1. If the sum and product of two quadratic surds are rational numbers, then the surds are conjugate surds.
2. If the sum and product of two quadratic surds are rational numbers, then the surds are _______ surds.
3. If the sum and product of two quadratic factors are rational numbers, then the factors are conjugate factors.
4. If the sum and product of two quadratic factors are a rational number, then the factors are _____ factors.
5. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).
6. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.
7. The sum of the squares of two numbers is 29, and their product is 10. Find the two numbers.
8. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –
(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9
9. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –
(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9
10. If the volumes of two vertical solid cylinders are equal, and their heights are in the ratio 1:2, then the ratio of the lengths of their radii will be –
(a) 1: √2 (b) √2:1 (c) 1:2 (d) 2:1
11. 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
12. If two angles of a triangle are 75° and \( \frac{\pi^c}{6} \), then what is the measure of the third angle?
(a) 75° (b) 60° (c) 65° (d) 70°
13. 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) \(\cfrac{10}{13}\)
14. In two concentric circles, a chord of the larger circle is tangent to the smaller circle. If the radii of the two circles are 10 cm and 4 cm respectively, then what is the length of the chord?
(a) 8 cm (b) 9 cm (c) 11 cm (d) 12 cm
15. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?
(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm
16. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?
(a) 145° (b) 55° (c) 110° (d) None of the above
17. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?
(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm
18. If the diameters of two cones are equal and the ratio of their slant heights is 5:7, then what is the ratio of their curved surface areas?
(a) 25:7 (b) 25:49 (c) 5:49 (d) 5:7
19. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?
(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27
20. If the volumes of two cones are in the ratio 2:3 and the ratio of their base radii is 1:2, then what is the ratio of their heights?
(a) 3:8 (b) 8:3 (c) 3:4 (d) 4:3
21. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?
(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)
22. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?
23. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?
24. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.
25. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?
26. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)
27. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?
28. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)
29. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.
30. If the angles of depression from a lighthouse to two ships located along the same straight line are 60° and 30°, and the nearer ship is 150 meters away from the lighthouse, then what is the distance of the farther ship from the lighthouse?
