The order of the differential equation whose general solution is \(y = e^x(a\cos x+b\sin x)\), where \(a\) and \(b\) are arbitrary constants, is:

Match List - I with List - II List - I | List - II A. \(\int\frac{1}{x+\sqrt{x}}dx\) | I. \(2\sqrt{x} + C\) B. \(\int\frac{e^{\log\sqrt{x}}}{x}dx\) | II. \(2(\sqrt{x}-1)e^{\sqrt{x}} + C\) C. \(\int\frac{dx}{4x^2-9}\) | III. \(2\log(\sqrt{x}+1) + C\) D. \(\int e^{\sqrt{x}}dx\) | I…

\(\frac{d}{dx}\left[\int_0^{2a} f(\sin 2x)\,dx\right] =\)

If \(y = e^{\log\sin^{-1}x} + e^{\log\cos^{-1}x}\),…

\(\int\tan x(\sec x-\tan x)\,dx =\)

If \(y = \log\left(\sec e^{x^2}\right)\), then \(\frac{dy}{dx} =\)

If \(\cos\alpha, \cos\beta, \cos\gamma\) are the direction cosines of a vector \(\vec{a}\), then the value of \(\cos2\alpha+\cos2\beta+\cos2\gamma\) is equal to:

For an objective function \(z = ax+by\) where \(a, b > 0\), the corner points of the feasible region determined by a set of constraints are \((0,20)\), \((10,10)\), \((30,30)\) and \((0,40)\). The condition on \(a\) and \(b\) such that max \(z\) occurs at both the points…

The value of \(\hat{i}\cdot(\hat{j}\times\hat{k})+\hat{j}\cdot(\hat{i}\times\hat{k})+\hat{k}\cdot(\hat{i}\times\hat{j})\) is