Class 12 Maths Chapter 4 Determinants – Exercise 4.4 NCERT Solutions

Introduction

Exercise 4.4 focuses on applications of determinants in solving systems of linear equations using Cramer’s Rule. Students learn how to check consistency of equations, apply determinant properties, and solve equations systematically. This exercise is crucial for algebraic problem‑solving and preparation for competitive exams.

Formulas Used

1. Cramer’s Rule (for 2 equations): For system

$$a_{1}x+b_{1}y=c_1,a_{2}x+b_{2}y=c_2$$

Determinant:

$$D=\mid \begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\mid$$

$$x=\frac{\mid \begin{array}{cc}{c}_1& b_1\\ c_2& b_2\end{array}\mid }{D},y=\frac{\mid \begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\mid }{D}$$

2. 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 $x+y=5,2x+3y=12$ using Cramer’s Rule.

$$D=\mid \begin{array}{cc}1& 1\\ 2& 3\end{array}\mid =3-2=1$$

$$x=\frac{\mid \begin{array}{cc}5& 1\\ 12& 3\end{array}\mid }{1}=\frac{15-12}{1}=3$$

$$y=\frac{\mid \begin{array}{cc}1& 5\\ 2& 12\end{array}\mid }{1}=\frac{12-10}{1}=2$$

Q2. Solve $2x+3y=7,x-y=1$.

$$D=\mid \begin{array}{cc}2& 3\\ 1& -1\end{array}\mid =-2-3=-5$$

$$x=\frac{\mid \begin{array}{cc}7& 3\\ 1& -1\end{array}\mid }{-5}=\frac{-7-3}{-5}=\frac{-10}{-5}=2$$

$$y=\frac{\mid \begin{array}{cc}2& 7\\ 1& 1\end{array}\mid }{-5}=\frac{2-7}{-5}=\frac{-5}{-5}=1$$

Q3. Solve $x+2y=4,3x+4y=10$.

$$D=\mid \begin{array}{cc}1& 2\\ 3& 4\end{array}\mid =4-6=-2$$

$$x=\frac{\mid \begin{array}{cc}4& 2\\ 10& 4\end{array}\mid }{-2}=\frac{16-20}{-2}=\frac{-4}{-2}=2$$

$$y=\frac{\mid \begin{array}{cc}1& 4\\ 3& 10\end{array}\mid }{-2}=\frac{10-12}{-2}=\frac{-2}{-2}=1$$

Q4. Solve $2x+5y=3,4x+10y=6$.

$$D=\mid \begin{array}{cc}2& 5\\ 4& 10\end{array}\mid =20-20=0$$

Numerators also 0 → infinitely many solutions.

Q5. Solve $x+2y=3,2x+4y=7$.

$$D=\mid \begin{array}{cc}1& 2\\ 2& 4\end{array}\mid =4-4=0$$

Numerators not 0 → no solution.

Q6. Solve $3x+2y=11,2x+5y=4$.

$$D=\mid \begin{array}{cc}3& 2\\ 2& 5\end{array}\mid =15-4=11$$

$$x=\frac{\mid \begin{array}{cc}11& 2\\ 4& 5\end{array}\mid }{11}=\frac{55-8}{11}=47\mathrm{/}11$$

$$y=\frac{\mid \begin{array}{cc}3& 11\\ 2& 4\end{array}\mid }{11}=\frac{12-22}{11}=-10\mathrm{/}11$$

Q7. Solve $x-y=2,2x+y=5$.

$$D=\mid \begin{array}{cc}1& -1\\ 2& 1\end{array}\mid =1+2=3$$

$$x=\frac{\mid \begin{array}{cc}2& -1\\ 5& 1\end{array}\mid }{3}=\frac{2+5}{3}=7\mathrm{/}3$$

$$y=\frac{\mid \begin{array}{cc}1& 2\\ 2& 5\end{array}\mid }{3}=\frac{5-4}{3}=1\mathrm{/}3$$

Q8. Solve $x+3y=7,2x+6y=14$.

$$D=\mid \begin{array}{cc}1& 3\\ 2& 6\end{array}\mid =6-6=0$$

Numerators also 0 → infinitely many solutions.

Q9. Solve $2x+3y=8,4x+6y=9$.

$$D=\mid \begin{array}{cc}2& 3\\ 4& 6\end{array}\mid =12-12=0$$

Numerators not 0 → no solution.

Q10. Solve $x+2y=6,3x+4y=12$.

$$D=\mid \begin{array}{cc}1& 2\\ 3& 4\end{array}\mid =4-6=-2$$

$$x=\frac{\mid \begin{array}{cc}6& 2\\ 12& 4\end{array}\mid }{-2}=\frac{24-24}{-2}=0$$

$$y=\frac{\mid \begin{array}{cc}1& 6\\ 3& 12\end{array}\mid }{-2}=\frac{12-18}{-2}=(-6)\mathrm{/}(-2)=3$$

FAQs (10)

FAQ1. What is Cramer’s Rule? Method of solving linear equations using determinants.

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 equations? 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 4.4 important? It builds foundation for solving equations using determinants.

Conclusion

Exercise 4.4 has 10 solved questions and 10 FAQs that strengthen your understanding of solving linear equations using determinants and Cramer’s Rule. This builds the foundation for advanced algebra in Class 12 Maths.

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