Amongst the following , the total number of compounds soluble in aqueous NaOH is
Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of nm is ____
The circle C1:x2+y2=3, with centre at O, intersects the parabola x2=2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2andC3atR2andR3, respectively. Suppose C2andC3 have equal radii 23 and centres at Q2 and Q3 respectively. If Q2 and Q3 lie on the y-axis, then (a)Q2Q3=12(b)R2R3=46(c)area of triangle OR2R3 is 62(d)area of triangle PQ2Q3is=42
Let a,b,andc be three non coplanar unit vectors such that the angle between every pair of them is 3π. If a×b+b×x=pa+qb+rc where p,q,r are scalars then the value of q2p2+2q2+r2 is
Let f:RRbe a differentiable function such that f(0),f(2π)=3andfprime(0)=1.If g(x)=∫x2π[fprime(t)cosect−cottcosectf(t)]dtforx(0,2π],then (lim)x0g(x)=
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?
Consider the circle x2+y2=9 and the parabola y2=8x. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.
Let f:R0,1 be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)? ex−∫0xf(t)sintdt (b) f(x)+∫02πf(t)sintdt(c)x−∫02π−xf(t)costdt (d) x9−f(x)