1. The maximum value of a data set is 520. What should be the minimum value for the range of the data to be 100?
(a) 120 (b) 320 (c) 420 (d) 500
2. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.
3. The simplest value of \(\cfrac{\cos 53^\circ}{\sin 37^\circ}\) is _____.
4. If \(5 + \sqrt{x} = \sqrt{2} - y\), then the value of \((x + y)\) is _________.
5. If the numerical values of the volume and curved surface area of a sphere are equal, then its radius is _____ units.
6. If two circles are externally tangent to each other, the maximum number of common tangents is _____.
7. If \(x-2\) and \(x+3\) have \(x\) as their mean proportional, then the value of \(x\) is _____.
8. The maximum number of tangents that can be drawn to a circle from an external point is _____.
9. If \(x\) is the geometric mean between \((a^2bc)\) and \((4bc)\), then the value of \(x\) is ______.
10. If \(x\) is the mean proportional between \((x + 2)\) and \((x - 3)\), then the value of \(x\) is ________.
11. If the numerical values of the volume and the curved surface area of a right circular cylinder are equal, then the diameter of the cylinder is ___________ units.
12. If two circles neither intersect nor touch each other, the maximum number of common tangents that can be drawn to them is _________.
13. If \( \sin(\theta - 30^\circ) = \frac{1}{2} \), then the value of \( \cos \theta \) is _____.
14. If the data arranged in ascending order is 8, 9, 12, 17, x+2, x+4, 30, 34, 39 and the median is 24, then the value of x is _____.
15. If the curved surface area of a right circular cylinder is \(c\) square units, the base radius is \(r\) units, and the volume is \(v\) cubic units, then the value of \(\frac{cr}{v}\) is = _____.
16. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is:
(a) 70° (b) 110° (c) 35° (d) 55°
17. 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
18. If the mode of the numbers 64, 60, 48, x, 43, 48, 43, 34 is 43. Then the value of \((x+3)\) is.
(a) 44 (b) 45 (c) 46 (d) 48
19. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -
(a) 60° (b) 30° (c) 120° (d) 90°
20. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is -
(a) 20° (b) 25° (c) 30° (d) 40°
21. ABC is a triangle. Find the value of \(\sin\left(\cfrac{B+C}{2}\right)\)
(a) sin\(\cfrac{A}{2}\) (b) cos\(\cfrac{A}{2}\) (c) sinA (d) cosA
22. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).
(a) 25° (b) 50° (c) 60° (d) 65°
23. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).
(a) 54° (b) 72° (c) 36° (d) 27°
24. The minimum and maximum values of \(a + b\sin\theta\) are 5 and 6 respectively. Find the values of \(a\) and \(b\).
(a) \(a=1, b=5\) (b) \(a=-5, b=1\) (c) \(a=5, b=1\) (d) \(a=-1, b=5\)
25. In triangle ABC, angle B is a right angle. The hypotenuse is \(\sqrt{15}\), and the sum of the other two sides is 4. What is the value of \((\cos A + \cos C)\)?
(a) \(\cfrac{8}{\sqrt{13}}\) (b) \(\cfrac{-8}{\sqrt{15}}\) (c) \(\cfrac{-8}{\sqrt{13}}\) (d) \(\cfrac{8}{\sqrt{15}}\)
26. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?
(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)
27. The length, width, and height of a room are 5 m, 4 m, and 3 m respectively. What is the maximum length of a rod that can be placed inside the room?
(a) \(5\sqrt2\) meters (b) \(7\) meters (c) \(8\sqrt2\) meters (d) \(10\sqrt2\) meters
28. A is in joint variation with B and C². If A = 144 when B = 4 and C = 3, then what is the value of the constant of variation?
(a) \(\frac{1}{4}\) (b) \(\frac{1}{2}\) (c) \(\frac{1}{3}\) (d) \(\frac{1}{5}\)
29. The numerical value of a sphere’s volume and surface area are equal. What is the numerical value of the sphere’s radius?
(a) 2 (b) 3 (c) 4 (d) 5
30. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?
(a) 0 (b) 1 (c) 0.1 (d) 0.01
