For every pair of continuous functions $f, g :\left[0, 1\right]\to R$ such that $\max \left\{f\left(x\right):x \in \left[0, 1\right]\right\}=\max \{g\left(x\right) :x \in \left[0, 1\right]\},$then the correct statement (s) is (are)

Consider three planes
\[
P_1: x-y+z=1, P_2: x+y-z=-1
\]
and \(P_3: x-3 y+3 z=2\)
Let \(L_1, L_2\) and \(L_3\) be the lines of intersection of the planes \(P_2\) and \(P_3, P_3\) and \(P_1, P_1\) and \(P_2\), respectively.
Statement 1 Atleast two of the lines \(L_1, L_2\) and \(L_3\) are non-parallel.
Statement 2 The three planes do not have a common point.

A factory has a total of three manufacturing units, \(M_1, M_2\), and \(M_3\), which produce bulbs independent of each other. The units \(M_1, M_2\), and \(M_3\) produce bulbs in the proportions of 2:2:1, respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_1, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_2\) is \(\frac{2}{5}\).
If a bulb is chosen randomly from the bulbs produced by \(M_3\), then the probability that it is defective is ____.

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

The locus of the orthocentre of the triangle formed by the lines \((1+p) x-p y+p(1+p)=0\), \((1+q) x-q y+q(1+q)=0\) and \(y=0\), where \(p \neq q\), is

Let \(z\) be a complex number such that the imaginary part of \(z\) is non-zero and \(a=z^{2}+z+1\) is real. Then a cannot take the value

Let \(a, b, c, p, q\) be real numbers.
Suppose, \(\alpha, \beta\) are the roots of the equation \(x^2+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^2+2 b x+c=0\), where \(\beta^2 \notin\{-1,0,1\}\).
Statement \(1\left(p^2-q\right)\left(b^2-a c\right) \geq 0\).
Statement \(2 b \neq p a\) or \(c \neq q a\).

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha \mathrm{a}\mathrm{n}\mathrm{d} \beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$ . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{p}$ is the linear momentum, $\overrightarrow{L}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

	LIST -I		LIST-II
A)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\beta \widehat{j}$	P)	$\overrightarrow{p}$
B)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\beta \sin \omega t\widehat{j}$	Q)	$\overrightarrow{L}$
C)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\sin \omega t\widehat{j}$	R)	$K$
D)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\frac{\beta }{2}{t}^2\widehat{j}$	S)	$U$
		T)	$E$

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2+3z+4z^2}{2-3z+4z^2}$ is a real number, then the value of ${\left|z\right|}^2$ is ______.

The compound (s) having peroxide linkage is(are) :

Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

The area of the region $\left\{\left(\mathrm{x},\mathrm{y}\right):\mathrm{x}\mathrm{y}\le 8,\mathrm{ }1\le \mathrm{y}\le {\mathrm{x}}^2\right\}$ is

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\)\(\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\). Find a least odd natural number \(n_0\), so that \(B_n>A_n, \forall n \geq n_0\).