List of top Physics Questions on distance between two points

If \(\alpha, \beta\) are the roots of the equation \(x^{2}-px+q=0\) and \(\alpha>0\), \(\beta>0\), then \[ \alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=(p+6\sqrt{q}+4q^{\frac{1}{4}}\sqrt{p+2\sqrt{q}})^{\kappa}, \] where \(\kappa\) is:
The parabola \(y=4-x^{2}\) has vertex \(P\). It intersects the \(x\)-axis at \(A\) and \(B\). If the parabola is translated from its initial position to a new position by moving its vertex along the line \(y=x+4\), so that it intersects the \(x\)-axis at \(B\) and \(C\), then the abscissa of \(C\) will be:
Let \[ \vec{r}=\sin x(\vec{a}\times\vec{b})+\cos y(\vec{b}\times\vec{c})+2(\vec{c}\times\vec{a}), \] where \(\vec{a},\vec{b}\) and \(\vec{c}\) are three non-coplanar vectors. It is given that \(\vec{r}\) is perpendicular to \((\vec{a}+\vec{b}+\vec{c})\). Then the possible value(s) of \((x^{2}+y^{2})\) is/are:
If \[ f(x)=x(1331x^{2}-3630x+3300), \] then for \[ a=\cos^{2}\left(\tan^{-1}\left(\sin(\cot^{-1}3)\right)\right): \]
The ends $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $OX, OY$ respectively. If the rectangle $OAPB$ is completed, then the locus of the foot of the perpendicular drawn from $P$ to $AB$ is:
A figure is bounded by the curves \(y=x^{2}+1\), \(y=0\), \(x=0\) and \(x=1\). The point at which a tangent should be drawn to the curve \(y=x^{2}+1\) for it to cut off a trapezium of the greatest area from the figure is:
Consider a square $ABCD$ of diagonal length $2a$. The square is folded along the diagonal $AC$ so that the plane of $\Delta ABC$ is perpendicular to the plane of $\Delta ADC$. In this case the shortest distance between $AB$ and $CD$ is:
Let all the points on the curve \[ x^{2}+y^{2}-10x=0 \] are reflected about the line \(y=x+3\). If the locus of the reflected points is in the form \[ x^{2}+y^{2}+gx+fy+c=0, \] then the value of \((g+f+c)\) is:
Consider the following ellipse: \[ \frac{x^{2}}{f(K^{2}+2K+5)}+\frac{y^{2}}{f(K+11)}=1, \] where \(f(x)\) is a positive decreasing function. Then the value (values) of \(K\) for which the major axis coincides with x-axis is:
Intercepts of the plane \(\vec{r}\cdot\vec{n}=d \ (\ne0)\) on the coordinate axes respectively are:
The minimum length of intercept on any tangent to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) cut by the circle \(x^{2}+y^{2}=25\) is:
If \[ f(x)=\frac{1+x}{1-x} \] and \(A\) is a matrix such that \(A^{3}=0\), then \[ f(A)= \]
\(t_n\) denotes the nth term of an A.P. and \[ t_p=\frac{1}{q}, \qquad t_q=\frac{1}{p}. \] Then which one of the following options is a root of the equation \[ (p+2q-3r)x^{2}+(q+2r-3p)x+(r+2p-3q)=0? \]
\(\theta\) elimination from the equations \[ x^{2}+y^{2}=\frac{x\cos3\theta+y\sin3\theta}{\cos^{3}\theta} =\frac{y\cos3\theta-x\sin3\theta}{\sin^{3}\theta} \] will be:
The expression \[ \sum_{K=1}^{32}(3K+2)\left\{\sum_{r=1}^{10}\left(\sin\frac{2r\pi}{11}-i\cos\frac{2r\pi}{11}\right)\right\}^{K} \] represents:
A vector given by \[ \vec{P}=f(t)\hat{i}+g(t)\hat{j}+\hat{k} \] moves in such a way that it is always parallel to the vector \[ \vec{Q}=-f^{\prime\prime}(t)\hat{i}+f^{\prime}(t)\hat{j}+\hat{k}. \] The magnitude of \(\vec{P}\) is:
Consider a function \(f(x)\) which has exactly two roots at \(x=a\). If \[ \lim_{x\rightarrow a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m \ (\ne0), \] then the value of \(\lambda\) is:
If \[ \sum_{r=1}^{\infty}\tan^{-1}\left(\frac{1}{2r^{2}}\right)=a, \] then \(\tan a\) is equal to:
Given \(P(x)=x^{4}+ax^{3}+bx^{2}+cx+d\) such that \(x=0\) is the only real root of \(P^{\prime}(x)=0\). If \(P(-1)\)
You measure two quantities as \( A = 1.0 \,m \pm 0.2 \,m \), \( B = 2.0 \,m \pm 0.2 \,m \). We should report the correct value for \( \sqrt{AB} \) as: