Paragraph:
Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Let \(g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t\). Which of the following is true ?

Paragraph:
Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(\operatorname{det}(A)\) is not divisible by \(p\), is

Consider the system of equations
where \(\quad a, b, c, d \in\{0,1\}\)
Statement 1 The probability that the system of equations has a unique solution, is \(3 / 8\).
Statement 2 The probability that the system of equations has a solution, is 1 .

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow(0,4)\) be functions defined by \(f(x)=\log _e\left(x^2+2 x+4\right)\), and \(g(x)=\frac{4}{1+e^{-2 x}}\)
Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\).
Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is ________ .

A straight line \(L\) through the point \((3,-2)\) is inclined at an angle \(60^{\circ}\) to the line \(\sqrt{3} x+y=1\). If \(L\) also intersects the \(X\)-axis, then the equation of \(L\) is

Let RS be the diameter of the circle $x^2+y^2=1$ where, $S$ is the point$(1, 0)$. Let $P$be a variable point (other than $R \& S$) on the circle and tangents to the circle at $S \& P$meet at the point$Q$. The normal to the circle at P intersects a line drawn through$Q$ parallel to $RS$ at point $E$ . Then, the locus of $E$ passes through the point (s):

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function such that its derivative $f^{\text{'}}$ is continuous and $f\left(\pi \right)=-6$. If $F:\left[0,\pi \right]\to \mathrm{ℝ}$ is defined by $F\left(x\right)={\int }_0^{x}f\left(t\right)dt,$ and if ${\int }_0^{\pi }\left(f^{\text{'}}\left(x\right)+F\left(x\right)\right)\cos x dx=2$, then the value of $f\left(0\right)$ is_______

The colour of the $X_2$ molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to -

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

Paragraph:

Let \(P Q\) be a focal chord of the parabola \(y^{2}=4 a x\). The tangents to the parabola at \(P\) and \(Q\) meet at a point lying on the line \(y=2 x+a, a>0\).


Question:

Length of chord \(P Q\) is

Paragraph :
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

The correct statement(s) pertaining to the adsorption of a gas on a solid surface is(are)