Q1. Find the slope of the line with inclination 30°.
Solution:
Slope m = tan θ
= tan 30° = 1/√3.
Q2. Write the equation of the x-axis.
Solution:
On the x-axis, y = 0.
Equation: y = 0.
Q3. Write the equation of the y-axis.
Solution:
On the y-axis, x = 0.
Equation: x = 0.
Q4. Find the y-intercept of the line y = -2x + 5.
Solution:
Compare with y = mx + c.
Here c = 5.
So y-intercept = 5.
Q5. Find the slope of the line 4x + 3y - 9 = 0.
Solution:
Rearrange: 3y = -4x + 9 → y = (-4/3)x + 3.
Slope m = -4/3.
Q6. Check if point (3, -1) lies on the line 2x - y = 7.
Solution:
Substitute x = 3, y = -1:
LHS = 2(3) - (-1) = 6 + 1 = 7.
= RHS.
So (3, -1) lies on the line.
Q7. Find the slope of the line joining A(2, -3) and B(-4, 5).
Solution:
Slope m = (y₂ - y₁) / (x₂ - x₁)
= (5 - (-3)) / (-4 - 2)
= 8 / -6 = -4/3.
Q8. Find the equation of the line passing through (0, -2) and slope 3.
Solution:
Use slope-intercept form: y = mx + c.
Here slope m = 3, y-intercept c = -2.
Equation: y = 3x - 2.
Q9. If a line passes through points P(1, 2) and Q(3, k), and slope is 1, find k.
Solution:
Slope m = (k - 2) / (3 - 1).
1 = (k - 2)/2.
k - 2 = 2 → k = 4.
Q10. Determine if the lines y = 2x + 3 and y = -½x + 1 are perpendicular.
Solution:
Slope of first line m₁ = 2.
Slope of second line m₂ = -½.
Product m₁ × m₂ = 2 × (-½) = -1.
Hence, lines are perpendicular.
Q11. Find the equation of the line passing through (2, -1) and (4, 3).
Solution:
Step 1: Find slope m = (3 - (-1)) / (4 - 2) = 4 / 2 = 2.
Step 2: Use point-slope form: y - y₁ = m(x - x₁).
Take point (2, -1): y - (-1) = 2(x - 2).
Step 3: Simplify: y + 1 = 2x - 4 → y = 2x - 5.
Equation: y = 2x - 5.
Q12. The points A(1, 2), B(3, 6), and C(5, k) are collinear. Find k.
Solution:
Step 1: Slope of AB = (6 - 2)/(3 - 1) = 4/2 = 2.
Step 2: Slope of BC = (k - 6)/(5 - 3) = (k - 6)/2.
Step 3: For collinearity, slopes equal → 2 = (k - 6)/2.
Step 4: Cross multiply: 4 = k - 6 → k = 10.
So, k = 10.
Q13. A line cuts the x-axis at (4, 0) and the y-axis at (0, -2). Find its equation.
Solution:
Step 1: Two points are (4, 0) and (0, -2).
Step 2: Find slope m = (0 - (-2)) / (4 - 0) = 2/4 = ½.
Step 3: Use point-slope form with (4, 0): y - 0 = ½(x - 4).
Step 4: Simplify: y = ½x - 2.
Equation: y = ½x - 2.
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