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need solution for RD Sharma maths class 12 chapter 5 Determinants exercise Fill in the blanks question 31

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Answer: 8

Hint: Here, we use basic concept of determinant of matrix.

Given: \left[\begin{array}{lll} x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h \end{array}\right] K=?

Solution: Here, we use the property of determinant if elements of row and column of determinant are expressed as sum of two (or more) terms, then it can be expressed as sum of two ( or more) determinant.

                Let’s take \Delta of determinant

               \Delta=\left[\begin{array}{ccc} x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h \end{array}\right]

                Let’s use property for C1

               \begin{aligned} &\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right] \\ &\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{lll} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right] \end{aligned}

                Property using for C2

               \Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]

                Property using for C3

               \Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]

                    +\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]

                    +\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]

              So, here we split in 8 parts.

              So, k = 8

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