Define \(f: R \rightarrow R\) by \(f(x)=\max \{x+1,1-x, 2\}\). Then, \(f\) 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 \(\alpha+\beta=\)

The cosine of the angle \(A\) of the triangle with vertices \(A(1,-1,2), B(6,11,2), C(1,2,6)\) is

If $f\left(f\left(0\right)\right)=0$, where $f\left(x\right)=x^2+ax+b,b\ne 0$ then $a+b=$

\(\tanh (\log x)=\)

If three points \(A, B\) and \(C\) have position vectors \((1, x, 3),(3,4,7)\) and \((y,-2,-5)\) respectively and if they are collinear, then \((x, y)\) is

If the slope of one of the lines in the pair of lines \(8 x^2+a x y+y^2=0\) is thrice the slope of the second line, then \(a=\)

Let \(f: R \rightarrow R\) be defined by \(f(x)=2 x+3\). If \(\alpha\), \(\beta\) are the roots of the equation \(f\left(x^2\right)-2 f\left(\frac{x}{2}\right)-1=0\), then \(\alpha^2+\beta^2=\)

Identify the reactions in which dichromate acts as an oxidising reagent.
I. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+6 \mathrm{Fe}^{2+}+14 \mathrm{H}^{+} \longrightarrow 2 \mathrm{Cr}^{3+}+6 \mathrm{Fe}^{3+}+7 \mathrm{H}_2 \mathrm{O}\)
II. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+2 \mathrm{OH}^{-} \longrightarrow 2 \mathrm{CrO}_4^{2-}+\mathrm{H}_2 \mathrm{O}\)
III. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+6 \mathrm{I}^{-}+14 \mathrm{H}^{+} \longrightarrow 2 \mathrm{Cr}^{3+}+3 \mathrm{I}_2+7 \mathrm{H}_2 \mathrm{O}\)
IV. \(\mathrm{Na}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{KCl} \longrightarrow \mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{NaCl}\)

A body is projected into vertical \(\mathrm{X}\)-Y plane with \(\mathrm{X}\)-axis along horizontal and Y-axis along vertical with an initial velocity \((10 \hat{i}+p \hat{j})\). If the maximum height reached by the body is \(50 \%\) of its range, then the value of \(\mathrm{p}\) is

The locus of the centers of the circles that are passing through the intersection of the circles \(x^2+y^2=1\) and \(x^2+y^2-2 x+y=0\) is

If \(a x^2+2 h x y-2 a y^2+3 x+15 y-9=0\) represents a pair of lines intersecting at \((1,1)\), then ah \(=\)