Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be a continuous function such that $f\left(0\right)=1$ and ${\int }_0^{\frac{\pi }{3}}f\left(t\right)dt=0$ Then which of the following statements is (are) TRUE?

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

A line L : $y=\mathit{\text{m}}x+3$ meets $y-\mathrm{a}\mathrm{x}\mathrm{\text{is}}$ at E(0,3) and the arc of the parabola $y^2=16x\text{,}0\le y\le 6$ at the point $F\left(x_0,y_0\right)$.The tangent to the parabola at $F\left(x_0,y_0\right)$ intersects the y-axis at $G\left(0,y_1\right)$.The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	m =	P.	$\frac{1}{2}$
B.	Maximum area of ΔEFG is	Q.	4
C.	y0​ =	R.	2
D.	y1 =	S.	1

Let \(f(x)=x^4+a x^3+b x^2+c\) be a polynomial with real coefficients such that \(f(1)=-9\). Suppose that \(i \sqrt{3}\) is a root of the equation \(4 x^3+3 a x^2+2 b x=0\), where \(i=\sqrt{-1}\). If \(\alpha_1, \alpha_2, \alpha_3\), and \(\alpha_4\) are all the roots of the equation \(f(x)=0\), then \(\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2\) is equal to ________.

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

Let \(f\) be a function defined on \(R\) (the set of all real numbers) such that \(f^{\prime}(x)=2010(x-2009)\) \((x-2010)^2(x-2011)^3(x-2012)^4\), for all \(x \in R\). If \(g\) is a function defined on \(R\) with values in the interval \((0, \infty)\) such that \(f(x)=\ln (g(x))\), for all \(x \in R\), then the number of points in \(R\) at which \(g\) has a local maximum 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 $D$ is

\(\mathrm{CuSO}_4\) decolourises on addition of \(\mathrm{KCN}\), the product is

A particle of mass $\text{0.2 kg}$ is moving in one dimension under a force that delivers a constant power $\text{0.5 W}$ to the particle. If the initial speed $\left({in\; ms}^{-1}\right)$ of the particle is zero, the speed $\left({in\; ms}^{-1}\right)$ after $\text{5 s}$ is

Consider a thin square sheet of side \(L\) and thickness \(t\), made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right angled triangular prism of refractive index $n=\sqrt{2}$ . Light undergoes total internal reflection in the prism at the face PR when $\alpha$ has a minimum value of ${45}^o.$ The angle $\theta$ of the prism is:

A thin spherical insulating shell of radius $\mathrm{R}$ carries a uniformly distributed charge such that the potential at its surface is ${\mathrm{V}}_0$ . A hole with a small area $\mathrm{\alpha }4\mathrm{\pi }{\mathrm{R}}^2\left(\mathrm{\alpha }\ll 1\right)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The empty space in this \(\mathrm{HCP}\) unit cell is