Consider the following statements Statement I If \(a_0+\frac{a_1}{2}+\frac{a_2}{3}+\ldots .+\frac{a_n}{n+1}=0\), where \(a_0, a_1, \ldots ., a_n\) are real numbers, then the polynomial \(a_0+a_1 x+a_2 x^2+\ldots .+a_n x^n\) has a zero in the interval \((0,1)\).Statement II If \(f:[a, b] \rightarrow \mathbf{R}\) is continuous on \([a, b]\) and \(f\) is differentiable in \((a, b)\), where \(a>0\) and if \(\frac{f(a)}{a}=\frac{f(b)}{b}\), then there exists \(c \in(a, b)\), such that \(c f^{\prime}(c)=f(c)\). Which one of the following options is true?

If \(\frac{1-10 i \cos \theta}{1-10 \sqrt{3} i \sin \theta}\) is purely real, then one of the values of \(\theta\) is

If \(\cot \left(\frac{A}{2}\right)=\sqrt{\frac{1+a}{1-a}} \cdot \cot \left(\frac{\theta}{2}\right)\), then \(\cos \theta=\)

\[ \begin{aligned} & \left.\sin ^2 \beta^{\circ}\right)+\sin ^2\left(6^{\circ}\right)+\sin ^2\left(9^{\circ}\right)+\ldots+\sin ^2\left(84^{\circ}\right) \ & +\sin ^2\left(87^{\circ}\right)+\sin ^2\left(90^{\circ}\right)= \end{aligned} \]

The point \(P(3,2)\) undergoes the following transformations successively (i) Reflection about the line \(y=x\) (ii) Translation to a distance of 3 units in the positive direction of \(X\)-axis (iii) Rotation through an angle \(\frac{\pi}{4}\) about the origin in the counter-cloclswise direction Then, the final position of that point is

If \(\mathbb{R}-(\alpha, \beta)\) is the range of \(\frac{x+3}{(x-1)(x+2)}\), then the sum of the intercepts of the line \(\alpha x+\beta y+1=0\) on the coordinate axes is

If \(f(x)=\frac{x-1}{e^x}\), then \(f^{\prime}(0)+f^{\prime \prime}(0)=\)

\[ \begin{aligned} & \text { Given } \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{H}_2 \mathrm{O}(l) ; \Delta \mathrm{H}=-285 \mathrm{~kJ} \ & \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(l) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-76.6 \mathrm{~kJ} \ & \mathrm{~N}_2(\mathrm{~g})+3 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-348.2 \mathrm{~kJ} \ & \text { Calculate the } \Delta \mathrm{H} \text { of } 2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \end{aligned} \] Calculate the \(\Delta \mathrm{H}\) of \(2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})\).

The number of moles of \(\mathrm{H}_2\), which is required to produce 10 moles of \(\mathrm{NH}_3\) in the following reaction is \(a \mathrm{H}_2(g)+b \mathrm{NO}_2(g) \longrightarrow c \mathrm{NH}_3(g)+d \mathrm{H}_2 \mathrm{O}(g)\)

What will be the minimum speed of the roller-coaster so that the passenger at the top, when becomes upside down, does not fall out? Consider the acceleration due to gravity as $10 \mathrm{m} {\mathrm{s}}^{-2},$ and the radius of curvature of the roller coaster is $10 \mathrm{m}.$

If \(\alpha\) is a multiple root of the equation \(x^5-6 x^4+11 x^3-\) \(2 x^2-12 x+8=0\) then \(3 \alpha^2-2 \alpha+1=\)

A random experiment is conducted five times. If the number of successes of the experiment follows binomial distribution such that the difference of mean and variance of the successes is \(\frac{5}{9}\), then the probability of getting atmost two successes is

A body mass of \(1 \mathrm{~kg}\), initially at rest explodes and breaks into three parts. The masses of the parts are in the ratio \(1: 1: 3\). The two pieces of equal mass fly off perpendicular to each other with a speed of \(30 \mathrm{~m} / \mathrm{s}\) each. The velocity of the heavier part in \(\mathrm{m} / \mathrm{s}\) is