If \(f(x)=|x-1|+|x-2|+|x-3|, 2 < x < 3\), then \(f\) is

If the roots of the equation \(x^3-a x^2+b x-c=0\) are in GP, then \(\frac{b^3}{a^3}\)

Thecoefficientof \(x^3\) in the expansion of \((1-2 x)^{\frac{1}{2}}(1+3 x)^{\frac{1}{3}}\) is

For real number \(x\), if the minimum value of \(f(x)=x^2+2 b x+2 c^2\) is greater than the maximum value of \(g(x)=-x^2-2 c x+b^2\), then

$OABC$ is a tetrahedron. If $D,E$ are the midpoints of $OA$ and $BC$ respectively, then $\overrightarrow{DE}=$

If $PQRST$ is a pentagon, then the resultant of forces $\overline{PQ},\overline{PT},\overline{QR},\overline{SR},\overline{TS}$ and $\overline{PS}$ is

PQR is a right angled isosceles triangle with right angle at \(\mathrm{P}(2,1)\). If the equation of the line \(Q R\) is \(2 x+y=3\), then the equation representing the pair of lines \(P Q\) and \(P R\) is

The magnetic field at the centre of a current carrying circular coil of radius \(R\) is \(B_c\) and the magnetic field at a point on its axis at a distance \(R\) from its centre is \(B_a\). The value of \(\frac{B_c}{B_a}\) is

If \(3 x^2-7 x+2=0\) and \(15 x^2-11 x+a=0\) have a common root and \(a\) is a positive real number, then the sum of the roots of the equation \(15 x^2-a x+7=0\), is

In the arrangement shown in the figure, the coefficient of friction between two blocks is 0.5. The force of friction between the two blocks is (Assume that the \(4 \mathrm{~kg}\) block is placed on a smooth horizontal surface.) (Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\).)

In a \(\triangle A B C, D, E\) and \(F\) respectively are the points of contact of the incircle with the sides \(A B, B C\) and \(C A\) such that \(A D=\alpha, B E=\beta\) and \(C F=\gamma\), then \(\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma}=\)

A certain volume of a gas at 300 K expands adiabatically until its volume is doubled. The resultant fall in temperature of the gas is nearly (The ratio of the specific heats of the gas \(=1.5\) )

\(\mathrm{K}_{\mathrm{c}}\) for the following reaction is 99.0
\(\mathrm{A}_2(\mathrm{~g}) \stackrel{\mathrm{T}(\mathrm{~K})}{\rightleftharpoons} \mathrm{B}_2(\mathrm{~g})\)
In a one litre flask, 2 moles of \(\mathrm{A}_2\) was heated to \(\mathrm{T}(\mathrm{K})\) and the above equilibrium is reached. The concentrations at equilibrium of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_1\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_2\left(\mathrm{~B}_2\right)\) respectively. Now, one mole of \(\mathrm{A}_2\) was added to flask and heated to \(\mathrm{T}(\mathrm{K})\) to establish the equilibrium again. The concentrations of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_4\left(\mathrm{~B}_2\right)\) respectively. What is the value of \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) in \(\mathrm{mol}^{-1}\) ?