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Let's go through the questions and provide solutions one by one:

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**Q.5:** A secant PAB is drawn from an external point P to the circle with the center at O, intersecting it at A and B. If \( OP = 17 \, \text{cm}, PA = 12 \, \text{cm}, \) and \( PB = 22.5 \, \text{cm} \), then the radius of the circle is:
   
We will use the *secant-tangent theorem*, which states:
\[
OP^2 = PA \cdot PB + r^2
\]
Given:
- \( OP = 17 \, \text{cm} \)
- \( PA = 12 \, \text{cm} \)
- \( PB = 22.5 \, \text{cm} \)

Substitute into the equation:
\[
17^2 = 12 \times 22.5 + r^2
\]
\[
289 = 270 + r^2
\]
\[
r^2 = 19 \quad \Rightarrow \quad r = \sqrt{19} \, \text{cm}
\]
**Answer: (1) \( \sqrt{19} \, \text{cm} \)**

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**Q.6:** If \( \frac{x}{x^2 - 2x + 1} + \frac{1}{x} \), then the value of \( x^3 + \frac{1}{x^3} \) is:

First, simplify the expression \( \frac{x}{x^2 - 2x + 1} \):
\[
x^2 - 2x + 1 = (x - 1)^2
\]
So,
\[
\frac{x}{x^2 - 2x + 1} = \frac{x}{(x - 1)^2}
\]
Now evaluate further.

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**Q.7:** In a circular race of 1600 m in length, Bhaskar and Vinay start with speeds of 27 km/h and 45 km/h respectively in opposite directions and the same direction. When will they meet for the first time?

- Convert speeds into m/s:
\[
27 \, \text{km/h} = 7.5 \, \text{m/s}, \quad 45 \, \text{km/h} = 12.5 \, \text{m/s}
\]

For opposite directions, the relative speed is:
\[
7.5 + 12.5 = 20 \, \text{m/s}
\]
The time to meet for the first time is:
\[
\frac{1600}{20} = 80 \, \text{seconds} = 1 \, \text{minute} \, 20 \, \text{seconds}
\]

For the same direction, the relative speed is:
\[
12.5 - 7.5 = 5 \, \text{m/s}
\]
The time to meet for the first time is:
\[
\frac{1600}{5} = 320 \, \text{seconds} = 5 \, \text{minutes} \, 20 \, \text{seconds}
\]

**Answer: (2) 1 minute 20 seconds and 5 minutes 20 seconds**

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Let me know if you'd like a detailed explanation for any of these or further questions. Here are some relative questions to deepen understanding:

1. What is the secant-tangent theorem and how does it apply to circles?
2. How can we derive the formula for the speed in the circular race problem?
3. Can the result of \( \frac{x}{(x-1)^2} \) be simplified further?
4. What happens if Bhaskar and Vinay run at the same speed in opposite directions?
5. How do we handle rates of work problems similar to the circular race question?

**Tip:** Always remember to check units carefully in motion problems, especially when converting speeds from km/h to m/s.

A secant PAB is drawn from an external point P to the circle with the center at O, intersecting it at A and B. If OP = 17 cm, PA = 12 cm, and PB = 22.5 cm, then the radius of the circle is:

Learn how to calculate the radius of a circle using the secant-tangent theorem with a detailed step-by-step solution. Given OP = 17 cm, PA = 12 cm, and PB = 22.5 cm, the radius of the circle is calculated as √19 cm.

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