Which of the following statement patterns is a contradiction? \[ \begin{aligned} S_1 &: (p \to q) \land (p \land \sim q) \\ S_2 &: [p \land (p \to q)] \to q \\ S_3 &: (p \lor q) \to \sim p \\ S_4 &: [p \land (p \to q)] \leftrightarrow q \end{aligned} \]
If \[ y = \cot^{-1}\!\left(\sqrt{\frac{1-\sin x}{1+\sin x}}\right), \quad \text{then find} \quad \frac{dy}{dx}. \]
If the L.M.V.T. holds for the function \( f(x) = x + \dfrac{1}{x} \), \(x \in [1,3]\), then \(c =\)
If\[ A = \begin{bmatrix} \cos\theta & -\sin\theta \\ -\sin\theta & -\cos\theta \\ \end{bmatrix} \] then \(A^{-1} =\)
If two moles of an ideal gas at $546\ \text{K}$ occupy a volume of $44.8\ \text{L}$, what is the pressure of ideal gas at $546\ \text{K}$? ($R = 0.0821\ \text{L atm mol}^{-1}\text{K}^{-1}$)
What is the type of hole occupied if limiting value of $\dfrac{r^+}{r^-}$ is in the range of $0.225$ to $0.414$ ?
The sum of oxidation number of all atoms in $\mathrm{S_2O_3^{2-}}$ ion is
What is the conductivity of 0.01 M NaCl solution if resistance and cell constant of NaCl solution are 375 ohms and $0.5\ \text{cm}^{-1}$ respectively at 298 K ?
