Question:
If $\begin{vmatrix} 0 & 4 & 2a \\ -2 & a & -1 \\ -1 & -2 & 5 \end{vmatrix} = -86$, then the sum of all possible values of $a$ is:
If $\begin{vmatrix} 0 & 4 & 2a \\ -2 & a & -1 \\ -1 & -2 & 5 \end{vmatrix} = -86$, then the sum of all possible values of $a$ is:
Show Hint
Whenever a question asks for the "sum of all possible values" of roots from a quadratic equation, you don't need to solve for the individual roots! Just use Vieta's formula: $\text{Sum} = -\frac{b}{a}$.
Updated On: Jul 4, 2026
- 4
- 5
- $-4$
- 9
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
To evaluate a \(3\times3\) determinant, we expand it along any row or column. Expanding along the first row is convenient here because it contains a zero.
Step 1: Expand the determinant along the first row.
The given determinant is \[ \begin{vmatrix} 0 & 4 & 2a\\ -2 & a & -1\\ -1 & -2 & 5 \end{vmatrix} =-86. \] Expanding along the first row, \[ 0\cdot \begin{vmatrix} a & -1\\ -2 & 5 \end{vmatrix} -4 \begin{vmatrix} -2 & -1\\ -1 & 5 \end{vmatrix} +2a \begin{vmatrix} -2 & a\\ -1 & -2 \end{vmatrix} =-86. \]
Step 2: Evaluate the \(2\times2\) determinants.
The first term is zero. For the second determinant, \[ \begin{vmatrix} -2 & -1\\ -1 & 5 \end{vmatrix} =(-2)(5)-(-1)(-1) =-10-1=-11. \] Hence, \[ -4(-11)=44. \] For the third determinant, \[ \begin{vmatrix} -2 & a\\ -1 & -2 \end{vmatrix} =(-2)(-2)-a(-1) =4+a. \] Therefore, \[ 2a(4+a)=8a+2a^2. \]
Step 3: Form the quadratic equation.
Substituting these values, \[ 44+8a+2a^2=-86. \] \[ 2a^2+8a+130=0. \] Dividing throughout by \(2\), \[ a^2+4a+65=0. \]
Step 4: Find the sum of all possible values of \(a\).
For the quadratic equation \[ a^2+4a+65=0, \] Vieta's formula gives \[ \text{Sum of roots} = -\frac{4}{1} =-4. \]
Step 1: Expand the determinant along the first row.
The given determinant is \[ \begin{vmatrix} 0 & 4 & 2a\\ -2 & a & -1\\ -1 & -2 & 5 \end{vmatrix} =-86. \] Expanding along the first row, \[ 0\cdot \begin{vmatrix} a & -1\\ -2 & 5 \end{vmatrix} -4 \begin{vmatrix} -2 & -1\\ -1 & 5 \end{vmatrix} +2a \begin{vmatrix} -2 & a\\ -1 & -2 \end{vmatrix} =-86. \]
Step 2: Evaluate the \(2\times2\) determinants.
The first term is zero. For the second determinant, \[ \begin{vmatrix} -2 & -1\\ -1 & 5 \end{vmatrix} =(-2)(5)-(-1)(-1) =-10-1=-11. \] Hence, \[ -4(-11)=44. \] For the third determinant, \[ \begin{vmatrix} -2 & a\\ -1 & -2 \end{vmatrix} =(-2)(-2)-a(-1) =4+a. \] Therefore, \[ 2a(4+a)=8a+2a^2. \]
Step 3: Form the quadratic equation.
Substituting these values, \[ 44+8a+2a^2=-86. \] \[ 2a^2+8a+130=0. \] Dividing throughout by \(2\), \[ a^2+4a+65=0. \]
Step 4: Find the sum of all possible values of \(a\).
For the quadratic equation \[ a^2+4a+65=0, \] Vieta's formula gives \[ \text{Sum of roots} = -\frac{4}{1} =-4. \]
Hence,
\[ \boxed{\text{The sum of all possible values of }a=-4.} \]Was this answer helpful?
0
0
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
View More Questions
Top CBSE CLASS XII Determinants Questions
- If \[ \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & 4 \\ 0 & 4 & 2a \end{vmatrix} = -86 \] then the sum of all possible values of \( a \) is
- Obtain the value of \[ \Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix} \] in terms of \(x, y, z\). Further, if \(\Delta = 0\) and \(x, y, z\) are non–zero real numbers, prove that \[ x^{-1} + y^{-1} + z^{-1} = -1 \]
- Obtain the value of \[ \Delta= \begin{vmatrix} 1+x & 1 & 1\\ 1 & 1+y & 1\\ 1 & 1 & 1+z \end{vmatrix} \] in terms of \(x\), \(y\) and \(z\).
Further, if \[ \Delta=0 \] and \(x\), \(y\) and \(z\) are non-zero real numbers, prove that \[ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-1. \]
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
View More Questions