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**Objective Type Questions (M.C.Q.)****Q.24. Ifthen value of x is****(a) 3 ****(b) Â± 3 ****(c) Â± 6 ****(d) 6****Ans.** (c)**Solution.**

Given that

â‡’

â‡’ 2x^{2} â€“ 40 = 18 + 14 â‡’ 2x^{2} = 32 + 40

â‡’ 2x^{2} = 72 â‡’ x^{2} = 36

âˆ´ x = Â± 6

Hence, the correct option is (c).**Q.25. The value of determinant****(a) a ^{3} + b^{3} + c^{3} **

Here, we have

C

(Taking a + b + c common from C

R

Expanding along C

â‡’ (a + b + c) (a - b) (b - c) (- 1)

â‡’ (a + b + c) (a - b) (c - b)

Hence, the correct option is (d).

Area of triangle with vertices (x

â‡’

â‡’

â‡’

3k = 9 â‡’ k = 3

Hence, the correct option is (b).

(b) (b â€“ c) (c â€“ a) (a â€“ b)

(c) (a + b + c) (b â€“ c) (c â€“ a) (a â€“ b)

(d) None of these

Let

(Taking (b â€“ a) common from C

C

(C

= (a â€“ b)

= 0

Hence, the correct option is (d).

Given that

C

â‡’

Taking 2 cos x + sin x common from C

â‡’

R

â‡’ (2 cos x + sin x) (cos x â€“ sin x)

Hence, the correct option is (c).

Let

C

Hence, the correct option is (a).

Solution.

We have f(t) =

Expanding along R

= cos t(t

= -t

âˆ´

â‡’

â‡’

Hence, the correct option is (a).

(d) 2âˆš3/4

Solution.

Given that:

C

Expanding along R

â‡’

but maximum value of sin 2Î¸ = 1

Hence, the correct option is (a).

Solution.

Given that:

Expanding along R

= (a â€“ b) [2a(a + c)] = (a â€“ b) Ã— 2a Ã— (a + c) â‰ 0

Expanding along R

= - (b - a) [( - 2b) (b - c)] = 2b(b â€“ a) (b â€“ c) â‰ 0

Expanding along R

= a(bc) â€“ b(ac)= abc â€“ abc = 0

Hence, the correct option is (c).

Solution.

We have,

Expanding along R

= 2(6 â€“ 5) â€“ Î»(0 â€“ 5) â€“ 3(0 â€“ 2)

= 2 + 5Î» + 6 = 8 + 5Î»

If A

âˆ´ 8 + 5Î» â‰ 0 so Î» â‰

Hence, the correct option is (d).

Solution.

If A and B are two invertible matrices then

(a) adj A = |A| Ã— A

(b) det (A)

(c) Also, (AB)

(d) (A + B)

âˆ´ (A + B)

Hence, the correct option is (d).

Solution.

Given that

Taking x, y and z common from R

R

Takingcommon from R

C

Expanding along R

â‡’

â‡’+ 1 = 0 and xyz â‰ 0 (x â‰ y â‰ z â‰ 0)

âˆ´ x^{â€“1} + y^{â€“1} + z^{â€“1} = â€“ 1

Hence, the correct option is (d).**Q.36. The value of the determinant****(a) 9x ^{2}(x + y) **

Solution.

Let

C

[Taking (3x + 3y) common from C

R

Expanding along C

â‡’ 3(x + y) (y

Hence, the correct option is (b).

Solution.

Given that,

Expanding along C

â‡’ (2a^{2} + 4) â€“ 2(â€“ 4a â€“ 20) = 86

â‡’ 2a^{2} + 4 + 8a + 40 = 86

â‡’ 2a^{2} + 8a + 4 + 40 â€“ 86 = 0

â‡’ 2a^{2} + 8a â€“ 42 = 0

â‡’ a^{2} + 4a â€“ 21 = 0

â‡’ a^{2} + 7a â€“ 3a â€“ 21 = 0

â‡’ a(a + 7) â€“ 3(a + 7) = 0

â‡’ (a â€“ 3) (a + 7) = 0

âˆ´ a = 3, â€“ 7

Required sum of the two numbers = 3 â€“ 7 = â€“ 4.

Hence, the correct option is (c).**Fill in the blanks****Q.38. If A is a matrix of order 3 Ã— 3, then |3A| = _______ .****Ans.**

We know that for a matrix of order 3 Ã— 3,

|KA| = K^{3} |A|

âˆ´ **Q.39. If A is invertible matrix of order 3 Ã— 3, then |A ^{â€“1} | _______ .**

We know that for an invertible matrix A of any order,

We have,

C

[applying (a + b)

(Taking 4 common from C

â‡’ 4 Ã— 0 = 0 (âˆµ C

Given that: cos 2Î¸ = 0

â‡’

âˆ´

The determinant can be written as

Expanding along C

For any square matrix A, (A

The order of a matrix is 3 Ã— 3

âˆ´ Total number of elements = 3 Ã— 3 = 9

Hence, the number of minors in the determinant is 9.

The sum of the products of elements of any row with the co-factors of corresponding elements is equal to the value of the determinant of the given matrix.

Let

Expanding along R

â‡’ a

(where M

We have,

Expanding along R

â‡’

â‡’ x(x

â‡’ x

x

The roots of the equation may be the factors of 126 i.e., 2 Ã— 7 Ã— 9

9 is given the root of the determinant put x = 2 in eq. (1)

(2)

Hence, x = 2 is the other root.

Now, put x = 7 in eq. (1)

(7)

Hence, x = 7 is also the other root of the determinant.

Let

C

Taking (z â€“ x) common from C

R

Taking (y â€“ z) common from R

= (z â€“ x) (y â€“ z) (xyz â€“ x + y) = (y â€“ z) (z â€“ x) (y â€“ x + xyz)

Given that

Taking (1 + x)

â‡’ (1 + x)

âˆ´ 0 = A + Bx + Cx

By comparing the like terms, we get A = 0.

Since (A

So, (A

If A is a non-singular square matrix, then for any non-zero

scalar â€˜aâ€˜, aA is invertible.

âˆ´

So, (aA) is inverse of

is true.

False.

Since = for a non-singular matrix.

True.

True.

Since

If A is a square matrix of order n

then

âˆ´

True.

Let

R

a, b, c are in A.P.

âˆ´ b â€“ a = c â€“ b â‡’ 2b = a + c

False.

Sincewhere n is the order of the square matrix.

True.

Let

Splitting up C

[âˆµ C

[Taking cos A and cos B common from C

= cos A cos B (0) [âˆµ C

= 0

True.

Let

Splitting up C

Splitting up C

Similarly by splitting C

Given that:

L.H.S.

C

[Taking 2 common from C

C

C

Splitting up C

= 2 Ã— 16 = 32

True.

Let

C

Expanding along C

= sin Î¸ cos Î¸ â€“ 0 = sin Î¸ cos Î¸

[Maximum value of sin 2Î¸ = 1]

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