If the solution of $\frac{dy}{dx}-y{\log }_{e}0.5=0, y\left(0\right)=1$, and $y\left(x\right)\to k$, as $x\to \infty$ then $k=$

If \(\vec{a}=t \vec{b}\) where \(t < 0\) is a scalar, then

A unit vector coplanar with \(\mathbf{i}+\mathbf{j}+3 \mathbf{k}\) and \(\mathbf{i}+3 \mathbf{j}+\mathbf{k}\) and perpendicular to \(\mathbf{i}+\mathbf{j}+\mathbf{k}\) is

If the projections of the line segment $\overrightarrow{AB}$ on $XY, YZ$ and $ZX$ planes are $\sqrt{15},\sqrt{46},7$ respectively, then the projection of $\overrightarrow{AB}$ on $y-$axis is

If the function $f\left(x\right)$, defined below, is continuous on the interval $[0,8]$, then $f\left(x\right)=\left\{\begin{array}{l}{x}^2+ax+b,0\le x<2\\ 3x+2,2\le x\le 4\\ 2ax+5b,4<x\le 8\end{array}\right.$

If $\int \frac{(x-1{)}^2}{{\left(x^2+1\right)}^2}dx={\tan }^{-1}\left(x\right)+g\left(x\right)+k$, then $g\left(x\right)$ is equal to

The values of a function \(f(x)\) at different values of \(x\) are as follows


Then, the approximate area (in square units) bounded by the curve \(y=f(x)\) and \(x\)-axis between \(x=0\) and 5 , using the Trapezoidal rule, is

If one root of the equation \(a x^3+b x+c=0\) is twice another root, then

Using the standard electrode potentials given below identify the correct statements from the following.
\[
\begin{aligned}
\mathrm{Fe}^{2+}+2 e^{-} \longrightarrow \mathrm{Fe} ; E^{\circ} & =-0.44 \mathrm{~V} \\
\mathrm{Cu}^{2+}+2 e^{-} \longrightarrow \mathrm{Cu} ; E^{\circ} & =+0.34 \mathrm{~V} \\
\mathrm{Ag}^{+}+e^{-} \longrightarrow \mathrm{Ag} ; E^{\circ} & =+0.80 \mathrm{~V}
\end{aligned}
\]
(i) Copper can displace iron from \(\mathrm{FeSO}_4\) solution.
(ii) Iron can displace copper from \(\mathrm{CuSO}_4\) solution.
(iii) Silver can displace copper from \(\mathrm{CuSO}_4\) solution.
(iv) Iron can displace silver from \(\mathrm{AgNO}_3\) solution.

If \(\alpha, \beta\) are acute angles such that \(\sin \beta=2 \sin \alpha\) and \(3 \cos \beta=2 \cos \alpha\), then \(\sec (\alpha+\beta)=\)

Let \(\pi_1\) be the plane determined by the vectors \(\hat{i}+\hat{j}\) and \(\hat{j}+\hat{k}, \pi_2\) be the plane determined by the vectors \(\hat{i}-\hat{j}\) and \(\hat{i}+\hat{j}-\hat{k}\). Let \(\vec{a}\) be a vector parallel to the line of intersection of \(\pi_1\) and \(\pi_2\). If \(|\vec{a}|=\sqrt{14}\), then \(|\vec{a} \cdot(\hat{i}+\hat{j}+\hat{k})|=\)

In the expansion of \((1+x)^n\) the coefficients of \(p\) th and \((p+1)\) th terms are respectively \(p\) and \(q\), then \(p+q\) is equal to

If \(I_n=\int(\log x)^n d x\), then \(I_n+n I_{n-1}=\)