1. AC = CE ABC is an equilateral triangle BC = AC = CE ∠BCA = 60° ∠BCE = 180° − 60° = 120° BC = CE ∠CBE = ∠CEB = \(\frac{180° − 120°}{2} = 30°\) 180° = \(\pi\) radians 120° = \(\frac{\pi × 120}{180} = \frac{2\pi}{3}\) radians 30° = \(\frac{\pi × 30}{180} = \frac{\pi}{6}\) radians Angles of triangle ACE in radians: \(\frac{2\pi}{3}\), \(\frac{\pi}{6}\), \(\frac{\pi}{6}\)
2. In triangle \( \triangle ABC \), AC = BC and side BC is extended up to point D. If \( \angle ACD = 144^\circ \), then find the radian measure of each angle of triangle ABC.
3. If ABCD is a cyclic quadrilateral and \(\angle\)A=120°, what is the measure of \(\angle\)C ?
(a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)
4. 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) -\(\cfrac{1}{3}\)
5. 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
6. If \( 2 \cos \theta = 1 \), what is the value of \( \theta \) ?
(a) 10° (b) 15° (c) 60° (d) 30°
7. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:
(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm
8. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be:
(a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm
9. The value of 1 radian is:
(a) In between 40° and 50° (b) less than 40° (c) In between 50° and 60° (d) greater than 60°
10. A tangent is drawn from an external point A to a circle centered at O, touching the circle at point B. Given: OB=5 cm, AO=13 cm, Find the length of AB.
(a) 12 cm (b) 13 cm (c) 6.5 cm (d) 6 cm
11. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\).
(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4
12. In the cyclic quadrilateral ABCD, if \(\angle\)A = 120°, then the measure of \(\angle\)C in a circular sense.
(a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)
13. The coefficient of \(x\) in the quadratic equation \(x+\cfrac{1}{x}=6\).
(a) 6 (b) -6 (c) 0 (d) 1
14. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.
(a) -8 (b) -1 (c) 3 (d) 8
15. 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°
16. If \(u_i = \cfrac{x_i - 35}{10}\), \(∑f_i u_i = 30\), and \(∑f_i = 60\), then the value of \(\bar{x}\) is –
(a) 40 (b) 20 (c) 80 (d) None of these
17. If \( \tan A \tan B = 1\), then the value of \( \tan \cfrac{(A+B)}{2} \) will be –
(a) 1 (b) √3 (c) \(\cfrac{1}{√3}\) (d) None of these
18. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –
(a) 30° (b) 60° (c) 90° (d) 120°
19. 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°
20. If \( \tan \theta \cos 60° = \cfrac{√3}{2} \), find the value of \(\sin(\theta - 15°)\)
(a) \(\cfrac{1}{√2}\) (b) 1 (c) √2 (d) 0
21. 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°
22. If \(x = \sqrt{7 + 4√3}\), find the value of \(x - \cfrac{1}{x}\).
(a) 2 (b) 2√3 (c) 4 (d) 2-√3
23. In the cyclic quadrilateral \(ABCD\), if \(\angle A = 120°\), find the measure of the angle \(\angle C\).
24. If \(2 + b\sqrt{3} = \cfrac{1}{2+\sqrt{3}}\), then \(b\) = ?
(a) 1 (b) -1 (c) 0 (d) 2
25. If \( r\cos\theta = 1 \) and \( r\sin\theta = \sqrt{3} \), then the value of \( \theta \) is—
(a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{6}\)
26. If ABCD is a cyclic quadrilateral and ∠A = 100°, then the measure of ∠C is—?
(a) 50° (b) 200° (c) 80° (d) 180°
27. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—
(a) 2 (b) 18 (c) 9 (d) 8
28. If \(tanα + cotα = 2\), then the value of \(tan^{13}α + cot^{13}α\) is—?
(a) 13 (b) 2 (c) 1 (d) 0
29. In triangle ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. AP = QC, AB = 12 cm, AQ = 2 cm. Find the length of CQ.
(a) 4 cm (b) 6 cm (c) 9 cm (d) None of the above
30. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).
(a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above
