1. Let us determine with respect to which power of the variable the equation \(x^6 - x^3 - 2 = 0\) becomes a quadratic equation.

2. Determine for which value of \(a\) the equation \((a - 2)x^2 + 3x + 5 = 0\) will not be a quadratic equation.

(a) \(a=0\) (b) \(a=2\) (c) \(a=4\) (d) \(a=-2\)

3. Let us determine for which value of \(a\) the equation \((a - 2)x^2 + 3x + 5 = 0\) will **not** be a quadratic equation.

4. A quadratic equation in one variable \(x\), with real coefficients, has equal coefficients for \(x^2\) and the constant term. Show with reasoning that any other quadratic equation whose roots are the reciprocals of the original equation's roots must be identical to the original equation.

5. A specific quadratic equation has equal coefficients for \(x^2\) and the constant term (\(x^0\)) — Show with reasoning that the equation whose roots are the reciprocals of this equation’s roots is identical to the original equation.

6. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

7. Without solving, determine all values of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) has real and equal roots.

8. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

9. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

10. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

11. The equation \(x^6+5x^3+6=0\) is quadratic in terms of \(x^3\).

12. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).

13. Divide 42 into two parts such that one part is equal to the square of the other part. — Form a quadratic equation with one variable from the given statement.

14. The distance between two stations is 300 km. A train travels from the first station to the second at a uniform speed. If the speed of the train had been 5 km/h more, it would have taken 2 hours less to reach the second station. — Form a quadratic equation with one variable from the given statement.

15. If the speed of the stream is 2 km/h, and Ratan Majhi takes 10 hours to go 21 km downstream and return the same distance upstream, — Form a quadratic equation with one variable from the given statement.

16. Let us calculate the value of \(k\) for which \(\frac{2}{3}\) is a root of the quadratic equation \(7x^2 + kx - 3 = 0\).

17. Let us calculate the value of \(k\) for which \(-a\) is a root of the quadratic equation \(x^2 + 3ax + k = 0\).

18. Calculate the value(s) of \(k\) for which the quadratic equation \(49x^2 + kx + 1 = 0\) will have real and equal roots.

19. For which value(s) of \(k\) will the quadratic equation \(3x^2 - 5x + 2k = 0\) have real and equal roots?

20. For which value(s) of \(k\) will the quadratic equation \(9x^2 - 24x + k = 0\) have real and equal roots?

21. Calculate the value(s) of \(k\) for which the quadratic equation \(2x^2 + 3x + k = 0\) will have real and equal roots.

22. Calculate the value(s) of \(k\) for which the quadratic equation \(x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0\) will have real and equal roots.

23. For which value(s) of \(k\) will the quadratic equation \((3k + 1)x^2 + 2(k + 1)x + k = 0\) have real and equal roots?

24. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).

25. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{α^2}{β} + \frac{β^2}{α}\).

26. Express the equation \(x^3 - 4x^2 - x + 1 = (x + 2)^3\) in the standard form of a quadratic equation and write the coefficients of \(x^2\), \(x\), and \(x^0\).

27. Write the value of \(k\) for which the quadratic equation \(9x^2 + 3kx + 4 = 0\) has real and equal roots.

28. Determine the value(s) of \(\theta\) within the range \(0^\circ \leq \theta \leq 90^\circ\) for which the equation \(2\cos^2\theta - 3\cos\theta + 1 = 0\) holds true.

29. Jayanta, Ajit, and Kunal jointly started a partnership business with a total capital of ₹15,000. At the end of the year, they earned profits of ₹800, ₹1,000, and ₹1,200 respectively. Profit ratio = 800 : 1000 : 1200 = 4 : 5 : 6 Total capital = ₹15,000 Sum of ratio parts = 4 + 5 + 6 = 15 Jayanta’s capital = ₹\(\frac{4}{15} × 15,000 = 4,000\) Ajit’s capital = ₹\(\frac{5}{15} × 15,000 = 5,000\) Kunal’s capital = ₹\(\frac{6}{15} × 15,000 = 6,000\)

30. Got it — sticking strictly to translation. Here's the English version without any extra commentary: If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 5x^2 - 3x + 6 = 0 \), then \( \alpha + \beta = -\frac{-3}{5} = \frac{3}{5} \) and \( \alpha\beta = \frac{6}{5} \) \(\therefore \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \) \( = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2} \) (Answer)