The locus of the point which is equidistant from the point \((1,1)\) and the line \(\mathrm{x}+\mathrm{y}+1=0\) is

If the \(2^{\text {nd }}, 3^{\text {rd }}\) and \(4^{\text {th }}\) terms in the expansion of \((x+a)^{\mathrm{n}}\) are \(96,216,216\) respectively and \(n\) is a positive integer then \(a+x=\)

Number of unit vectors of the form $a\hat{\mathrm{i}}+b\hat{\mathrm{j}}+c\hat{\mathrm{k}},$ where $a, b, c\in W$ is

If \(f(x)=k x^3-9 x^2+9 x+3(k>0)\) is increasing for all \(x\), then

The equation \(6 x^4-5 x^3+13 x^2-5 x+6=0\) will have

If the circle \(x^2+y^2+8 x-4 y+c=0\) touches the circle \(x^2+y^2+2 x+4 y-11=0\) externally and cuts the circle \(x^2+y^2-6 x+8 y+k=0\) orthogonally, then \(\mathrm{k}=\)

Suppose \(P\) and \(Q\) lie on \(3 x+4 y-4=0\) and \(5 x-y-4=0\) respectively. If the mid-point of \(P Q\) is \((1,5)\), then the slope of the line passing through \(P\) and \(Q\) is

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the number of points where the function \(y=[x]+|1-x|,-1 \leq x \leq 3\) is not differentiable, is

The radius of an atomic nucleus of mass number $64$ is $4.8$ fermi. Then the mass number of another atomic nucleus of radius $6$ fermi is

Match the following lists.
\begin{array}{|c|c|c|c|}
\hline & List I & & List II \\
\hline A & \begin{array}{l}\text { Zeroth law of } \\
\text { thermodynamics }\end{array} & 1 & \begin{array}{l}\text { Direction of flow of } \\
\text { heat }\end{array} \\
\hline B & \begin{array}{l}\text { First law of } \\
\text { thermodynamics }\end{array} & \| & Work done is zero \\
\hline\underline{\mathrm{C}} & Free expansion of a gas & III & Thermal equilibrium \\
\hline D & \begin{array}{l}\text { Second law of } \\
\text { Thermodynamics }\end{array} & IV & \begin{array}{l}\text { Law of conservation } \\
\text { of energy }\end{array} \\
\hline
\end{array}
The correct answer is
\(\begin{array}{lccccccccc} & \text { A } & \text { B } & \text { C } & \text { D } \end{array}\)

Particles and their anti-particles have

The distance between the lines represented by \(x^2+2 x y+y^2-8 m x-8 m y-9 m^2=0\) is

If \(\alpha\) and \(\beta\) are the roots of the equation \(2 x^2+6 x+k=0\), then the maximum value of \(\left[\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right]\) when \(k < 0\) is