MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

General Term of Binomial Expansion

Prove the fol...

Question

Prove the following identities:

$|\begin{matrix} 2 y & y - z - x & 2 y \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| = {(x + y + z)}^{3}$

Open in App

Solution

$LHS = |\begin{matrix} 2 y & y - z - x & 2 y \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| = |\begin{matrix} 2 y + 2 z + x - y - z & y - z - x + 2 z + 2 x & 2 y + z - x - y + 2 x \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| [Applying R_{1} \to R_{1} + R_{2} + R_{3}] = |\begin{matrix} x + y + z & x + y + z & x + y + z \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| = (x + y + z) |\begin{matrix} 1 & 1 & 1 \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| [Taking (x + y + z) common from R_{1}] = (x + y + z) |\begin{matrix} 0 & 1 & 1 \\ 0 & 2 z & z - x - y \\ - x - y - z & 2 x & 2 x \end{matrix}| [Applying C_{1} \to C_{1} - C_{2}] = {(x + y + z)}^{2} |\begin{matrix} 0 & 1 & 1 \\ 0 & 2 z & z - x - y \\ - 1 & 2 x & 2 x \end{matrix}| [Taking (x + y + z) common from C_{1}] = {(x + y + z)}^{2} [- 1 (z - x - y - 2 z)] [Expanding along first column] = {(x + y + z)}^{3} = RHS ∴ |\begin{matrix} 2 y & y - z - x & 2 y \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix}| = {(x + y + z)}^{3}$

Suggest Corrections

thumbs-up

0

Similar questions

Q. Prove the following identities:

|\begin{matrix} y + z & z & y \\ z & z + x & x \\ y & x & x + y \end{matrix}| = 4 x y z

Q. Simplify the following:

(x - 2 y) (y - 3 x) + (x + y) (x - 3 y) - (y - 3 x) (4 x - 5 y)

Q. Prove the following identities:

|\begin{matrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{matrix}| = a^{2} (a + x + y + z)

Q. Show that
(1)

{(x + y + z)}^{3} > 27 (y + z - x) (z + x - y) (x + y - z)

x y z > (y + z - x) (z + x - y) (x + y - z)

Q. Prove the following identities:

(y + z)^{2} (z + x)^{2} (x + y)^{2} = \sum x^{4} (y + z)^{2} + 2 (\sum y z)^{3} - 2 x^{2} y^{2} z^{2}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

General Term

MATHEMATICS

Watch in App

explore_more_icon

Explore more

General Term of Binomial Expansion

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon