Let \(x \in R\) and \(\log _2 x>0\). Then, the vectors \(\mathbf{A}=\left(2, \log _2 x, s\right)\) and \(\mathbf{B}=\left(\log _2 x, s, \log _2 x\right)\) include an acute angle if

The area of the region (in sq. units) enclosed between the curves \(y=|x|, y=[x]\) and the ordinates \(x=-1, x=0, x=1\) is

The quadratic equation whose roots are the coordinates of the circumcentre of the triangle formed by the points \((-2,-1),(6,-1)(2,5)\) is

The value of ' \(k\) ' for which the equation \(x^2-4 x y-y^2+6 x+2 y+k=0\) represents a pair of straight lines is equal to ........

Let \(C\) be the centre of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) and \(P\) be a point on it. If the tangent at \(P\) to the hyperbola meets the straight lines \(b x-a y=0\) and \(b x+a y=0\) respectively in \(Q\) and \(R\), then \(C Q . C R=\)

Variance of the following discrete frequency distribution is

Class Interval	0–2	2–4	4–6	6–8	8–10
Frequency	2	3	5	3	2

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

If \(\Delta\) is the area of the triangle formed by the positive \(x\)-axis and the normal and tangent to the circle \(x^2+y^2=4\) at \((1, \sqrt{3})\), then \(\Delta\) is equal to

\(\mathrm{XeF}_2\) is hydrolyzed in the presence of small amount of water. What are the gaseous products formed?

The solution of the differential equation \(\frac{d y}{d x}-2 y \tan 2 x=e^x \sec 2 x\) is

The distance between pair of parallel lines represented by
\[
x^2+2 x y+y^2-8 a x-8 a y-9 a^2=0
\]
is. units

If the line joining \(A(1,3,4)\) and \(B\) is divided by the point \((-2,3,5)\) in the ratio \(1: 3\), then \(B\) is

Gypsum is added to clinker during cement manufacture to