If A \((1,0,2), \mathrm{B}(2,1,0), \mathrm{C}(2,-5,3), \mathrm{D}(0,3,2)\) are four points and the point of intersection of the lines AB and CD is \(\mathrm{P}(a, b, c)\), then \(a+b+c=\)

If the position vectors of the vertices \(A, B\) and \(C\) of \(\triangle A B C\) are \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}},-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) respectively, then \(\angle B=\)

The length of the intercept on the line $4x-3y-10=0$ by the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}-20=0$ is

The equation of the circle passing through the points of intersection of the circles \(x^2+y^2-2 p x=0\) and \(x^2+y^2-2 q y=0\) and having its centre on \(\frac{x}{p}-\frac{y}{q}=2\), is

The transformed equation of \(x^2-y^2+2 x+4 y=0\) when the origin is shifted to the point \((-1,2)\) is

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets
\(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is
\(\begin{array}{llll} & \text { List-I } & & \text { List-II } \\ \text { (i) } & P & \text { (A) } & (0,0) \\ \text { (ii) } & Q & \text { (B) } & (6,0) \\ \text { (iii) } & R & \text { (C) } & (-2,1) \\ \text { (iv) } & S & \text { (D) }(-6,0) \\ & & \text { (E) }(-6,-3) \\ & & \text { (F) }(-6,3)\end{array}\)
(i)
(ii)
(iii)
(iv)

. If \(\alpha, \beta\) are the roots of the equation \(x^2-2 x+4=0\), then \(\alpha^n+\beta^n=\ldots \ldots\) \(x \cos \left(\frac{n \pi}{3}\right)\) for any \(n \in \mathbf{N}\)

If \(\int \frac{x^4+1}{x^2+1} d x=A x^3+B x^2+C x+D \operatorname{Tan}^{-1} x+E\), then \(A+B+C+D=\)

Among the following the correct statements are
I. \(\mathrm{LiH}, \mathrm{BeH}_2\) and \(\mathrm{MgH}_2\) are saline hydrides with significant covalent character
II. Saline hydrides are volatile
III. Electron - precise hydrides are Lewis bases
IV. The formula for chromium hydride is \(\mathrm{CrH}\) The correct option is

If \(\alpha\) is the maximum value and \(\beta\) is the minimum value of \(\cos ^2 \frac{x}{4}+\sin \frac{x}{4}\), \(x \in R\), then \(\alpha-\beta=\)

A large charged plane having surface charge density \(4.9 \times 10^{-6} \mathrm{C} \mathrm{m}^{-2}\) lies in the \(\mathrm{x}-\mathrm{y}\) plane. A circular plane of radius of \(1 \mathrm{~cm}\) is lying completely in the region where \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) coordinates are all positive. When the plane's normal makes an angle \(60^{\circ}\) with the z-axis, the electric flux through the circular plane is \(\left(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right)\)

If \(x-4=0\) is the radical axis of two orthogonal circles out of which one is \(x^2+y^2=36\), then the centre of the other circles is

If the equation of the plane passing through the point \((2,-1,3)\) and perpendicular to each of the planes \(3 x-2 y+z=8\) and \(x+y+z=6\) is \(l x+m y+n z=1\), then \(4 m+2 n-31=\)