Solving Linear Equations PYQs
Practice Class 12 CBSE Board Previous Year Questions (2014-2026)
Q1
2024 (Set 65/1/1)
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Solve the following system of linear equations using matrix method: 5 Marks
$2x - 3y + 5z = 11$
$3x + 2y - 4z = -5$
$x + y - 2z = -3$
Let $A = \begin{pmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{pmatrix}, X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, B = \begin{pmatrix} 11 \\ -5 \\ -3 \end{pmatrix}$.
Step 1: $|A| = 2(0) + 3(-2) + 5(1) = -1$.
Step 2: $adj(A) = \begin{pmatrix} 0 & -1 & 2 \\ 2 & -9 & 23 \\ 1 & -5 & 13 \end{pmatrix}$.
Step 3: $A^{-1} = \frac{1}{-1} \begin{pmatrix} 0 & -1 & 2 \\ 2 & -9 & 23 \\ -1 & 5 & 13 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{pmatrix}$.
Step 4: $X = A^{-1}B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.
x=1, y=2, z=3
Q2
2023 (Case Study)
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Three schools A, B and C decided to award their selected students for three values of Honesty (x), Punctuality (y) and Obedience (z). School A decided to award ₹11,000 for the three values to 2, 3 and 4 students respectively. School B decided to award ₹10,700 to 3, 2 and 1 students respectively. School C decided to award ₹9,100 to 1, 3 and 2 students respectively. 4 Marks
(i) Represent the above information in terms of a matrix equation $AX=B$.
(ii) Find $|A|$.
(iii) Find the award money for each value (x, y, z).
(i) Represent the above information in terms of a matrix equation $AX=B$.
(ii) Find $|A|$.
(iii) Find the award money for each value (x, y, z).
i) Matrix equation: $\begin{pmatrix} 2 & 3 & 4 \\ 3 & 2 & 1 \\ 1 & 3 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 11000 \\ 10700 \\ 9100 \end{pmatrix}$.
ii) $|A| = 2(4-3) - 3(6-1) + 4(9-2) = 2(1) - 3(5) + 4(7) = 2 - 15 + 28 = 15$.
iii) Solving $X = A^{-1}B$ gives $x = 2100, y = 1000, z = 1200$.
Honesty: ₹2100, Punctuality: ₹1000, Obedience: ₹1200
Q3
2024 (MCQ)
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For what value of $k$, the system of equations $x + 2y = 3$ and $5x + ky = 7$ has a unique solution? 1 Mark
For a unique solution, $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
$\frac{1}{5} \neq \frac{2}{k} \Rightarrow k \neq 10$.
k ≠ 10
Q4
2026 Sample Paper
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A farmer produces three types of crops X, Y and Z. The selling price per unit is ₹2.50, ₹1.50 and ₹1.00 respectively. The cost price per unit is ₹2.00, ₹1.00 and ₹0.50 respectively. The annual sale (in units) is given by the matrix $P = \begin{pmatrix} 10000 & 2000 & 18000 \\ 6000 & 20000 & 8000 \end{pmatrix}$. 4 Marks
(i) Find the total revenue from each market using matrix algebra.
(ii) Find the total profit from each market using matrix algebra.
(i) Find the total revenue from each market using matrix algebra.
(ii) Find the total profit from each market using matrix algebra.
i) Revenue matrix $= P \times S$, where $S = \begin{pmatrix} 2.50 \\ 1.50 \\ 1.00 \end{pmatrix}$.
Market I: $10000(2.5) + 2000(1.5) + 18000(1) = 25000 + 3000 + 18000 = ₹46000$.
Market II: $6000(2.5) + 20000(1.5) + 8000(1) = 15000 + 30000 + 8000 = ₹53000$.
ii) Profit matrix $= P \times (S - C)$, where $S-C = \begin{pmatrix} 0.50 \\ 0.50 \\ 0.50 \end{pmatrix}$.
Market I Profit: $(10000+2000+18000) \times 0.5 = 30000 \times 0.5 = ₹15000$.
Market II Profit: $(6000+20000+8000) \times 0.5 = 34000 \times 0.5 = ₹17000$.
Market I Profit: ₹15000, Market II Profit: ₹17000