RSM12-P-IV-T(M)-MA-2
1.
Find the complexl' numbers satisfying the equation 2|z|2 +z2 - 5 + i V3 =0.
2
Using the equation z8 +1 = 0 prove that
f
TUY
771 371Y 5 TTv cos 0 - cos — cos 9 - c o s — | . 8 A 8
cos48=8 cos 9 - cos - cos 9 - cos — 8A 8
[10]
[10]
3
If Zi ^ z2 = 0 and Zi z2 +z3 z4 = 0 then prove that the points representing z1t z2, z3, z4 are concyclic. [10]
4.
Show that the number of ways of selecting n objects out of 3n objects, of which n are alike and rest are different is 22n_1 + ^ — ^ j - . n! (n - 1Jf
[10]
5.
There are 6n seats in a row. In how many ways n persons can sit such that between any two person there are at least three seats vacant. If n is even then in how many ways they can sit such that each person has exactly one neighbour. [10]
6.
Find the distance of the point p(i + j + k) in the plane n which passes through the points A(2i + j + k), B(I + 2j + k) and c(i + j + 2k). Also find the vector of the foot of the perpendicular from P on the plane.
position [10]
7.
Let ABC and PQR be any two perpendiculars from the points A, concurrent. Using vector methods P, Q, R to BC, CA, AB respectively
triangles in the same plane. Assume that the B, C to the sides QR, RP, PQ respectively are or otherwise, prove that the perpendiculars from are also concurrent. [10]
8.
Four students after selection in IIT-JEE decide to join anyone of the four NT Kanpur, Delhi, Bombay or Kharagpur. They discuss the placement position after passing and find the degree of attractiveness for joining in proportional to numbers 8, 9, 10, 12. Find the probability that each will join a different NT. [10]
9.
Watson received a telegram from Sherlock Holmes to catch a specified train from London to Ciifton. He reached the platform where the specified train was ready to leave. The train consisted of p carriages each of which will hold q passengers and in all (pq - m ) passengers had occupied their seats. If waston is equally likely to get any vacant palace, find the chance that he will travel in the same carriage with Sherlock Holmes. [10]
10.
If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement, find the probability that the roots of the equation x2+ px + q = 0 are real and distinct. [10]
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