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 \((30,30)\) and \((0,40)\) is:

If \(\theta\) is the angle between any two vectors \(\vec{a}\) and \(\vec{b}\), then \(|\vec{a}\cdot\vec{b}| = |\vec{a}\times\vec{b}|\) for \(\theta\) equal to

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

If \(A = \begin{bmatrix}2&1&0\\3&1&2\\0&4&-1\end{bmatrix}\), then \(|\text{adj}(A)|\) is equal to

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

If \(A = \begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}\), then:

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…

Match List - I with List - II List - I | List - II A. The range of \(\sin^{-1}x\) is | I. \(\left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right]\) B. The range of \(\tan^{-1}x\) is | II. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) C. The range of \(\text{cosec}^{-1}x\)…

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: