CBSE Questions for Class 12 Commerce Maths Determinants Quiz 8 - MCQExams.com

If $${x}^{a}{y}^{b}={e}^{m},{x}^{c}{y}^{d}={e}^{n}, { \triangle  }_{ 1 }=\begin{vmatrix} m & b \\ n & d \end{vmatrix},{ \triangle  }_{ 2 }=\begin{vmatrix} a & m \\ c & n \end{vmatrix}$$ and $${ \triangle  }_{ 3 }=\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ the value of $$x$$ and $$y$$ are respectively
  • $$\dfrac { { \triangle }_{ 1 } }{ { \triangle }_{ 3 } }$$ and $$\dfrac { { \triangle }_{ 2 } }{ { \triangle }_{ 3 } }$$
  • $$\dfrac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } }$$ and $$\dfrac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } }$$
  • $$\log { \left( \dfrac { { \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right) } and\log { \left( \dfrac { { \triangle }_{ 2 } }{ { \triangle }_{ 3 } } \right) }$$
  • $${ e }^{ { \triangle }_{ 1 }/{ \triangle }_{ 3 } }\,\ and\,\ { e }^{ { \triangle }_{ 2 }/{ \triangle }_{ 3 } }$$
The value of $$\dfrac { 1 }{ x-y } \begin{vmatrix} 1 & 0 & 0 \\ 3 & { x }^{ 3 } & 1 \\ 5 & { y }^{ 3 } & 1 \end{vmatrix}$$ is-
  • $$x+y$$
  • $${x}^{2}-xy+{y}^{2}$$
  • $${x}^{2}+xy+{y}^{2}$$
  • $${x}^{3}-{y}^{3}$$
One of the roots of $$\begin{vmatrix}x + a & b & c\\ a & x + b & c\\ a & b & x + c\end{vmatrix} = 0$$ is
  • $$abc$$
  • $$a + b + c$$
  • $$-(a + b + c)$$
  • $$-abc$$
Three straight lines $$2x+11y-5=0, 24x+7y-20=0$$ and $$4x-3y-2=0$$
  • form a triangle
  • are only congruent
  • are concurrent with one line bisecting the angle between the other two
  • None of above
If $$\Delta = \begin{vmatrix} x & 2y-z & -z \\ y & 2x-z & -z \\ y & 2y-z & \quad 2x-2y-z \end{vmatrix}$$,then
  • x-y is a factor of $$\Delta$$
  • $$(x-y)^2$$ is a factor of $$\Delta$$
  • (x-y)^3 is a factor of $$\Delta$$
  • $$\Delta$$ is independent of z
For distinct numbers $$a,b,c,x,y,z\ \epsilon R$$ if $${ \Delta  }_{ 1 }\left| \begin{matrix} \left( a-x \right) ^{ 2 } & \left( b-x \right) ^{ 2 } & \left( c-x \right) ^{ 2 } \\ \left( a-y \right) ^{ 2 } & \left( b-y \right) ^{ 2 } & \left( c-y \right) ^{ 2 } \\ \left( a-z \right) ^{ 2 } & \left( b-z \right) ^{ 2 } & \left( c-z \right) ^{ 2 } \end{matrix} \right| { \Delta  }_{ 2 }\left| \begin{matrix} (ax+1)^{ 2 } & (bx+1)^{ 2 } & (cx+1)^{ 2 } \\ (ay+1)^{ 2 } & (by+1)^{ 2 } & (cy+1)^{ 2 } \\ (az+1)^{ 2 } & (bz+1)^{ 2 } & (cz+1)^{ 2 } \end{matrix} \right| $$ then $$\frac { { \Delta  }_{ 1 }^{ 2 } }{ { \Delta  }_{ 2 }^{ 2 } } +\frac { { \Delta  }_{ 2 }^{ 2 } }{ { \Delta  }_{ 1 }^{ 2 } } =$$
  • $$\dfrac {5}{4}$$
  • $$\dfrac {10}{3}$$
  • $$\dfrac {1}{4}$$
  • $$None\ of\ these$$
Let $$\Delta =$$$$\begin{vmatrix} sin\theta cos \phi & sin\theta sin\phi & cos\theta \\ cos\theta cos\phi & cos\theta sin\phi & -sin\theta \\ -sin\theta sin\phi & sin\theta cos\phi & 0\end{vmatrix}$$, then
  • $$\Delta$$ is independent of $$\theta$$
  • $$\Delta$$ is independent of $$\phi$$
  • $$\Delta$$ is a constant
  • none of these
The determinant $$\Delta = \begin{vmatrix} a^ 2(1+x) & ab & ac \\ ab & b^ 2(1+x) & bc \\ ac & bc & c^ 2(1+x) \end{vmatrix}$$ is divisible by
  • $$(x + 3)$$
  • $$(1 + x)^2$$
  • $$x^2$$
  • $$(x^2 + 1)$$
The determinant $$\Delta = \begin{vmatrix} b & c & b\alpha+c \\ c & d & c\alpha+d \\ b\alpha+c & c\alpha+d & a\alpha^3-c\alpha \end{vmatrix} $$ is equal to zero if 
  • b,c,d are in A.P
  • b,c,d are in G.P
  • b,c,d are in H.P
  • $$\alpha$$ is a root of $$ax^3 - bx^2 - 3cx - d = 0$$
Find the values of $$a$$ and $$b$$ so that the points $$( a, b, 3),( 2, 0, -1)$$ and $$(1, -1, -3)$$ are collinear.
  • $$a=4,b=2$$
  • $$a=0,b=2$$
  • $$a=4,b=-2$$
  • $$a=-4,b=-2$$
$$\Delta =\left| \begin{matrix} 0 & i-100 & i-500 \\ 100-i & 0 & 1000-i \\ 500-i & i-1000 & 0 \end{matrix} \right|$$ is equal to
  • $$100$$
  • $$500$$
  • $$1000$$
  • $$0$$
Let $$A =\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$$ and suppose that det.(A) =2 then the det.(B) equals, where $$B =\begin{vmatrix} 4x & 2a & -p \\ 4y & 2b & -q \\ 4z & 2c & -t \end{vmatrix}$$ 
  • $$det(B) = -2$$
  • $$det(B) = -8$$
  • $$det(B) = -16$$
  • $$det(B) = 8$$
The value of determinant $$\begin{vmatrix} a^ 2 & a & 1 \\ cos(nx) & cos(n+1)x & cos(n+2)x \\ sin(nx) & sin(n+1)x & sin(n+2)x \end{vmatrix}$$ is independent of 
  • n
  • a
  • x
  • a , n and x
In $$\begin{vmatrix} 1 & 2 & 7 \\ 3 & 7 & -5 \\ -1 & 4 & 3 \end{vmatrix}$$, cofactor of $$2=___________$$ and cofactor of $$-1=___________.$$
  • $$-4,-59$$
  • $$4,-59$$
  • $$-4,59$$
  • $$59,4$$
If $$f(x) = \left|
\begin{array}{111}
x-3 & 2x^2 -18 & 3x^3 -81\\
x-5 & 2x^2-50 & 4x^3-500\\
1 & 2 & 3 \\
\end {array}
\right|
$$ then $$f(1). f(3) + f(3) .f(5) + f(5) .f(1)$$ is equal to-
  • $$f(1)$$
  • $$f(3)$$
  • $$f(1) + f(3)$$
  • $$f(1) + f(5)$$
The points $$({X}_{1},{Y}_{1})$$, $$({X}_{2},{Y}_{2})$$, $$({X}_{1},{Y}_{2})$$ and $$({X}_{2},{Y}_{1})$$ are always
  • Collinear
  • Concyclic
  • Vertices of a square
  • Vertices of rectangle
If $$a,b,c$$ are non-zeros, then the system of equation : $$(\alpha+a)x+\alpha+\alpha z=0$$; $$\alpha x+(a+b)y+\alpha z=0$$; $$\alpha x+\alpha y+(\alpha +c)z=0$$ has a non-trivial solution if
  •  $$\dfrac{1}{\alpha}=-(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c})$$
  • $${\alpha}^{-1}=a+b+c$$
  • $$\alpha+a+b+c=1$$
  • None of the above
If $$p{\lambda ^4} + p{\lambda ^3} + p{\lambda ^2} + s\lambda  + t = $$ $$\left| {\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda } & {\lambda  + 1} & {\lambda  + 3}\\{\lambda  + 1} & {2 - \lambda } & {\lambda  - 4}\\{\lambda  - 3} & {\lambda  + 4} & {3\lambda }\end{array}} \right|$$, then value of t is 
  • $$16$$
  • $$18$$
  • $$17$$
  • $$19$$
State whether following statement is true or false.
If $$A$$ is a square matrix of order $$n$$, then $$|Adj (Adj \,A)| $$ is of order $$(n^2)$$.
  • True
  • False
$$A=\left[ \begin{matrix} 5 & 5\alpha  & \alpha  \\ 0 & \alpha  & 5\alpha  \\ 0 & 0 & 5 \end{matrix} \right] $$; If $$\left| { A }^{ 2 } \right| =25$$, then $$\left| \alpha  \right| =$$
  • $$5$$
  • $$5^{2}$$
  • $$1$$
  • $$\dfrac{1}{5}$$
the following relation is $$\begin{vmatrix} a & a + b & a + b + c \\ 2a & 3a + 2b & 4a + 3b + 2c = a^3 \\ 3a & 6a + 2b & 10a + 6b + 3c \end{vmatrix} = a
^3$$ 
  • True
  • False
If $$ \begin{vmatrix} \lambda^2 + 3 \lambda & \lambda -1 & \lambda +3 \\ \lambda + 1 & 2- \lambda & \lambda - 4 \\ \lambda-3 & \lambda + 4 & 3 \lambda \end{vmatrix} = p \lambda^4 + q \lambda^3 + r \lambda^2 + s \lambda + t $$ then $$ t = $$
  • $$16$$
  • $$17$$
  • $$18$$
  • $$19$$
$$\begin{vmatrix}\dfrac{1}{a} &bc &a^2 \\ \dfrac{1}{b} &ca & b^2\\ \dfrac{1}{c} & ab & c^2\end{vmatrix}$$ is equal to -
  • $$(a - b)(b-c) ( c-a)$$
  • $$abc(a-b)(b-c)(c-a)$$
  • $$0$$
  • 1
$$\begin{vmatrix} log e & log e^2 & log e^3\\ log e^2 & log e^3 & log e^4\\ log e^3 & log e^4 & log e^5\end{vmatrix}=$$?
  • $$0$$
  • $$1$$
  • $$4$$ log e
  • $$5$$ log e
If $$f(x)=\begin{vmatrix} 1 & x & x+1\\ 2x & x(x-1) & (x+1)x\\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1)\end{vmatrix}$$ then $$f(100)$$ is equal to?
  • $$0$$
  • $$1$$
  • $$100$$
  • $$-100$$
the value of the determinant of order $$3$$ remains unchanged if its rows and columns are interchanged. that statement is ___
  • True
  • False
If none of $$a, b, c$$ is zero,  Whether the given equation  $$\begin{vmatrix} -bc & { b }^{ 2 }+bc & { c }^{ 2 }+bc \\ { a }^{ 2 }+ac & -ac & { c }^{ 2 }+ac \\ { a }^{ 2 }+ab & { b }^{ 2 }+ab & -ab \end{vmatrix}={ \left( bc+ca+ab \right)  }^{ 3 }$$ is ?
  • True
  • False
The value of the determinant $$\left| \begin{matrix} { b }^{ 2 }-ab & b-c & bc-ac \\ ab-{ b }^{ 2 } & a-b & { b }^{ 2 }-ab \\ bc-ac & c-a & ab-{ b }^{ 2 } \end{matrix} \right| $$
  • $$abc$$
  • $$a+b+c$$
  • $$0$$
  • $$ab+bc+ca$$
if $$a>0$$ and discriminant of $${ax}^{2}+{2bx}+{c}$$ is $$-ve$$, then $$\left| \begin{matrix} a & b & ax+b \\ b & c & bx+c \\ ax+b & bx+c & 0 \end{matrix} \right|$$ is
  • $$+ve$$
  • $${(ac-b^2)(ax^2+2bx+c)}$$
  • $$-ve$$
  • $$0$$
$$\left| \begin{matrix} \sqrt { 13 } +\sqrt { 3 }  & 2\sqrt { 5 }  & \sqrt { 5 }  \\ \sqrt { 15 } +\sqrt { 26 }  & 5 & \sqrt { 10 }  \\ 3+\sqrt { 65 }  & \sqrt { 15 }  & 5 \end{matrix} \right| =$$ 
  • $$15\sqrt{2}-25\sqrt{3}$$
  • $$15\sqrt{5}-25\sqrt{6}$$
  • $$25\sqrt{2}-15\sqrt{3}$$
  • $$0$$
The value of $$\left| \begin{matrix} 1+w & { w }^{ 2 } & -w \\ 1+{ w }^{ 2 } & w & -{ w }^{ 2 } \\ { w }^{ 2 }+w & w & -{ w }^{ 2 } \end{matrix} \right| $$ is equal to 
  • $$0$$
  • $$2 \omega$$
  • $$2 {\omega}^{2}$$
  • $$-3 {\omega}^{2}$$
If the points $$A(at^{2}_{1},2at_{1}), B(at^{2}_{2},2at_{2})$$ and $$C(\alpha,0)$$ are collinear, then $$t_{1} t_{2}$$ equals
  • $$2$$
  • $$-1$$
  • $$1$$
  • $$None\ of\ these$$
Let $$F(x)=$$$$\left | \left|  \right| \begin{matrix}1  &1+sin\ x  &1+sin\ x+cos\ x \\  2 &3+2\ sin\ x  &4+3\ sin\ x+2\ cos\ x  \\  3&6+3\ sin\ x&10+6\ sin\ x+3\ cos\ x  \end{matrix} \right |$$  then $$F'\ \left ( \dfrac{\pi}{2} \right )$$ is equal to
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
If $$A = \left[ {\begin{array}{*{20}{c}}a&0&0\\0&a&0\\0&0&a\end{array}} \right]$$ then find the value of $$\left| A \right|\left| {adjA} \right|$$
  • $${a^3}$$
  • $${a^6}$$
  • $${a^9}$$
  • a
If $$\begin{vmatrix} 1+x & 2 & 3 \\ 1 & 2+x & 3 \\ 1 & 2 & 3+x \end{vmatrix}=0$$ then $$x=$$
  • $$1$$
  • $$-1$$
  • $$-6$$
  • $$6$$
If $$A = {\left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right]_{3 \times 2}}$$ then determinant $$\left( {A{A^T}} \right)$$ is equal to
  • $$0$$
  • $${a^2} + {b^2} + {c^2}$$
  • $${p^2} + {q^2} + {r^2}$$
  • $${p^2} + {q^2}$$
If $${t_{1,}}{t_2}\,$$ and $${t_3}$$ distinct. and the points $$\left( {{t_1}.2a{t_1} + a{t_1}^3} \right).\left( {{t_2}.2a{t_2} + a{t_2}^3} \right),\left( {{t_3}.2a{t_3} + a{t_3}^3} \right)$$ are collinear, then $${t_1} + {t_2} + {t_3} = $$
  • $$t_1 t_2 t_3 =-1$$
  • $$t_1 +t_2 +t_3 =t_1 t_2 t_3$$
  • $$t_1 +t_2 +t_3 =0$$
  • $$t_1 +t_2 +t_3 =-1$$
Using properties of determinants it can be proved
$$\begin{vmatrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \end{vmatrix}=4abc$$
  • True
  • False
If $$\Delta  = \left| {\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}} \right|$$ for $$x \ne 0,\,y \ne 0$$ then $$\Delta $$ is
  • Divisible by neither $$x$$ nor $$y$$
  • Divisible by both $$x$$ and $$y$$
  • Divisible by $$x$$ but not $$y$$
  • Divisible by $$y$$ but not $$x$$
If $$A=\quad \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix}, B=(adj\quad A)$$ and $$C=5A$$, then $$\cfrac { \left| adj\quad B \right|  }{ \left| C \right|  } $$ is equal to
  • $$5$$
  • $$25$$
  • $$-1$$
  • $$1$$
Find the value of the determinant $$\begin{vmatrix} 1 & 0 & 0 \\ 2 & \cos { x }  & \sin { x }  \\ 3 & \sin { x }  & \cos { x }  \end{vmatrix}$$.
  • $$\cos{2x}$$
  • $$1$$
  • $$0$$
  • $$\sin{2x}$$
If the points $$(k, 2-2k)$$, $$(1-k, 2k)$$ and $$(-k-4, 6-2k)$$ be collinear, the number of possible values of $$k$$ are 
  • $$4$$
  • $$2$$
  • $$1$$
  • $$3$$
If $$A=\begin{bmatrix} 1 & 2 & -2 \\ -2 & 2 & 1 \\ 2 & 1 & 2 \end{bmatrix}$$ then $${A}^{-1}$$=
  • $$A$$
  • $$\dfrac{1}{9} {A}^{T}$$
  • $$\dfrac{1}{9}A$$
  • $$\dfrac{1}{9}{A}^{-1}$$
The determinant $$\begin{bmatrix} b_{1}+c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\ b_{2}+c_{2} & c_{2}+a_{2} & a_{2}+b_{2} \\ b_{3}+c_{3} & c_{3}+a_{3} & a_{3}+b_{3}\end{bmatrix}=$$_____
  • $$\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$
  • $$2\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$
  • $$3\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$
  • $$4\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$
In a triangle ABC, with usual notations, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix}=0,$$ then $$4sin^2A+24sin^2B+36sin^2C$$ is equal to 
  • $$48$$
  • $$50$$
  • $$44$$
  • $$34$$
If adj $$A=\begin{bmatrix}20 & -20 \\ 10 & 10 \end{bmatrix}$$ , then $$|A|=$$..... 
  • 400
  • 200
  • $$\pm 20$$
  • 0
29 If $$z=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ where 0, I are 2x2 null and identity matrix then det $$\left( \left[ z \right]  \right) $$ is  _______________.
  • 1
  • -1
  • 0
  • None of these
Sum of the real roots of the equation $$\begin{vmatrix} 1 & 4 & 20\\ 1 & -2 & 5 \\ 1 & 2x & 5{x}^{2} \end{vmatrix}=0$$ is
  • $$-2$$
  • $$-1$$
  • $$0$$
  •  $$1$$
The cofactor of the element $$4$$ in the determinant $$\begin{vmatrix} 1 & 3 & 5 & 1\\ 2 & 3 & 4 & 2\\ 8 & 0 & 1 & 1\\ 0 & 2 & 1 & 1\end{vmatrix}$$ is?
  • $$4$$
  • $$10$$
  • $$-10$$
  • $$-4$$
Find the values of $$x$$ if, $$\left| \begin{matrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^{ 2 } \end{matrix} \right| =0$$
  • $$-1, 2$$
  • $$-1, -2$$
  • $$1, -2$$
  • $$1, 2$$
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