Types of lines we can construct in xy-plane:
A. Vertical lines
B. Horizontal lines
C. Lines with +ve slope and +ve y-intercept
D. Lines with +ve slope and -ve y-intercept
E. Lines with -ve slope and +ve y-intercept
F. Lines with -ve slope and -ve y-intercept

NOTE: I'm going to use A, B, C, D, E and F for the respective lines

Total lines = $35 = A + B + C + D + E + F$

Given: $C + D = 15$
Now, C = $\frac{1}{3}(15) = 5$
Thus, $D = 15 - 5 = 10$

So, $35 = 0 + B + 5 + 10 + E + F$

Now, 23 lines have a y-intercept less than or equal to zero i.e. $B + D + F = 23$
Plugging $D = 10$ from above;

$B + 10 + F = 23$
$B + F = 13$

Qs: Number of lines containing no point in the Quadrant I
This means all lines which have a -ve slope and -ve y-intercept i.e. Lines marked as $F$

Since, $B + F = 13$
Even if we assume that all the lines which are marked as $B$ have y-intercept as $0$ (in order to maximise $F$), still we have $F$ (lines with -ve slope and -ve y-intercept) as $13$ only

Col. A: 13
Col. B: 15

Col. A < Col. B
Hence, option B

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