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For the following questions answer them individually
Let M and N be two $$3 \times 3$$ matrices such that MN = NM. Further, if $$M \neq N^2$$ and $$M^2 = N^4$$, then
For every pair of continuous functions $$f, g: [0, 1] \rightarrow R$$ such that
$$max \left\{f(x) : x \in [0, 1]\right\} = max \left\{{g(x) : x \in [0, 1]}\right\}$$
the correct statement(s) is(are) :
Let $$f: (0, \infty) \rightarrow R$$ be given by
$$f(x) = \int_{\frac{1}{x}}^{x} e^{-(t + \frac{1}{t})} \frac {dt}{t}.$$
Then
Let $$f: [a, b] \rightarrow [1, \infty)$$ be a continuous function and let $$g: R \rightarrow R$$ be defined as
$$g(x) = \begin{cases}0 & if x < a,\\\int_{a}^{x}f(t)dt & if a\leq x \leq b,\\\int_{a}^{b}f(t)dt & if x > b. \end{cases}$$
Then
Let $$f:(-\frac{\pi}{2}, \frac{\pi}{2}) \rightarrow R$$ be given by
$$f(x) = (\log (\sec x + \tan x))^3.$$
Then
From a point $$P(\lambda, \lambda, \lambda)$$ perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that $$\angle QPR$$ is a right angle, then the possible value(s) of $$\lambda$$ is(are)
Let $$\overrightarrow{x}, \overrightarrow{y}$$ and $$\overrightarrow{z}$$ be three vectors each of magnitude $$\sqrt 2$$ and the angle between each pair of them is $$\frac{\pi}{3}.$$ If $$\overrightarrow{a}$$ is a nonzero vector perpendicular to $$\overrightarrow{x}$$ and $$\overrightarrow{y} \times \overrightarrow{z}$$ and $$\overrightarrow{b}$$ is a nonzero vector perpendicular to $$\overrightarrow{y}$$ and $$\overrightarrow{z} \times \overrightarrow{x}$$, then
A circle S passes through the point (0, 1) and is orthogonal to the circles $$(x - 1)^2 + y^2 = 16$$ and $$x^2 + y^2 = 1.$$ Then
Let M be a $$2 \times 2$$ symmetric matrix with integer entries. Then M is invertible if
