both sides determinant value we get,

or

or taking R.H.S.,

or, we have then

Therefore

Hence the correct answer is B.

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#### Q : 16      If    , verify that . Hence find .

Given matrix: ;

To show:

Finding each term:

So now we have,

Now finding the inverse of A;

Post-multiplying by  as,

...................(1)

Now,

From equation (1) we get;

Hence inverse of A is :

#### Q : 15     For the matrix     Show that      Hence, find . .

Given matrix: ;

To show:

Finding each term:

So now we have,

Now finding the inverse of A;

Post-multiplying by  as,

...................(1)

Now,

From equation (1) we get;

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#### Q : 14      For the matrix  , find the numbers  and  such that .

Given  then we have the relation

So, calculating each term;

therefore  ;

So, we have equations;

and

We get .

#### Q : 13            If   ​ , show that  . Hence find

Given  then we have to show the relation

So, calculating each term;

therefore  ;

Hence .

[Post multiplying by , also ]

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#### Q : 12        Let      and  .  Verify that  .

We have  and .

then calculating;

Finding the inverse of AB.

Calculating the cofactors fo AB:

Then we have adj(AB):

and |AB| = 61(67) - (-87)(-47) = 4087-4089 = -2

Therefore we have inverse:

.....................................(1)

Now, calculating inverses of A and B.

|A| = 15-14 = 1   and |B| = 54- 56 = -2

and

therefore we have

and

Now calculating .

........................(2)

From (1) and (2) we get

Hence proved.

#### Q : 11       Find the inverse of each of the matrices (if it exists).

Given the matrix :

To find the inverse we have to first find adjA then as we know the relation:

So, calculating |A| :

Now, calculating the cofactors terms and then adjA.

So, we have

Therefore inverse of A will be: