Linear Inequalities In One Variable

The determinant can be equal to zero in the following cases:

Case (i): Two values are zero

If two of the values are zero, there are 4 possible pairs: (a and b), (b and d), (d and c), or (c and a). Each case can occur in ways, so the total number of such cases is:

Case (ii): Any three values are zero

If any three of the values are zero, the number of such cases is given by the combination , which is:

Case (iii): All four values are zero

If all four values are zero, there is only 1 such case:

Case (iv): All four values are non-zero but the same

If all four values are non-zero and the same, there are 7 possible values for them:

Case (v): Two values are alike and the other two are alike (non-zero)

If two values are alike and the other two are alike, there are such cases. This gives:

Calculating the number of invertible matrices:

The total number of invertible matrices is given by the total number of cases minus the non-invertible cases:

Thus, the number of invertible matrices is 3780.

Given :
The position vector of point P is given by:
The position vector of point R is given by:
Now, let's find the position vector of the point that divides PR internally in the ratio 5:4. This can be done using the section formula:
4M=5+45R+4P​




This is the same as 3b+2d​, which confirms that the option (B) is indeed correct. It represents a point that divides PR internally in the ratio 5:4.
So, the correct option is (B): is the position vector of a point that divides PR internally in the ratio 5:4