The sum of the coefficients in the expansion of \(\left(1+x+x^2\right)^n\) is

The order and the degree of the differential equation \(y=p x+\sqrt{a^2 p^2+b^2}\), ( where \(p=\frac{d y}{d x}\) ) are respectively .

The quadrilateral formed by the pairs of line \(x y+x+y+1=0, x y+3 x+3 y+9=0\) is

If \(\frac{x^2}{a}+\frac{2 x y}{h}+\frac{y^2}{b}=0\) represents a pair of straight lines such that the slope of one of the lines is twice the other, then \(\frac{a b}{h^2}=\)

If \(x^{2019} \cdot y^{2020}=(x+y)^{4039}\), then \(\frac{d y}{d x}=\)

If \(f: R \rightarrow R\) is defined by \(f(x)=[2 x]-2[x]\) for \(x \in R\), then the range of \(f\) is (Here \([x]\) denotes the greatest integer not exceeding \(x\) )

The orthogonal projection vector of \(\vec{a}=2 \hat{i}+3 \hat{j}+3 \hat{k}\) on \(\vec{b}=\hat{i}-2 \hat{j}+\hat{k}\) is

For the matrix \(A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right] \cdot A^{-1}=\)

For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{\mathbf{i}}+x \hat{\mathbf{j}}+x \hat{\mathbf{k}}, x \hat{\mathbf{i}}+(x-v) \hat{\mathbf{j}}+x \hat{\mathbf{k}}\),
\(x \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(x-w) \hat{\mathbf{k}}\) and
\((x-1) \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}+(x-1) \hat{\mathbf{k}}\) are coplanar, then

The abscissae of two points \(P, Q\) are the roots of the equation \(2 x^2+4 x-7=0\) and their ordinates are the roots of the equation \(3 x^2-12 x-1=0\). Then the centre of the circle with \(P Q\) as a diameter is

If \(\int \frac{\cos (13 x)-\cos (14 x)}{1+2 \cos (9 x)} d x=\frac{\sin (4 x)}{a}-\frac{\sin (5 x)}{b}\) \(+c\), then \(a^b=\)

The standard reduction potentials of \(\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Ag}\) are respectively \(-0.76,0.34\) and \(0.8 \mathrm{~V}\). The following cells were constructed :
(1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}\)
(2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}\)
(3) \(\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}\)
What is the correct order of \(E_{\text {cell }}^{\circ}\) of these cells?

If \(\log y\) is an integrating factor of \(\frac{d x}{d y}+P(y) x=Q(y)\), then \(\mathrm{P}(\mathrm{y})=\)