1. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

2. If the roots of the equation \(ax^2 + bx + 35 = 0\) are \(-5\) and \(-7\), find the values of \(a\) and \(b\).

3. If the roots of the equation \(x^2 + ax + b = 0\) are \(-2\) and \(3\), then find the values of \(a\) and \(b\).

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

4. If \(x^2 - px + 12 = (x - 3)(x - a)\) is an identity, then the values of \(a\) and \(p\) respectively are?

(a) \(a = 4, p = 7\) (b) \(a = 7, p = 4\) (c) \(a = 4, p =-7\) (d) \(a =-4, p = 7\)

5. If \(x^2 + ax + b = (x+2)(x-2)\), find the values of \(a\) and \(b\).

(a) 1,2 (b) 2,1 (c) 0,4 (d) 0,-4

6. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).

7. The three values of a variable are \(4, 5,\) and \(7\), and their frequencies are \(p - 2, p + 1,\) and \(p - 1\) respectively. If the combined (weighted) mean of the variable is \(5.4\), what is the value of \(p\)?

(a) 1 (b) 2 (c) 3 (d) 4

8. If the quadratic equation \(ax^2 + 7x + b = 0\) has two roots \(\frac{2}{3}\) and \(-3\), then let us find the values of \(a\) and \(b\).

9. Given that \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), where \(0^\circ \leq A + B \leq 90^\circ\) and \(A > B\), determine the values of angles \(A\) and \(B\).

10. If \( \cos(A - B) = \cfrac{\sqrt{3}}{2} \) and \( \cos(A + B) = \cfrac{1}{2} \), then find the smallest positive values of both \(A\) and \(B\).

11. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

12. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

13. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

14. The values of \( \cos 54^\circ \) and \( \sin 36^\circ \) are equal.

15. Find the values of \( a \) and \( b \) if the roots of the equation \( ax^2+bx+35=0 \) are (-5) and (-7).

16. The values of \( \cos 54° \) and \( \sin 36° \) are equal.

17. The values of \(\cos 36^\circ\) and \(\sin 54^\circ\) are equal.

18. Let us check whether the values \(\frac{5}{6}\) and \(\frac{4}{3}\) can be roots of the equation \(x + \frac{1}{x} = \frac{13}{6}\).

19. In a cuboid-shaped room, the length, breadth, and height are respectively \(a\), \(b\), and \(c\) units. Given \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), calculate the length of the longest rod that can be placed inside the room.

20. The values of \(\cos 54^\circ\) and \(\sin 36^\circ\) are equal.

21. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

22. If the measures of three angles of a quadrilateral are respectively \(\cfrac{ \pi}{3}, \cfrac{5\pi}{6}\), and 90°, write the degree and radian values of the fourth angle.

23. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

24. If a cuboid has \(x\) edges and \(y\) faces, what is the minimum value of \(a\) such that \((x + y + a)\) becomes a perfect square?

25. 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 | ---