For the following questions answer them individually
Let A be a $$3 \times 3$$ real matrix such that $$A\begin{pmatrix}1\\0\\1\end{pmatrix} = 2\begin{pmatrix}1\\0\\1\end{pmatrix}$$, $$A\begin{pmatrix}-1\\0\\1\end{pmatrix} = 4\begin{pmatrix}-1\\0\\1\end{pmatrix}$$, $$A\begin{pmatrix}0\\1\\0\end{pmatrix} = 2\begin{pmatrix}0\\1\\0\end{pmatrix}$$. Then, the system $$(A - 3I)\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$$ has
If $$a = \sin^{-1}(\sin 5)$$ and $$b = \cos^{-1}(\cos 5)$$, then $$a^2 + b^2$$ is equal to
If the function $$f: (-\infty, -1] \rightarrow [a, b]$$ defined by $$f(x) = e^{x^3 - 3x + 1}$$ is one-one and onto, then the distance of the point $$P(2b + 4, a + 2)$$ from the line $$x + e^{-3}y = 4$$ is:
Consider the function $$f: (0, \infty) \rightarrow R$$ defined by $$f(x) = e^{-|\log_e x|}$$. If $$m$$ and $$n$$ be respectively the number of points at which $$f$$ is not continuous and $$f$$ is not differentiable, then $$m + n$$ is
Let $$f, g: [0, \infty) \rightarrow R$$ be two functions defined by $$f(x) = \int_{-x}^{x}(|t| - t^2)e^{-t^2}dt$$ and $$g(x) = \int_{0}^{x^2}t^{1/2}e^{-t}dt$$. Then the value of $$9(f(\sqrt{\log_e 9}) + g(\sqrt{\log_e 9}))$$ is equal to
The area of the region enclosed by the parabola $$y = 4x - x^2$$ and $$3y = (x - 4)^2$$ is equal to
The temperature $$T(t)$$ of a body at time $$t = 0$$ is $$160°F$$ and it decreases continuously as per the differential equation $$\frac{dT}{dt} = -K(T - 80)$$, where $$K$$ is positive constant. If $$T(15) = 120°F$$, then $$T(45)$$ is equal to
Let $$(\alpha, \beta, \gamma)$$ be mirror image of the point $$(2, 3, 5)$$ in the line $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$. Then $$2\alpha + 3\beta + 4\gamma$$ is equal to
The shortest distance between lines $$L_1$$ and $$L_2$$, where $$L_1: \frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+4}{2}$$ and $$L_2$$ is the line passing through the points $$A(-4, 4, 3)$$, $$B(-1, 6, 3)$$ and perpendicular to the line $$\frac{x-3}{-2} = \frac{y}{3} = \frac{z-1}{1}$$, is
A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
