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Worksheet: Equation of a Line | Mathematics Class 10 ICSE PDF Download

Section A – Very Short Answer Questions 

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.

Section B – Short Answer Questions 

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.

Section C – Long Answer Questions 

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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FAQs on Worksheet: Equation of a Line - Mathematics Class 10 ICSE

1. What is the general form of the equation of a line?
Ans. The general form of the equation of a line in a two-dimensional space is given by the equation Ax + By + C = 0, where A, B, and C are constants, and (x, y) are the coordinates of any point on the line.
2. How do you find the slope of a line given two points?
Ans. The slope (m) of a line can be found using the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two distinct points on the line. This formula represents the change in y over the change in x.
3. What are the different forms of the equation of a line?
Ans. The equation of a line can be represented in several forms: 1. Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept. 2. Point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line. 3. Standard form: Ax + By = C, where A, B, and C are integers.
4. How can you determine if two lines are parallel?
Ans. Two lines are parallel if they have the same slope. In other words, if the slopes of two lines represented by the equations y = m₁x + b₁ and y = m₂x + b₂ are equal (m₁ = m₂), then the lines are parallel.
5. What is the significance of the y-intercept in the equation of a line?
Ans. The y-intercept is the point at which the line crosses the y-axis. In the slope-intercept form of the equation (y = mx + b), the value of b represents the y-intercept. It indicates the value of y when x is zero and is crucial for graphing the line.
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