Match the chemical substances in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the 4×4 matrix given in the ORS.
Late a∈Rand let f:Rbe given by f(x)=x5−5x+a,thenf(x)has three real roots if a>4f(x)has only one real roots if a>4f(x)has three real roots if a<−4f(x)has three real roots if −4<a<4
Let Tbe the line passing through the points P(−2,7)and Q(2,−5). Let F1be the set of all pairs of circles (S1,S2)such that Tis tangent to S1at Pand tangent to S2at Q, and also such that S1and S2touch each other at a point, say, M. Let E1be the set representing the locus of Mas the pair (S1,S2)varies in F1. Let the set of all straight lines segments joining a pair of distinct points of E1and passing through the point R(1,1)be F2. Let E2be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?The point (−2,7)lies in E1(b) The point (54,57)does NOT lie in E2(c) The point (21,1)lies in E2(d) The point (0,23)does NOT lie in E1
A box B1, contains 1 white ball, 3 red balls and 2 black balls. Another box B2, contains 2 white balls, 3 red balls and 4 black balls. A third box B3, contains 3 white balls, 4 red balls and 5 black balls.
Let f(x)=∣1−x∣1−x(1+∣1−x∣)cos(1−x1) for x=1. Then: (A)(lim)n→1−f(x) does not exist (B)(lim)n→1+f(x) does not exist (C)(lim)n→1−f(x)=0 (D)(lim)n→1+f(x)=0
Let Obe the origin, and OXxOY,OZbe three unit vectors in the direction of the sides QR, RP, PQ, respectively of a triangle PQR.If the triangle PQR varies, then the minimum value of cos(P+Q)+cos(Q+R)+cos(R+P)is:−23 (b) 35 (c) 23 (d) −35
Let f:(0,∞)R be given by f(x)=∫x1xte−(t+t1)dt, then (a)f(x) is monotonically increasing on [1,∞)(b)f(x) is monotonically decreasing on (0,1)(c)f(2x) is an odd function of x on R