Assuming 2s-2p mixing is NOT operative, the paramagnetic species among the following is
For any positive integer n, define fn:(0,∞)→Ras fn(x)=j=1∑ntan−1(1+(x+j)(x+j−1)1)for all x∈(0,∞).Here, the inverse trigonometric function tan−1xassumes values in (−2π,2π)˙Then, which of the following statement(s) is (are) TRUE?j=1∑5tan2(fj(0))=55(b) j=1∑10(1+fj′(0))sec2(fj(0))=10(c) For any fixed positive integer n, (lim)x→∞tan(fn(x))=n1(d) For any fixed positive integer n, (lim)x→∞sec2(fn(x))=1
Let f:R→(0,∞) and g:R→R be twice differentiable functions such that f" and g" are continuous functions on R. suppose fprime(2)=g(2)=0,f(2)=0 and g′(2)=0, If x→2limf′(x)g′(x)f(x)g(x)=1 then
Let △PQR be a triangle. Let a=QR,b=RP and c=PQ. If ∣a∣=12,∣∣b∣∣=43 and b.c=24, then which of the following is (are) true ?
A line L : y = mx + 3 meets y-axis at E (0, 3) and the arc of the parabola y2=16x 0≤y≤6 at the point art F(x0,y0). The tangent to the parabola at F(X0,Y0) intersects the y-axis at G(0,y). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum P) m= Q) = Maximum area of △EFG is (R) y0= (S) y1=
A farmer F1has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1)and R(2, 0). From this land, a neighbouring farmer F2takes away the region which lies between the side PQand a curve of the form y=xn (n>1). If the area of the region taken away by the farmer F2is exactly 30% of the area of PQR, then the value of nis _______.
Let f[0,1]→R (the set of all real numbers be a function.Suppose the function f is twice differentiable, f(0)=f(1)=0,and satisfies f′(x)–2f′(x)+f(x)≤ex,x∈[0,1].Which of the following is true for 0<x<1?
Let Sbe the set of all column matrices [b1b2b3]such that b1,b2,b3∈Rand the system of equations (in real variable)−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each [b1b2b3]∈S?(a) x+2y+3z=b1,4y+5z=b2and x+2y+6z=b3(b) x+y+3z=b1,5x+2y+6z=b2and −2x−y−3z=b3(c) −x+2y−5z=b1,2x−4y+10z=b2and x−2y+5z=b3(d) x+2y+5z=b1,2x+3z=b2and x+4y−5z=b3
let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P1:x+2y−z+1=0 and P2:2x−y+z−1=0, Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following points lie(s) on M?