Get Answers to all your Questions

header-bg qa

Need Solution for R.D.Sharma Maths Class 12 Chapter 5 determinants  Exercise 5.4 Question 12 Maths Textbook Solution.

Answers (1)

best_answer

Answer:x=-1, y=-5 \text { and } z=8

Hint: Use Cramer’s rule to solve a system of linear equations

Given:

\begin{aligned} &x-4 y-z=11 \\ &2 x-5 y+2 z=39 \\ &-3 x+2 y+z=1 \end{aligned}

Solution:

First take coefficient of variables x, y and z.

|\mathrm{A}|=\left|\begin{array}{ccc} 1 & -4 & -1 \\ 2 & -5 & 2 \\ -3 & 2 & 1 \end{array}\right|                                                                        \because(Taking first row for solving determinant)

\begin{aligned} &=1(-5-4)+4(2+6)-1(4-15) \\ &=1(-9)+4(8)-1(-11) \\ &=-9+32+11 \\ &=34 \end{aligned}

Now for x, the x column is replaced with constant column i.e.

\begin{aligned} D_{x} &=\left|\begin{array}{ccc} 11 & -4 & -1 \\ 39 & -5 & 2 \\ 1 & 2 & 1 \end{array}\right| \\ &=11(-5-4)+4(39-2)-1(78+5) \\ &=11(-9)+4(37)-1(83) \\ &=-99+148-83 \\ &=34 \end{aligned}

If we are solving for y, the y column is replaced with constant column i.e.

\begin{aligned} \mathrm{D}_{\mathrm{x}} &=\left|\begin{array}{ccc} 1 & 11 & -1 \\ 2 & 39 & 2 \\ -3 & 1 & 1 \end{array}\right| \\ &=1(39-2)-11(2+6)-1(2+117) \\ &=1(37)-11(8)-1(119) \\ &=37-88-119 \\ &=-170 \end{aligned}

If we are solving for z, the z column is replaced with constant column i.e.

\begin{aligned} \mathrm{D}_{z} &=\left|\begin{array}{ccc} 1 & -4 & 11 \\ 2 & -5 & 39 \\ -3 & 2 & 1 \end{array}\right| \\ &=1(-5-78)+4(2+117)+11(4-15) \end{aligned}

\begin{aligned} &=-83+4(119)+11(-11) \\ &=272 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{-34}{34}=-1 \\ &\Rightarrow y=\frac{D_{y}}{D}=\frac{-170}{34}=-5 \\ &\Rightarrow z=\frac{D_{z}}{D}=\frac{272}{34}=8 \end{aligned}

Concept: Determinant solving of 3 x 3 matrix (Cramer’s rule)

Note: Cramer’s rule will give us unique solution to a system of equations, if it exists. However, if the system has no solution or an infinitive number of solutions that is determinant is zero.

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads