\(\frac{\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 89^{\circ}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\ldots+\cos 44^{\circ}\right)+1}=\)

For the curve \(y=5 x-2 x^3\), if \(x\) increases at the rate of 2 units/sec, the rate of change in the slope of the curve at \(x=3\) is ......../sec

\(\begin{aligned} & \sin ^4 \frac{\pi}{8}+\sin ^4 \frac{2 \pi}{8}+\sin ^4 \frac{3 \pi}{8}+\sin ^4 \frac{4 \pi}{8} \\ & +\sin ^4 \frac{5 \pi}{8}+\sin ^4 \frac{6 \pi}{8}+\sin ^4 \frac{7 \pi}{8}=\end{aligned}\)

If \(y=x \operatorname{Tan}^{-1}\left(\frac{x}{y}\right)\), then \(\frac{d y}{d x}=\)

If \(\cos x+\sin x=\frac{1}{2}\) and \(0 < x < \pi\), then \(\tan x=\)

The area of the parallelogram, whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k},$ is equal to

Air is discharging from a large spherical balloon at the rate of 4 cubic meters per minute. Then, the rate at which the surface area is shrinking when the radius of the balloon is \(8 \mathrm{~m}\), is

Which of the following reactions are not feasible?
I. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CH}_2 \frac{\mathrm{HCl}}{\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{CO}\right)_2 \mathrm{O}_2} \longrightarrow \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl}\)
II. \(\mathrm{C}_6 \mathrm{H}_6+\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl} \stackrel{\mathrm{AlCl}_3}{\longrightarrow} \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}\left(\mathrm{CH}_3\right)_2\)
III. \(\mathrm{CH}_3 \mathrm{CHClCH}_2 \mathrm{Cl} \underset{\Delta}{\stackrel{\mathrm{KOH}}{\longrightarrow}} \mathrm{CH}_3 \mathrm{C} \equiv \stackrel{+}{\mathrm{CH}}\)
IV. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CHCH}_3 \stackrel{\mathrm{KMnO}_4 / \mathrm{H}^{+}}{\longrightarrow} \mathrm{CH}_3 \mathrm{COOH}\)

The enamel present on teeth becomes much harder due to the conversion of \(\left[3 \mathrm{Ca}_3\left(\mathrm{PO}_4\right)_2, \mathrm{X}\right]\) into \(\left[3 \mathrm{Ca}_3\left(\mathrm{PO}_4\right)_2, \mathrm{Y}\right]\). What are X and Y ?

An element (atomic weight \(=250 \mathrm{u}\) ) crystallises in a simple cubic lattice. If the density of the unit cell is \(7.2 \mathrm{~g} \mathrm{~cm}^{-3}\), what is the radius (in \(Å\) ) of the atom of the element?
\(\left(\mathrm{N}=6.02 \times 10^{23} \mathrm{~mol}^{-1}\right)\)

The value of
\(\left\{x \in \mathbb{R} \mid\left[\log (1.6)^{1-x^2}-(0.625)^{6(1+x)}\right] \in \mathbb{R}\right\}\) is

If \(Q\) is the inverse of \(A\), when \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right]\) and \(10 \times Q=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & x \\ 1 & -2 & 3\end{array}\right]\), find \(x=\)

\(\tan x+\frac{\cos x}{1+\sin x}=\)