If \( \left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| = \left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right| \), then value of x is
3
± 3
± 6
6
The value of determinant
\( \left| \begin{matrix} a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c \end{matrix} \right| \)
a³ + b³ + c³
3bc
a³ + b³ + c³ − 3abc
none of these
The area of a triangle with vertices (−3,0), (3,0) and (0,k) is 9 sq. units. The value of k will be
9
3
−9
6
The determinant
\( \left| \begin{matrix} b^2-ab & b-c & bc-ac \\ ab-a^2 & a-b & b^2-ab \\ bc-ac & c-a & ab-a^2 \end{matrix} \right| \)
equals
abc (b−c)(c−a)(a−b)
(b−c)(c−a)(a−b)
(a+b+c)(b−c)(c−a)(a−b)
None of these
The number of distinct real roots of
\( \left| \begin{matrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{matrix} \right| = 0 \)
in the interval \( -\dfrac{\pi}{4} \le x \le \dfrac{\pi}{4} \)
0
2
1
3
If A, B and C are angles of a triangle, then the determinant
\( \left| \begin{matrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{matrix} \right| \)
0
−1
1
None of these
Let \( f(t)= \left| \begin{matrix} \cos t & 1 \\ 2\sin t & 2t \end{matrix} \right| \). Then \( \lim_{t→0} \dfrac{f(t)}{t^2} \) is equal to
0
−1
2
3
The maximum value of
\( \left| \begin{matrix} 1 & 1 \\ 1+\cos\theta & 1+\sin\theta \end{matrix} \right| \)
(\(\theta\) is real)
1/2
\( \dfrac{\sqrt{3}}{2} \)
√2
\( \dfrac{2\sqrt{3}}{4} \)
If \( f(x) = \left| \begin{matrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{matrix} \right| \), then
f(a) = 0
f(b) = 0
f(0) = 0
f(1) = 0
If \( A = \begin{pmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{pmatrix} \), then \( A^{-1} \) exists if
\( \lambda = 2 \)
\( \lambda \neq 2 \)
\( \lambda = -2 \)
None of these
If A and B are invertible matrices, then which of the following is not correct?
adj A = |A| A^{-1}
det(A^{-1}) = [det(A)]^{-1}
(AB)^{-1} = B^{-1} A^{-1}
(A + B)^{-1} = B^{-1} + A^{-1}
If x, y, z are all different from zero and
\( \left| \begin{matrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{matrix} \right| = 0 \), then value of \( x^{-1} + y^{-1} + z^{-1} \) is
xyz
x^{-1} y^{-1} z^{-1}
−x−y−z
−1
The value of the determinant
\( \left| \begin{matrix} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{matrix} \right| \)
9x²(x+y)
9y²(x+y)
3y²(x+y)
7x²(x+y)
There are two values of a which make determinant \( \Delta = \left| \begin{matrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{matrix} \right| = 86 \). Sum of these numbers is
4
5
−4
9