1. If O is the circumcenter of triangle ABC and \(\angle\)OAB = 50°, then the value of \(\angle\)ACB will be .............

2. The equilateral triangle XYZ is inscribed in a circle. If O is the center of the circle, what is the measure of \(\angle\)YOZ?

(a) 60° (b) 30° (c) 90° (d) 120°

3. In triangle ABC, the circumcenter is O. It is given that \(\angle\)BAC = 85° and \(\angle\)BCA = 55°. Find the value of \(\angle\)OAC.

4. O is the circumcenter of \(\triangle\)ABC. If \(\angle\)BAC = 50°, then determine the value of \(\angle\)OBC.

(a) 30° (b) 60° (c) 40° (d) 50°

5. In triangle ABC, the circumcenter is \(O\) and \(\angle\)OAB = 50°. Determine the value of \(\angle\)ACB.

6. O is the center of a circle, and AB is its diameter. ABCD is a cyclic quadrilateral where AB\(\parallel\)DC, and \(\angle\)BAC = 75°. The value of \(\angle\)BCD will be-

(a) 60° (b) 45° (c) 75° (d) 50°

7. In \(\triangle\)ABC, the circumcenter is \(O\) and \(\angle\)BAC = \(65^\circ\). Determine the value of \(\angle\)OBC.

8. O is the circumcenter of triangle ABC. If \(\angle BAC = 85^\circ\) and \(\angle BCA = 55^\circ\), then determine the value of \(\angle OAC\).

9. 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

10. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –

(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

11. ABCD is a cyclic trapezium, where AD \(\parallel\) BC. If \(\angle\)ABC = 70°, then the measure of \(\angle\)BCD will be—?

(a) 110° (b) 80° (c) 70° (d) 120°

12. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

13. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)

14. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

15. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be—

(a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

16. The radius of the circumcircle of an equilateral triangle with side length \(5\sqrt{3}\) cm will be —

(a) 6 cm (b) 5 cm (c) 4 cm (d) 3 cm

17. O is the circumcenter of triangle ABC. If \(\angle\)BAC = 85° and \(\angle\)BCA = 70°, then what is the measure of \(\angle\)OAC?

(a) \(65^o\) (b) \(42\cfrac{1}{2}^o\) (c) \(50^o\) (d) \(25^o\)

18. The circumcenter of triangle \(\triangle\)ABC lies outside the triangle. If the largest angle of the triangle is \(\angle\)BAC, then:

(a) \(\angle\)BAC = 90° (b) \(\angle\)BAC<90° (c) \(\angle\)BAC > 90° (d) \(\angle\)BAC = \(\angle\)ACB = \(\angle\)ABC

19. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

20. If the assumed mean is 22, class width is 10, total frequency is 80, and the value of \(\sum{f_iu_i}\) is 16, then the actual (or true) mean will be —

(a) 23 (b) 24 (c) 25 (d) 26

21. If the mean of the numbers \(x_1, x_2, x_3, x_4, ..., x_n\) is \(\bar{x}\), then the value of \((x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + ... + (x_n - \bar{x})\) will be —

(a) 0 (b) 1 (c) 3 (d) 5

22. If the data arranged in ascending order is 6, 9, 11, 12, x+2, x+3, 17, 20, 21, 24 and the median is 21, then the value of x will be —

(a) 13.5 (b) 15.5 (c) 18.5 (d) 21.5

23. From a statistical frequency distribution table, the given values are: \(n = 84,\) \(L = 30,\) \(f = 30,\) \(h = 30,\) \(cf = 40\) Therefore, the median of the distribution will be —

(a) 30 (b) 32 (c) 34 (d) 36

24. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.

25. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

26. In your uncle's factory, the value of a machine depreciates at a rate of 10% per year. If the current value of the machine is ₹6000, what will its value be after 3 years?

27. The current value of a printing machine is ₹1,80,000. If the annual depreciation rate is 10%, what will be the value of the machine after 3 years?

28. The expression \(x^4 - 2x^2 + k\) will be a perfect square when the value of \(k\) is:

(a) \(\cfrac{1}{4}\) (b) 0 (c) \(\cfrac{1}{2}\) (d) 1

29. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

30. If the current price of a machine is ₹\(b\) and its value depreciates by \(r\%\) every year, then after \(n\) years, the value of the machine will be ₹\(b\left(1 - \cfrac{r}{100}\right)^n\).