The atomic masses of He and Ne are 4 and 20 a.m.u., respectively. The value of the de Broglie wavelength of He gas at −73∘C is ''M'' times that of the de Broglie wavelength of Ne at 727∘C. M is
The number of real solutions of the equation sin−1(i=1∑∞xi+1−xi=1∑∞(2x)i)=2π−cos−1(i=1∑∞(−2x)i−i=1∑∞(−x)i)lying in the interval (−21,21)is ____. (Here, the inverse trigonometric function =sin−1xand cos−1xassume values in [2π,2π]and [0, π], respectively.)
Â·If the normals of the parabola y2=4x drawn at the end points of its latus rectum are tangents to the circle (x−3)2(y+2)2=r2 , then the value of r2 is
A curve passes through the point (1,6π) . Let the slope of the curve at each point (x,y) be xy+sec(xy),x>0. Then the equation of the curve is
Let f:[21,1]→R (the set of all real numbers) be a positive, non-constant, and differentiable function such that fprime(x)<2f(x)andf(21)=1 . Then the value of ∫211f(x)dx lies in the interval (a)(2e−1,2e) (b) (3−1,2e−1)(c)(2e−1,e−1) (d) (0,2e−1)
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 △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 ?
Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 27 on y-axis is (are)
For every twice differentiable function f:R→[−2, 2]with (f(0))2+(fprime(0))2=85, which of the following statement(s) is (are) TRUE?There exist r, s∈Rwhere r<s, such that fis one-one on the open interval (r, s)(b) There exists x0∈(−4, 0)such that ∣∣fprime(x0)∣∣≤1(c) (lim)x→∞f(x)=1(d) There exists α∈(−4, 4)such that f(α)+f(α)=0and fprime(α)=0