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Cramer’s Rule - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Cramer’s law is considered one the most difficult concept.

  • 46 Questions around this concept.

Solve by difficulty

QuestionEASY

The number of values of k, for which the system of equations :

(k+1)x+8y=4k

kx+(k+3)y=3k-1

has no solution, is:

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Consider the system of linear equations

x_{1}+2x_{2}+x_{3}=3

2x_{1}+3x_{2}+x_{3}=3

3x_{1}+5x_{2}+2x_{3}=1

The system has

View Solution

The system of linear equations

x +λy −z = 0
λx − y − z = 0
x + y − λz = 0

has a non-trivial solution for:

View Solution

The set of all values of \lambda for which the system of linear equations :

2x_{1}-2x_{2}+x_{3}=\lambda x_{1}

2x_{1}-3x_{2}+2x_{3}=\lambda x_{2}

-x_{1}+2x_{2}\; \; \; =\lambda x_{3}

has a non-trivial solution,

View Solution

If S is the set of distinct values of ‘b’ for which the following system of linear equations

x+y+z=1

x+ay+z=1

ax+by+z=0

has no solution, then S is:

View Solution

Concepts Covered - 1

Cramer’s law

Cramer’s law

For the system of equations in two variables:

\\\mathrm{Let \; a_1x +b_1y = c_1\; and \; a_2x + b_2y = c_2, where} \\\mathrm{\frac{a_1}{a_2}\neq\frac{b_1}{b_2}} \\\\\mathrm{On \; solving \; this \; equation \; by \; cross \; multiplication, we \; get} \\\mathrm{\frac{x}{b_2c_1-b_1c_2}=\frac{y}{a_1c_2-a_2c_1}=\frac{1}{a_1b_2-a_2b_1}} \\\mathrm{or \; \frac{x}{\begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix}}=\frac{y}{\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix}}=\frac{1}{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}}} \\\mathrm{or \; x=\frac{\begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}}, y=\frac{\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}}}

We can observe that the first column in the numerator of x is of constants and 2nd column in the numerator of y is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variables where the third column in the numerator of the value of z will be constant and the denominator will be formed by the value of coefficients of the variables.

 

For the system of equations in three variables:

\\\mathrm{Let \;us\; consider\; the \; system\; of \;equations} \\\mathrm{a_1x+b_1y +c_1z =d_1 \;\;...(i)} \\\mathrm{a_2x+b_2y +c_2z =d_2\;\;\;...(ii)} \\\mathrm{a_3x+b_3y +c_3z =d_3\;\;\;...(iii)} \\\mathrm{then \; \Delta,\; which\; will\; be \;determinant\; of\; coefficient\; of \;variables, will\; be } \\\mathrm{\Delta = \begin{vmatrix} a_1 & b_1 & c_1\\ a_2& b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}}\\ \Delta_1\; numerator\; of\; x \;is: \\\mathrm{\Delta_1= \begin{vmatrix} d_1 & b_1 &c_1 \\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3 \end{vmatrix}} \\Similarly\; \Delta_2 = \begin{vmatrix} a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3 \end{vmatrix}\; and \; \mathrm{\Delta_3 = \begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3 \end{vmatrix}}

 

i) If \Delta \neq0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are  \\\mathrm{x=\frac{\Delta_1}{\Delta}, y=\frac{\Delta_2}{\Delta}, z=\frac{\Delta_3}{\Delta}}

ii) If \Delta =0, and any of \Delta_1\neq 0 \; or \;\Delta_2\neq 0 \; or \;\Delta_3\neq 0

Then the system of equations is inconsistent and hence no solution exists.

iii) If all \Delta =\Delta_1=\Delta_2=\Delta_3= 0 then

System of equations is consistent and it has an infinite number of solutions (except when all three equations represent parallel planes, in which case there is no solution)

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Cramer’s law

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