If the angles \(A, B\) and \(C\) of a triangle are in an arithmetic progression and if \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively, then the value of the expression \(\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A\) is

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of ${\lambda }^2$ is

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2-\mathrm{x}-1=0,$ with $\mathrm{\alpha }>\mathrm{\beta }.$ For all positive integers $\mathrm{n},$ define
${\mathrm{a}}_{\mathrm{n}}=\frac{{\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}}{\mathrm{\alpha }-\mathrm{\beta }},\mathrm{n}\ge 1$
${\mathrm{b}}_1=1$ and ${\mathrm{b}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}+1},\mathrm{n}\ge 2.$
Then which of the following options is/are correct?

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\), where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is

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_______

Let \(f(x)=x^2\) and \(g(x)=\sin x\) for all \(x \in R\). Then, the set of all \(x\) satisfying \((\) fogogof \()(x)=(\operatorname{gogof})(x)\), where \((f \circ g)(x)=f(g(x))\) is

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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 ?

A single slit diffraction experiment is performed to determine the slit width using the equation, \(\frac{b d}{D}=m \lambda\), where \(b\) is the slit width, \(D\) the shortest distance between the slit and the screen, \(d\) the distance between the \(m^{\text {th }}\) diffraction maximum and the central maximum, and \(\lambda\) is the wavelength. \(D\) and \(d\) are measured with scales of least count of 1 cm and 1 mm, respectively. The values of \(\lambda\) and \(m\) are known precisely to be 600 nm and 3, respectively. The maximum absolute error (in \(\mu \mathrm{m}\)) in the value of \(b\) estimated using the diffraction maximum that occurs for \(m=3\) with \(d=5 \mathrm{~mm}\) and \(D=1 \mathrm{~m}\) is ______.

Let \(S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.\) and \(\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}\). If the area of the region \(S\) is \(\alpha \sqrt{2}\), then \(\alpha\) is equal to

For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant $\mathrm{K}$ in terms of change in entropy is described by:

If \(f(x)=\min \left\{1, x^2, x^3\right\}\), then

Assuming that Hund's rule is violated, the bond order and magnetic nature of the diatomic molecule \(\mathrm{B}_2\) is

The solubility of a salt of weak acid $\left(\mathrm{AB}\right)\mathrm{ }\mathrm{at}\mathrm{ }\mathrm{pH}3\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{Y}\times {10}^{-3}\mathrm{ }\mathrm{mol}\mathrm{ }{\mathrm{L}}^{-1}$ . The value of $\mathrm{Y}$ is ____. (Given that the value of solubility product of $\mathrm{AB}\left({\mathrm{K}}_{\mathrm{sp}}\right)=2\times {10}^{-10}\mathrm{ }$, and the value of ionization constant of $\mathrm{HB}\left(\mathrm{Ka}\right)=1\times {10}^{-8}$)