KVPY Government Jobs Exams

Let A and B be any two n × n matrices such that the following conditions hold : AB = BA and there exist
positive integers k and l such that Ak = I (the identity matrix) and Bl = 0 (the zero matrix). Then-
(A) A + B = I (B) det (AB) = 0
(C) det (A + B) ≠ 0 (D) (A + B)m = 0 for some integer m

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KVPY Other Question

The locus of the point P = (a, b) where a, b are real numbers such that the roots of x^3 + ax^2 + bx + a = 0 are in
arithmetic progression is-
(A) an ellipse
(B) a circle
(C) a parabola whose vertex in on the y-axis
(D) a parabola whose vertex is on the x-axis The smallest possible positive slope of a line whose y-intercept is 5 and which has a common point with the
ellipse 9x^2 + 16y^2 = 144 is-