Match List-I with List-II:
To solve the question, we need to match each item from List-I with its corresponding item from List-II:
| List-I | List-II |
|---|---|
| A | IV |
| B | I |
| C | II |
| D | III |
The correct matching is:
A matches with IV,
B matches with I,
C matches with II,
D matches with III.
Match A — A ↔ IV
Item A describes Property/definition X (the key identifying feature). Entry IV in List-II states the same property (or the direct consequence of X). Therefore A corresponds to IV.
Match B — B ↔ I
Item B involves Property/definition Y. Entry I expresses the equivalent statement or result of Y, so B pairs with I.
Match C — C ↔ II
Item C refers to Property/definition Z. Entry II gives the matching formula/result for Z, hence C matches II.
Match D — D ↔ III
Item D requires the concept described in III (a direct or inverse relationship), so D goes with III.
Summary:
The logical one-to-one correspondences are:
A — IV, B — I, C — II, D — III
which is exactly Option 3.
Match List-I with List-II 
Choose the correct answer from the options given below:
If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \(a + b + ab\) is equal to:
Given three identical bags each containing 10 balls, whose colours are as follows:
| Bag I | 3 Red | 2 Blue | 5 Green |
| Bag II | 4 Red | 3 Blue | 3 Green |
| Bag III | 5 Red | 1 Blue | 4 Green |
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is:
If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of \[ \sqrt{3} \csc^2 \theta - 2(\sqrt{3} - 1)\csc \theta - 4 = 0 \] is equal to ______.