1. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?
2. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct?
(a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)
3. If \(a^2 b^2 + 1 = 2ab\), then which of the following is correct?
(a) \(a-b \propto b\) (b) \(a-b \propto a\) (c) \(a \propto b\) (d) \(a \propto \cfrac{1}{b}\)
4. If \(3x = 5\sin\theta\) and \(4y = 5\cos\theta\), then which of the following relations is correct?
(a) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=1\) (b) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=0\) (c) \(\cfrac{9x^2}{25}-\cfrac{16y^2}{25}=1\) (d) \(\cfrac{16x^2}{25}+\cfrac{9y^2}{25}=1\)
5. If \( \cos\theta = p \) and \( \cot\theta = q \), then which of the following relationships is true?
(a) \(\cfrac{1}{p^2}+\cfrac{1}{q^2}=1\) (b) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=1\) (c) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=0\) (d) \(\cfrac{1}{q^2}+\cfrac{1}{p^2}=1\)
6. If two circles with radii \(r_1\) and \(r_2\) touch each other externally, and the distance between their centers is \(d\), then which of the following is correct?
(a) \(r_1+d=r_2\) (b) \(r_2+d=r_1\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2=d\)
7. If \((x^2 + y^2) \propto (x^2 - y^2)\), then which of the following is correct?
(a) \(x\propto y\) (b) \(xy=\) constant (c) \(x^2\propto y\) (d) \(y^2\propto x\)
8. If the combined average of 5, 15, 22, x, y, 25, and z is 14, then which of the following is correct?
(a) x+y+z=42 (b) x+y+z=31 (c) x+y=z (d) x+y=2z
9. If \(a \propto b^3\) and \(c \propto \sqrt{a}\), then which of the following is correct?
(a) \(b \propto c^2\) (b) \(b \propto c^3\) (c) \(b^3 \propto c^3\) (d) \(b^3 \propto c^2\)
10. If \(a \propto \frac{1}{b}\) and \(b \propto \frac{1}{c}\), then which of the following is correct?
(a) \(a \propto c\) (b) \(a \propto \cfrac{1}{c}\) (c) \(a^2 \propto c\) (d) \(a \propto c^2\)
11. If \(x\) and \(y\) are the factors of \(30\), then which of the following is correct –
(a) \(x \propto y\) (b) \(x^2 \propto y\) (c) \(x \propto \cfrac{1}{y}\) (d) \(x \propto y^2\)
12. In triangle ABC, angle B is a right angle. If a circle is drawn with AC as the diameter and it intersects AB at point D, then which of the following statements is correct— (i) AB > AD (ii) AB = AD (iii) AB < AD
13. 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
14. 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
15. If \(A\) is the total surface area and \(V\) is the volume of a sphere, which of the following is correct?
(a) \(A^3=36πV^2\) (b) \(A^2=4V^2\) (c) \(A^3=36V^2\) (d) \(A^3=36V^3\)
16. If \(∑f_i(x_i - a) = 400\), \(∑f_i = 50\), and \(a\) (assumed mean) = 52, then the value of the combined mean \(\bar{x}\) is –
(a) 52 (b) 60 (c) 80 (d) 90
17. If \(∑f_i d_i = 400\), \(∑f_i = 50\), and \(a =\) assumed mean \(= 52\), then the value of the combined mean is –
(a) 52 (b) 60 (c) 80 (d) 55
18. 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
19. 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
20. If \( \sec\theta = m \) and \( \tan\theta = n \), then which of the following
(a) \(m=n\) (b) \(m\gt n\) (c) \(m\lt n\) (d) None of the above
21. 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
22. 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
23. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?
24. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?
25. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.
26. If \(u_i = \frac{x_i - 30}{10}\), \(∑f_i = 50\), and \(∑u_i f_i = 25\), then what is the value of \(\bar{x}\)?
27. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]
28. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).
(a) 1 (b) 2 (c) 3 (d) 4
29. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?
30. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?
