1. 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) \(xyz=1\)

2. If \(\cfrac{x}{y} \propto (x + y)\) and \(\cfrac{y}{x} \propto (x - y)\), then show that \((x^2 - y^2)\) is constant.

3. If \(\frac{x}{y} \propto x + y\) and \(\frac{y}{x} \propto x - y\), then show that \((x^2 - y^2)\) is a constant quantity.

4. What is the fourth proportional of \((x + y)\), \((x^2 - y^2)\), and \((x^2 + xy + y^2)\)?

(a) \(x^3-y^3\) (b) \(x^3+y^3\) (c) \(x^2-y^2\) (d) \(x^2+y^2\)

5. 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) \(b^3 \propto c^2\)

6. 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) \(a \propto \cfrac{1}{b}\)

7. 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) \(a \propto c^2\)

8. 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?

9. If \(tanθ = 0\), then which of the following is correct?

(a) sinθ=0 (b) cosθ=0 (c) cotθ=0 (d) None of these

10. 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) \(r_1-r_2=d\)

11. 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

12. If \(\left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y}\), then show that \((x^2 + y^2) \propto xy\).

13. 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) \(\cfrac{16x^2}{25}+\cfrac{9y^2}{25}=1\)