Class 9 Maths Chapter 13 End-of-Chapter Exercises Solutions

The line passes through the two given points (0, 0) and (2, 6).

The formula for the slope (m) of a straight line passing through (x₁, y₁) and (x₂, y₂) is:

Slope = Vertical rise / Horizontal run = Δy / Δx = y₂ − y₁x₂ − x₁

m = 6 − 02 − 0 = 62 = 3

Explanation: The slope is 3. This means that for every 1 unit increase in x, the value of y increases by 3 units (the rate of change is 3 cm per minute).

The y-intercept is the point where the line crosses the y-axis, which corresponds to x = 0.

From the graph and the equation y = 3x, when x = 0, y = 3(0) = 0.

Significance: A y-intercept of 0 indicates that the line passes directly through the origin. In the context of the water tank, it signifies that at time x = 0 (before filling begins), the water level is 0 cm (the tank starts empty).

The relationship between the water level y (in cm) and the time x (in minutes) is given by:

(a) Water level after 5 minutes:

(b) Time to reach a height of 21 cm:

A point lies on the line if its coordinates satisfy the equation y = 3x.

Substitute the x-coordinate x = 4 into the equation:

For the point to lie on the line, its y-coordinate must be 12.

However, the given point has y = 10, not 12.

Since 10 ≠ 12, the point does not satisfy the equation.

The line x = 3 is a vertical line consisting of all points whose x-coordinate is 3.

Substitute x = 3 into the equation y = 3x:

The given conversion equation is:

To draw the graph, determine three convenient points by choosing multiples of 5 for C:

Plot these points on the coordinate plane with Celsius (C) on the horizontal axis and Fahrenheit (F) on the vertical axis, and draw a straight line through them.

Subtract 32 from both sides:

Multiply both sides by 59:

C = 63 × 59 = 7 × 5 = 35

When C = 0:

When F = 0:

C = −32 × 59 = −1609 ≈ −17.78°C

Let the common numerical temperature be t, so that F = C = t.

Substitute into the conversion equation:

Multiply both sides by 5:

Therefore, −40° is the temperature which is numerically the same in both scales:

We need to draw both lines on the same coordinate graph and find their point of intersection.

Plot (0, 6) and (3, 0), then draw the first straight line.

Rearrange the equation:

Plot (0, −2) and (1, 0), then draw the second straight line.

From the graph, the two straight lines intersect at the point (2, 2).

The two equations shown on the front and back cover pages of the textbook are:

15x − 10y − 115 = 0 ⇒ 15x − 10y = 115

Subtract equation (1) from equation (2):

The point of intersection of the two lines is (9, 2).

The x-intercept is the point where the line crosses the x-axis. At every point on the x-axis, y = 0.

Dividing by m, provided m ≠ 0:

Let the number of calling minutes per month be x.

The monthly cost of Plan A is 50 + 0.20x.

The monthly cost of Plan B is 30 + 0.30x.

Plan A is cheaper for more than 200 minutes.

Plan B is cheaper for fewer than 200 minutes.

At 200 minutes, both plans cost ₹90.

A fixed slope determines the steepness and direction of a line, but the line can be moved to different positions while retaining that slope.

When both the slope and a point are fixed, the line's direction and position are determined.

Only one straight line can pass through the given point with that slope.

For a pair of linear equations to have a unique solution:

Write both equations in standard form:

For infinitely many solutions:

From the first equality:

From the second equality:

The first equation is:

The second equation is:

For both equations to represent the same line, the coefficients and constant must be in the same ratio.

Multiplying equation (1) by 2 gives:

For equation (2) to be this same equation:

When k = 3, equation (2) becomes 2x + 4y = 6, which is exactly twice equation (1).

For this line-based model, slope is calculated as:

Robot 1

Robot 1 starts from (0, 0), moves 3 units right and 4 units up.

Its slope is:

Since the line passes through the origin, its equation is:

Robot 2

Robot 2 starts at (10, 0), moves 1 unit right and 2 units up.

Its slope is:

Using the starting point (10, 0), its equation is:

At the intersection, both equations have the same x- and y-coordinates. Therefore,

Multiplying both sides by 3:

Answer:

(30, 40)

The two line-based paths intersect at (30, 40).

Robot 1

Robot 1 starts at (3, 0), moves 5 units right and 5 units up.

Its slope is:

Its line passes through (3, 0), so:

Robot 2

Robot 2 starts at (7, 0), moves 5 units right and 3 units down.

Its slope is:

Using its starting point (7, 0):

To find where the two extended straight lines intersect, equate equations (1) and (2):

Multiplying both sides by 5:

The extended lines meet at:

(4.5, 1.5)

However, Robot 2 starts at x = 7 and moves to the right. It will never reach x = 4.5. The intersection lies behind its starting point.

Answer under the straight-line path interpretation:

The two forward-moving line paths do not intersect. Their extended straight lines would meet at (4.5, 1.5), but Robot 2 never reaches that point.

Let the first reading be a minutes after 2 o’clock. Fifteen minutes later, the second reading is b minutes after 3 o’clock.

From 2 o’clock to 3 o’clock is 60 minutes, so:

Rearranging:

Interpreting the intended relation as a = 6b:

Substitute equation (2) into equation (1):

The first reading was 54 minutes after 2 o'clock, or 2:54. Fifteen minutes later, the second reading was 3:09.

Let the tens digit be x and the units digit be y.

The sum of the digits is 15:

The original number is 10x + y, and the reversed number is 10y + x.

The reversed number exceeds the original number by 9:

Substitute into equation (1):

Given:

ABCD is a cyclic quadrilateral with angles:

Theorem / Geometric Property:

The sum of either pair of opposite angles of a cyclic quadrilateral is 180° (supplementary angles):

∠A + ∠C = 180° and ∠B + ∠D = 180°

Multiply equation (2) by 3 to eliminate y:

3 × (4x + y) = 3 × 160

Subtract equation (1) from equation (3):

Let the usual speed of the train be v km/h and the usual time be t hours.

Expanding and simplifying:

vt − 6v + 6t − 36 = vt

Expanding:

vt + 6v − 4t − 24 = vt

Substitute equation (1) into equation (2):

Distance:

d = vt = 24 × 30 = 720 km

Let the present age of the father be x years and the present sum of the ages of his four children be y years.

The father's age equals the sum of the children's ages:

After 20 years, the sum of the children's ages will increase by:

After 20 years, the father's age will be x + 20, and the sum of the children's ages will be y + 80.

According to the question: