CBSE Questions for Class 12 Commerce Maths Determinants Quiz 11 - MCQExams.com

If $$D_r = \begin{vmatrix} r & x & n(n+1)/2 \\ 2r-1 & y & n^{ 2 } \\ 3r-2 & z & n(3n-1)/2 \end{vmatrix} $$, then $$\sum^n_{r \times 1} \, D_r$$ is equal to
  • $$\dfrac{1}{6} n (n + 1)(2n + 1)$$
  • $$\dfrac{1}{4} n^2 (n + 1)^2$$
  • $$0$$
  • None of these
 $$\Delta  = \left| \matrix{  1 + {a^2} + {a^4}\;\;1 + ab + {a^2}{b^2}\;\;1 + ac + {a^2}{c^2} \hfill \cr   1 + ab + {a^2}{b^2}\;\;1 + {b^2} + {b^4}\;\;1 + bc + {b^2}{c^2} \hfill \cr   1 + ac + {a^2}{c^2}\;\;1 + bc + {b^2}{c^2}\;\;1 + {c^2} + {c^4} \hfill \cr}  \right|is\;equal\;to$$                  
  • $${\left( {a + b + c} \right)^6}$$
  • $${\left( {a - b} \right)^2}{\left( {b - c} \right)^2}{\left( {c - a} \right)^2}$$
  • 4 (a-b)(b-c)(c-a)
  • None of these
If in the determinant $$\Delta =\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\end{vmatrix}$$, $$A_1, B_1, C_1$$ etc., be the co-factors of $$a_1, b_1, c_1$$ etc., then which of the following relations is incorrect?
  • $$a_1A_1+b_1B_1+c_1C_1=\Delta$$
  • $$a_2A_2+b_2B_2+c_2C_2=\Delta$$
  • $$a_3A_3+b_3B_3+c_3C_3=\Delta$$
  • $$a_1A_2+b_1B_2+c_1C_2=\Delta$$
Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix} \, and \, U_1, U_2, U_3$$ be column matrices satisfying $$AU_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , AU_2 = \begin{bmatrix} 2 \\ 3 \\ 6 \end{bmatrix} , AU_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$$. If U is $$3 \times 3$$ matrix, whose columns are $$U_1, U_2, U_3$$. then |U| is 
  • $$-11$$
  • $$-3$$
  • $$\dfrac{3}{2}$$
  • $$2$$
$$\begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix}=$$
  • $$xyz\left( \dfrac { 1 }{ x } +\dfrac { 1 }{ y } +\dfrac { 1 }{ z } \right) $$
  • $$xyz$$
  • $$1+\dfrac { 1 }{ x } +\dfrac { 1 }{ y } +\dfrac { 1 }{ z }$$
  • $$\dfrac { 1 }{ x } +\dfrac { 1 }{ y } +\dfrac { 1 }{ z }$$
If $$\triangle_{r}=\begin{vmatrix} { 2 }^{ r-1 } & 2.{ 3 }^{ r-1 } & 4.{ 5 }^{ r-1 } \\ x & y & z \\ { 2 }^{ n-1 } & { 3 }^{ n-1 } & { 5 }^{ n-1 } \end{vmatrix}$$, then $$\sum_{r=1}^{n}(\triangle_{r})$$ is equal to
  • $$xyz$$
  • $$1$$
  • $$-1$$
  • $$0$$
$$\begin{vmatrix} a^2
+2a & 2 a + 1 & 1 \\ 2a+1 & a+2 & 1 \\ 3 & 3 & 1
\end{vmatrix} =$$
  • $$(a-1)^2$$
  • $$(a-1)^3$$
  • $$(a-1)^4$$
  • $$2(a-1)$$
Let k be a positive real number and let 
A = $$\begin{bmatrix} 2k-1 & 2\sqrt{k} & 2\sqrt{ k}  \\ 2\sqrt{ k}  & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix}$$
B = $$\begin{bmatrix} 0 & 2k - 1 & \sqrt{ k}  \\ 1- 2\sqrt{ k}  & 0 & -2k \\ 2\sqrt{k} & -\sqrt{k} & 0 \end{bmatrix}$$
If $$det(Adj(A)) + det(Adj(B))$$ = 2 then [k] is equal to
  • 4
  • 6
  • 0
  • 1
If $$\theta \varepsilon R,$$ then the determinant $$\Delta =\begin{vmatrix} \sin { \theta  }  & \cos { \theta  }  & \sin { 2\theta  }  \\ \sin { \left( \theta +\dfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta +\dfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta +\dfrac { 4\pi  }{ 3 }  \right)  }  \\ \sin { \left( \theta -\dfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta -\dfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta -\dfrac { 4\pi  }{ 3 }  \right)  }  \end{vmatrix}=$$
  • $$-\sin {\theta} - \cos {\theta}$$
  • $$\sin {2\theta}$$
  • $$1+\sin {2\theta} - \cos {2\theta}$$
  • $$None\ of\ these$$
If $$\theta \epsilon R$$, then the $$determinant $$  $$\Delta$$ $$ $$ =\begin{vmatrix} \sin { \theta  }  & \cos { \theta  }  & \sin { 2\theta  }  \\ \sin { \left( \theta +\cfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta +\cfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta +\cfrac { 2\pi  }{ 3 }  \right)  }  \\ \sin { \left( \theta -\cfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta -\cfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta -\cfrac { 2\pi  }{ 3 }  \right)  }  \end{vmatrix}= 
  • $$-\sin { \theta } -\cos { \theta } $$
  • $$\sin { 2\theta } $$
  • $$1+\sin { 2\theta } -\cos { 2\theta } $$
  • None of these
Solve $$\Delta=\begin{vmatrix} \sqrt { 13 } +\sqrt { 3 }  & 2\sqrt { 5 }  & \sqrt { 5 }  \\ \sqrt { 15 } +\sqrt { 26 }  & 5 & \sqrt { 10 }  \\ 3+\sqrt { 65 }  & \sqrt { 15 }  & 5 \end{vmatrix}=$$
  • $$15\sqrt { 2 } -25\sqrt { 3 } $$
  • $$25\sqrt { 3 } -15\sqrt { 2 } $$
  • $$3\sqrt { 5 } $$
  • $$-15\sqrt { 2 } +7\sqrt { 3 } $$
Let $$\left[ \begin{matrix} \cos ^{ -1 }{ x }  & \cos ^{ -1 }{ y }  & \cos ^{ -1 }{ z }  \\ \cos ^{ -1 }{ y }  & \cos ^{ -1 }{ z }  & \cos ^{ -1 }{ x }  \\ \cos ^{ -1 }{ z }  & \cos ^{ -1 }{ x }  & \cos ^{ -1 }{ y }  \end{matrix} \right] $$ such that $$|A|=0$$, then maximum value of $$x+y+z$$ is
  • $$3$$
  • $$0$$
  • $$1$$
  • $$2$$
Matrix $$A = \left| {\begin{array}{*{20}{c}}x & 3 & 2\\1 & y & 4\\2 & 2 & z\end{array}} \right|$$, if $$xyz=60$$ and $$8x+4y+3z=20$$, then $$a(adjA)$$ is equal to 
  • $$\left| {\begin{array}{*{20}{c}}
    {64} & 0 & 0\\
    0 & {64} & 0\\
    0 & 0 & {64}
    \end{array}} \right|$$
  • $$\left| {\begin{array}{*{20}{c}}
    {88} & 0 & 0\\
    0 & {88} & 0\\
    0 & 0 & {88}
    \end{array}} \right|$$
  • $$\left| {\begin{array}{*{20}{c}}
    {68} & 0 & 0\\
    0 & {68} & 0\\
    0 & 0 & {68}
    \end{array}} \right|$$
  • $$\left| {\begin{array}{*{20}{c}}
    {34} & 0 & 0\\
    0 & {34} & 0\\
    0 & 0 & {34}
    \end{array}} \right|$$
The number of distinct values of a $$2 \times 2$$ determinant whose entries are from set $$\{-1, 0, 1\}$$ is
  • $$4$$
  • $$6$$
  • $$5$$
  • $$3$$
$$f\left( x \right) = \left| {\begin{array}{*{20}{c}}{x - 2}&{{{\left( {x - 1} \right)}^2}}&{{x^3}}\\{x - 1}&{{x^2}}&{{{\left( {x + 1} \right)}^3}}\\x&{{{\left( {x + 1} \right)}^2}}&{{{\left( {x + 2} \right)}^3}}\end{array}} \right|$$
  • $$0$$
  • $$2$$
  • $$-2$$
  • None of these
If $$\left( \omega \neq 1 \right)$$  is a cubic root of unity then $$ \left| \begin{matrix} 1 & 1+i+{ \omega  }^{ 2 } & { { \omega  } }^{ 2 } \\ 1-i & -1 & { \omega  }^{ 2 }-1 \\ -i & -1+\omega -i & -1 \end{matrix} \right|$$ equals-
  • $$0$$
  • $$1$$
  • $$i$$
  • $$\omega$$
If A is a square matrix of order 3, then $$\left| Adj(Adj{  A }^{ 2 }) \right| =$$
  • $${ \left| A \right| }^{ 2 }$$
  • $${ \left| A \right| }^{ 4 }$$
  • $${ \left| A \right| }^{ 8 }$$
  • $${ \left| A \right| }^{ 16 }$$
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-
  • $$0$$
  • $$1$$
  • $$\omega$$
  • $$\omega ^{2}$$
If $$|A|$$ denotes the value of the determinant of the square matrix $$A$$ order $$3$$, then $$|-2A|=$$
  • $$-8|A|$$
  • $$8|A|$$
  • $$-2|A|$$
  • None of these
$$f(x) = \begin{vmatrix}2\cos x & 1 & 0\\ x - \dfrac {\pi}{2} & 2\cos x & 1\\ 0 & 1 & 2\cos x\end{vmatrix}\Rightarrow f'(x) =$$
  • $$0$$
  • $$2$$
  • $$\pi/2$$
  • $$\pi - 6$$
State whether the statement is true/false.

If $$\mathbf { A } ( \mathbf { x } )$$  $$= \left[ \begin{array} { c c c } { \cos x } & { - \sin x } & { 0 } \\ { \sin x } & { \cos x } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right]$$, then adj $$[ \mathrm { A } ( \mathrm { x } ) ] = \mathrm { A } ( - \mathrm { x } )$$.
  • True
  • False
If $$a + b + c = 0$$  one root of $$\left| {\begin{array}{*{20}{c}}{a - x}&c&b\\c&{b - x}&a\\b&a&{c - x}\end{array}} \right|$$ =0 is
  • $$x = 1$$
  • $$x = 2$$
  • $$x = {a^2} + {b^2} + {c^2}$$
  • $$x = 0$$
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:
  • $$a+b+c=0,x+y+z=0$$
  • $$a+b+c=0,x=y=z$$
  • $$a=b=c,x+y+z=0$$
  • None of these
If $$\left| \begin{array} { r r r } { x } & { 2 } & { x } \\ { x ^ { 2 } } & { x } & { 0 } \\ { x } & { x } & { 8 } \end{array} \right|$$ = $$A x ^ { 4 } + B x ^ { 3 } + c x ^ { 2 } + D x + E$$ , then the value of $$5 A + 4 B + 2 C + 2 D + E$$ is equal to
  • $$-11$$
  • $$17$$
  • $$-17$$
  • 0
The maximum and minimum values of $$(3\times 3)$$ determinant whose elements belong to $$\left\{ 0,1,2,3 \right\} $$ is
  • $$\pm 9$$
  • $$\pm 15$$
  • $$\pm 54$$
  • $$\pm 32$$
The value of $$\begin{vmatrix} 1 & 1 & 1 \\ { \left( { 2 }^{ x }+{ 2 }^{ -x } \right)  }^{ 2 } & { \left( { 3 }^{ x }+{ 3 }^{ -x } \right)  }^{ 2 } & { \left( { 5 }^{ x }+{ 5 }^{ -x } \right)  }^{ 2 } \\ { \left( { 2 }^{ x }-{ 2 }^{ -x } \right)  }^{ 2 } & { \left( { 3 }^{ x }-{ 3 }^{ -x } \right)  }^{ 2 } & { \left( { 5 }^{ x }-{ 5 }^{ -x } \right)  }^{ 2 } \end{vmatrix}$$ is equal to
  • $$0$$
  • $$30^{x}$$
  • $$30^{-x}$$
  • $$None\ of\ these$$
If the points $$A(x, 2), B(-3, -4)$$ and $$C(7, -5)$$ are collinear, then the value of $$x$$ is :
  • $$-63$$
  • $$63$$
  • $$60$$
  • $$-60$$
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
  • $$a,b,c$$are in A.P.
  • $$a,b,c$$ are in G.P.
  • $$a,b,c$$ are in H.P.
  • None of these
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  
  • $$1+x$$
  • $$(1+x)^{2}$$
  • $$x^{2}$$
  • $$none\ of\ these$$
If $$A=\begin{bmatrix} 2 & 1 & -1 \\ 0 & 1 & 4 \\ 0 & 0 & 3 \end{bmatrix}$$, then $$tr(adj(adj\ A))$$ is equal  to
  • $$18$$
  • $$24$$
  • $$36$$
  • $$48$$
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  
  • $$(i)$$
  • $$(ii)$$
  • $$(iii) and (iv)$$
  • $$None$$ $$of$$ $$these$$
If $$A=\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix}$$, then the value of $$|A||adj A|$$ is 
  • $$a^{9}$$
  • $$a^{5}$$
  • $$a^{6}$$
  • $$a^{27}$$
If $$A=\left[ \begin{matrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{matrix} \right]$$, then $$det(adj(adj A))$$
  • $$(14)^{4}$$
  • $$(14)^{3}$$
  • $$(14)^{2}$$
  • $$(14)^{1}$$
$$A=\begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}$$ and A (adj A)=KI, then the value of 'K' is ...
  • 2
  • -2
  • 10
  • -10
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 
  • 0
  • 1
  • 2
  • 3
Let $$f\left( x \right) = {\sin ^{ - 1}}\left( {\tan x} \right) + {\cos ^{ - 1}}\left( {\cot x} \right)$$ then
  • $$f(x)= \frac {\pi}{2}$$ wherever defined
  • domain of $$f(x)$$ is $$x= n \pi \pm \frac {\pi}{4}, n \in 1$$
  • period $$f(x)$$ is $$\frac {\pi}{2}$$
  • $$f(x)$$ in many one function
If $$A = \begin{bmatrix} 4 & 2 \\ 3 & 4 \end{bmatrix}$$ then |adj A| is equal to 
  • 16
  • 10
  • 6
  • none of these
Let $$A$$ be a non-singular matrix of order $$n$$ nad $$\left|A\right|=K$$, then $$\left(adj A\right)^{-1}$$ is 
  • $$\dfrac{A}{K}$$
  • $$K^{n-1}\left(adj A\right)$$
  • $$K^{n-2}A$$
  • $$KA$$
Which of the following values of  $$\alpha $$ satisfy the equation

$$\left| \begin{array}{ll} { (1+\alpha )^{ { 2 } } } & { (1+2\alpha )^{ { 2 } } } & { (1+3\alpha )^{ { 2 } } } \\ { (2+\alpha )^{ { 2 } } } & { (2+2\alpha )^{ { 2 } } } & { (2+3\alpha )^{ { 2 } } } \\ { (3+\alpha )^{ { 2 } } } & { (3+2\alpha )^{ { 2 } } } & { (3+3\alpha )^{ { 2 } } } \end{array} \right| =-648\alpha $$ ?
  • $$-4$$
  • $$9$$
  • $$-9$$
  • $$4$$
Let $$A =[a_{ij}]$$ be a $$3 \times 3 $$ matrix whose determinant is $$5$$. Then the determinant of the matrix $$B = [ 2^{i-j} a_{ij} ]$$ is
  • $$5$$
  • $$10$$
  • $$20$$
  • $$40$$
If $$A = \begin{bmatrix}1 & -1 & 2\\ 3 & 0 & -2\\ 1 & 0 & 3\end{bmatrix}$$, value of $$|A(adj \,A)|$$:
  • $$11$$
  • $$11^2$$
  • $$11^3$$
  • $$-11$$
If $$\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{matrix} \right]$$ then $$|adj\ (adj\ A)|$$ is equal to
  • $$18^{3}$$
  • $$18^{2}$$
  • $$18^{4}$$
  • $$18^{6}$$
If $$A=\begin{bmatrix} 1 & -2 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$$, then $$A.adj(a)=$$
  • $$\begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$$
  • $$\begin{bmatrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{bmatrix}$$
  • $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
  • $$\begin{bmatrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{bmatrix}$$
If $$A$$ is a square matrix of order $$n$$ and $$|A|=D$$ and $$|adj A|=D^{\prime}$$, then
  • $$DD^{\prime}=D^{2}$$
  • $$DD^{-1}=D^{-1}$$
  • $$DD^{\prime}=D^{n-1}$$
  • $$DD^{\prime}=D^{n}$$
If the points (k, 2 - 2k) (1 - k, 2k) and (-k -4, 6 -2x) be collinear the possible values of k are
  • -$$\dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
  • 1
  • - 1
If $$A=\begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$$, then $$adj(3A^{2}+12A)$$ is equal to:
  • $$\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}$$
  • $$\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}$$
  • $$\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}$$
  • $$\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}$$
Let $${\Delta _{\text{o}}} = $$ $$\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] $$ and let $${\Delta _1}$$ denote the determinant formed by the cofactors of elements of $${\Delta _0}$$ and $${\Delta _2}$$ denote the determinant formed by the cofactor of $${\Delta _1},$$ similarly $${\Delta _n}$$ denotes the determinant formed by the cofactors of $${\Delta _{n - 1}}$$ then the determinant value of $${\Delta _n}$$ is
  • $${\Delta _0}^{2n}\;$$
  • $${\Delta _0}^{{2^n}}\;$$
  • $${\Delta _0}^{{n^2}}\;$$
  • $${\Delta ^2}_0\;$$
$$P = \left[ {\begin{array}{*{20}{c}}1&\alpha &3\\1&3&3\\2&4&4\end{array}} \right]$$ is the adjoint of a $$3 \times 3$$ matrix A and $$\left| A \right| = 4,$$ then $$\alpha $$ is equal to 
  • $$4$$
  • $$11$$
  • $$5$$
  • $$0$$
If  $$A = \left( \begin{array} { l l } { 1 } & { 2 } \\ { 3 } & { 5 } \end{array} \right), $$  then the value of the determinant  $$\left| A ^ { 2009 } - 5 A ^ { 2008 } \right|$$  is
  • $$- 6$$
  • $$- 5$$
  • $$- 4$$
  • $$4$$
  • $$6$$
If $$A=\begin{bmatrix} { a }_{ 1 } & { a }_{ 2 } & { a }_{ 3 } \\ { b }_{ 1 } & { b }_{ 2 } & { b }_{ 3 } \\ { c }_{ 1 } & { c }_{ 2 } & { c }_{ 3 } \end{bmatrix}$$ and $$A_i,B_i,C_i$$ are cofactors of $$a_i,b_i,c_i$$ then $$a_1B_1+a_2B_2+a_3B_3=$$
  • 0
  • |A|
  • $$|A|^2$$
  • 2|A|
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers