Linear Equation Problem Set 1 - SSC 10th Important Syllabus
Question 1: What is the degree of a linear equation in two variables?
- A) 0
- B) 1 ✓ Correct
- C) 2
- D) 3
Explanation: The degree of any linear equation is always 1. An equation of the general form $ax+by+c=0$ (where $a$ and $b$ are not simultaneously zero) has a highest power of 1 for its variables.
Question 2: Solve the simultaneous equations $x+y=5$ and $x-y=1$.
- A) $(2,3)$
- B) $(3,2)$ ✓ Correct
- C) $(4,1)$
- D) $(1,4)$
Solution:
Let equation (1) be: $x+y=5$
Let equation (2) be: $x-y=1$
Adding equations (1) and (2):
$(x+y)+(x-y)=5+1$
$2x=6$
$x=3$Substitute $x=3$ into equation (1):
$3+y=5$
$y=5-3$
$y=2$Therefore, the solution $(x,y)$ is $(3,2)$.
Question 3: Find the value of determinant $D$ for the simultaneous equations $3x-4y=10$ and $4x+3y=5$ using Cramer's Rule.
- A) $7$
- B) $-7$
- C) $25$ ✓ Correct
- D) $-25$
Solution:
The given equations are in the standard form $ax+by=c$.
For equation 1: $a_1=3$, $b_1=-4$, $c_1=10$
For equation 2: $a_2=4$, $b_2=3$, $c_2=5$To find the determinant $D$, we use the coefficients of $x$ and $y$:
$$D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$$
$$D = (a_1 \times b_2) - (b_1 \times a_2)$$
$$D = (3 \times 3) - (-4 \times 4)$$
$$D = 9 - (-16)$$
$$D = 9 + 16$$
$$D = 25$$
Question 4: What is the condition for two linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ to have infinitely many solutions?
- A) $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
- B) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
- C) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ ✓ Correct
- D) $a_1b_2 - a_2b_1 = 1$
Note: When the ratios of the coefficients of $x$, $y$, and the constant terms are all equal ($\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$), the two equations represent the exact same line graphically. Therefore, any point on the line is a valid solution, resulting in infinitely many solutions.
