Home/Class 12/Maths/

Question and Answer

If a, b, c, are in A.P, then the determinant
\(\left|\begin{array}{ccc} {x+2} & {x+3} & {x+2 a} \\ {x+3} & {x+4} & {x+2 b} \\ {x+4} & {x+5} & {x+2 c} \end{array}\right|\) is
  • 1
  • 2x
  • x
  • 0
loading
settings
Speed
00:00
04:56
fullscreen
If a, b, c, are in A.P, then the determinant
\(\left|\begin{array}{ccc} {x+2} & {x+3} & {x+2 a} \\ {x+3} & {x+4} & {x+2 b} \\ {x+4} & {x+5} & {x+2 c} \end{array}\right|\) is
  • 1
  • 2x
  • x
  • 0

Answer

Let \(\Delta=\left|\begin{array}{lll} {x+2} & {x+3} & {x+2 a} \\ {x+3} & {x+4} & {x+2 b} \\ {x+4} & {x+5} & {x+2 c} \end{array}\right|\) 
Since a, b, c are in A.P.
\(\therefore\) 2b = a + c
\(\Delta=\left|\begin{array}{ccc} {x+2} & {x+3} & {x+2 a} \\ {x+3} & {x+4} & {x+(a+c)} \\ {x+4} & {x+5} & {x+2 c} \end{array}\right|\) 
Applying Elementary Row transformations
\(R_1 \rightarrow R_1 - R_2,~~ and~~ R_3\rightarrow  R_3 - R_2\)
\(\Delta=\left|\begin{array}{ccc} {-1} & {-1} & {a-c} \\ {\mathrm{x}+3} & {\mathrm{x}+4} & {\mathrm{x}+\mathrm{a}+\mathrm{c}} \\ {1} & {1} & {\mathrm{c}-\mathrm{a}} \end{array}\right|\) 
\(R_1 \rightarrow R_1+R_3\)
\(\Delta=\left|\begin{array}{ccc} {0} & {0} & {0} \\ {\mathrm{x}+3} & {\mathrm{x}+4} & {\mathrm{x}+\mathrm{a}+\mathrm{c}} \\ {1} & {1} & {\mathrm{c}-\mathrm{a}} \end{array}\right|\)
[In a determinant if all elements of a row is 0 then the value of determinant is 0]
So, here all the elements of first row (\(R_1\)) are zero.
\(\therefore\) \(\Delta=0\)
Therefore the choice is: D
To Keep Reading This Answer, Download the App
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play
To Keep Reading This Answer, Download the App
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play
Correct44
Incorrect0
Watch More Related Solutions
By using properties of determinants, show that \(\left| {\begin{array}{*{20}{c}} {{a^2} + 1}&{ab}&{ac} \\ {ab}&{{b^2} + 1}&{bc} \\ {ca}&{cb}&{{c^2} + 1} \end{array}} \right| = 1 + {a^2} + {b^2} + {c^2}\)
Examine the consistency of the system of equation x + 2y = 2; 2x + 3y = 3
Find the inverse of the matrix (if it exists) given \(\left[ {\begin{array}{*{20}{c}} { - 1}&5 \\ { - 3}&2 \end{array}} \right]\)
Examine the consistency of the system of equation \(2x - y = 5;\,\,x + y = 4\)
For the matrix \(A = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right]\), find numbers a and b such that A2 + aA + bI = 0.
Let A = \(\left[\begin{array}{ccc} {1} & {2} & {1} \\ {2} & {3} & {1} \\ {1} & {1} & {5} \end{array}\right]\). verify that (A–1)–1 = A
Evaluate \(\left| {\begin{array}{*{20}{c}} 0&1&2 \\ { - 1}&0&{ - 3} \\ { - 2}&3&0 \end{array}} \right|\)
Using the property of determinant and without expanding prove that \(\left| {\begin{array}{*{20}{c}} x&a&{x + a} \\ y&b&{y + b} \\ z&c&{z + c} \end{array}} \right| = 0\)
Evaluate the determinant \(\left| {\begin{array}{*{20}{c}} {{x^2} - x + 1}&{x - 1} \\ {x + 1}&{x + 1} \end{array}} \right|\)
Write minors and cofactors of the element of \(\left| {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right|\)

Load More