Real part of $(\cos 4+i\sin 4+1{)}^{2020}$ is_______

The set of values of \(x\) for which the inequalities \(x^2-3 x-10 < 0, \quad 10 x-x^2-16>0 \quad\) hold simultaneously, is

If the arithmetic mean of the following frequency table is 50 , then
\begin{array}{|c|c|}
\hline Class & Frequency \\
\hline 0-20 & 17 \\
20-40 & f_1 \\
40-60 & 32 \\
60-80 & f_2 \\
80-100 & 19 \\
\hline & \mathbf{1 2 0} \\
\hline
\end{array}

Which of the following functions are odd?
I. \(f(x)=x\left(\frac{e^x-1}{e^x+1}\right)\)
II. \(f(x)=k^x+k^{-x}+\cos x\)
III. \(f(x)=\log \left(x+\sqrt{x^2+1}\right)\)

In a \(\triangle A B C,(a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}\) is equal to

\(\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|\) is not equal to

If \(\alpha, \beta, \gamma\) are the roots of the equation \(\mathrm{x}^3+\mathrm{px}^2+\mathrm{qx}+\mathrm{r}=0\), then \(\alpha^3+\beta^3+\gamma^3=\)

After the coordinate axes are rotated through an angle \(\frac{\pi}{4}\) in the anti clockwise direction without shifting the origin, if the equation \(x^2+y^2-2 x-4 y-20=0\) transforms to \(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) in the new coordinate system, then \(\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=\)

A is a point on the circle with radius 8 and centre at \(O\). A particle P is moving on the circumference of the circle starting from A. M. is the foot of the perpendicular from \(P\) on \(O A\) and \(\angle \mathrm{POM}=\theta\). When \(O M=4\) and \(\frac{d \theta}{d t}=6\) radians \(/ \mathrm{sec}\), then the rate of change of PM is (in units/sec)

Two digits are selected at random from the digits 1 through 9. If their sum is even, then the probability that both are odd is

If \(3 x+2 \sqrt{2} y+k=0\) is a normal to the hyperbola \(4 x^2-9 y^2-36=0\) making positive intercepts on both the axes, then \(\mathrm{k}=\)

If \(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) are coplanar, then \(\lambda\) is equal to :

Names of units of some physical quantities are given in List-I and their dimensions formulae are given in List-Il. Match the correct pairs in the lists
\(\begin{array}{ll|ll}
\hline & \text{ List-I } & \text{ List-II } \\
\hline A. & \mathrm{Pa}-\mathrm{s} & (i) & {\left[\mathrm{L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]} \\
\hline B. & \mathrm{NmK}^{-1} & (ii) & {\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]} \\
\hline C. & \mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1} & (iii) & {\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]} \\
\hline D. & \mathrm{Wm}^{-1} \mathrm{~K}^{-1} & (iv) & {\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]} \\
\hline
\end{array}\)
\(\begin{array}{llll}A & B & C & D\end{array}\)