Determinants Test

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Determinants Test - 66

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Weekly Quiz Competition

Question 1
1 / -0

If $$1, \omega, \omega^{2}$$ are the roots of unity then $$ \triangle =\left| \begin{matrix} 1 & { \omega }^{ n } & { \omega }^{ 2n } \\ { \omega }^{ n } & { \omega }^{ 2n } & 1 \\ { \omega }^{ 2n } & 1 & { \omega }^{ n } \end{matrix} \right|$$ is equal to-

A
$$0$$

B
$$1$$

C
$$\omega$$

D
$$\omega ^{2}$$

Solution
Question 2
1 / -0

If a matrix $$\begin{bmatrix} { \left( x-a \right) }^{ 2 } & { \left( x-b \right) }^{ 2 } & { \left( x-c \right) }^{ 2 } \\ { \left( y-a \right) }^{ 2 } & { \left( y-b \right) }^{ 2 } & { \left( y-c \right) }^{ 2 } \\ { \left( z-a \right) }^{ 2 } & { \left( z-b \right) }^{ 2 } & { \left( z-c \right) }^{ 2 } \end{bmatrix}$$ is a zero matrix, then $$a,b,c,x,y,z$$ are connected by:

A
$$a+b+c=0,x+y+z=0$$

B
$$a+b+c=0,x=y=z$$

C
$$a=b=c,x+y+z=0$$

D
None of these

Solution
Question 3
1 / -0

The determinant $$\Delta=\left| \begin{matrix} { a }^{ 2 }\left( 1+x \right) & ab & ac \\ ab & { b }^{ 2 }\left( 1+x \right) & bc \\ ac & bc & { c }^{ 2 }\left( 1+x \right) \end{matrix} \right| $$ is divisible by

A
$$1+x$$

B
$$(1+x)^{2}$$

C
$$x^{2}$$

D
$$none\ of\ these$$

Solution
Question 4
1 / -0

The determinant $$\left| \begin{matrix} a & b & a\alpha +b \\ b & c & b\alpha +c \\ a\alpha +b & b\alpha +c & 0 \end{matrix} \right|$$ is equal to zero, if

A
$$a,b,c$$are in A.P.

B
$$a,b,c$$ are in G.P.

C
$$a,b,c$$ are in H.P.

D
None of these

Solution
Question 5
1 / -0

If the points $$A(x, 2), B(-3, -4)$$ and $$C(7, -5)$$ are collinear, then the value of $$x$$ is :

A
$$-63$$

B
$$63$$

C
$$60$$

D
$$-60$$

Solution

Given points are $$A(x,2)\: B(-3,-4) \: C(7, -5)$$
$$(x_{1},y_{1})=(x,2),(x_{2},y_{2})= (-3,-4)(x_{3},y_{3})=(7,-5)$$
$$x_{1}[y_{2}-y_{3}]+x_{2}[y_{3}-y_{1}]+x_{3}[y_{1}-y_{2}]=0$$
$$x[(-4)-(-5)]+(-3)[(-5)-2]+7[2-(-4)]=0$$
$$x(-4+5)-3(-5-2)+7(2+4)=0$$
$$x(1)-3(-7)+7(6)=0$$
$$x+21+42=0$$
$$x+63=0$$
$$x=-63$$
$$\therefore $$ The value of x is $$-63$$.
Question 6
1 / -0

Which of the following is/are true ?
(i) Adjoint of a symmetric matrix is symmetric
(ii) Adjoint of a unit matrix is a unit matrix
(iii) A(adj A)=(adj A) A= [A]f and
(iv) Adjoint of a diagonal matrix is a diagonal matrix

A
$$(i)$$

B
$$(ii)$$

C
$$(iii) and (iv)$$

D
$$None$$ $$of$$ $$these$$

Solution
Question 7
1 / -0

If $$A=\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix}$$, then the value of $$|A||adj A|$$ is

A
$$a^{9}$$

B
$$a^{5}$$

C
$$a^{6}$$

D
$$a^{27}$$

Solution
Question 8
1 / -0

If $$A=\left[ \begin{matrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{matrix} \right]$$, then $$det(adj(adj A))$$

A
$$(14)^{4}$$

B
$$(14)^{3}$$

C
$$(14)^{2}$$

D
$$(14)^{1}$$

Solution
Question 9
1 / -0

Let $$P\left( x \right) =\begin{vmatrix} x & -3+4i & 3-4i \\ x & -7i & 5+6i \\ x & 7-2i & -7-2i \end{vmatrix}$$
The number of values of x for which $$P\left( x \right) =0$$ is

A
0

B
1

C
2

D
3
Question 10
1 / -0

Let $$A$$ be a non-singular matrix of order $$n$$ nad $$\left|A\right|=K$$, then $$\left(adj A\right)^{-1}$$ is

A
$$\dfrac{A}{K}$$

B
$$K^{n-1}\left(adj A\right)$$

C
$$K^{n-2}A$$

D
$$KA$$

Solution