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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:

View Solution

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)

Study it with Videos

Cramer’s law

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