The circle \(x^2+y^2-6 x-10 y+p=0\) neither intersects nor touch the coordinate axes and the point \((1,4)\) lies inside the circle. Then the range of possible values of ' \(p\) ' is

In a \(\triangle A B C\) (shown in the figure below). State whether the following are true or false


(i) \(\mathbf{A B}+\mathbf{B C}+\mathbf{C A}=\mathbf{0}\)
(ii) \(\mathbf{A B}+\mathbf{B C}-\mathbf{A C}=\mathbf{0}\)
(iii) \(\mathbf{A B}-\mathbf{C B}+\mathbf{C A}=\mathbf{0}\)
(iv) \(\mathbf{A B}+\mathbf{B C}-\mathbf{C A}=\mathbf{0}\)

\(\lim _{x \rightarrow 0} x^2 \sin \frac{\pi}{x}\) is equal to

If the function \(f(x)=\sin x-\cos ^2 x\) is defined on the interval \([-\pi, \pi]\), then \(f\) is strictly increasing in the interval

If \(\theta\) is the angle between the tangents from \((-1,0)\) to the circle \(x^2+y^2-5 x+4 y-2=0\), then \(\theta\) is equal to

\(\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}\) is equal to

The locus of the point of intersection of the tangents at the
endpoints of normal chords of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is

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

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 \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

Identify \(X, Y\) and \(Z\) in the following reaction.
\[
\begin{array}{ccc}
& 2 \mathrm{CH}_3 \mathrm{Cl}+X \underset{570 \mathrm{~K}}{\longrightarrow} \\
X & Y & Z
\end{array}
\]

\(X O Z\)-plane divides the join of \((2,3,1)\) and \((6,7,1)\) in the ratio:

\(\int \frac{x-1}{(x+1) \sqrt{x^3+x^2+x}} d x=\)

If the function \(f(x)=x^3+2 p x^2+27 x+16\) is strictly increasing for all \(x \in R\), then the range of \(p\) is