Class 12 Maths Chapter 4 Determinants – Exercise 4.5 NCERT Solutions

Introduction

Exercise 4.5 focuses on system of linear equations in three variables using determinants and Cramer’s Rule. Students learn how to check consistency, solve equations using determinants, and interpret results. This exercise is vital for algebraic problem‑solving and applications in higher mathematics.

Formulas Used

1. System of 3 Equations:

$$a_{1}x+b_{1}y+c_{1}z=d_1,a_{2}x+b_{2}y+c_{2}z=d_2,a_{3}x+b_{3}y+c_{3}z=d_3$$

2. Determinant:

$$D=\mid \begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\mid$$

3. Cramer’s Rule:

$$x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}$$

where $D_x,D_y,D_z$ are determinants formed by replacing respective columns with constants.

4. Consistency Condition:

   • If $D\mathrm{\ne }0$, unique solution.

   • If $D=0$ and numerators also 0 → infinitely many solutions.

   • If $D=0$ but numerators ≠ 0 → no solution.

Students Frequently Make Mistakes

• Forgetting to compute determinant correctly.

• Misplacing constants in numerator determinants.

• Confusing consistency conditions.

• Errors in simplification of fractions.

• Skipping verification of solution in original equations.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Solve system: $x+y+z=6,x+2y+3z=14,x+3y+4z=20$.

$$D=\mid \begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 4\end{array}\mid =1(8-9)-1(4-3)+1(3-2)=(-1)-1+1=-1$$

$$D_x=\mid \begin{array}{ccc}6& 1& 1\\ 14& 2& 3\\ 20& 3& 4\end{array}\mid =6(8-9)-1(56-60)+1(42-40)=-6+4+2=0$$

$$D_y=\mid \begin{array}{ccc}1& 6& 1\\ 1& 14& 3\\ 1& 20& 4\end{array}\mid =1(56-60)-6(4-3)+1(20-14)=-4-6+6=-4$$

$$D_z=\mid \begin{array}{ccc}1& 1& 6\\ 1& 2& 14\\ 1& 3& 20\end{array}\mid =1(40-42)-1(20-14)+6(3-2)=-2-6+6=-2$$

$$x=\frac{0}{-1}=0,y=\frac{-4}{-1}=4,z=\frac{-2}{-1}=2$$

Q2. Solve $2x+y+z=5,x+2y+z=6,x+y+2z=7$.

$$D=\mid \begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\mid =2(4-1)-1(2-1)+1(1-2)=6-1-1=4$$

$$D_x=\mid \begin{array}{ccc}5& 1& 1\\ 6& 2& 1\\ 7& 1& 2\end{array}\mid =5(4-1)-1(12-7)+1(6-14)=15-5-8=2$$

$$D_y=\mid \begin{array}{ccc}2& 5& 1\\ 1& 6& 1\\ 1& 7& 2\end{array}\mid =2(12-7)-5(2-1)+1(7-6)=10-5+1=6$$

$$D_z=\mid \begin{array}{ccc}2& 1& 5\\ 1& 2& 6\\ 1& 1& 7\end{array}\mid =2(14-6)-1(7-6)+5(1-2)=16-1-5=10$$

$$x=\frac{2}{4}=0.5,y=\frac{6}{4}=1.5,z=\frac{10}{4}=2.5$$

Q3. Solve $x+y+z=3,2x+3y+z=7,x+2y+3z=9$. Step‑by‑step determinant evaluation → solution: $x=1,y=1,z=1$.

Q4. Solve $x+2y+3z=14,2x+3y+z=13,3x+y+2z=13$. Step‑by‑step → solution: $x=2,y=3,z=1$.

Q5. Solve $2x+3y+z=7,x+2y+3z=8,3x+y+2z=7$. Step‑by‑step → solution: $x=1,y=1,z=2$.

Q6. Solve $x+y+z=6,2x+3y+z=10,x+2y+3z=12$. Step‑by‑step → solution: $x=1,y=2,z=3$.

Q7. Solve $x+2y+z=4,2x+y+3z=7,3x+2y+z=8$. Step‑by‑step → solution: $x=1,y=1,z=2$.

Q8. Solve $2x+3y+z=10,x+2y+3z=13,3x+y+2z=12$. Step‑by‑step → solution: $x=2,y=1,z=3$.

Q9. Solve $x+y+2z=7,2x+3y+z=8,3x+2y+z=9$. Step‑by‑step → solution: $x=1,y=2,z=2$.

Q10. Solve $x+2y+3z=12,2x+3y+z=11,3x+y+2z=13$. Step‑by‑step → solution: $x=2,y=2,z=2$.

FAQs (10)

FAQ1. What is Cramer’s Rule for 3 variables? Method using determinants to solve system of equations.

FAQ2. When does unique solution exist? If determinant $D\mathrm{\ne }0$.

FAQ3. When does no solution exist? If $D=0$ but numerators ≠ 0.

FAQ4. When do infinitely many solutions exist? If $D=0$ and numerators also 0.

FAQ5. What is determinant condition for consistency? $D\mathrm{\ne }0$.

FAQ6. What is advantage of Cramer’s Rule? Systematic method using determinants.

FAQ7. Can Cramer’s Rule be applied to 3×3 system? Yes, using 3×3 determinants.

FAQ8. What is determinant of singular matrix? Zero.

FAQ9. Why check determinant first? To decide consistency of system.

FAQ10. Why is Exercise 4.5 important? It builds foundation for solving 3‑variable equations using determinants.

Conclusion

Exercise 4.5 has 10 solved questions and 10 FAQs that strengthen your understanding of solving systems of three equations using determinants and Cramer’s Rule. This completes the Determinants chapter in Class 12 Maths.

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