CBSE Questions for Class 12 Commerce Maths Determinants Quiz 5 - MCQExams.com

If $$A$$ is any square matrix of order $$2$$, then  $$adj \left ( adj A \right )$$ =
  • $$A$$
  • $$\det A$$
  • $$A^{-1}$$
  • None of these
If $$\displaystyle A= \begin{bmatrix}0 &i-\sin x  &i- \cos x\\sin x -i &0  &\sin x-i \\cos x-i &- \sin x+ i  &0 \end{bmatrix}$$ then $$\displaystyle \left | A \right |$$ equals
  • $$0$$
  • $$\displaystyle \sin x $$
  • $$\displaystyle \cos x $$
  • $$1$$
If $$\Delta =\begin{vmatrix}
a & 5-i & 7+i\\
5+i & b & 3+i\\
7-i & 3-i & c
\end{vmatrix}$$, then $$\Delta $$ is always
  • real
  • imaginary
  • $$0$$
  • None of these
  • Both (A) & (R) are individually true & (R) is correct explanation of (A),
  • Both (A)& (R) are individually true but (R) is not the correct (proper) explanation of (A).
  • (A)is true but (R) is false,
  • (A)is false but (R ) is true.
$$\displaystyle A=\begin{bmatrix}1 &-1  &1 \\2
 &1  &-3 \\1  &1  &1 \end{bmatrix}$$ and $$\displaystyle B= \begin{bmatrix}4 &2  &2 \\-5
 &0  &\alpha \\1  &-2  &3 \end{bmatrix}$$ If $$B$$ is the adjoint of $$A$$ then $$\displaystyle \alpha$$ equals
  • $$2$$
  • $$-1$$
  • $$-2$$
  • $$5$$
 If $$(-3, 11), (6, 2)$$ and $$($$$$k$$$$, 4)$$ are  collinear points, then $$k$$ is equal to
  • 8
  • -8
  • 4
  • -4
The points $$X(-1, 3), Y (8, -3)$$ and $$Z(2, 1)$$
  • are collinear
  • not decided
  • cannot be plotted
  • are not collinear
If the points $$\displaystyle \left(\frac{2}{5}, \frac{1}{3}\right), \left(\frac{1}{2} , k \right)$$ and $$\displaystyle \left(\frac{4}{5}, 0 \right)$$ are collinear then find the value of k
  • $$\displaystyle k = \frac{1}{2}$$
  • $$\displaystyle k = \frac{1}{4}$$
  • $$\displaystyle k = \frac{1}{5}$$
  • $$\displaystyle k = -\frac{1}{2}$$
The points $$A(7, 8), B(-5, 2)$$ and $$ C(3, 6) $$ 
  • are not collinear
  • cannot be plotted
  • are not defined
  • are collinear
If every element of third order determinant of $$\displaystyle \Delta $$ is multiplied by 5 then value of new determinant equals to,
  • $$\displaystyle \Delta $$
  • $$\displaystyle 5 \Delta $$
  • $$\displaystyle 25 \Delta $$
  • $$\displaystyle 125 \Delta $$
If $$\omega \neq 1$$ is a complex cube root of unity, and
$$x+iy=\begin{vmatrix}
1 & i & -\omega \\
-i & 1 & \omega ^{2}\\
\omega  & -\omega ^{2} & 1
\end{vmatrix}$$
then
  • $$x=-1, y=0$$
  • $$x=1, y=-1$$
  • $$x=1, y=1$$
  • none of these
The points $$(a, b + c), (b, c + a)$$ and $$(c, a + b)$$ are
  • vertices of an equilateral triangle
  • concyclic
  • vertices of a right angled triangle
  • collinear
The points $$(k, 2-2k)$$, $$(-k+1, 2k)$$ and $$(-4-k, 6-2k)$$ are collinear for
  • all values of k
  • $$k=-1$$
  • $$k=1/2$$
  • no value of k.
If $$\triangle ABC$$ is not a right triangle, then value of
$$\Delta =\begin{vmatrix}
\tan A & 1 & 1\\
1 & \tan B & 1\\
1 & 1 & \tan C
\end{vmatrix}$$
is
  • $$-1$$
  • $$2$$
  • $$3$$
  • $$0$$
If $$\displaystyle \Delta =\begin{vmatrix}
0 &b-a  &c-a \\
a-b &0  &c-b \\
a-c &b-c  &0
\end{vmatrix}$$ then $$\displaystyle \Delta $$ is equal to
  • $$\displaystyle a+b+c$$
  • $$\displaystyle -(a+b+c)$$
  • $$\displaystyle abc $$
  • $$\displaystyle 0$$
If points $$(x,0)$$, $$(0,y)$$ and $$(1,1)$$ are collinear then the relation is-
  • $$x+y=1$$
  • $$x+y=xy$$
  • $$x+y+1=0$$
  • $$x+y+xy=0$$
If $$a, b, c> 1$$, $$\Delta =\begin{vmatrix}
\log _{a}\left ( abc \right ) & \log _{a}b & \log _{a}c\\
\log _{b}\left ( abc \right ) & 1 & \log _{b}c\\
\log _{c}\left ( abc \right ) & \log _{c}b & 1
\end{vmatrix}$$ is
  • $$0$$
  • $$\log _{a}b+\log _{b}c+\log _{c}a$$
  • $$\log _{abc}\left ( a+b+c \right )$$
  • none of these
For which value of '$$k$$' the points $$(7, -2), (5, 1), (3, k)$$ are collinear?
  • $$-4$$
  • $$4$$
  • $$8$$
  • $$-8$$
If $$(3,2)$$, $$(4,k)$$ and $$(5,3)$$ are collinear, then $$k$$ is equal to:
  • $$\dfrac { 3 }{ 2 }$$
  • $$\dfrac { 2 }{ 5 }$$
  • $$\dfrac { 5 }{ 2 }$$
  • $$\dfrac { 3 }{ 5 }$$
Let $$A$$ be a non-singular matrix. Then $$\left| adjA \right| $$ is equal to
  • $${ \left| A \right|  }^{ n }$$
  • $${ \left| A \right|  }^{ n-1 }$$
  • $${ \left| A \right|  }^{ n-2 }$$
  • None of these
The value of the determinant $$\begin{vmatrix} -a& b & c\\ a & -b &c \\ a & b &-c \end{vmatrix}$$ is equal to
  • $$0$$
  • $$(a-b)(b-c)(c-a)$$
  • $$(a+b)(b+c)(c+a)4abc$$
  • $$4abc$$
If $$A=\begin{bmatrix} 0&\sin \alpha   & \sin \alpha\sin \beta \\-\sin \alpha &0  &\cos \alpha \sin \beta  \\-\sin \alpha \sin \beta  &-\cos \alpha \cos \beta   &0 \end{bmatrix}$$, then which of the following is true?
  • $$|A|$$ is independent of $$\alpha $$ and $$\beta $$.
  • $$A^{-1}$$ depends only on $$\alpha $$.
  • $$A^{-1}$$ depends only on $$\beta $$.
  • None of these
If $$D_p=\begin{vmatrix} p& 15 & 8\\ p^2 & 35 & 9\\ p^3 & 25 & 10\end{vmatrix}$$, then $$D_1+D_2+D_3+D_4+D_5$$ is equal to
  • 0
  • 25
  • 625
  • none of these
$$\displaystyle a_{0}$$ equals
  • 0
  • 1
  • 2
  • 3
The minors and cofactors of -4 and 9 in determinant $$\begin{vmatrix} -1 & -2 & 3 \\ -4 & -5 & -6 \\ -7 & 8 & 9\end{vmatrix}$$ are respectively
  • $$42, 42 ; 3, 3$$
  • $$-42, 42 ; -3, -3$$
  • $$42, -42 ; 3, -3$$
  • $$42, 3 ; 42, 3$$
If $$A=\begin{bmatrix} 4 & 2 \\ 3 & 3 \end{bmatrix}$$, then adj (adj $$A$$) is equal to
  • $$\begin{bmatrix} 3 & -2 \\ -3 & 4 \end{bmatrix}$$
  • $$\begin{bmatrix} 4 & 2 \\ 3 & 3 \end{bmatrix}$$
  • $$6\begin{bmatrix} 4 & 2 \\ 3 & 3 \end{bmatrix}$$
  • None of these
A set of points which lie on same line are called as
  • collinear
  • non-collinear
  • concurrent
  • none of these
If $$A=\begin{bmatrix} 0& 0 & 1\\  0& 1 & 0\\ 1 &0 & 0\end{bmatrix}$$, then
  • AdjA is zero matrix
  • $$AdjA=\begin{bmatrix} 0& 0 & -1\\ 0& -1 & 0\\ -1 &0 & 0\end{bmatrix}$$
  • $$A^{-1} = A$$
  • $$A^2 = I$$
The point with the co-ordinates $$(2a,\ 3a),\ (3b,\ 2b)$$ & $$(c,\ c)$$ are collinear _____________ .
  • for no value of $$a,\ b,\ c$$
  • for all values of $$a,\ b,\ c$$
  • if $$a,\ \dfrac {c}{5},\ b$$ are in $$H.P$$.
  • if $$a,\ \dfrac {2}{5}c,\ b$$ are in $$H.P$$.
If lines AB, AC, AD and AE are parallel to a line then_______
  • A, B, C, D, E are collinear points
  • A, B, C, D, E are noncolinear points
  • AB & AC are parallel and AD & AE are perpendicular
  • None of these
Three lines $$px + qy + r = 0, qx + ry + p = 0$$ and $$rx + py + q = 0$$ are concurrent if
  • $$p + q + r = 0$$
  • $$p^2 + q^2 + r^2 = pr + qr + pq$$
  • $$p^3 + q^3 + r^3 = 3pqr$$
  • none of these
The adjoint of the matrix $$\begin{bmatrix} 1 & -2 & 3 \\ 0 & 2 & -1 \\ -4 & 5 & 2 \end{bmatrix}$$ is:
  • $$\begin{bmatrix} 9 & 19 & -4 \\ 4 & 14 & 1 \\ 8 & 3 & 2 \end{bmatrix}$$
  • $$\begin{bmatrix} 9 & 4 & 8 \\ 19 & 14 & 3 \\ -4 & 1 & 2 \end{bmatrix}$$
  • $$\begin{bmatrix} 9 & -19 & -4 \\ -4 & 14 & -1 \\ 8 & -3 & 2 \end{bmatrix}$$
  • none of these
If $$A$$ and $$B$$ are square matrices of same orders then adj. $$(AB)$$ equals
  • adj $$A$$. adj $$B$$
  • adj $$B$$/adj $$A$$
  • adj $$A+$$ adj $$B$$
  • adj $$A-$$adj $$B$$
If cofactor of 2x in the determinant $$\begin{vmatrix}x & 1 & -2 \\ 1 & 2x & x-1 \\ x-1 & x & 0\end{vmatrix}$$ is zero, then $$x$$ equals to
  • 0
  • 2
  • 1
  • -1
If $$A = (a_{ij})$$ is a $$4\times 4$$ matrix and $$C_{ij}$$ is the co-factor of the element $$a_{ij}$$ in Det (A), then the expression $$a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14}$$ equals
  • $$0$$
  • $$-1$$
  • $$1$$
  • $$Det. (A)$$
Consider the determinant $$\Delta=\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}$$
$$M_{ij} =$$ Minor of the element of $$i^{th}$$ row & $$j^{th}$$ column.
$$C_{ij} =$$ Cofactor of element of $$i^{th}$$ row & $$j^{th}$$ column.
$$a_3M_{13} - b_3M_{23} + c_3M_{33}$$ is equal to
  • $$0$$
  • $$4\Delta$$
  • $$2\Delta$$
  • $$\Delta$$
Let $$A = [a_{ij}]_{n\times n}$$ be a square matirx and let $$c_{ij}$$ be cofactor of $$a_{ij}$$ in A. If $$C = [c_{ij}]$$, then
  • $$|C|=|A|$$
  • $$|C|=|A|^{n-1}$$
  • $$|C|=|A|^{n-2}$$
  • none of these
If in the determinant $$\Delta=\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}, A_i, B_i, C_i$$ etc. be the co-factors of $$a_i, b_i, c_i$$ etc., then which of the following relations is incorrect?
  • $$a_1A_1 + b_1B_1 + c_1C_1=\Delta$$
  • $$a_2A_2 + b_2B_2 + c_2C_2=\Delta$$
  • $$a_3A_3 + b_3B_3 + c_3C_3=\Delta$$
  • $$a_1A_2 + b_1B_2 + c_1C_2=\Delta$$
Consider the determinant $$\Delta=\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}$$
$$M_{ij} =$$ Minor of the element of $$i^{th}$$ row & $$j^{th}$$ column.
$$C_{ij} =$$ Cofactor of element of $$i^{th}$$ row & $$j^{th}$$ column.
Value of $$b_1.C_{31} + b_2.C_{32} + b_3.C_{33}$$ is
  • $$0$$
  • $$\Delta$$
  • $$2\Delta$$
  • $$\Delta^2$$
If $$A+B+C=\pi$$, then $$\begin{vmatrix}sin(A+B+C) & sin B & cos C\\ -sinB & 0 & tan A\\ cos(A+B) & -tanA & 0\end{vmatrix}$$ equals
  • 0
  • 2 sin B tan A cos C
  • 1
  • none of these
If $$f(x)=\begin{vmatrix} cos x & 1 & 0 \\ 1 & cos x & 1 \\ 0 & 1 & cos x\end{vmatrix}$$ the $$f'(\dfrac \pi 3)$$ equals
  • $$\dfrac {11\sqrt 3}{8}$$
  • $$\dfrac {5\sqrt 3}{8}$$
  • $$-\dfrac {5\sqrt 3}{8}$$
  • none of these
If $$x, y, z$$ are positive numbers, then value of the determinant $$\begin{vmatrix}1 & log_xy & log_xz \\ log_yx & 1 & log_yz\\ log_zx & log_zy & 1\end{vmatrix}$$ is equal to
  • $$0$$
  • $$3$$
  • $$log xyz$$
  • none
If $$\Delta =\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$$ and $$A_2, B_2, C_2$$ are respectively cofactors of $$a_2, b_2, c_2$$ then $$a_1A_2 + b_1B_2 + c_1C_2$$ is equal to
  • $$-\Delta$$
  • 0
  • $$\Delta$$
  • none of these
If [a] denotes the greatest integer less than or equal to a and $$-1 \leq x < 0, 0 \leq y < 1, 1 \leq z < 2$$, then $$\begin{vmatrix}[x]+1 & [y] & [z] \\ [x] & [y]+1 & [z] \\ [x] & [y] & [z]+1\end{vmatrix}$$ is equal to
  • $$[x]$$
  • $$[y]$$
  • $$[z]$$
  • None of these
If $$A =$$$$\displaystyle \begin{bmatrix}\alpha  & 2\\ 2 & \alpha \end{bmatrix}$$ and $$\displaystyle \left | A \right |^{3}=125$$ then the value of $$\displaystyle \alpha $$ is
  • $$\displaystyle \pm 1$$
  • $$\displaystyle \pm 2$$
  • $$\displaystyle \pm 3$$
  • $$\displaystyle \pm 5$$
If the points $$A(1, 2) ,\ O (0, 0)$$ and $$C (a, b)$$ are collinear then 
  • $$a = b$$
  • $$a = 2b$$
  • $$2a = b$$
  • $$a = -b$$
The points (-a ,-b), (0, 0) (a, b) and $$\displaystyle (a^{2},ab)$$ are
  • Collnear
  • Vertices of parallelogram
  • Vertices of a rectangle
  • None of these
If $$A=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$, then $$A(Adj. A)$$ equals-
  • $$\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$$
  • $$\begin{bmatrix} 0 & 10 \\ 10 & 0 \end{bmatrix}$$
  • $$\begin{bmatrix} 10 & 1 \\ 1 & 10 \end{bmatrix}$$
  • none of these
If $$A=\begin{bmatrix} 1 & -2 & 3 \\ 4 & 0 & -1 \\ -3 & 1 & 5 \end{bmatrix}$$, then $${(adj. A)}_{23}$$ is equal to
  • $$13$$
  • $$-13$$
  • $$5$$
  • $$-5$$
If the points $$(5, 1),\  (1, p)\  \& \  (4, 2)$$ are collinear then the value of p will be 
  • $$1$$
  • $$5$$
  • $$2$$
  • $$-2$$
0:0:1


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