1. If the circular (radian) measure of an angle is \( \frac{7\pi}{12} \), what is its value in the sexagesimal (degree) system?
(a) 90° (b) 105° (c) 135° (d) 160°
2. If \(20^{-x} = \dfrac{1}{7}\), then the value of \((20)^{2x}\) is:
(a) \(\frac{1}{49}\) (b) 7 (c) 49 (d) 1
3. If \(\log_{10} (7x - 5) = 2\), then the value of \(x\) is:
(a) 10 (b) 12 (c) 15 (d) 18
4. If the roots of the quadratic equation \(5x^2+13x+k=0\) are reciprocals of each other, then the value of \(k\) is:
(a) 3 (b) 4 (c) 5 (d) -5
5. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), find the values of the two angles in degrees and radians.
6. 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) -\(\cfrac{1}{3}\)
7. The median of the data set 6,7,8,8,9,15,10,15,20,19,25,24,216,7,8,8,9,15,10,15,20,19,25,24,21 is:
(a) 10 (b) 15 (c) 9 (d) 19
8. 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
9. If \( 2 \cos \theta = 1 \), what is the value of \( \theta \) ?
(a) 10° (b) 15° (c) 60° (d) 30°
10. 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°
11. If the total interest on a principal amount in 4 years is \(\cfrac{1}{5}\) of the principal, the annual rate of interest is:
(a) 4% (b) 5% (c) 10% (d) 20%
12. The median of 8, 15, 10, 11, 7, 9, 11, 13, 16 is:
(a) 15 (b) 10 (c) 11.5 (d) 11
13. The value of (sin43°cos47° +cos43°sin47°) is:
(a) 0 (b) 1 (c) sin4° (d) cos4°
14. 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
15. If the data arranged in ascending order, 8, 9, 12, 17, x+2, x+4, 30, 31, 34, 39 has a median of 24, then the value of x -
(a) 22 (b) 21 (c) 20 (d) 24
16. The value of (√125 – √5) is
(a) √120 (b) √80 (c) √100 (d) 5√5
17. If 16, 15, 17, 16, 15, x, 19, 17, 14 have a mode of 15, then the value of x is-
(a) 15 (b) 16 (c) 17 (d) 19
18. 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°
19. 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
20. 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
21. 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°
22. 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
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. If \(x = \sqrt{7 + 4√3}\), find the value of \(x - \cfrac{1}{x}\).
(a) 2 (b) 2√3 (c) 4 (d) 2-√3
25. If the mode of the data set 16, 15, 17, 16, 15, x, 19, 17, 14 is 15, then the value of x is—
26. 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) \(\cfrac{π}{6}\)
27. At an annual simple interest rate of 12%, if the ratio of principal to interest after \(x\) years is 25:24, what is the value of \(x\)?
(a) 8 (b) 10 (c) 12 (d) 5
28. The value of \(\cfrac{7π}{12}\) in the sexagesimal system is—?
(a) 115° (b) 150° (c) 135° (d) 105°
29. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—
(a) 2 (b) 18 (c) 9 (d) 8
30. If \(tanα + cotα = 2\), then the value of \(tan^{13}α + cot^{13}α\) is—?
(a) 13 (b) 2 (c) 1 (d) 0
