The corner points of the bounded feasible region determined by a set of constraints in an LPP are P(0, 5) Q(3, 5), R(5, 0) and S(4,1). If the objective function is \(z=ax+2by\), where, \(a,b>0\), then the condition on \(a\) and \(b\) such that the maximum value of \(z\) occurs at Q and S is

The area of the region bounded by the curves \(y=x\) and \(y=x^3\) is:

The projection of the vector \(5\hat i+\hat j-3\hat k\) on the vector \(\hat i+2\hat j-3\hat k\) is

\(\int e^{-x}(\cot x+\operatorname{cosec}^2x)\,dx=\)

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

If \(\begin{vmatrix}p-a&0&c-r\\0&q-b&c-r\\a&b&r\end{vmatrix}=0\) then the value of \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\) is

Match List-I with List-II List-I | List-II (A) \(\tan^{-1}\frac{2}{11}+\tan^{-1}\frac{7}{24}\) | (I) \(\frac{3\pi}{4}\) (B) \(\tan^{-1}2+\tan^{-1}3\) | (II) \(\pi\) (C) \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3\) | (III) \(\tan^{-1}\frac{1}{2}\) (D)…

\(\int \frac{e^{2x}-1}{e^{2x}+1}\,dx=\)

Match List-I with List-II List-I | List-II (A) The number of possible matrices of order \(3\times3\) with each entry 1 or 0 | (I) \(2^4\) (B) The number of possible matrices of order \(2\times3\) with each entry 1 or 0 | (II) \(2^9\) (C) The number of possible matrices of order \…