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 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) \(r_1-r_2=d\)

3. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

4. If the combined average of the data set 7, 5, 2, 10, x, 6, 4, y, 8, 1 is 5, then what is the value of \(x + y\)?

5. If the average of the numbers 6, 7, x, 8, y, and 14 is 9, then

(a) x+y=21 (b) x+y=19 (c) x-y=21 (d) x-y=19

6. If the average of the numbers 6, 7, x, 8, y, and 16 is 9, then—?

(a) x + y = 21 (b) x + y = 17 (c) x - y = 21 (d) x - y = 19

7. If the combined average of the expressions ???? − 3 , ???? − 1 , 7 , ???? , 2 ???? − 1 , and 3 ???? − 5 is 3.5, then what will be the median of these expressions?

(a) 3.5 (b) 4.5 (c) 5 (d) 5.5

8. If a cuboid has number of faces = x, number of edges = y, number of vertices = z, and number of diagonals = p, then what is the value of (x − y + z + p)?

9. If the average of the first four numbers out of five is 26, and the average of the last four numbers is 25, then find the difference between the first and the last number.

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

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

12. 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?

13. If the combined mean of the following data is 15, then calculate the value of \(p\):

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