Let x = x(y) be the solution of the differential equation \(2ye^{\frac{x}{y^2}}dx+(y^2−4xe^{\frac{x}{y^2}})dy=0 \)such that x(1) = 0. Then, x(e) is equal to
(Where E is the electromotive force)Which of the above half cells would be preferred to be used as reference electrode?
Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of \(\int_{0}^{\frac{\pi}{2}} y \,dx\)is equal to :
Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \(\frac{x^2}{a^2}−\frac{y^2}{b^2}=1\)Let e′ and l′ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If \(e^2=\frac{11}{14}l\) and \((e^′)^2=\frac{11}{8}l^′\)then the value of 77a + 44b is equal to :
Let,\(\vec{a}=a\hat{i}+2\hat{j}−\hat{k}\) and \(\vec{b}=−2\hat{i}+α\hat{j}+\hat{k}\), where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15(α^2+4)}\) , then the value of \(2|\vec{a}|^2+(\vec{a}⋅\vec{b})|\vec{b}|^2 \)is equal to :
The value of \(\lim_{{n \to \infty}} 6\tan\left\{\sum_{{r=1}}^{n} \tan^{-1}\left(\frac{1}{{r^2+3r+3}}\right)\right\}\)is equal to :
