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TT School of Management institute involved in teaching, training and research. Currently it has 37 faculty
members. They are involved in three jobs: teaching, training and research. Each faculty member working
with TT School of Management has to be involved in at least one of the three job mentioned above:

A maximum number of faculty members are involved in training. Among them, a number of faculty
members are having additional involvement in the research.

The number of faculty members in research alone is double the number of faculty members involved
in all the three jobs.

17 faculty members are involved in teaching.

The number of faculty members involved in teaching alone is less than the number of faculty members
involved in research alone.

Ten faculty members involved in the teaching are also involved in at least one more job.

80. After some time, the faculty members who were involved in all the three tasks were asked to with
drawl from one task. As a result, one of the faculty members each opted out of teaching and
research, while remaining ones involved in all the three tasks opted out of training. Which one of the
following statements, then necessarily follows:
(A) The least number of faculty members is now involved in teaching.
(B) More faculty members are now associated with training as compared to research.
(C) More faculty members are now involved in teaching as compared to research.
(D) None of the above.

81. Based on the information given above, the minimum number of faculty members involved in both
training and teaching, but not in research is:
(A) 1 (B) 3 (C) 4 (D) 5

Approach plz ?

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What are the last two digits of 9483^67483?
approach plz ?
Using Cauchy Schwartz equation:

(x^2 + 4y^2 + 9z^2) * (1 + 1/2^2 + 1/3^2) > (x+y+z)^2

(x^2 + 4y^2 + 9z^2) * (49/36) > 14^2
x^2 + 4y^2 + 9z^2 > 144


least value of x^2 + 4y^2 + 9z^2 if x+y+z=14

approach plz

if the roots of the equation x^10 - 1 = 0 are 1 ,a,b,c, ...

find the value of (1-a)(1-b)(1-c) ...
approach plz .......

In how many ways can 4 persons be chosen from a row of 10 persons such that no two persons chosen are sitting next to each other?

(a) 210 (b) 35 (c) 72 (d) None of these

approach plz

Basically we are looking for sum of all natural numbers less than 90 and coprime to 90

So, it will be E(90)*90/2 = 24*90/2 = 1080

=> Sum of all such x/90 = 1080/90 = 12

for sum why euler is multiply by 90/2 ...plz explain...

find the sum of all rational numbers x/90 which are less than 1 and such that HCF(x,90)=1


approach plz

easy one :)

find the no of real roots of x^4 +x^2 = 0


approach plz ?...