12. In study of a certain university, the following data were collected. There were 30 students
taking mathematics, 28 students taking physics and 43 taking English. 14 students were
taking mathematics and physics, 15 students were taking mathematics and English and 19
students were taking physics and English, 6 students were taking all the three subjects.
a. How many students were taking physics only?
b. How many students were taking mathematics and English but not physics?
c. How many students were taking exactly one of these subjects?
d. Draw a Venn diagram.
13. Find an odd natural number x such that LCM(x,40) = 1400.
14. Two incandescent lamps are turned on at the same time. One blinks every 4 seconds and
the other blinks every 6 seconds. In 60 seconds, how many times will they blink at the
same time?

a. To find the number of students taking physics only, we need to subtract the students taking physics and other subjects from the total number of students taking physics.

Total students taking physics = 28

Students taking physics and mathematics = 14

Students taking physics and English = 19

Students taking all three subjects = 6

So, the number of students taking physics only = Total students taking physics - (Students taking physics and mathematics + Students taking physics and English - Students taking all three subjects)

= 28 - (14 + 19 - 6) = 28 - 27 = 1

Therefore, there is 1 student taking physics only.

b. To find the number of students taking mathematics and English but not physics, we subtract the students taking all three subjects from the total number of students taking mathematics and English.

Students taking mathematics and English = 15

Students taking all three subjects = 6

So, the number of students taking mathematics and English but not physics = Students taking mathematics and English - Students taking all three subjects

= 15 - 6 = 9

Therefore, there are 9 students taking mathematics and English but not physics.

c. To find the number of students taking exactly one of these subjects, we need to add up the students taking each subject individually and subtract those taking more than one subject.

Students taking only mathematics = Total students taking mathematics - (Students taking mathematics and physics + Students taking mathematics and English - Students taking all three subjects)

= 30 - (14 + 15 - 6) = 30 - 23 = 7

Similarly, students taking only physics = 1 (from part a)

Students taking only English = 9 (from part b)

So, the total number of students taking exactly one subject = 7 (mathematics) + 1 (physics) + 9 (English) = 17

Therefore, there are 17 students taking exactly one of these subjects.

d. I can't draw a diagram directly, but you can visualize it using the provided information. Draw three overlapping circles representing mathematics, physics, and English. Label the intersections with the given numbers: 14 in the overlap of math and physics, 15 in the overlap of math and English, 19 in the overlap of physics and English, and 6 in the center where all three overlap.

For question 13, to find an odd natural number x such that LCM(x, 40) = 1400, we can observe that 1400 is divisible by 40. Therefore, we can set x = 1400.

For question 14, the least common multiple of 4 and 6 is 12. So, both lamps will blink at the same time every 12 seconds. In 60 seconds, they will blink at the same time 60 / 12 = 5 times.