List of top Physics Questions

Let \(f:(0,1)\rightarrow(0,1)\) be a bijective differentiable function such that \(f^{\prime}(x)\ne0 \ \forall x\in(0,1)\) and \(f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}\). Suppose for all \(x\), \[ \lim_{t\rightarrow x}\frac{\int_{0}^{t}\sqrt{1-(f(s))^{2}}\,ds-\int_{0}^{x}\sqrt{1-(f(s))^{2}}\,ds}{f(t)-f(x)}=f(x) \] Then the value of \(f\left(\frac{1}{4}\right)\) belongs to:
Let \[ \det A=\left|\begin{matrix}l&m&n\\ p&q&r\\ 1&1&1\end{matrix}\right|. \] If \[ (l-m)^{2}+(p-q)^{2}=9, \] \[ (m-n)^{2}+(q-r)^{2}=16, \] \[ (n-l)^{2}+(r-p)^{2}=25, \] then the value of \((\det A)^{2}\) is:
If for two real numbers \(a,b\) with \(|a|\le1\) and \(|b|\le1\), \[ \frac{1}{3}+\frac{\sin^{-1}a+\sin^{-1}b}{4}+\frac{(\sin^{-1}a+\sin^{-1}b)^{2}}{16}+\frac{(\sin^{-1}a+\sin^{-1}b)^{3}}{64}+\dots=\frac{2(8-3\pi)}{3(16+3\pi)}, \] then the value of \[ \sin^{-1}(a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}}) \] is:
Let us define the power of a matrix \(A\) as the maximum \(m\in\mathbb{Z}^{+}\) such that \(A^{m}=I\). For two matrices \(A\) and \(B\) if \(A^{5}=I\) and \(ABA^{-1}=B^{2}\), then the power of the matrix \(B\) is between:
Suppose \(A\) is denoted the set of all numbers between 1 and 700 which are divisible by 3 and let \(B\) is denoted the set of all numbers between 1 and 300 which are divisible by 7. If \(C=\{(a,b)\mid a\in A,b\in B, a\ne b \text{ and } a+b=\text{even number}\}\), then order of \(C\) is:
The number of 3-digit numbers of the form $xyz$ with $x<y$, $z<y$ and $x\ne0$ is:
If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f(e^{x})+f(\ln|x|)$ is:
The equation \[ |x+1|^{\log_{x+1}(3+2x-x^{2})}=(x-3)|x| \] has:
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:
The least positive value of \(a\) for which the equation \[ \int_{0}^{x}(t^{2}-8t+13)dt=x\sin\frac{a}{x} \] has a solution is:
If \[ \int_{0}^{1}\left(\sum_{r=1}^{2013}\frac{x}{x^{2}+r^{2}}\right)\left(\prod_{r=1}^{2013}(x^{2}+r^{2})\right)dx =\frac{1}{2}\left[\left(\prod_{r=1}^{2013}(1+r^{2})\right)-K\right], \] then \(K\) is:
For a real number \(y\), consider \([y]\) denotes the greatest integer less than or equal to \(y\). If \[ f(x)=\frac{\tan(\pi[x-\pi])}{1+[x]^{2}}, \] then:
The term independent of \(x\) in the expansion of \[ \left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15} \] is equal to:
The total number of polynomials of the form \[ x^{3}+ax^{2}+bx+c \] which are divisible by \(x^{2}+1\), where \(a,b,c\in\{1,2,3,\dots,10\}\) is:
The value of the integral \[ \int\frac{\left(\sqrt[3]{x+\sqrt{2-x^{2}}}\right)\left(\sqrt[6]{1-x\sqrt{2-x^{2}}}\right)}{\sqrt[3]{1-x^{2}}}\,dx \] for \(x\in(0,1)\) is:
The solution of the differential equation \[ 2x^{2}y\frac{dy}{dx}=\tan(x^{2}y^{2})-2xy^{2}, \] given \(y(1)=\sqrt{\frac{\pi}{2}}\) 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:
Let domain and range of \(f(x)\) and \(g(x)\) is \([0,\infty)\). If \(f(x)\) is an increasing function, \(g(x)\) is a decreasing function, \(h(x)=f\{g(x)\}\), \(h(0)=0\) and \(p(x)=h(x^{3}-2x^{2}+2x)-h(4)\) then for all \(x\in(0,2)\):
Let 10 Bags \(B_{1},B_{2},\dots,B_{10}\) which contain 21, 22, \dots, 30 different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is:
Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in (0,90)$; (where $[\cdot]$ denotes the greatest integer function) is:
If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\), \(\vec{c}=\hat{i}+2\hat{j}-\hat{k}\) then the value of \(\left|\begin{matrix}\vec{a}\cdot\vec{a}&\vec{a}\cdot\vec{b}&\vec{a}\cdot\vec{c}\\ \vec{b}\cdot\vec{a}&\vec{b}\cdot\vec{b}&\vec{b}\cdot\vec{c}\\ \vec{c}\cdot\vec{a}&\vec{c}\cdot\vec{b}&\vec{c}\cdot\vec{c}\end{matrix}\right|\) is equal to:
The general solution of the equation \(\sin^{100}x-\cos^{100}x=1\) 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:
Let \(a_{1},a_{2},a_{3},\dots\) are in G.P. such that \(n>m\), \(a_{n}>a_{m}\) and \(a_{1}+a_{n}=66\), \(a_{2}\cdot a_{n-1}=128\). If \(\sum_{r=1}^{n}a_{r}=126\), then \(n\) is: