1. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then the value of \(x\) is —

(a) \(\pm 1\) (b) \(\pm \cfrac{1}{2}\) (c) \(\pm \cfrac{1}{\sqrt2}\) (d) \(\pm \cfrac{1}{\sqrt3}\)

2. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

3. The English translation is: "If \(x^2 = \sin^2 30° + 4\cot^2 45° - \sec^2 60°\), determine the value of \(x\)."

4. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

5. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

6. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

7. If the product of the roots of the equation \(x^2 – 5x + k = 12\) is \(–3\), find the value of \(k\).

8. If \(x^2 - \frac{5}{6}x + \frac{1}{6} = 0\), then the value of \(x\) is -

(a) \(\cfrac{1}{2}, \cfrac{1}{3}\) (b) 2, 3 (c) 3, 4 (d) 3, 4

9. The value of \(x\) in the equation \(2x^2 - 7x + 3 = 0\) is -

(a) \(4, \cfrac{1}{3}\) (b) \(3, 3\) (c) \(3, \cfrac{1}{2}\) (d) \(\cfrac{1}{2}, \cfrac{1}{4}\)

10. If the sum of the roots of the equation \(x^2 - (k + 6)x + 2(2k - 1) = 0\) is half of their product, then what is the value of \(k\)?

(a) 6 (b) 7 (c) 1 (d) 5

11. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), determine the value of \(k\).

(a) -2 (b) -8 (c) 8 (d) 12

12. If one of the roots of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), determine the value of \(q\).

13. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\). Let me know if you'd like a step-by-step solution next. I'm ready when you are.

14. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

15. If one root of the equation \(x^2 + ax + 3 = 0\) is 1, find the value of \(a\).

16. If one root of the equation \(x^2 - (2 + b)x + 6 = 0\) is 2, write the value of the other root.

17. If one of the roots of each of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), find the value of \(q\).

18. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

19. If \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 25 - x^2\) is an equation, find the value of \(x\).

20. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

21. If one root of the equation \(x^2 - (2 + b)x + 6 = 0\) is 2, then the value of the other root is..........

22. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

23. If the median of the following data is 32 and \(x + y = 100\), determine the values of \(x\) and \(y\). | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|-------|-------| | Frequency | 10 | x | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | y | 10 | ---

24. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0

25. The coefficient of \(x\) in the quadratic equation \(x+\cfrac{1}{x}=6\).

(a) 6 (b) -6 (c) 0 (d) 1

26. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –

(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

27. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

(a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

28. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?

(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above

29. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

30. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)?

(a) 1 (b) \(\cfrac{2}{\sqrt3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)