Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
Show that each of the relation R in the set A = { x ∈ Z : 0 ≤ x ≤ 12}, given by I. R={(a,b):Ia-bI is a multiple of 4} II. R={(a,b):a=b}is an equivalence relation. Find the set of all elements related to 1 in each case.
\(∫\frac {xdx}{(x-1)(x-2)}\ equals\)
For all real values of \(x\), the minimum value of \(\frac{1-x+x^{2}}{1+x+x^{2}}\) is
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \(\frac{8}{27}\) of the volume of the sphere.
\(\frac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}}\),\(x∈[0,1]\)
\(∫\frac {dx}{x(x^2+1)} \ equals \)
The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)(x3+xcosx+tan5x+1)dx is
Show that \(\int_{0}^{a}\)ƒ(x)g(x)dx=2\(\int_{0}^{a}\)ƒ(x)dx,if f and g are defined as ƒ(x)=ƒ(a-x)and g(x)+g(a-x)=4
\(\int \sqrt{1+x^2}dx\) is equal to
The maximum value of\( [x(x-1)+1]^{\frac{1}{3}},0≤x≤1\) is
Find the maximum value of 2x3−24x+107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find the intervals in which the function f given by f(x)=x3+\(\frac{1}{x^3}\),x≠0 is (i) increasing (ii) decreasing
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \(tan^{-1}\sqrt{2}.\)
Show that the right circular cone of least curved surface and given volume has an altitude equal to\(\sqrt{2}\) time the radius of the base.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i)f(x)=x3,x∈[-2,2] (ii) f(x)=sin x+cos x,x∈[0,π] (iii) f(x)=4x-1/2x2,x∈[-2,\(\frac{9}{2}\)] (iv) f(x)=(x-1)2+3,x∈[-3,1]
Prove that the following functions do not have maxima or minima: (i) f(x) = ex (ii) g(x) = logx (iii) h(x) = x3 + x2+x+1