Let \(\mathrm{A}(2,3), \mathrm{B}(3,-1)\) and \(\mathrm{C}(-3,2)\) be three points. If the centre of the circle passing through \(A, B\) and \(C\) is \((h, k)\), then \(2 k-4 h=\)

The direction cosines of two lines are connected by the relations \(l+m-n=0\) and \(l m-2 m n+n l=0\). If \(\theta\) is the acute angle between those lines then \(\cos \theta=\)

\[
\int \frac{\sqrt{x^2+1}\left[\log \left(x^2+1\right)-2 \log x\right]}{x^4} d x=
\]

The direction cosines of two lines are \(\left\langle\frac{\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}\right\rangle\) and \(\left\langle\frac{-\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}\right\rangle\). Then the angle between the lines is equal to

If \(f: R \rightarrow R\) is defined by
\(\begin{aligned}
& f(x)=\left\{\begin{array}{ccc}
\frac{x-2}{x^2-3 x+2} & \text { if } & x \in R-\{1,2\} \\
2 & \text { if } & x=1 \\
1 & \text { if } & x=2
\end{array}\right. \text { then } \\
& \lim _{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}=
\end{aligned}\)

\(\int_0^1 x \operatorname{Sin}^{-1} x d x=\)

If two angles of \(\triangle A B C\) are \(45^{\circ}\) and \(60^{\circ}\), then the ratio of the smallest and the greatest sides are

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

The gravitational potential energy of a system of three masscs \(\mathrm{m}, 2 \mathrm{~m}\) and \(3 \mathrm{~m}\) pläced at three vertices of equilateral triangle of side ' \(a\) ' is

The point of intersection of the lines represented by \(\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})\) and \(\mathbf{r}=(-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})\) is

If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{b}}=+\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and the orthogonal projection of \(\overrightarrow{\mathbf{b}}\) on \(\overrightarrow{\mathbf{a}}\) is \(\frac{4}{3}(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})\), then \(\lambda\) is equal to

\(\lim _{x \rightarrow 2} \frac{\sqrt{1+4 x}-\sqrt{3+3 x}}{x^3-8}=\)

If \(\frac{x^2+x+1}{x^2+2 x+1}=A+\frac{B}{x+1}+\frac{C}{(x+1)^2}\), then \(A-B\) is equal to