If \(|\mathbf{a}|=3,|\mathbf{b}|=4\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(120^{\circ}\), then \(|4 \mathbf{a}+3 \mathbf{b}|\) is equal to

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)\) is a normal to the circle \((x+1)^2+(y+1)^2=1\), then \(a^2\) lies in the interval

If the equation of the circle whose radius is 3 units and which touches internally the circle \(x^2+y^2-4 x-6 y-12\) \(=0\) at the point \((-1,-1)\) is \(x^2+y^2+p x+q y+r=0\), then \(p+q-r=\)

If \(f(a)=\log \left|\frac{1-a}{1+a}\right|\) for \(a \neq\{-1,1\}\), then the set of values of all ' \(a\) ', for which \(f\left(\frac{2 a}{1+a^2}\right)>0\) is

\(\sum_{k=1}^n k(k+1)(k+2) \ldots(k+r-1)=\)

If the vertices of $\Delta \mathrm{ABC}$ are $A=(2,3,5), B=(-1,3,2), C=(3,5,-2),$ then the area of the $\Delta \mathrm{ABC}$ (in sq. units) is

If $i{z}^3+z^2-z+i=0$ (where $z$ is a complex number), then the value of $\left|z\right|$ is

From the following data
\[
\begin{aligned}
& \mathrm{CH}_3 \mathrm{OH}(l)+\frac{3}{2} \mathrm{O}_2(g) \longrightarrow \mathrm{CO}_2(g)+2 \mathrm{H}_2 \mathrm{O}(l) \\
& \Delta_r H^{\circ}=-726 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
& \mathrm{H}_2(g)+\frac{1}{2} \mathrm{O}_2(g) \longrightarrow \mathrm{H}_2 \mathrm{O}(l) \text {; } \\
& \Delta_r \mathrm{H}^{\circ}=-286 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
& \mathrm{C}(\text { graphite })+\mathrm{O}_2(\mathrm{~g}) \longrightarrow \mathrm{CO}_2(\mathrm{~g}) \text {; } \\
& \Delta_r \mathrm{H}^{\circ}=-393 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
&
\end{aligned}
\]
The standard enthalpy of formation of \(\mathrm{CH}_3 \mathrm{OH}(l)\) in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is

If \(m\) and \(M\) respectively denote the minimum and maximum of \(f(x)=(x-1)^2+3\) for \(x \in[-3,1]\), then the ordered pair \((m, M)\) is equal to

If \(m_1\) and \(m_2\) are the roots of the equation
\(x^2+(\sqrt{3}+2) x+(\sqrt{3}-1)=0,\)
then the area of the triangle formed by the lines \(y=m_1 x, y=m_2 x\) and \(y=c\)

Let $a,b,c$ be such that $(b+c)\ne 0$ and

$\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|$$+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ {\left(-1\right)}^{n+2}a& {\left(-1\right)}^{n-1}b& {\left(-1\right)}^{n}c\end{array}\right|=0$

Then the value of $n$ is

If \(x\) is real and \(\alpha, \beta\) are maximum and minimum values of \(\frac{x^2-x+1}{x^2+x+1}\) respectively then \(\alpha+\beta=\)

\(\int \frac{5 \tan x}{(\tan x)-2} d x=\alpha x+\beta \log |\sin x-2 \cos x|\) \(+\gamma\), then \(\alpha-\beta=\)