1. In triangle ∆ABO, the circumcenter is C; If \(\angle\)CAB=50° then what is \(\angle\)AOB =?

(a) 50° (b) 100° (c) 40° (d) 80°

2. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

3. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

4. If the average of \( x_1, x_2, x_3, \dots, x_n \) is \( \bar{x} \), then what is the average of \( ax_1, ax_2, ax_3, \dots, ax_n \) ?

(a) \(\bar{x}\) (b) \(a\bar{x}\) (c) \(n\bar{x}\) (d) None of these

5. The volumes of two perpendicular circular rings are equal. The ratio of their heights is 4:9, then what will be the ratio of their radii?

(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9

6. If the ratio of principal to interest-principal in a year is 8:9, then what is the annual simple interest rate?

(a) 12\(\cfrac{1}{2}\)% (b) 15\(\cfrac{1}{2}\)% (c) 12% (d) 15%

7. If the interest on 4000 rupees for 4 years is 1600 rupees, then what is the interest rate?

(a) 5% (b) 10% (c) 12\(\cfrac{1}{2}\)% (d) 15%

8. If the annual simple interest on a principal is \(\cfrac{1}{9}\) of it, then what will be the accumulated amount in 4 years if the total sum becomes 1326 rupees?

(a) ₹ 1080 (b) ₹ 1120 (c) ₹ 918 (d) ₹ 750

9. If a principal amount becomes 560 rupees in 3 years and 600 rupees in 5 years, then what is the principal amount?

(a) ₹ 400 (b) ₹ 500 (c) ₹ 480 (d) ₹ 540

10. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

11. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?

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

12. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

13. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

14. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?

(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm

15. If the sides of a triangle are in the ratio \(\sqrt{7} : \sqrt{3} : 2\), then what type of triangle is it?

(a) right angle. (b) equilateral (c) isosceles (d) obtuse-angle

16. If a right-angled quadrilateral has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \(x - y + z\)?

(a) 8 (b) 6 (c) 2 (d) 12

17. If a right-angled quadrilateral has \(a\) number of vertices, \(b\) number of edges, and \(c\) number of faces, then what is the value of \(2a - b + 3c\)?

(a) 16 (b) 18 (c) 20 (d) 22

18. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°?

(a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)

19. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?

(a) 1 (b) 1 (c) 0 (d) None of the above

20. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?

(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)

21. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?

(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

22. If \(a + b + c = 0\), then what is the value of \(\cfrac{a^3 + b^3 + c^3}{abc} - 3\)?

(a) 1 (b) 0 (c) -1 (d) None of the above

23. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)?

(a) 0 (b) 1 (c) 2 (d) 3

24. If a cuboid has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \((x - y + z)^2\)?

(a) 2 (b) 4 (c) 6 (d) None of the above

25. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^6}\)?

(a) -1 (b) 1 (c) 2 (d) None of the above

26. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)?

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

27. If PQ is a tangent to a circle centered at O and touches the circle at point C, then what is the value of \(\angle\)BCP?

(a) 90\(^o\) (b) 80\(^o\) (c) 70\(^o\) (d) None of the above

28. 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

29. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

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

30. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above