Question 2

Let's represent each situation as a quadratic equation.

Step 1 — Rectangular Plot Dimensions

We need to find the length and breadth of the plot. Let the breadth of the rectangular plot be $x$ metres. The length is one more than twice its breadth. So, the length is $(2x + 1)$ metres. The area of the plot is given as $528 \text{ m}^2$. The area of a rectangle is length multiplied by breadth.

$x(2x + 1) = 528$

Let's expand the left side.

$2x^2 + x = 528$

Now, we move all terms to one side to form a quadratic equation.

$2x^2 + x - 528 = 0$

$\boxed{2x^2 + x - 528 = 0}$

This equation represents the situation for the rectangular plot.

Step 2 — Consecutive Integers Product

We need to find two consecutive positive integers. Let the first positive integer be $x$. Since they are consecutive, the next integer will be $x + 1$. Their product is given as 306.

$x(x + 1) = 306$

Let's expand the left side of the equation.

$x^2 + x = 306$

Now, we move 306 to the left side to get the standard quadratic form.

$x^2 + x - 306 = 0$

$\boxed{x^2 + x - 306 = 0}$

This equation represents the situation for the consecutive integers.

Step 3 — Rohan's Age

We need to find Rohan's present age. Let Rohan's present age be $x$ years. Rohan's mother is 26 years older than him. So, his mother's present age is $(x + 26)$ years. We consider their ages 3 years from now. Rohan's age after 3 years will be $(x + 3)$ years. His mother's age after 3 years will be $(x + 26 + 3)$, which is $(x + 29)$ years. The product of their ages 3 years from now will be 360.

$(x + 3)(x + 29) = 360$

Let's expand the left side using the distributive property.

$x^2 + 29x + 3x + 87 = 360$

Combine the like terms.

$x^2 + 32x + 87 = 360$

Now, we move 360 to the left side to form the quadratic equation.

$x^2 + 32x + 87 - 360 = 0$

$x^2 + 32x - 273 = 0$

$\boxed{x^2 + 32x - 273 = 0}$

This equation represents the situation for Rohan's age.

Step 4 — Train Speed

We need to find the speed of the train. Let the uniform speed of the train be $x \text{ km/h}$. The distance traveled is 480 km. The time taken to cover this distance is $\frac{\text{Distance}}{\text{Speed}}$.

$\text{Time} = \frac{480}{x} \text{ hours}$

If the speed had been 8 km/h less, the new speed would be $(x - 8) \text{ km/h}$. The new time taken would be $\frac{480}{x - 8} \text{ hours}$. This new time is 3 hours more than the original time.

$\frac{480}{x - 8} = \frac{480}{x} + 3$

Let's rearrange the equation to isolate the terms with $x$.

$\frac{480}{x - 8} - \frac{480}{x} = 3$

Now, we find a common denominator for the left side.

$\frac{480x - 480(x - 8)}{x(x - 8)} = 3$

Let's expand the numerator and the denominator.

$\frac{480x - 480x + 3840}{x^2 - 8x} = 3$

Simplify the numerator.

$\frac{3840}{x^2 - 8x} = 3$

Multiply both sides by $(x^2 - 8x)$.

$3840 = 3(x^2 - 8x)$

$3840 = 3x^2 - 24x$

Finally, move all terms to one side to form the quadratic equation.

$0 = 3x^2 - 24x - 3840$

$3x^2 - 24x - 3840 = 0$

$\boxed{3x^2 - 24x - 3840 = 0}$

This equation represents the situation for the train's speed.

Answer

(i) The length and breadth of the plot satisfy the equation $2x^2 + x - 528 = 0$. (ii) The two consecutive positive integers satisfy the quadratic equation $x^2 + x - 306 = 0$. (iii) Rohan's present age satisfies the quadratic equation $x^2 + 32x - 273 = 0$. (iv) The speed of the train satisfies the quadratic equation $3x^2 - 24x - 3840 = 0$.